Daily Papers

Algorithms for online learning typically require one or more boundedness assumptions: that the domain is bounded, that the losses are Lipschitz, or both. In this paper, we develop a new setting for online learning with unbounded domains and non-Lipschitz losses. For this setting we provide an algorithm which guarantees R_{T}(u)le tilde O(G|u|T+L|u|^{2}T) regret on any problem where the subgradients satisfy |g_{t}|le G+L|w_{t}|, and show that this bound is unimprovable without further assumptions. We leverage this algorithm to develop new saddle-point optimization algorithms that converge in duality gap in unbounded domains, even in the absence of meaningful curvature. Finally, we provide the first algorithm achieving non-trivial dynamic regret in an unbounded domain for non-Lipschitz losses, as well as a matching lower bound. The regret of our dynamic regret algorithm automatically improves to a novel L^{*} bound when the losses are smooth.

Robustness against adversarial attacks and distribution shifts is a long-standing goal of Reinforcement Learning (RL). To this end, Robust Adversarial Reinforcement Learning (RARL) trains a protagonist against destabilizing forces exercised by an adversary in a competitive zero-sum Markov game, whose optimal solution, i.e., rational strategy, corresponds to a Nash equilibrium. However, finding Nash equilibria requires facing complex saddle point optimization problems, which can be prohibitive to solve, especially for high-dimensional control. In this paper, we propose a novel approach for adversarial RL based on entropy regularization to ease the complexity of the saddle point optimization problem. We show that the solution of this entropy-regularized problem corresponds to a Quantal Response Equilibrium (QRE), a generalization of Nash equilibria that accounts for bounded rationality, i.e., agents sometimes play random actions instead of optimal ones. Crucially, the connection between the entropy-regularized objective and QRE enables free modulation of the rationality of the agents by simply tuning the temperature coefficient. We leverage this insight to propose our novel algorithm, Quantal Adversarial RL (QARL), which gradually increases the rationality of the adversary in a curriculum fashion until it is fully rational, easing the complexity of the optimization problem while retaining robustness. We provide extensive evidence of QARL outperforming RARL and recent baselines across several MuJoCo locomotion and navigation problems in overall performance and robustness.

Generative Adversarial Networks (GANs) have achieved remarkable results in the task of generating realistic natural images. In most successful applications, GAN models share two common aspects: solving a challenging saddle point optimization problem, interpreted as an adversarial game between a generator and a discriminator functions; and parameterizing the generator and the discriminator as deep convolutional neural networks. The goal of this paper is to disentangle the contribution of these two factors to the success of GANs. In particular, we introduce Generative Latent Optimization (GLO), a framework to train deep convolutional generators using simple reconstruction losses. Throughout a variety of experiments, we show that GLO enjoys many of the desirable properties of GANs: synthesizing visually-appealing samples, interpolating meaningfully between samples, and performing linear arithmetic with noise vectors; all of this without the adversarial optimization scheme.

Recent advances in maximizing mutual information (MI) between the source and target have demonstrated its effectiveness in text generation. However, previous works paid little attention to modeling the backward network of MI (i.e., dependency from the target to the source), which is crucial to the tightness of the variational information maximization lower bound. In this paper, we propose Adversarial Mutual Information (AMI): a text generation framework which is formed as a novel saddle point (min-max) optimization aiming to identify joint interactions between the source and target. Within this framework, the forward and backward networks are able to iteratively promote or demote each other's generated instances by comparing the real and synthetic data distributions. We also develop a latent noise sampling strategy that leverages random variations at the high-level semantic space to enhance the long term dependency in the generation process. Extensive experiments based on different text generation tasks demonstrate that the proposed AMI framework can significantly outperform several strong baselines, and we also show that AMI has potential to lead to a tighter lower bound of maximum mutual information for the variational information maximization problem.

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for the ability of these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, and neural network theory, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new algorithm, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep neural network training, and provide preliminary numerical evidence for its superior performance.

Gaussian process (GP) regression provides a strategy for accelerating saddle point searches on high-dimensional energy surfaces by reducing the number of times the energy and its derivatives with respect to atomic coordinates need to be evaluated. The computational overhead in the hyperparameter optimization can, however, be large and make the approach inefficient. Failures can also occur if the search ventures too far into regions that are not represented well enough by the GP model. Here, these challenges are resolved by using geometry-aware optimal transport measures and an active pruning strategy using a summation over Wasserstein-1 distances for each atom-type in farthest-point sampling, selecting a fixed-size subset of geometrically diverse configurations to avoid rapidly increasing cost of GP updates as more observations are made. Stability is enhanced by permutation-invariant metric that provides a reliable trust radius for early-stopping and a logarithmic barrier penalty for the growth of the signal variance. These physically motivated algorithmic changes prove their efficacy by reducing to less than a half the mean computational time on a set of 238 challenging configurations from a previously published data set of chemical reactions. With these improvements, the GP approach is established as, a robust and scalable algorithm for accelerating saddle point searches when the evaluation of the energy and atomic forces requires significant computational effort.

Building local surrogates to accelerate stationary point searches on potential energy surfaces spans decades of effort. Done correctly, surrogates can reduce the number of expensive electronic structure evaluations by roughly an order of magnitude while preserving the accuracy of the underlying theory, with the gain depending on oracle cost, search distance, and the availability of analytical forces. We present a unified Bayesian optimization view of minimization, single-point saddle searches, and double-ended path searches: all three share one six-step surrogate loop and differ only in the inner optimization target and the acquisition criterion. The framework uses Gaussian process regression with derivative observations, inverse-distance kernels, and active learning, and we develop optional extensions for production use, including farthest-point sampling with the Earth Mover's Distance, MAP regularization, an adaptive trust radius, and random Fourier features for scaling. Accompanying pedagogical Rust code demonstrates that all three applications use the same Bayesian optimization loop, bridging the gap between theoretical formulation and practical execution.

Concept shift is a prevailing problem in natural tasks like medical image segmentation where samples usually come from different subpopulations with variant correlations between features and labels. One common type of concept shift in medical image segmentation is the "information imbalance" between label-sparse samples with few (if any) segmentation labels and label-dense samples with plentiful labeled pixels. Existing distributionally robust algorithms have focused on adaptively truncating/down-weighting the "less informative" (i.e., label-sparse in our context) samples. To exploit data features of label-sparse samples more efficiently, we propose an adaptively weighted online optimization algorithm -- AdaWAC -- to incorporate data augmentation consistency regularization in sample reweighting. Our method introduces a set of trainable weights to balance the supervised loss and unsupervised consistency regularization of each sample separately. At the saddle point of the underlying objective, the weights assign label-dense samples to the supervised loss and label-sparse samples to the unsupervised consistency regularization. We provide a convergence guarantee by recasting the optimization as online mirror descent on a saddle point problem. Our empirical results demonstrate that AdaWAC not only enhances the segmentation performance and sample efficiency but also improves the robustness to concept shift on various medical image segmentation tasks with different UNet-style backbones.

We consider the distributionally robust optimization (DRO) problem with spectral risk-based uncertainty set and f-divergence penalty. This formulation includes common risk-sensitive learning objectives such as regularized condition value-at-risk (CVaR) and average top-k loss. We present Prospect, a stochastic gradient-based algorithm that only requires tuning a single learning rate hyperparameter, and prove that it enjoys linear convergence for smooth regularized losses. This contrasts with previous algorithms that either require tuning multiple hyperparameters or potentially fail to converge due to biased gradient estimates or inadequate regularization. Empirically, we show that Prospect can converge 2-3times faster than baselines such as stochastic gradient and stochastic saddle-point methods on distribution shift and fairness benchmarks spanning tabular, vision, and language domains.

Gradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal first-order methods, allows to obtain in a black-box fashion new zeroth-order algorithms for non-smooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to first-order methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddle-point problems, and distributed optimization.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

In this paper, we study a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. While restart strategies are commonly employed in first-order methods to achieve optimal convergence under strong convexity, they introduce structural complexity and practical overhead, making algorithm design and nesting cumbersome. To address this, we propose a restart-free stochastic gradient sliding algorithm that eliminates the need for explicit restart phases when the simple nonsmooth component is strongly convex. Through a novel and carefully designed parameter selection strategy, we prove that the proposed algorithm achieves an ε-solution with only O(log(1ε)) gradient evaluations for the smooth component and O(1ε) stochastic subgradient evaluations for the nonsmooth component, matching the optimal complexity of existing multi-phase (restart-based) methods. Moreover, for the case where the nonsmooth component is structured, allowing the overall problem to be reformulated as a bilinear saddle-point problem, we develop a restart-free accelerated stochastic gradient sliding algorithm. We show that the resulting method requires only O(log(1ε)) gradient computations for the smooth component while preserving an overall iteration complexity of O(1{sqrtε}) for solving the corresponding saddle-point problems. Our work thus provides simpler, restart-f

Variational inequalities in general and saddle point problems in particular are increasingly relevant in machine learning applications, including adversarial learning, GANs, transport and robust optimization. With increasing data and problem sizes necessary to train high performing models across various applications, we need to rely on parallel and distributed computing. However, in distributed training, communication among the compute nodes is a key bottleneck during training, and this problem is exacerbated for high dimensional and over-parameterized models. Due to these considerations, it is important to equip existing methods with strategies that would allow to reduce the volume of transmitted information during training while obtaining a model of comparable quality. In this paper, we present the first theoretically grounded distributed methods for solving variational inequalities and saddle point problems using compressed communication: MASHA1 and MASHA2. Our theory and methods allow for the use of both unbiased (such as Randk; MASHA1) and contractive (such as Topk; MASHA2) compressors. New algorithms support bidirectional compressions, and also can be modified for stochastic setting with batches and for federated learning with partial participation of clients. We empirically validated our conclusions using two experimental setups: a standard bilinear min-max problem, and large-scale distributed adversarial training of transformers.

Two-point zeroth order methods are important in many applications of zeroth-order optimization, such as robotics, wind farms, power systems, online optimization, and adversarial robustness to black-box attacks in deep neural networks, where the problem may be high-dimensional and/or time-varying. Most problems in these applications are nonconvex and contain saddle points. While existing works have shown that zeroth-order methods utilizing Omega(d) function valuations per iteration (with d denoting the problem dimension) can escape saddle points efficiently, it remains an open question if zeroth-order methods based on two-point estimators can escape saddle points. In this paper, we show that by adding an appropriate isotropic perturbation at each iteration, a zeroth-order algorithm based on 2m (for any 1 leq m leq d) function evaluations per iteration can not only find epsilon-second order stationary points polynomially fast, but do so using only Oleft(d{mepsilon^{2}psi}right) function evaluations, where psi geq Omegaleft(epsilonright) is a parameter capturing the extent to which the function of interest exhibits the strict saddle property.

