Daily Papers

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

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.

We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily specialized to accelerated SGD under the strong growth condition. In this special case, our analysis reduces the dependence on the strong growth constant from rho to rho as compared to prior work. This improvement is comparable to a square-root of the condition number in the worst case and address criticism that guarantees for stochastic acceleration could be worse than those for SGD.

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.

Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate O(1/{T}^2) with T the number of steps, improving upon the O(1/T) rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate O(1/T), outperforming the rate O(1/T) for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates ell_2-accurate score estimates, and does not require log-concavity or smoothness on the target distribution.

Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In this paper we propose a novel, robust and accelerated stochastic optimizer that relies on two key elements: (1) an accelerated Nesterov-like Stochastic Differential Equation (SDE) and (2) its semi-implicit Gauss-Seidel type discretization. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively in the case of the minimization of a quadratic function. This analysis allows us to come up with an optimal learning rate in terms of the convergence rate while ensuring the stability of NAG-GS. This is achieved by the careful analysis of the spectral radius of the iteration matrix and the covariance matrix at stationarity with respect to all hyperparameters of our method. Further, we show that NAG- GS is competitive with state-of-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models such as the logistic regression model, the residual networks models on standard computer vision datasets, Transformers in the frame of the GLUE benchmark and the recent Vision Transformers.

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent work has shown that these costs can be reduced by line search methods that iteratively adjust the stepsize. We propose an alternative approach to stochastic line search by using a new algorithm based on forward step model building. This model building step incorporates second-order information that allows adjusting not only the stepsize but also the search direction. Noting that deep learning model parameters come in groups (layers of tensors), our method builds its model and calculates a new step for each parameter group. This novel diagonalization approach makes the selected step lengths adaptive. We provide convergence rate analysis, and experimentally show that the proposed algorithm achieves faster convergence and better generalization in well-known test problems. More precisely, SMB requires less tuning, and shows comparable performance to other adaptive methods.

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.

Despite the success of adaptive time-stepping in ODE simulation, it has so far seen few applications for Stochastic Differential Equations (SDEs). To simulate SDEs adaptively, methods such as the Virtual Brownian Tree (VBT) have been developed, which can generate Brownian motion (BM) non-chronologically. However, in most applications, knowing only the values of Brownian motion is not enough to achieve a high order of convergence; for that, we must compute time-integrals of BM such as int_s^t W_r , dr. With the aim of using high order SDE solvers adaptively, we extend the VBT to generate these integrals of BM in addition to the Brownian increments. A JAX-based implementation of our construction is included in the popular Diffrax library (https://github.com/patrick-kidger/diffrax). Since the entire Brownian path produced by VBT is uniquely determined by a single PRNG seed, previously generated samples need not be stored, which results in a constant memory footprint and enables experiment repeatability and strong error estimation. Based on binary search, the VBT's time complexity is logarithmic in the tolerance parameter varepsilon. Unlike the original VBT algorithm, which was only precise at some dyadic times, we prove that our construction exactly matches the joint distribution of the Brownian motion and its time integrals at any query times, provided they are at least varepsilon apart. We present two applications of adaptive high order solvers enabled by our new VBT. Using adaptive solvers to simulate a high-volatility CIR model, we achieve more than twice the convergence order of constant stepping. We apply an adaptive third order underdamped or kinetic Langevin solver to an MCMC problem, where our approach outperforms the No U-Turn Sampler, while using only a tenth of its function evaluations.

We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate O(1/k^2) on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of O(1/k) on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.