Real-world datasets exhibit imbalances of varying types and degrees. Several techniques based on re-weighting and margin adjustment of loss are often used to enhance the performance of neural networks, particularly on minority classes. In this work, we analyze the class-imbalanced learning problem by examining the loss landscape of neural networks trained with re-weighting and margin-based techniques. Specifically, we examine the spectral density of Hessian of class-wise loss, through which we observe that the network weights converge to a saddle point in the loss landscapes of minority classes. Following this observation, we also find that optimization methods designed to escape from saddle points can be effectively used to improve generalization on minority classes. We further theoretically and empirically demonstrate that Sharpness-Aware Minimization (SAM), a recent technique that encourages convergence to a flat minima, can be effectively used to escape saddle points for minority classes. Using SAM results in a 6.2\% increase in accuracy on the minority classes over the state-of-the-art Vector Scaling Loss, leading to an overall average increase of 4\% across imbalanced datasets. The code is available at: https://github.com/val-iisc/Saddle-LongTail.

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of alpha-coherent function for which we provide convergence analysis. We show that for strictly alpha-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in alpha-coherent class of functions.

In centralized settings, it is well known that stochastic gradient descent (SGD) avoids saddle points and converges to local minima in nonconvex problems. However, similar guarantees are lacking for distributed first-order algorithms. The paper studies distributed stochastic gradient descent (D-SGD)--a simple network-based implementation of SGD. Conditions under which D-SGD avoids saddle points and converges to local minima are studied. First, we consider the problem of computing critical points. Assuming loss functions are nonconvex and possibly nonsmooth, it is shown that, for each fixed initialization, D-SGD converges to critical points of the loss with probability one. Next, we consider the problem of avoiding saddle points. In this case, we again assume that loss functions may be nonconvex and nonsmooth, but are smooth in a neighborhood of a saddle point. It is shown that, for any fixed initialization, D-SGD avoids such saddle points with probability one. Results are proved by studying the underlying (distributed) gradient flow, using the ordinary differential equation (ODE) method of stochastic approximation, and extending classical techniques from dynamical systems theory such as stable manifolds. Results are proved in the general context of subspace-constrained optimization, of which D-SGD is a special case.

In the Lagrange-Newton method, where Newton's method is applied to a Lagrangian function that includes equality constraints, all stationary points are saddle points. It is therefore not possible to use a line-search method based on the value of the objective function; instead, the line search can operate on merit functions. In this report, we explore an alternative line-search method which is applicable to this case; it particulary addresses the damping of the step length in tight valleys. We propose a line-search criterion based on the divergence of the field of Newton step vectors. The visualization of the criterion for two-dimensional test functions reveals a network of ravines with flat bottom at the zero points of the criterion. The ravines are typically connected to stationary points. To traverse this ravine network in order to approach a stationary point, a zigzag strategy is devised. Numerical experiments demonstrate that the novel line-search strategy succeeds from most starting points in all test functions, but only exhibits the desired damping of the step length in some situations. At the present stage it is therefore difficult to appraise the utility of this contribution.

We investigate a primal-dual (PD) method for the saddle point problem (SPP) that uses a linear approximation of the primal function instead of the standard proximal step, resulting in a linearized PD (LPD) method. For convex-strongly concave SPP, we observe that the LPD method has a suboptimal dependence on the Lipschitz constant of the primal function. To fix this issue, we combine features of Accelerated Gradient Descent with the LPD method resulting in a single-loop Accelerated Linearized Primal-Dual (ALPD) method. ALPD method achieves the optimal gradient complexity when the SPP has a semi-linear coupling function. We also present an inexact ALPD method for SPPs with a general nonlinear coupling function that maintains the optimal gradient evaluations of the primal parts and significantly improves the gradient evaluations of the coupling term compared to the ALPD method. We verify our findings with numerical experiments.

Training large neural networks (NNs) requires optimizing high-dimensional data-dependent loss functions. The optimization landscape of these functions is often highly complex and textured, even fractal-like, with many spurious local minima, ill-conditioned valleys, degenerate points, and saddle points. Complicating things further is the fact that these landscape characteristics are a function of the data, meaning that noise in the training data can propagate forward and give rise to unrepresentative small-scale geometry. This poses a difficulty for gradient-based optimization methods, which rely on local geometry to compute updates and are, therefore, vulnerable to being derailed by noisy data. In practice,this translates to a strong dependence of the optimization dynamics on the noise in the data, i.e., poor generalization performance. To remediate this problem, we propose a new optimization procedure: Rolling Ball Optimizer (RBO), that breaks this spatial locality by incorporating information from a larger region of the loss landscape in its updates. We achieve this by simulating the motion of a rigid sphere of finite radius rolling on the loss landscape, a straightforward generalization of Gradient Descent (GD) that simplifies into it in the infinitesimal limit. The radius serves as a hyperparameter that determines the scale at which RBO sees the loss landscape, allowing control over the granularity of its interaction therewith. We are motivated by the intuition that the large-scale geometry of the loss landscape is less data-specific than its fine-grained structure, and that it is easier to optimize. We support this intuition by proving that our algorithm has a smoothing effect on the loss function. Evaluation against SGD, SAM, and Entropy-SGD, on MNIST and CIFAR-10/100 demonstrates promising results in terms of convergence speed, training accuracy, and generalization performance.

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones~--~by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

Most value function learning algorithms in reinforcement learning are based on the mean squared (projected) Bellman error. However, squared errors are known to be sensitive to outliers, both skewing the solution of the objective and resulting in high-magnitude and high-variance gradients. To control these high-magnitude updates, typical strategies in RL involve clipping gradients, clipping rewards, rescaling rewards, or clipping errors. While these strategies appear to be related to robust losses -- like the Huber loss -- they are built on semi-gradient update rules which do not minimize a known loss. In this work, we build on recent insights reformulating squared Bellman errors as a saddlepoint optimization problem and propose a saddlepoint reformulation for a Huber Bellman error and Absolute Bellman error. We start from a formalization of robust losses, then derive sound gradient-based approaches to minimize these losses in both the online off-policy prediction and control settings. We characterize the solutions of the robust losses, providing insight into the problem settings where the robust losses define notably better solutions than the mean squared Bellman error. Finally, we show that the resulting gradient-based algorithms are more stable, for both prediction and control, with less sensitivity to meta-parameters.

The primal dual hybrid gradient algorithm (PDHG), which is also known as the Arrow-Hurwicz method, is a fundamental algorithm for saddle point problems especially in imaging. It also inspires a great number of influential algorithms such as the stochastic PDHG and the Chambolle-Pock's primal dual algorithm. In the literature, convergence theory of the PDHG is established only when some more restrictive conditions are additionally assumed, and it is proved that the PDHG with any constant step sizes could diverge for generic setting of convex saddle point problems. The Chambolle-Pock's primal dual algorithm, as an influential variant of the PDHG, is thus widely used due to its provable convergence theory and competitive numerical performance. However, step sizes of the Chambolle-Pock's primal dual algorithm are inherently bounded by its associated matrix spectrum, and this restriction could limit its computational capacity structurally. To address these limitations both in theory and practice, we propose a class of adaptive primal dual hybrid gradient algorithms for generic convex saddle point problems in this paper. By exploiting the prediction-correction algorithmic framework, the global convergence theory of the proposed schemes can be determined only by the average spectrum of the underlying matrix, and it thus leads to a potential acceleration. The numerical experiment on the assignment problem illustrates the superior numerical performance of the proposed method.

We consider function optimization as a sequential decision making problem under budget constraint. This constraint limits the number of objective function evaluations allowed during the optimization. We consider an algorithm inspired by a continuous version of a multi-armed bandit problem which attacks this optimization problem by solving the tradeoff between exploration (initial quasi-uniform search of the domain) and exploitation (local optimization around the potentially global maxima). We introduce the so-called Simultaneous Optimistic Optimization (SOO), a deterministic algorithm that works by domain partitioning. The benefit of such approach are the guarantees on the returned solution and the numerical efficiency of the algorithm. We present this machine learning approach to optimization, and provide the empirical assessment of SOO on the CEC'2014 competition on single objective real-parameter numerical optimization test-suite.

We study the problem of black-box optimization of a function f of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after n evaluations is at most a factor of sqrt(ln n) away from the error of the best known optimization algorithms using the knowledge of the smoothness.

We present a numerical framework to treat infinite time horizon spatially distributed optimal control problems via the associated canonical system derived by Pontryagin's Maximum Principle. The basic idea is to consider the canonical system in two steps. First we perform a bifurcation analysis of canonical steady states using the continuation and bifurcation package pde2path, yielding a number of so called flat and patterned canonical steady states. In a second step we link pde2path to the two point boundary value problem solver TOM to study time dependent canonical paths to steady states having the so called saddle point property. As an example we consider a shallow lake model with diffusion.

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate O(k^{-1/4}). As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

We present a new algorithm to optimize distributions defined implicitly by parameterized stochastic diffusions. Doing so allows us to modify the outcome distribution of sampling processes by optimizing over their parameters. We introduce a general framework for first-order optimization of these processes, that performs jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical guarantees on the performance of our method, as well as experimental results demonstrating its effectiveness in real-world settings.

We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a (delta,epsilon)-stationary point from O(epsilon^{-4}delta^{-1}) stochastic gradient queries to O(epsilon^{-3}delta^{-1}), which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to online learning, after which our results follow from standard regret bounds in online learning. For deterministic and second-order smooth objectives, applying more advanced optimistic online learning techniques enables a new complexity of O(epsilon^{-1.5}delta^{-0.5}). Our techniques also recover all optimal or best-known results for finding epsilon stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

Solving a linear system Ax=b is a fundamental scientific computing primitive for which numerous solvers and preconditioners have been developed. These come with parameters whose optimal values depend on the system being solved and are often impossible or too expensive to identify; thus in practice sub-optimal heuristics are used. We consider the common setting in which many related linear systems need to be solved, e.g. during a single numerical simulation. In this scenario, can we sequentially choose parameters that attain a near-optimal overall number of iterations, without extra matrix computations? We answer in the affirmative for Successive Over-Relaxation (SOR), a standard solver whose parameter omega has a strong impact on its runtime. For this method, we prove that a bandit online learning algorithm--using only the number of iterations as feedback--can select parameters for a sequence of instances such that the overall cost approaches that of the best fixed omega as the sequence length increases. Furthermore, when given additional structural information, we show that a contextual bandit method asymptotically achieves the performance of the instance-optimal policy, which selects the best omega for each instance. Our work provides the first learning-theoretic treatment of high-precision linear system solvers and the first end-to-end guarantees for data-driven scientific computing, demonstrating theoretically the potential to speed up numerical methods using well-understood learning algorithms.