Zeroth-order optimization addresses problems where gradient information is inaccessible or impractical to compute. While most existing methods rely on first-order approximations, incorporating second-order (curvature) information can, in principle, significantly accelerate convergence. However, the high cost of function evaluations required to estimate Hessian matrices often limits practical applicability. We present the subspace-based approximate Hessian (ZO-SAH) method, a zeroth-order optimization algorithm that mitigates these costs by focusing on randomly selected two-dimensional subspaces. Within each subspace, ZO-SAH estimates the Hessian by fitting a quadratic polynomial to the objective function and extracting its second-order coefficients. To further reduce function-query costs, ZO-SAH employs a periodic subspace-switching strategy that reuses function evaluations across optimization steps. Experiments on eight benchmark datasets, including logistic regression and deep neural network training tasks, demonstrate that ZO-SAH achieves significantly faster convergence than existing zeroth-order methods.

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}).

In this book chapter, we briefly describe the main components that constitute the gradient descent method and its accelerated and stochastic variants. We aim at explaining these components from a mathematical point of view, including theoretical and practical aspects, but at an elementary level. We will focus on basic variants of the gradient descent method and then extend our view to recent variants, especially variance-reduced stochastic gradient schemes (SGD). Our approach relies on revealing the structures presented inside the problem and the assumptions imposed on the objective function. Our convergence analysis unifies several known results and relies on a general, but elementary recursive expression. We have illustrated this analysis on several common schemes.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every d iterations, where d is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor d.

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.

Second-order optimization has been developed to accelerate the training of deep neural networks and it is being applied to increasingly larger-scale models. In this study, towards training on further larger scales, we identify a specific parameterization for second-order optimization that promotes feature learning in a stable manner even if the network width increases significantly. Inspired by a maximal update parameterization, we consider a one-step update of the gradient and reveal the appropriate scales of hyperparameters including random initialization, learning rates, and damping terms. Our approach covers two major second-order optimization algorithms, K-FAC and Shampoo, and we demonstrate that our parameterization achieves higher generalization performance in feature learning. In particular, it enables us to transfer the hyperparameters across models with different widths.

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.

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 several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For the case of neural network weight matrices, we propose maintaining only the per-row and per-column sums of these moving averages, and estimating the per-parameter second moments based on these sums. We demonstrate empirically that this method produces similar results to the baseline. Secondly, we show that adaptive methods can produce larger-than-desired updates when the decay rate of the second moment accumulator is too slow. We propose update clipping and a gradually increasing decay rate scheme as remedies. Combining these methods and dropping momentum, we achieve comparable results to the published Adam regime in training the Transformer model on the WMT 2014 English-German machine translation task, while using very little auxiliary storage in the optimizer. Finally, we propose scaling the parameter updates based on the scale of the parameters themselves.

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.

We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and Adam. The main disadvantage of traditional second order methods is their heavier per iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in ADAHESSIAN, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that ADAHESSIAN achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of Adam. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that ADAHESSIAN: (i) achieves 1.80%/1.45% higher accuracy on ResNets20/32 on Cifar10, and 5.55% higher accuracy on ImageNet as compared to Adam; (ii) outperforms AdamW for transformers by 0.13/0.33 BLEU score on IWSLT14/WMT14 and 2.7/1.0 PPL on PTB/Wikitext-103; (iii) outperforms AdamW for SqueezeBert by 0.41 points on GLUE; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of ADAHESSIAN is comparable to first order methods, and that it exhibits robustness towards its hyperparameters.

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

Approximate second-order optimization methods often exhibit poorer generalization compared to first-order approaches. In this work, we look into this issue through the lens of the loss landscape and find that existing second-order methods tend to converge to sharper minima compared to SGD. In response, we propose Sassha, a novel second-order method designed to enhance generalization by explicitly reducing sharpness of the solution, while stabilizing the computation of approximate Hessians along the optimization trajectory. In fact, this sharpness minimization scheme is crafted also to accommodate lazy Hessian updates, so as to secure efficiency besides flatness. To validate its effectiveness, we conduct a wide range of standard deep learning experiments where Sassha demonstrates its outstanding generalization performance that is comparable to, and mostly better than, other methods. We provide a comprehensive set of analyses including convergence, robustness, stability, efficiency, and cost.

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.