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

We propose a method that achieves near-optimal rates for smooth stochastic convex optimization and requires essentially no prior knowledge of problem parameters. This improves on prior work which requires knowing at least the initial distance to optimality d0. Our method, U-DoG, combines UniXGrad (Kavis et al., 2019) and DoG (Ivgi et al., 2023) with novel iterate stabilization techniques. It requires only loose bounds on d0 and the noise magnitude, provides high probability guarantees under sub-Gaussian noise, and is also near-optimal in the non-smooth case. Our experiments show consistent, strong performance on convex problems and mixed results on neural network training.

We consider the optimization problem of the form min_{x in R^d} f(x) triangleq E_{xi} [F(x; xi)], where the component F(x;xi) is L-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most O( L^4 d^{3/2} epsilon^{-4} + Delta L^3 d^{3/2} delta^{-1} epsilon^{-4}) stochastic zeroth-order oracle complexity to find a (delta,epsilon)-Goldstein stationary point of objective function, where Delta = f(x_0) - inf_{x in R^d} f(x) and x_0 is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to O(L^3 d^{3/2} epsilon^{-3}+ Delta L^2 d^{3/2} delta^{-1} epsilon^{-3}).

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

Optimal transport is an important tool in machine learning, allowing to capture geometric properties of the data through a linear program on transport polytopes. We present a single-loop optimization algorithm for minimizing general convex objectives on these domains, utilizing the principles of Sinkhorn matrix scaling and mirror descent. The proposed algorithm is robust to noise, and can be used in an online setting. We provide theoretical guarantees for convex objectives and experimental results showcasing it effectiveness on both synthetic and real-world data.

We consider minimizing functions for which it is expensive to compute the (possibly stochastic) gradient. Such functions are prevalent in reinforcement learning, imitation learning and adversarial training. Our target optimization framework uses the (expensive) gradient computation to construct surrogate functions in a target space (e.g. the logits output by a linear model for classification) that can be minimized efficiently. This allows for multiple parameter updates to the model, amortizing the cost of gradient computation. In the full-batch setting, we prove that our surrogate is a global upper-bound on the loss, and can be (locally) minimized using a black-box optimization algorithm. We prove that the resulting majorization-minimization algorithm ensures convergence to a stationary point of the loss. Next, we instantiate our framework in the stochastic setting and propose the SSO algorithm, which can be viewed as projected stochastic gradient descent in the target space. This connection enables us to prove theoretical guarantees for SSO when minimizing convex functions. Our framework allows the use of standard stochastic optimization algorithms to construct surrogates which can be minimized by any deterministic optimization method. To evaluate our framework, we consider a suite of supervised learning and imitation learning problems. Our experiments indicate the benefits of target optimization and the effectiveness of SSO.

This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.

This paper focuses on the problem of constrained stochastic optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient free. Under convexity and smoothness assumption, we show that the proposed algorithm converges to the optimal objective function at a rate Oleft(1/T^{1/3}right), where T denotes the iteration count. In particular, the primal sub-optimality gap is shown to have a dimension dependence of Oleft(d^{1/3}right), which is the best known dimension dependence among all zeroth order optimization algorithms with one directional derivative per iteration. For non-convex functions, we obtain the Frank-Wolfe gap to be Oleft(d^{1/3}T^{-1/4}right). Experiments on black-box optimization setups demonstrate the efficacy of the proposed algorithm.

Data sparsity is a common issue to train machine learning tools such as neural networks for engineering and scientific applications, where experiments and simulations are expensive. Recently physics-constrained neural networks (PCNNs) were developed to reduce the required amount of training data. However, the weights of different losses from data and physical constraints are adjusted empirically in PCNNs. In this paper, a new physics-constrained neural network with the minimax architecture (PCNN-MM) is proposed so that the weights of different losses can be adjusted systematically. The training of the PCNN-MM is searching the high-order saddle points of the objective function. A novel saddle point search algorithm called Dual-Dimer method is developed. It is demonstrated that the Dual-Dimer method is computationally more efficient than the gradient descent ascent method for nonconvex-nonconcave functions and provides additional eigenvalue information to verify search results. A heat transfer example also shows that the convergence of PCNN-MMs is faster than that of traditional PCNNs.

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings: they either assume some knowledge of the problem parameters, impose strong global Lipschitz conditions, or fail to give bounds that hold with high probability. We provide a comprehensive analysis of this basic method without any of these limitations, in both the convex and non-convex (smooth) cases, that additionally supports a general ``affine variance'' noise model and provides sharp rates of convergence in both the low-noise and high-noise~regimes.

We study the problem of optimizing a function under a budgeted number of evaluations. We only assume that the function is locally smooth around one of its global optima. The difficulty of optimization is measured in terms of 1) the amount of noise b of the function evaluation and 2) the local smoothness, d, of the function. A smaller d results in smaller optimization error. We come with a new, simple, and parameter-free approach. First, for all values of b and d, this approach recovers at least the state-of-the-art regret guarantees. Second, our approach additionally obtains these results while being agnostic to the values of both b and d. This leads to the first algorithm that naturally adapts to an unknown range of noise b and leads to significant improvements in a moderate and low-noise regime. Third, our approach also obtains a remarkable improvement over the state-of-the-art SOO algorithm when the noise is very low which includes the case of optimization under deterministic feedback (b=0). There, under our minimal local smoothness assumption, this improvement is of exponential magnitude and holds for a class of functions that covers the vast majority of functions that practitioners optimize (d=0). We show that our algorithmic improvement is borne out in experiments as we empirically show faster convergence on common benchmarks.

We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex non-smooth function, while the denominator part is a convex or concave function. This problem is difficult to solve since it is non-convex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. The proposed methods iteratively solve a one-dimensional subproblem globally, and they are guaranteed to converge to coordinate-wise stationary points. In the case of a convex denominator, under a weak locally bounded non-convexity condition, we prove that the optimality of coordinate-wise stationary point is stronger than that of the standard critical point and directional point. Under additional suitable conditions, CD methods converge Q-linearly to coordinate-wise stationary points. In the case of a concave denominator, we show that any critical point is a global minimum, and CD methods converge to the global minimum with a sublinear convergence rate. We demonstrate the applicability of the proposed methods to some machine learning and signal processing models. Our experiments on real-world data have shown that our method significantly and consistently outperforms existing methods in terms of accuracy.

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are sub-optimal. Instead, these problems have been shown to satisfy certain generalized-smooth conditions, which have not been well understood in the existing literature. In this paper, we propose a notion of alpha-symmetric generalized-smoothness that extends the existing notions and covers many important functions such as high-order polynomials and exponential functions. We study the fundamental properties and establish descent lemmas for the functions in this class. Then, to solve such a large class of nonconvex problems, we design a special deterministic normalized gradient descent algorithm that achieves the optimal iteration complexity O(epsilon^{-2}), and also prove that the popular SPIDER variance reduction algorithm achieves the optimal sample complexity O(epsilon^{-3}) in the stochastic setting. Our results show that solving generalized-smooth nonconvex problems is as efficient as solving smooth nonconvex problems.

Exploiting partial first-order information in a cyclic way is arguably the most natural strategy to obtain scalable first-order methods. However, despite their wide use in practice, cyclic schemes are far less understood from a theoretical perspective than their randomized counterparts. Motivated by a recent success in analyzing an extrapolated cyclic scheme for generalized variational inequalities, we propose an Accelerated Cyclic Coordinate Dual Averaging with Extrapolation (A-CODER) method for composite convex optimization, where the objective function can be expressed as the sum of a smooth convex function accessible via a gradient oracle and a convex, possibly nonsmooth, function accessible via a proximal oracle. We show that A-CODER attains the optimal convergence rate with improved dependence on the number of blocks compared to prior work. Furthermore, for the setting where the smooth component of the objective function is expressible in a finite sum form, we introduce a variance-reduced variant of A-CODER, VR-A-CODER, with state-of-the-art complexity guarantees. Finally, we demonstrate the effectiveness of our algorithms through numerical experiments.

Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t.~a Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We prove a faster linear convergence result when a Polyak-{\L}ojasiewicz (P{\L}) condition holds. To our knowledge, this work is the first to provide non-asymptotic convergence guarantees -- variance-reduced or not -- for a cyclic block coordinate method in general composite (smooth + nonsmooth) nonconvex settings. Our experimental results demonstrate the efficacy of the proposed cyclic scheme in training deep neural nets.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter sigma > 0. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be O(sigma)-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as O(sigma)-approximation of the original BO, we propose first-order algorithms that find an epsilon-stationary solution by optimizing the penalty formulation with sigma = O(epsilon). When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an epsilon-stationary point of the penalty function, using in total O(epsilon^{-3}) and O(epsilon^{-7}) accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with O(1) samples per iteration, and achieves the improved oracle-complexity of O(epsilon^{-3}) and O(epsilon^{-5}), respectively.

We consider in this paper the problem of optimal experiment design where a decision maker can choose which points to sample to obtain an estimate hatβ of the hidden parameter β^{star} of an underlying linear model. The key challenge of this work lies in the heteroscedasticity assumption that we make, meaning that each covariate has a different and unknown variance. The goal of the decision maker is then to figure out on the fly the optimal way to allocate the total budget of T samples between covariates, as sampling several times a specific one will reduce the variance of the estimated model around it (but at the cost of a possible higher variance elsewhere). By trying to minimize the ell^2-loss E [lVerthatβ-β^{star}rVert^2] the decision maker is actually minimizing the trace of the covariance matrix of the problem, which corresponds then to online A-optimal design. Combining techniques from bandit and convex optimization we propose a new active sampling algorithm and we compare it with existing ones. We provide theoretical guarantees of this algorithm in different settings, including a O(T^{-2}) regret bound in the case where the covariates form a basis of the feature space, generalizing and improving existing results. Numerical experiments validate our theoretical findings.

Assessing sampling uncertainty in extremum estimation can be challenging when the asymptotic variance is not analytically tractable. Bootstrap inference offers a feasible solution but can be computationally costly especially when the model is complex. This paper uses iterates of a specially designed stochastic optimization algorithm as draws from which both point estimates and bootstrap standard errors can be computed in a single run. The draws are generated by the gradient and Hessian computed from batches of data that are resampled at each iteration. We show that these draws yield consistent estimates and asymptotically valid frequentist inference for a large class of regular problems. The algorithm provides accurate standard errors in simulation examples and empirical applications at low computational costs. The draws from the algorithm also provide a convenient way to detect data irregularities.

Optimization is a ubiquitous modeling tool and is often deployed in settings which repeatedly solve similar instances of the same problem. Amortized optimization methods use learning to predict the solutions to problems in these settings, exploiting the shared structure between similar problem instances. These methods have been crucial in variational inference and reinforcement learning and are capable of solving optimization problems many orders of magnitudes times faster than traditional optimization methods that do not use amortization. This tutorial presents an introduction to the amortized optimization foundations behind these advancements and overviews their applications in variational inference, sparse coding, gradient-based meta-learning, control, reinforcement learning, convex optimization, optimal transport, and deep equilibrium networks. The source code for this tutorial is available at https://github.com/facebookresearch/amortized-optimization-tutorial.

We study the problem of approximating stationary points of Lipschitz and smooth functions under (varepsilon,delta)-differential privacy (DP) in both the finite-sum and stochastic settings. A point w is called an alpha-stationary point of a function F:R^drightarrowR if |nabla F(w)|leq alpha. We provide a new efficient algorithm that finds an Obig(big[sqrt{d}{nvarepsilon}big]^{2/3}big)-stationary point in the finite-sum setting, where n is the number of samples. This improves on the previous best rate of Obig(big[sqrt{d}{nvarepsilon}big]^{1/2}big). We also give a new construction that improves over the existing rates in the stochastic optimization setting, where the goal is to find approximate stationary points of the population risk. Our construction finds a Obig(1{n^{1/3}} + big[sqrt{d}{nvarepsilon}big]^{1/2}big)-stationary point of the population risk in time linear in n. Furthermore, under the additional assumption of convexity, we completely characterize the sample complexity of finding stationary points of the population risk (up to polylog factors) and show that the optimal rate on population stationarity is tilde Thetabig(1{n}+sqrt{d}{nvarepsilon}big). Finally, we show that our methods can be used to provide dimension-independent rates of Obig(1{n}+minbig(big[sqrt{rank}{nvarepsilon}big]^{2/3},1{(nvarepsilon)^{2/5}}big)big) on population stationarity for Generalized Linear Models (GLM), where rank is the rank of the design matrix, which improves upon the previous best known rate.

Accurate determination of transition states is central to an understanding of reaction kinetics. Double-endpoint methods where both initial and final states are specified, such as the climbing image nudged elastic band (CI-NEB), identify the minimum energy path between the two and thereby the saddle point on the energy surface that is relevant for the given transition, thus providing an estimate of the transition state within the harmonic approximation of transition state theory. Such calculations can, however, incur high computational costs and may suffer stagnation on exceptionally flat or rough energy surfaces. Conversely, methods that only require specification of an initial set of atomic coordinates, such as the minimum mode following (MMF) method, offer efficiency but can converge on saddle points that are not relevant for transition of interest. Here, we present an adaptive hybrid algorithm that integrates the CI-NEB with the MMF method so as to get faster convergence to the relevant saddle point. The method is benchmarked for the Baker-Chan (BC) saddle point test set using the PET-MAD machine-learned potential as well as 59 transitions of a heptamer island on Pt(111) from the OptBench benchmark set. A Bayesian analysis of the performance shows a reduction in energy and force calculations of 57% [95% CrI: -64%, -50%] relative to CI-NEB for the BC set, while a 31% mean reduction is found for the transitions of the heptamer island. These results establish this hybrid method as a highly effective tool for high-throughput automated chemical discovery of atomic rearrangements.

First-order optimization (FOO) algorithms are pivotal in numerous computational domains such as machine learning and signal denoising. However, their application to complex tasks like neural network training often entails significant inefficiencies due to the need for many sequential iterations for convergence. In response, we introduce first-order optimization expedited with approximately parallelized iterations (OptEx), the first framework that enhances the efficiency of FOO by leveraging parallel computing to mitigate its iterative bottleneck. OptEx employs kernelized gradient estimation to make use of gradient history for future gradient prediction, enabling parallelization of iterations -- a strategy once considered impractical because of the inherent iterative dependency in FOO. We provide theoretical guarantees for the reliability of our kernelized gradient estimation and the iteration complexity of SGD-based OptEx, confirming that estimation errors diminish to zero as historical gradients accumulate and that SGD-based OptEx enjoys an effective acceleration rate of Omega(N) over standard SGD given parallelism of N. We also use extensive empirical studies, including synthetic functions, reinforcement learning tasks, and neural network training across various datasets, to underscore the substantial efficiency improvements achieved by OptEx.

We study infinite-horizon average-reward Markov decision processes (AMDPs) in the context of general function approximation. Specifically, we propose a novel algorithmic framework named Local-fitted Optimization with OPtimism (LOOP), which incorporates both model-based and value-based incarnations. In particular, LOOP features a novel construction of confidence sets and a low-switching policy updating scheme, which are tailored to the average-reward and function approximation setting. Moreover, for AMDPs, we propose a novel complexity measure -- average-reward generalized eluder coefficient (AGEC) -- which captures the challenge of exploration in AMDPs with general function approximation. Such a complexity measure encompasses almost all previously known tractable AMDP models, such as linear AMDPs and linear mixture AMDPs, and also includes newly identified cases such as kernel AMDPs and AMDPs with Bellman eluder dimensions. Using AGEC, we prove that LOOP achieves a sublinear mathcal{O}(poly(d, sp(V^*)) Tbeta ) regret, where d and beta correspond to AGEC and log-covering number of the hypothesis class respectively, sp(V^*) is the span of the optimal state bias function, T denotes the number of steps, and mathcal{O} (cdot) omits logarithmic factors. When specialized to concrete AMDP models, our regret bounds are comparable to those established by the existing algorithms designed specifically for these special cases. To the best of our knowledge, this paper presents the first comprehensive theoretical framework capable of handling nearly all AMDPs.

A unified framework for first-order optimization algorithms fornonconvex unconstrained optimization is proposed that uses adaptivelypreconditioned gradients and includes popular methods such as full anddiagonal AdaGrad, AdaNorm, as well as adpative variants of Shampoo andMuon. This framework also allows combining heterogeneous geometriesacross different groups of variables while preserving a unifiedconvergence analysis. A fully stochastic global rate-of-convergenceanalysis is conducted for all methods in the framework, with andwithout two types of momentum, using reasonable assumptions on thevariance of the gradient oracle and without assuming boundedstochastic gradients or small enough stepsize.

In multi-fidelity optimization, biased approximations of varying costs of the target function are available. This paper studies the problem of optimizing a locally smooth function with a limited budget, where the learner has to make a tradeoff between the cost and the bias of these approximations. We first prove lower bounds for the simple regret under different assumptions on the fidelities, based on a cost-to-bias function. We then present the Kometo algorithm which achieves, with additional logarithmic factors, the same rates without any knowledge of the function smoothness and fidelity assumptions, and improves previously proven guarantees. We finally empirically show that our algorithm outperforms previous multi-fidelity optimization methods without the knowledge of problem-dependent parameters.

We study the problem of constrained efficient global optimization, where both the objective and constraints are expensive black-box functions that can be learned with Gaussian processes. We propose CONFIG (CONstrained efFIcient Global Optimization), a simple and effective algorithm to solve it. Under certain regularity assumptions, we show that our algorithm enjoys the same cumulative regret bound as that in the unconstrained case and similar cumulative constraint violation upper bounds. For commonly used Matern and Squared Exponential kernels, our bounds are sublinear and allow us to derive a convergence rate to the optimal solution of the original constrained problem. In addition, our method naturally provides a scheme to declare infeasibility when the original black-box optimization problem is infeasible. Numerical experiments on sampled instances from the Gaussian process, artificial numerical problems, and a black-box building controller tuning problem all demonstrate the competitive performance of our algorithm. Compared to the other state-of-the-art methods, our algorithm significantly improves the theoretical guarantees, while achieving competitive empirical performance.

Neural networks trained with gradient descent often learn solutions of increasing complexity over time, a phenomenon known as simplicity bias. Despite being widely observed across architectures, existing theoretical treatments lack a unifying framework. We present a theoretical framework that explains a simplicity bias arising from saddle-to-saddle learning dynamics for a general class of neural networks, incorporating fully-connected, convolutional, and attention-based architectures. Here, simple means expressible with few hidden units, i.e., hidden neurons, convolutional kernels, or attention heads. Specifically, we show that linear networks learn solutions of increasing rank, ReLU networks learn solutions with an increasing number of kinks, convolutional networks learn solutions with an increasing number of convolutional kernels, and self-attention models learn solutions with an increasing number of attention heads. By analyzing fixed points, invariant manifolds, and dynamics of gradient descent learning, we show that saddle-to-saddle dynamics operates by iteratively evolving near an invariant manifold, approaching a saddle, and switching to another invariant manifold. Our analysis also disentangles data-induced and initialization-induced saddle-to-saddle dynamics. In particular, the former leads to low-rank weights while the latter to sparse weights. Equipped with the theory, we predict the effects of data distribution and weight initialization on the duration and number of plateaus in learning. Overall, our theory offers a framework for understanding when and why gradient descent progressively learns increasingly complex solutions.

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an epsilon-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

Recently, the study of linear misspecified bandits has generated intriguing implications of the hardness of learning in bandits and reinforcement learning (RL). In particular, Du et al. (2020) show that even if a learner is given linear features in R^d that approximate the rewards in a bandit or RL with a uniform error of varepsilon, searching for an O(varepsilon)-optimal action requires pulling at least Omega(exp(d)) queries. Furthermore, Lattimore et al. (2020) show that a degraded O(varepsilond)-optimal solution can be learned within poly(d/varepsilon) queries. Yet it is unknown whether a structural assumption on the ground-truth parameter, such as sparsity, could break the varepsilond barrier. In this paper, we address this question by showing that algorithms can obtain O(varepsilon)-optimal actions by querying O(varepsilon^{-s}d^s) actions, where s is the sparsity parameter, removing the exp(d)-dependence. We then establish information-theoretical lower bounds, i.e., Omega(exp(s)), to show that our upper bound on sample complexity is nearly tight if one demands an error O(s^{delta}varepsilon) for 0<delta<1. For deltageq 1, we further show that poly(s/varepsilon) queries are possible when the linear features are "good" and even in general settings. These results provide a nearly complete picture of how sparsity can help in misspecified bandit learning and provide a deeper understanding of when linear features are "useful" for bandit and reinforcement learning with misspecification.

Learning to optimize has emerged as a powerful framework for various optimization and machine learning tasks. Current such "meta-optimizers" often learn in the space of continuous optimization algorithms that are point-based and uncertainty-unaware. To overcome the limitations, we propose a meta-optimizer that learns in the algorithmic space of both point-based and population-based optimization algorithms. The meta-optimizer targets at a meta-loss function consisting of both cumulative regret and entropy. Specifically, we learn and interpret the update formula through a population of LSTMs embedded with sample- and feature-level attentions. Meanwhile, we estimate the posterior directly over the global optimum and use an uncertainty measure to help guide the learning process. Empirical results over non-convex test functions and the protein-docking application demonstrate that this new meta-optimizer outperforms existing competitors.