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 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.

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.

An increasing number of studies have focused on stochastic first-order methods (SFOMs) under heavy-tailed gradient noises, which have been observed in the training of practical deep learning models. In this paper, we focus on two types of gradient noises: one is sub-Weibull noise, and the other is noise under the assumption that it has a bounded p-th central moment (p-BCM) with pin (1, 2]. The latter is more challenging due to the occurrence of infinite variance when pin (1, 2). Under these two gradient noise assumptions, the in-expectation and high-probability convergence of SFOMs have been extensively studied in the contexts of convex optimization and standard smooth optimization. However, for weakly convex objectives-a class that includes all Lipschitz-continuous convex objectives and smooth objectives-our understanding of the in-expectation and high-probability convergence of SFOMs under these two types of noises remains incomplete. We investigate the high-probability convergence of the vanilla stochastic subgradient descent (SsGD) method under sub-Weibull noises, as well as the high-probability and in-expectation convergence of clipped SsGD under the p-BCM noises. Both analyses are conducted in the context of weakly convex optimization. For weakly convex objectives that may be non-convex and non-smooth, our results demonstrate that the theoretical dependence of vanilla SsGD on the failure probability and number of iterations under sub-Weibull noises does not degrade compared to the case of smooth objectives. Under p-BCM noises, our findings indicate that the non-smoothness and non-convexity of weakly convex objectives do not impact the theoretical dependence of clipped SGD on the failure probability relative to the smooth case; however, the sample complexity we derived is worse than a well-known lower bound for smooth optimization.

This paper studies the fitting of Hessian or its inverse with stochastic Hessian-vector products. A Hessian fitting criterion, which can be used to derive most of the commonly used methods, e.g., BFGS, Gaussian-Newton, AdaGrad, etc., is used for the analysis. Our studies reveal different convergence rates for different Hessian fitting methods, e.g., sublinear rates for gradient descent in the Euclidean space and a commonly used closed-form solution, linear rates for gradient descent on the manifold of symmetric positive definite (SPL) matrices and certain Lie groups. The Hessian fitting problem is further shown to be strongly convex under mild conditions on a specific yet general enough Lie group. To confirm our analysis, these methods are tested under different settings like noisy Hessian-vector products, time varying Hessians, and low precision arithmetic. These findings are useful for stochastic second order optimizations that rely on fast, robust and accurate Hessian estimations.

We present a novel approach to accelerate stochastic gradient descent (SGD) by utilizing curvature information obtained from Hessian-vector products or finite differences of parameters and gradients, similar to the BFGS algorithm. Our approach involves two preconditioners: a matrix-free preconditioner and a low-rank approximation preconditioner. We update both preconditioners online using a criterion that is robust to stochastic gradient noise and does not require line search or damping. To preserve the corresponding symmetry or invariance, our preconditioners are constrained to certain connected Lie groups. The Lie group's equivariance property simplifies the preconditioner fitting process, while its invariance property eliminates the need for damping, which is commonly required in second-order optimizers. As a result, the learning rate for parameter updating and the step size for preconditioner fitting are naturally normalized, and their default values work well in most scenarios. Our proposed approach offers a promising direction for improving the convergence of SGD with low computational overhead. We demonstrate that Preconditioned SGD (PSGD) outperforms SoTA on Vision, NLP, and RL tasks across multiple modern deep-learning architectures. We have provided code for reproducing toy and large scale experiments in this paper.

We propose an abstract framework to establish the convergence of the iterates of stochastic versions of a broad range of monotone operator splitting methods in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The proposed analysis involves a reduced inclusion model with two operators. At each iteration, stochastic approximations to points in the graphs of these two operators are used to form the update. The results are applied to derive the almost sure and L^2 convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations.