In this paper, we propose a generic optimization approach for challenging objective functions that finds applications in various statistical problems. We focus on objective functions with two parameter blocks of one amenable to analytic optimization, and another that is irregular or computationally expensive. To address this setting, we propose the Voronoi-Elitism Genetic Algorithm (VEGA), a derivative-free optimization method that embeds geometric information into genetic search. The proposed algorithm retains elite candidates and constructs Voronoi-based neighborhoods around them, whose crossover and self-adaptive mutation balance exploitation of promising solutions with exploration of under-covered regions. We study the high dimensional behavior of genetic search by analyzing distance concentration, and the effects of population size and shrinking mutation, which shows that the algorithm improves spatial coverage and yields sharper distance bounds under limited computational budgets. Simulation studies are conducted to compare VEGA with two genetic-type algorithms competitors in finite samples. A real data application on Stack Exchange activity data further illustrates its ability to identify stable structural changes, implying the algorithm is computationally flexible for high-dimensional, derivative-free optimization and applicable for various statistical problems.

This paper investigates the global convergence of stepsized Newton methods for convex functions. We propose several simple stepsize schedules with fast global convergence guarantees, up to O (k^{-3}), nearly matching lower complexity bounds Omega (k^{-3.5}) of second-order methods. For cases with multiple plausible smoothness parameterizations or an unknown smoothness constant, we introduce a stepsize backtracking procedure that ensures convergence as if the optimal smoothness parameters were known.

Policy optimization methods are powerful algorithms in Reinforcement Learning (RL) for their flexibility to deal with policy parameterization and ability to handle model misspecification. However, these methods usually suffer from slow convergence rates and poor sample complexity. Hence it is important to design provably sample efficient algorithms for policy optimization. Yet, recent advances for this problems have only been successful in tabular and linear setting, whose benign structures cannot be generalized to non-linearly parameterized policies. In this paper, we address this problem by leveraging recent advances in value-based algorithms, including bounded eluder-dimension and online sensitivity sampling, to design a low-switching sample-efficient policy optimization algorithm, LPO, with general non-linear function approximation. We show that, our algorithm obtains an varepsilon-optimal policy with only O(text{poly(d)}{varepsilon^3}) samples, where varepsilon is the suboptimality gap and d is a complexity measure of the function class approximating the policy. This drastically improves previously best-known sample bound for policy optimization algorithms, O(text{poly(d)}{varepsilon^8}). Moreover, we empirically test our theory with deep neural nets to show the benefits of the theoretical inspiration.

During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central alpha-th moment for alpha in (1,2] in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization.

In this work, we consider and analyze the sample complexity of model-free reinforcement learning with a generative model. Particularly, we analyze mirror descent value iteration (MDVI) by Geist et al. (2019) and Vieillard et al. (2020a), which uses the Kullback-Leibler divergence and entropy regularization in its value and policy updates. Our analysis shows that it is nearly minimax-optimal for finding an varepsilon-optimal policy when varepsilon is sufficiently small. This is the first theoretical result that demonstrates that a simple model-free algorithm without variance-reduction can be nearly minimax-optimal under the considered setting.

The annealing phase is a pivotal convergence stage in LLM pre-training that ultimately determines final model quality. However, effectively selecting training data during this phase remains a key challenge. Current strategies rely on empirical heuristics, such as domain filtering or context extension, which lack a principled grounding in optimization theory. In this work, we characterize the annealing phase through the lens of the loss landscape's spectral geometry. We argue that optimal convergence requires gradient updates to satisfy heterogeneous constraints across different eigen-directions. Building on this insight, we formulate data selection as a problem of satisfying these directional constraints. To this end, we propose DiReCT (Directionally-Restrained Constrained Training), a novel framework that reformulates sample selection in the annealing stage as a constrained optimization problem. By imposing explicit directional constraints on per-sample gradients based on the spectral properties of the Hessian, DiReCT identifies samples that align with the optimal curvature-aware descent path. Extensive experiments across various model scales demonstrate that DiReCT consistently achieves state-of-the-art performance. For future research, code is available at https://github.com/xuyj233/Direct.

Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep learning systems). In this setting, the optimal O(1/t) convergence rate for solving smooth monotone variational inequalities is achieved by the Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of an extra gradient step per iteration (which can become quite substantial in deep learning applications), several algorithms have been proposed as surrogates to Extra-Gradient with a single oracle call per iteration. In this paper, we develop a synthetic view of such algorithms, and we complement the existing literature by showing that they retain a O(1/t) ergodic convergence rate in smooth, deterministic problems. Subsequently, beyond the monotone deterministic case, we also show that the last iterate of single-call, stochastic extra-gradient methods still enjoys a O(1/t) local convergence rate to solutions of non-monotone variational inequalities that satisfy a second-order sufficient condition.

The task of locating first order saddle points on high-dimensional surfaces describing the variation of energy as a function of atomic coordinates is an essential step for identifying the mechanism and estimating the rate of thermally activated events within the harmonic approximation of transition state theory. When combined directly with electronic structure calculations, the number of energy and atomic force evaluations needed for convergence is a primary issue. Here, we describe an efficient implementation of Gaussian process regression (GPR) acceleration of the minimum mode following method where a dimer is used to estimate the lowest eigenmode of the Hessian. A surrogate energy surface is constructed and updated after each electronic structure calculation. The method is applied to a test set of 500 molecular reactions previously generated by Hermez and coworkers [J. Chem. Theory Comput. 18, 6974 (2022)]. An order of magnitude reduction in the number of electronic structure calculations needed to reach the saddle point configurations is obtained by using the GPR compared to the dimer method. Despite the wide range in stiffness of the molecular degrees of freedom, the calculations are carried out using Cartesian coordinates and are found to require similar number of electronic structure calculations as an elaborate internal coordinate method implemented in the Sella software package. The present implementation of the GPR surrogate model in C++ is efficient enough for the wall time of the saddle point searches to be reduced in 3 out of 4 cases even though the calculations are carried out at a low Hartree-Fock level.

Significant recent work has studied the ability of gradient descent to recover a hidden planted direction θ^star in S^{d-1} in different high-dimensional settings, including tensor PCA and single-index models. The key quantity that governs the ability of gradient descent to traverse these landscapes is the information exponent k^star (Ben Arous et al., (2021)), which corresponds to the order of the saddle at initialization in the population landscape. Ben Arous et al., (2021) showed that n gtrsim d^{max(1, k^star-1)} samples were necessary and sufficient for online SGD to recover θ^star, and Ben Arous et al., (2020) proved a similar lower bound for Langevin dynamics. More recently, Damian et al., (2023) showed it was possible to circumvent these lower bounds by running gradient descent on a smoothed landscape, and that this algorithm succeeds with n gtrsim d^{max(1, k^star/2)} samples, which is optimal in the worst case. This raises the question of whether it is possible to achieve the same rate without explicit smoothing. In this paper, we show that Langevin dynamics can succeed with n gtrsim d^{ k^star/2 } samples if one considers the average iterate, rather than the last iterate. The key idea is that the combination of noise-injection and iterate averaging is able to emulate the effect of landscape smoothing. We apply this result to both the tensor PCA and single-index model settings. Finally, we conjecture that minibatch SGD can also achieve the same rate without adding any additional noise.

Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is quasar-convexity, a non-convex generalization of convexity that subsumes convex functions. Existing algorithms for minimizing quasar-convex functions in the stochastic setting have either high complexity or slow convergence, which prompts us to derive a new class of stochastic methods for optimizing smooth quasar-convex functions. We demonstrate that our algorithms have fast convergence and outperform existing algorithms on several examples, including the classical problem of learning linear dynamical systems. We also present a unified analysis of our newly proposed algorithms and a previously studied deterministic algorithm.

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which n component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter L. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is mathcal{O}( n + nLvarepsilon^{-1}), which improves upon existing methods by a factor up to n. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a O(1/k^2) and O(1/k) rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves O(1/k^2) and mathcal{O}(1/k^2) rates. We then provide a matching complexity lower bound to establish that Theta(1/k^2) is indeed the optimal accelerated rate.

In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The key innovation of our analysis is a novel notion called Gradient-Normalized Smoothness, which characterizes the maximum radius of a ball around the current point that yields a good relative approximation of the gradient field. Our theory establishes a natural intrinsic connection between Hessian approximation and the linearization of the gradient. Importantly, Gradient-Normalized Smoothness does not depend on the specific problem class of the objective functions, while effectively translating local information about the gradient field and Hessian approximation into the global behavior of the method. This new concept equips approximate second-order algorithms with universal global convergence guarantees, recovering state-of-the-art rates for functions with H\"older-continuous Hessians and third derivatives, quasi-self-concordant functions, as well as smooth classes in first-order optimization. These rates are achieved automatically and extend to broader classes, such as generalized self-concordant functions. We demonstrate direct applications of our results for global linear rates in logistic regression and softmax problems with approximate Hessians, as well as in non-convex optimization using Fisher and Gauss-Newton approximations.

In this paper we provide a novel analytical perspective on the theoretical understanding of gradient-based learning algorithms by interpreting consensus-based optimization (CBO), a recently proposed multi-particle derivative-free optimization method, as a stochastic relaxation of gradient descent. Remarkably, we observe that through communication of the particles, CBO exhibits a stochastic gradient descent (SGD)-like behavior despite solely relying on evaluations of the objective function. The fundamental value of such link between CBO and SGD lies in the fact that CBO is provably globally convergent to global minimizers for ample classes of nonsmooth and nonconvex objective functions, hence, on the one side, offering a novel explanation for the success of stochastic relaxations of gradient descent. On the other side, contrary to the conventional wisdom for which zero-order methods ought to be inefficient or not to possess generalization abilities, our results unveil an intrinsic gradient descent nature of such heuristics. This viewpoint furthermore complements previous insights into the working principles of CBO, which describe the dynamics in the mean-field limit through a nonlinear nonlocal partial differential equation that allows to alleviate complexities of the nonconvex function landscape. Our proofs leverage a completely nonsmooth analysis, which combines a novel quantitative version of the Laplace principle (log-sum-exp trick) and the minimizing movement scheme (proximal iteration). In doing so, we furnish useful and precise insights that explain how stochastic perturbations of gradient descent overcome energy barriers and reach deep levels of nonconvex functions. Instructive numerical illustrations support the provided theoretical insights.

We study the problem of global maximization of a function f given a finite number of evaluations perturbed by noise. We consider a very weak assumption on the function, namely that it is locally smooth (in some precise sense) with respect to some semi-metric, around one of its global maxima. Compared to previous works on bandits in general spaces (Kleinberg et al., 2008; Bubeck et al., 2011a) our algorithm does not require the knowledge of this semi-metric. Our algorithm, StoSOO, follows an optimistic strategy to iteratively construct upper confidence bounds over the hierarchical partitions of the function domain to decide which point to sample next. A finite-time analysis of StoSOO shows that it performs almost as well as the best specifically-tuned algorithms even though the local smoothness of the function is not known.