Shampoo is one of the leading approximate second-order optimizers: a variant of it has won the MLCommons AlgoPerf competition, and it has been shown to produce models with lower activation outliers that are easier to compress. Yet, applying Shampoo currently comes at the cost of significant computational slowdown, due to its expensive internal operations. In this paper, we take a significant step to address this shortcoming by proposing \method (for Distributed Accelerated SHampoo), a faster implementation of Distributed Shampoo based on two main new techniques: First, we show that preconditioner blocks can be stacked into 3D tensors to significantly improve GPU utilization; second, we introduce the Newton-DB iteration and the Chebyshev polynomial approximations as novel and faster approaches for computing the inverse matrix roots required by Shampoo. Along with these algorithmic contributions, we provide a first in-depth analysis of how matrix scaling critically affects Shampoo convergence. On the practical side, our GPU-aware implementation achieves up to 4.83times faster optimizer steps compared to the well-optimized Distributed Shampoo, while Newton-DB attains the lowest validation perplexity per iteration among all tested methods. Our code is available at https://github.com/IST-DASLab/DASH.

We introduce a novel framework for efficient sampling from complex, unnormalised target distributions by exploiting multiscale dynamics. Traditional score-based sampling methods either rely on learned approximations of the score function or involve computationally expensive nested Markov chain Monte Carlo (MCMC) loops. In contrast, the proposed approach leverages stochastic averaging within a slow-fast system of stochastic differential equations (SDEs) to estimate intermediate scores along a diffusion path without training or inner-loop MCMC. Two algorithms are developed under this framework: MultALMC, which uses multiscale annealed Langevin dynamics, and MultCDiff, based on multiscale controlled diffusions for the reverse-time Ornstein-Uhlenbeck process. Both overdamped and underdamped variants are considered, with theoretical guarantees of convergence to the desired diffusion path. The framework is extended to handle heavy-tailed target distributions using Student's t-based noise models and tailored fast-process dynamics. Empirical results across synthetic and real-world benchmarks, including multimodal and high-dimensional distributions, demonstrate that the proposed methods are competitive with existing samplers in terms of accuracy and efficiency, without the need for learned models.

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.

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.

We introduce a new class of algorithms, Stochastic Generalized Method of Moments (SGMM), for estimation and inference on (overidentified) moment restriction models. Our SGMM is a novel stochastic approximation alternative to the popular Hansen (1982) (offline) GMM, and offers fast and scalable implementation with the ability to handle streaming datasets in real time. We establish the almost sure convergence, and the (functional) central limit theorem for the inefficient online 2SLS and the efficient SGMM. Moreover, we propose online versions of the Durbin-Wu-Hausman and Sargan-Hansen tests that can be seamlessly integrated within the SGMM framework. Extensive Monte Carlo simulations show that as the sample size increases, the SGMM matches the standard (offline) GMM in terms of estimation accuracy and gains over computational efficiency, indicating its practical value for both large-scale and online datasets. We demonstrate the efficacy of our approach by a proof of concept using two well known empirical examples with large sample sizes.

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.

This paper presents a comprehensive study on the unified module for accelerating stable-diffusion processes, specifically focusing on the lcm-lora module. Stable-diffusion processes play a crucial role in various scientific and engineering domains, and their acceleration is of paramount importance for efficient computational performance. The standard iterative procedures for solving fixed-source discrete ordinates problems often exhibit slow convergence, particularly in optically thick scenarios. To address this challenge, unconditionally stable diffusion-acceleration methods have been developed, aiming to enhance the computational efficiency of transport equations and discrete ordinates problems. This study delves into the theoretical foundations and numerical results of unconditionally stable diffusion synthetic acceleration methods, providing insights into their stability and performance for model discrete ordinates problems. Furthermore, the paper explores recent advancements in diffusion model acceleration, including on device acceleration of large diffusion models via gpu aware optimizations, highlighting the potential for significantly improved inference latency. The results and analyses in this study provide important insights into stable diffusion processes and have important ramifications for the creation and application of acceleration methods specifically, the lcm-lora module in a variety of computing environments.