Minimax problems are notoriously challenging to optimize. However, we demonstrate that the two-timescale extragradient can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under a mild condition. This work surpasses all previous results as we eliminate a crucial assumption that the Hessian, with respect to the maximization variable, is nondegenerate.

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an epsilon-stationary solution of the bilevel problem after epsilon^{-7/2}, epsilon^{-5/2}, and epsilon^{-3/2} iterations (each iteration using O(1) samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to epsilon^{-5/2}, epsilon^{-4/2}, and epsilon^{-3/2}, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

Gaussian processes (GP) are a well studied Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to high-dimensional functions, as their per-iteration time and space cost is at least quadratic in the number of dimensions d and iterations t. Given a set of A alternatives to choose from, the overall runtime O(t^3A) is prohibitive. In this paper we introduce BKB (budgeted kernelized bandit), a new approximate GP algorithm for optimization under bandit feedback that achieves near-optimal regret (and hence near-optimal convergence rate) with near-constant per-iteration complexity and remarkably no assumption on the input space or covariance of the GP. We combine a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching based on leverage score sampling, and we prove that randomly sampling inducing points based on their posterior variance gives an accurate low-rank approximation of the GP, preserving variance estimates and confidence intervals. As a consequence, BKB does not suffer from variance starvation, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most O(d_{eff}) points, where d_{eff} is the effective dimension of the explored space, which is typically much smaller than both d and t. This greatly reduces the dimensionality of the problem, thus leading to a O(TAd_{eff}^2) runtime and O(A d_{eff}) space complexity.

This paper presents a new approach and algorithm for solving a class of constrained Bi-Level Optimization (BLO) problems in which the lower-level problem involves constraints coupling both upper-level and lower-level variables. Such problems have recently gained significant attention due to their broad applicability in machine learning. However, conventional gradient-based methods unavoidably rely on computationally intensive calculations related to the Hessian matrix. To address this challenge, we begin by devising a smooth proximal Lagrangian value function to handle the constrained lower-level problem. Utilizing this construct, we introduce a single-level reformulation for constrained BLOs that transforms the original BLO problem into an equivalent optimization problem with smooth constraints. Enabled by this reformulation, we develop a Hessian-free gradient-based algorithm-termed proximal Lagrangian Value function-based Hessian-free Bi-level Algorithm (LV-HBA)-that is straightforward to implement in a single loop manner. Consequently, LV-HBA is especially well-suited for machine learning applications. Furthermore, we offer non-asymptotic convergence analysis for LV-HBA, eliminating the need for traditional strong convexity assumptions for the lower-level problem while also being capable of accommodating non-singleton scenarios. Empirical results substantiate the algorithm's superior practical performance.

Preconditioned gradient methods are among the most general and powerful tools in optimization. However, preconditioning requires storing and manipulating prohibitively large matrices. We describe and analyze a new structure-aware preconditioning algorithm, called Shampoo, for stochastic optimization over tensor spaces. Shampoo maintains a set of preconditioning matrices, each of which operates on a single dimension, contracting over the remaining dimensions. We establish convergence guarantees in the stochastic convex setting, the proof of which builds upon matrix trace inequalities. Our experiments with state-of-the-art deep learning models show that Shampoo is capable of converging considerably faster than commonly used optimizers. Although it involves a more complex update rule, Shampoo's runtime per step is comparable to that of simple gradient methods such as SGD, AdaGrad, and Adam.

We propose an adaptive step size with an energy approach for a suitable class of preconditioned gradient descent methods. We focus on settings where the preconditioning is applied to address the constraints in optimization problems, such as the Hessian-Riemannian and natural gradient descent methods. More specifically, we incorporate these preconditioned gradient descent algorithms in the recently introduced Adaptive Energy Gradient Descent (AEGD) framework. In particular, we discuss theoretical results on the unconditional energy-stability and convergence rates across three classes of objective functions. Furthermore, our numerical results demonstrate excellent performance of the proposed method on several test bed optimization problems.

Proximal constraints are fundamental to the stability of the Large Language Model reinforcement learning. While the canonical clipping mechanism in PPO serves as an efficient surrogate for trust regions, we identify a critical bottleneck: fixed bounds strictly constrain the upward update margin of low-probability actions, disproportionately suppressing high-advantage tail strategies and inducing rapid entropy collapse. To address this, we introduce Band-constrained Policy Optimization (BandPO). BandPO replaces canonical clipping with Band, a unified theoretical operator that projects trust regions defined by f-divergences into dynamic, probability-aware clipping intervals. Theoretical analysis confirms that Band effectively resolves this exploration bottleneck. We formulate this mapping as a convex optimization problem, guaranteeing a globally optimal numerical solution while deriving closed-form solutions for specific divergences. Extensive experiments across diverse models and datasets demonstrate that BandPO consistently outperforms canonical clipping and Clip-Higher, while robustly mitigating entropy collapse.

Diffusion models (DMs) have achieved state-of-the-art generative performance but suffer from high sampling latency due to their sequential denoising nature. Existing solver-based acceleration methods often face significant image quality degradation under a low-latency budget, primarily due to accumulated truncation errors arising from the inability to capture high-curvature trajectory segments. In this paper, we propose the Ensemble Parallel Direction solver (dubbed as EPD-Solver), a novel ODE solver that mitigates these errors by incorporating multiple parallel gradient evaluations in each step. Motivated by the geometric insight that sampling trajectories are largely confined to a low-dimensional manifold, EPD-Solver leverages the Mean Value Theorem for vector-valued functions to approximate the integral solution more accurately. Importantly, since the additional gradient computations are independent, they can be fully parallelized, preserving low-latency sampling nature. We introduce a two-stage optimization framework. Initially, EPD-Solver optimizes a small set of learnable parameters via a distillation-based approach. We further propose a parameter-efficient Reinforcement Learning (RL) fine-tuning scheme that reformulates the solver as a stochastic Dirichlet policy. Unlike traditional methods that fine-tune the massive backbone, our RL approach operates strictly within the low-dimensional solver space, effectively mitigating reward hacking while enhancing performance in complex text-to-image (T2I) generation tasks. In addition, our method is flexible and can serve as a plugin (EPD-Plugin) to improve existing ODE samplers.

Large Language Models (LLMs) have achieved great success in many real-world applications, especially the one serving as the cognitive backbone of Multi-Agent Systems (MAS) to orchestrate complex workflows in practice. Since many deployment scenarios preclude MAS workflow modifications and its performance is highly sensitive to the input prompts, prompt optimization emerges as a more natural approach to improve its performance. However, real-world prompt optimization for MAS is impeded by three key challenges: (1) the need of sample efficiency due to prohibitive evaluation costs, (2) topology-induced coupling among prompts, and (3) the combinatorial explosion of the search space. To address these challenges, we introduce MASPOB (Multi-Agent System Prompt Optimization via Bandits), a novel sample-efficient framework based on bandits. By leveraging Upper Confidence Bound (UCB) to quantify uncertainty, the bandit framework balances exploration and exploitation, maximizing gains within a strictly limited budget. To handle topology-induced coupling, MASPOB integrates Graph Neural Networks (GNNs) to capture structural priors, learning topology-aware representations of prompt semantics. Furthermore, it employs coordinate ascent to decompose the optimization into univariate sub-problems, reducing search complexity from exponential to linear. Extensive experiments across diverse benchmarks demonstrate that MASPOB achieves state-of-the-art performance, consistently outperforming existing baselines.

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to induce an optimistic SSP problem whose associated value iteration scheme is guaranteed to converge. We prove that EB-SSP achieves the minimax regret rate O(B_{star} S A K), where K is the number of episodes, S is the number of states, A is the number of actions, and B_{star} bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of B_{star}, nor of T_{star}, which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of T_{star} is available) where the regret only contains a logarithmic dependence on T_{star}, thus yielding the first (nearly) horizon-free regret bound beyond the finite-horizon MDP setting.

We propose a new variant of AMSGrad, a popular adaptive gradient based optimization algorithm widely used for training deep neural networks. Our algorithm adds prior knowledge about the sequence of consecutive mini-batch gradients and leverages its underlying structure making the gradients sequentially predictable. By exploiting the predictability and ideas from optimistic online learning, the proposed algorithm can accelerate the convergence and increase sample efficiency. After establishing a tighter upper bound under some convexity conditions on the regret, we offer a complimentary view of our algorithm which generalizes the offline and stochastic version of nonconvex optimization. In the nonconvex case, we establish a non-asymptotic convergence bound independently of the initialization. We illustrate the practical speedup on several deep learning models via numerical experiments.

Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second order statistics of the data, are far less prevalent despite strong theoretical properties, due to their prohibitive computation, memory and communication costs. In an attempt to bridge this gap between theoretical and practical optimization, we present a scalable implementation of a second-order preconditioned method (concretely, a variant of full-matrix Adagrad), that along with several critical algorithmic and numerical improvements, provides significant convergence and wall-clock time improvements compared to conventional first-order methods on state-of-the-art deep models. Our novel design effectively utilizes the prevalent heterogeneous hardware architecture for training deep models, consisting of a multicore CPU coupled with multiple accelerator units. We demonstrate superior performance compared to state-of-the-art on very large learning tasks such as machine translation with Transformers, language modeling with BERT, click-through rate prediction on Criteo, and image classification on ImageNet with ResNet-50.

We study convergence lower bounds of without-replacement stochastic gradient descent (SGD) for solving smooth (strongly-)convex finite-sum minimization problems. Unlike most existing results focusing on final iterate lower bounds in terms of the number of components n and the number of epochs K, we seek bounds for arbitrary weighted average iterates that are tight in all factors including the condition number kappa. For SGD with Random Reshuffling, we present lower bounds that have tighter kappa dependencies than existing bounds. Our results are the first to perfectly close the gap between lower and upper bounds for weighted average iterates in both strongly-convex and convex cases. We also prove weighted average iterate lower bounds for arbitrary permutation-based SGD, which apply to all variants that carefully choose the best permutation. Our bounds improve the existing bounds in factors of n and kappa and thereby match the upper bounds shown for a recently proposed algorithm called GraB.

We investigate a general formulation for clustering and transductive few-shot learning, which integrates prototype-based objectives, Laplacian regularization and supervision constraints from a few labeled data points. We propose a concave-convex relaxation of the problem, and derive a computationally efficient block-coordinate bound optimizer, with convergence guarantee. At each iteration,our optimizer computes independent (parallel) updates for each point-to-cluster assignment. Therefore, it could be trivially distributed for large-scale clustering and few-shot tasks. Furthermore, we provides a thorough convergence analysis based on point-to-set maps. Were port comprehensive clustering and few-shot learning experiments over various data sets, showing that our method yields competitive performances, in term of accuracy and optimization quality, while scaling up to large problems. Using standard training on the base classes, without resorting to complex meta-learning and episodic-training strategies, our approach outperforms state-of-the-art few-shot methods by significant margins, across various models, settings and data sets. Surprisingly, we found that even standard clustering procedures (e.g., K-means), which correspond to particular, non-regularized cases of our general model, already achieve competitive performances in comparison to the state-of-the-art in few-shot learning. These surprising results point to the limitations of the current few-shot benchmarks, and question the viability of a large body of convoluted few-shot learning techniques in the recent literature.

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to numerically implement the proximal step. The most challenging step, in this setup, is to evaluate functions involving density explicitly, such as entropy, in terms of samples. This paper builds on the recent works with a slight but crucial difference: we propose to utilize a variational formulation of the objective function formulated as maximization over a parametric class of functions. Theoretically, the proposed variational formulation allows the construction of gradient flows directly for empirical distributions with a well-defined and meaningful objective function. Computationally, this approach replaces the computationally expensive step in existing methods, to handle objective functions involving density, with inner loop updates that only require a small batch of samples and scale well with the dimension. The performance and scalability of the proposed method are illustrated with the aid of several numerical experiments involving high-dimensional synthetic and real datasets.

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

Computing a Gaussian process (GP) posterior has a computational cost cubical in the number of historical points. A reformulation of the same GP posterior highlights that this complexity mainly depends on how many unique historical points are considered. This can have important implication in active learning settings, where the set of historical points is constructed sequentially by the learner. We show that sequential black-box optimization based on GPs (GP-Opt) can be made efficient by sticking to a candidate solution for multiple evaluation steps and switch only when necessary. Limiting the number of switches also limits the number of unique points in the history of the GP. Thus, the efficient GP reformulation can be used to exactly and cheaply compute the posteriors required to run the GP-Opt algorithms. This approach is especially useful in real-world applications of GP-Opt with high switch costs (e.g. switching chemicals in wet labs, data/model loading in hyperparameter optimization). As examples of this meta-approach, we modify two well-established GP-Opt algorithms, GP-UCB and GP-EI, to switch candidates as infrequently as possible adapting rules from batched GP-Opt. These versions preserve all the theoretical no-regret guarantees while improving practical aspects of the algorithms such as runtime, memory complexity, and the ability of batching candidates and evaluating them in parallel.

Bayesian optimization (BO) has established itself as a leading strategy for efficiently optimizing expensive-to-evaluate functions. Existing BO methods mostly rely on Gaussian process (GP) surrogate models and are not applicable to (doubly-stochastic) Gaussian Cox processes, where the observation process is modulated by a latent intensity function modeled as a GP. In this paper, we propose a novel maximum a posteriori inference of Gaussian Cox processes. It leverages the Laplace approximation and change of kernel technique to transform the problem into a new reproducing kernel Hilbert space, where it becomes more tractable computationally. It enables us to obtain both a functional posterior of the latent intensity function and the covariance of the posterior, thus extending existing works that often focus on specific link functions or estimating the posterior mean. Using the result, we propose a BO framework based on the Gaussian Cox process model and further develop a Nystr\"om approximation for efficient computation. Extensive evaluations on various synthetic and real-world datasets demonstrate significant improvement over state-of-the-art inference solutions for Gaussian Cox processes, as well as effective BO with a wide range of acquisition functions designed through the underlying Gaussian Cox process model.

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of O left(varepsilon^{-(3α-2){(α-1)}} right) for finding an varepsilon-stationary point, where αin(1,2] denotes the heavy-tail index. For the inexact-polar setting with σ_1=0, we also provide guarantees that do not require prior knowledge of α. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.

We introduce Feasible Learning (FL), a sample-centric learning paradigm where models are trained by solving a feasibility problem that bounds the loss for each training sample. In contrast to the ubiquitous Empirical Risk Minimization (ERM) framework, which optimizes for average performance, FL demands satisfactory performance on every individual data point. Since any model that meets the prescribed performance threshold is a valid FL solution, the choice of optimization algorithm and its dynamics play a crucial role in shaping the properties of the resulting solutions. In particular, we study a primal-dual approach which dynamically re-weights the importance of each sample during training. To address the challenge of setting a meaningful threshold in practice, we introduce a relaxation of FL that incorporates slack variables of minimal norm. Our empirical analysis, spanning image classification, age regression, and preference optimization in large language models, demonstrates that models trained via FL can learn from data while displaying improved tail behavior compared to ERM, with only a marginal impact on average performance.

We study variance-dependent regret bounds for Markov decision processes (MDPs). Algorithms with variance-dependent regret guarantees can automatically exploit environments with low variance (e.g., enjoying constant regret on deterministic MDPs). The existing algorithms are either variance-independent or suboptimal. We first propose two new environment norms to characterize the fine-grained variance properties of the environment. For model-based methods, we design a variant of the MVP algorithm (Zhang et al., 2021a). We apply new analysis techniques to demonstrate that this algorithm enjoys variance-dependent bounds with respect to the norms we propose. In particular, this bound is simultaneously minimax optimal for both stochastic and deterministic MDPs, the first result of its kind. We further initiate the study on model-free algorithms with variance-dependent regret bounds by designing a reference-function-based algorithm with a novel capped-doubling reference update schedule. Lastly, we also provide lower bounds to complement our upper bounds.

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast O(n^2) per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein W_1, W_2 distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

The Information Bottleneck (IB) method frequently suffers from unstable optimization, characterized by abrupt representation shifts near critical points of the IB trade-off parameter, beta. In this paper, I introduce a novel approach to achieve stable and convex IB optimization through symbolic continuation and entropy-regularized trajectories. I analytically prove convexity and uniqueness of the IB solution path when an entropy regularization term is included, and demonstrate how this stabilizes representation learning across a wide range of eta values. Additionally, I provide extensive sensitivity analyses around critical points (beta) with statistically robust uncertainty quantification (95% confidence intervals). The open-source implementation, experimental results, and reproducibility framework included in this work offer a clear path for practical deployment and future extension of my proposed method.

We propose an extension of Thompson sampling to optimization problems over function spaces where the objective is a known functional of an unknown operator's output. We assume that queries to the operator (such as running a high-fidelity simulator or physical experiment) are costly, while functional evaluations on the operator's output are inexpensive. Our algorithm employs a sample-then-optimize approach using neural operator surrogates. This strategy avoids explicit uncertainty quantification by treating trained neural operators as approximate samples from a Gaussian process (GP) posterior. We derive regret bounds and theoretical results connecting neural operators with GPs in infinite-dimensional settings. Experiments benchmark our method against other Bayesian optimization baselines on functional optimization tasks involving partial differential equations of physical systems, demonstrating better sample efficiency and significant performance gains.

Recent Meta-learning for Black-Box Optimization (MetaBBO) methods harness neural networks to meta-learn configurations of traditional black-box optimizers. Despite their success, they are inevitably restricted by the limitations of predefined hand-crafted optimizers. In this paper, we present Symbol, a novel framework that promotes the automated discovery of black-box optimizers through symbolic equation learning. Specifically, we propose a Symbolic Equation Generator (SEG) that allows closed-form optimization rules to be dynamically generated for specific tasks and optimization steps. Within Symbol, we then develop three distinct strategies based on reinforcement learning, so as to meta-learn the SEG efficiently. Extensive experiments reveal that the optimizers generated by Symbol not only surpass the state-of-the-art BBO and MetaBBO baselines, but also exhibit exceptional zero-shot generalization abilities across entirely unseen tasks with different problem dimensions, population sizes, and optimization horizons. Furthermore, we conduct in-depth analyses of our Symbol framework and the optimization rules that it generates, underscoring its desirable flexibility and interpretability.

Recently, the impressive empirical success of policy gradient (PG) methods has catalyzed the development of their theoretical foundations. Despite the huge efforts directed at the design of efficient stochastic PG-type algorithms, the understanding of their convergence to a globally optimal policy is still limited. In this work, we develop improved global convergence guarantees for a general class of Fisher-non-degenerate parameterized policies which allows to address the case of continuous state action spaces. First, we propose a Normalized Policy Gradient method with Implicit Gradient Transport (N-PG-IGT) and derive a mathcal{O}(varepsilon^{-2.5}) sample complexity of this method for finding a global varepsilon-optimal policy. Improving over the previously known mathcal{O}(varepsilon^{-3}) complexity, this algorithm does not require the use of importance sampling or second-order information and samples only one trajectory per iteration. Second, we further improve this complexity to mathcal{mathcal{O} }(varepsilon^{-2}) by considering a Hessian-Aided Recursive Policy Gradient ((N)-HARPG) algorithm enhanced with a correction based on a Hessian-vector product. Interestingly, both algorithms are (i) simple and easy to implement: single-loop, do not require large batches of trajectories and sample at most two trajectories per iteration; (ii) computationally and memory efficient: they do not require expensive subroutines at each iteration and can be implemented with memory linear in the dimension of parameters.

We study the deterministic global optimization of trained Gaussian process posterior mean functions over hyperrectangular domains. Although the posterior mean function has a compact closed-form representation, its global optimization is challenging because it remains nonlinear and nonconvex. Existing exact deterministic approaches become increasingly difficult to scale as the number of training data points grows, leading to approximation-based methods that improve tractability by optimizing a modified (inexact) objective. In this work, we propose PALM-Mean, a piecewise-analytic lower-bounding framework embedded in reduced-space spatial branch-and-bound. At each node, kernel terms that are locally important are replaced by a sign-aware piecewise-linear relaxation in an appropriate scalar distance variable, while the remaining terms are bounded analytically in closed form. We show this hybrid approach yields a valid lower bound for the posterior mean, while limiting the size of the branch-and-bound subproblems. We establish validity of the node lower bounds and varepsilon-global convergence of the resulting algorithm. Computational results on synthetic benchmarks and real-world application problems show that PALM-Mean improves scalability relative to representative general-purpose deterministic global solvers, particularly as the number of training data points increases.

Plateaus, where an agent's performance stagnates at a suboptimal level, are a common problem in deep on-policy RL. Focusing on PPO due to its widespread adoption, we show that plateaus in certain regimes arise not because of known exploration, capacity, or optimization challenges, but because sample-based estimates of the loss eventually become poor proxies for the true objective over the course of training. As a recap, PPO switches between sampling rollouts from several parallel environments online using the current policy (which we call the outer loop) and performing repeated minibatch SGD steps against this offline dataset (the inner loop). In our work we consider only the outer loop, and conceptually model it as stochastic optimization. The step size is then controlled by the regularization strength towards the previous policy and the gradient noise by the number of samples collected between policy update steps. This model predicts that performance will plateau at a suboptimal level if the outer step size is too large relative to the noise. Recasting PPO in this light makes it clear that there are two ways to address this particular type of learning stagnation: either reduce the step size or increase the number of samples collected between updates. We first validate the predictions of our model and investigate how hyperparameter choices influence the step size and update noise, concluding that increasing the number of parallel environments is a simple and robust way to reduce both factors. Next, we propose a recipe for how to co-scale the other hyperparameters when increasing parallelization, and show that incorrectly doing so can lead to severe performance degradation. Finally, we vastly outperform prior baselines in a complex open-ended domain by scaling PPO to more than 1M parallel environments, thereby enabling monotonic performance improvement up to one trillion transitions.

Reinforcement Learning, particularly through policy gradient methods, has played a central role in enabling reasoning capabilities of Large Language Models. However, the optimization stability of policy gradients in this setting remains understudied. As a result, existing implementations often resort to conservative hyperparameter choices to ensure stability, which requires more training samples and increases computational costs. Hence, developing models for reliably tracking the underlying optimization dynamics and leveraging them into training enables more sample-efficient regimes and further unleashes scalable post-training. We address this gap by formalizing the stochastic optimization problem of policy gradients with explicit consideration of second-order geometry. We propose a tractable computational framework that tracks and leverages curvature information during policy updates. We further employ this framework to design interventions in the optimization process through data selection. The resultant algorithm, Curvature-Aware Policy Optimization (CAPO), identifies samples that contribute to unstable updates and masks them out. Theoretically, we establish monotonic improvement guarantees under realistic assumptions. On standard math reasoning benchmarks, we empirically show that CAPO ensures stable updates under aggressive learning regimes where baselines catastrophically fail. With minimal intervention (rejecting fewer than 8% of tokens), CAPO achieves up to 30x improvement in sample efficiency over standard GRPO for LLM reasoning.

We introduce Riemannian Lyapunov Optimizers (RLOs), a family of optimization algorithms that unifies classic optimizers within one geometric framework. Unlike heuristic improvements to existing optimizers, RLOs are systematically derived from a novel control-theoretic framework that reinterprets optimization as an extended state discrete-time controlled dynamical system on a Riemannian parameter manifold. Central to this framework is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: rapid alignment of the speed state to a target graph, followed by controlled evolution within it. We formalize this by constructing a strict Lyapunov function that certifies convergence to a target manifold. This perspective yields a constructive ``optimizer generator" that not only recovers classic algorithms but enables the principled design of RLOs. We validate our theory via geometric diagnostics and demonstrate that grounding optimizer design in control theory yields state-of-the-art performance in large-scale benchmarks. Overall, RLOs bridge control theory and modern machine learning optimization, providing a unified language and a systematic toolkit for designing stable, effective optimizers.

We present a feasibility-seeking approach to neural network training. This mathematical optimization framework is distinct from conventional gradient-based loss minimization and uses projection operators and iterative projection algorithms. We reformulate training as a large-scale feasibility problem: finding network parameters and states that satisfy local constraints derived from its elementary operations. Training then involves projecting onto these constraints, a local operation that can be parallelized across the network. We introduce PJAX, a JAX-based software framework that enables this paradigm. PJAX composes projection operators for elementary operations, automatically deriving the solution operators for the feasibility problems (akin to autodiff for derivatives). It inherently supports GPU/TPU acceleration, provides a familiar NumPy-like API, and is extensible. We train diverse architectures (MLPs, CNNs, RNNs) on standard benchmarks using PJAX, demonstrating its functionality and generality. Our results show that this approach is as a compelling alternative to gradient-based training, with clear advantages in parallelism and the ability to handle non-differentiable operations.

Algorithms for min-max optimization and variational inequalities are often studied under monotonicity assumptions. Motivated by non-monotone machine learning applications, we follow the line of works [Diakonikolas et al., 2021, Lee and Kim, 2021, Pethick et al., 2022, B\"ohm, 2022] aiming at going beyond monotonicity by considering the weaker negative comonotonicity assumption. In particular, we provide tight complexity analyses for the Proximal Point, Extragradient, and Optimistic Gradient methods in this setup, closing some questions on their working guarantees beyond monotonicity.

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. Y is to be predicted upon observing a (possibly high dimensional) random vector X by means of a predictive function f(X) as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function m(x)=E[Ymid X=x]. Under classic smoothness conditions combined with the assumption that the tails of Y-m(X) are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

We study optimization problems on Hadamard manifolds, motivated by recent advances in geometric approaches to optimization on curved spaces, particularly those involving the structure of Busemann functions. We introduce a projection based variant of the proximal point algorithm, termed the Busemann hybrid projection proximal point algorithm, which replaces Euclidean hyperplanes with horospheres defined via convex Busemann functions. The algorithm performs projections in closed form using the gradients of these functions, resulting in a geometrically intrinsic scheme that requires no tangent space linear solves. We allow for inexact subgradient evaluations and prove global convergence under controlled inexactness, with a relative error level strictly below one. We establish a Fejér type descent and sublinear complexity with a rate proportional to the inverse square root of the iteration count, and show that the exact variant coincides with the classical Riemannian proximal point algorithm. The framework clarifies the role of Busemann based subdifferentials in optimization on spaces of nonpositive curvature.

Optimization (PPO) has been positioned by recent literature as the canonical method for the RL part of RLHF. PPO performs well empirically but has a heuristic motivation and handles the KL-divergence constraint used in LM-RLHF in an ad-hoc manner and suffers form reward oscillations, entropy collapse, value function drift, and sudden policy divergence that require frequent restarts and extensive hyperparameter tuning. In this paper, we develop a new pure on policy actor-critic RL method for the LM-RLHF setting. We present SAFE (Stable Alignment Finetuning with Entropy-aware control),a novel RLHF algorithm that combines a Double Soft-Min Critic for pessimistic value estimation with a new multi-layer stabilization framework combining entropy-gated KL regulation, and PID-controlled adaptive thresholds. Unlike standard PPO's symmetric KL penalties, SAFE distinguishes high-entropy exploration from low-entropy mode collapse and adjusts penalties dynamically based on reward velocity. Experiments on a 3B parameter model show SAFE achieves +5.15\% training-average reward than PPO (0.725 vs 0.689), negligible reward crashes, and superior KL control than ppo . Our method adds minimal computational overhead and provides an interpretable, crash-resistant RLHF framework that maintains aggressive learning speed while ensuring stable long-horizon optimization suitable for production deployment. Code is available at https://github.com/ryyzn9/SAFE

Variance reduction (VR) techniques have contributed significantly to accelerating learning with massive datasets in the smooth and strongly convex setting (Schmidt et al., 2017; Johnson & Zhang, 2013; Roux et al., 2012). However, such techniques have not yet met the same success in the realm of large-scale deep learning due to various factors such as the use of data augmentation or regularization methods like dropout (Defazio & Bottou, 2019). This challenge has recently motivated the design of novel variance reduction techniques tailored explicitly for deep learning (Arnold et al., 2019; Ma & Yarats, 2018). This work is an additional step in this direction. In particular, we exploit the ubiquitous clustering structure of rich datasets used in deep learning to design a family of scalable variance reduced optimization procedures by combining existing optimizers (e.g., SGD+Momentum, Quasi Hyperbolic Momentum, Implicit Gradient Transport) with a multi-momentum strategy (Yuan et al., 2019). Our proposal leads to faster convergence than vanilla methods on standard benchmark datasets (e.g., CIFAR and ImageNet). It is robust to label noise and amenable to distributed optimization. We provide a parallel implementation in JAX.

Training neural networks requires optimizing a loss function that may be highly irregular, and in particular neither convex nor smooth. Popular training algorithms are based on stochastic gradient descent with momentum (SGDM), for which classical analysis applies only if the loss is either convex or smooth. We show that a very small modification to SGDM closes this gap: simply scale the update at each time point by an exponentially distributed random scalar. The resulting algorithm achieves optimal convergence guarantees. Intriguingly, this result is not derived by a specific analysis of SGDM: instead, it falls naturally out of a more general framework for converting online convex optimization algorithms to non-convex optimization algorithms.

We investigate safe multi-agent reinforcement learning, where agents seek to collectively maximize an aggregate sum of local objectives while satisfying their own safety constraints. The objective and constraints are described by {\it general utilities}, i.e., nonlinear functions of the long-term state-action occupancy measure, which encompass broader decision-making goals such as risk, exploration, or imitations. The exponential growth of the state-action space size with the number of agents presents challenges for global observability, further exacerbated by the global coupling arising from agents' safety constraints. To tackle this issue, we propose a primal-dual method utilizing shadow reward and κ-hop neighbor truncation under a form of correlation decay property, where κ is the communication radius. In the exact setting, our algorithm converges to a first-order stationary point (FOSP) at the rate of Oleft(T^{-2/3}right). In the sample-based setting, we demonstrate that, with high probability, our algorithm requires mathcal{O}left(ε^{-3.5}right) samples to achieve an ε-FOSP with an approximation error of O(φ_0^{2κ}), where φ_0in (0,1). Finally, we demonstrate the effectiveness of our model through extensive numerical experiments.

Bayesian optimisation is a popular, surrogate model-based approach for optimising expensive black-box functions. Given a surrogate model, the next location to expensively evaluate is chosen via maximisation of a cheap-to-query acquisition function. We present an epsilon-greedy procedure for Bayesian optimisation in batch settings in which the black-box function can be evaluated multiple times in parallel. Our epsilon-shotgun algorithm leverages the model's prediction, uncertainty, and the approximated rate of change of the landscape to determine the spread of batch solutions to be distributed around a putative location. The initial target location is selected either in an exploitative fashion on the mean prediction, or -- with probability epsilon -- from elsewhere in the design space. This results in locations that are more densely sampled in regions where the function is changing rapidly and in locations predicted to be good (i.e close to predicted optima), with more scattered samples in regions where the function is flatter and/or of poorer quality. We empirically evaluate the epsilon-shotgun methods on a range of synthetic functions and two real-world problems, finding that they perform at least as well as state-of-the-art batch methods and in many cases exceed their performance.

In this paper, we study the predict-then-optimize problem where the output of a machine learning prediction task is used as the input of some downstream optimization problem, say, the objective coefficient vector of a linear program. The problem is also known as predictive analytics or contextual linear programming. The existing approaches largely suffer from either (i) optimization intractability (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii) requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach to the problem called maximum optimality margin which designs the machine learning loss function by the optimality condition of the downstream optimization. The max-margin formulation enjoys both computational efficiency and good theoretical properties for the learning procedure. More importantly, our new approach only needs the observations of the optimal solution in the training data rather than the objective function, which makes it a new and natural approach to the inverse linear programming problem under both contextual and context-free settings; we also analyze the proposed method under both offline and online settings, and demonstrate its performance using numerical experiments.

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow O(1{T}) last-iterate convergence rate to a Nash equilibrium. While the O(1{T}) rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal O(T) regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal O(1{T}) last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an O(log T) individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art O(T) bound.