We study first-order methods for convex optimization problems with functions f satisfying the recently proposed ell-smoothness condition ||nabla^{2}f(x)|| le ellleft(||nabla f(x)||right), which generalizes the L-smoothness and (L_{0},L_{1})-smoothness. While accelerated gradient descent AGD is known to reach the optimal complexity O(L R / varepsilon) under L-smoothness, where varepsilon is an error tolerance and R is the distance between a starting and an optimal point, existing extensions to ell-smoothness either incur extra dependence on the initial gradient, suffer exponential factors in L_{1} R, or require costly auxiliary sub-routines, leaving open whether an AGD-type O(ell(0) R / varepsilon) rate is possible for small-varepsilon, even in the (L_{0},L_{1})-smoothness case. We resolve this open question. Leveraging a new Lyapunov function and designing new algorithms, we achieve O(ell(0) R / varepsilon) oracle complexity for small-varepsilon and virtually any ell. For instance, for (L_{0},L_{1})-smoothness, our bound O(L_0 R / varepsilon) is provably optimal in the small-varepsilon regime and removes all non-constant multiplicative factors present in prior accelerated algorithms.

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.

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

Diffusion models have demonstrated remarkable success in generative tasks, yet their iterative denoising process results in slow inference, limiting their practicality. While existing acceleration methods exploit the well-known U-shaped similarity pattern between adjacent steps through caching mechanisms, they lack theoretical foundation and rely on simplistic computation reuse, often leading to performance degradation. In this work, we provide a theoretical understanding by analyzing the denoising process through the second-order Adams-Bashforth method, revealing a linear relationship between the outputs of consecutive steps. This analysis explains why the outputs of adjacent steps exhibit a U-shaped pattern. Furthermore, extending Adams-Bashforth method to higher order, we propose a novel caching-based acceleration approach for diffusion models, instead of directly reusing cached results, with a truncation error bound of only \(O(h^k)\) where h is the step size. Extensive validation across diverse image and video diffusion models (including HunyuanVideo and FLUX.1-dev) with various schedulers demonstrates our method's effectiveness in achieving nearly 3times speedup while maintaining original performance levels, offering a practical real-time solution without compromising generation quality.

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide mathcal{O}(kappa) accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.

This work proposes a Momentum-Enabled Kronecker-Factor-Based Optimizer Using Rank-1 updates, called MKOR, that improves the training time and convergence properties of deep neural networks (DNNs). Second-order techniques, while enjoying higher convergence rates vs first-order counterparts, have cubic complexity with respect to either the model size and/or the training batch size. Hence they exhibit poor scalability and performance in transformer models, e.g. large language models (LLMs), because the batch sizes in these models scale by the attention mechanism sequence length, leading to large model size and batch sizes. MKOR's complexity is quadratic with respect to the model size, alleviating the computation bottlenecks in second-order methods. Because of their high computation complexity, state-of-the-art implementations of second-order methods can only afford to update the second order information infrequently, and thus do not fully exploit the promise of better convergence from these updates. By reducing the communication complexity of the second-order updates as well as achieving a linear communication complexity, MKOR increases the frequency of second order updates. We also propose a hybrid version of MKOR (called MKOR-H) that mid-training falls backs to a first order optimizer if the second order updates no longer accelerate convergence. Our experiments show that MKOR outperforms state -of-the-art first order methods, e.g. the LAMB optimizer, and best implementations of second-order methods, i.e. KAISA/KFAC, up to 2.57x and 1.85x respectively on BERT-Large-Uncased on 64 GPUs.

This work studies the convergence of two-timescale stochastic approximations (SA), a class of iterative algorithms that update two sets of parameters in fast and slow timescales respectively. Notable examples of two-timescale SA in reinforcement learning (RL) include temporal difference learning with gradient correction (TDC) and actor-critic methods. Previously, the stability (i.e., boundedness) and convergence of two-timescale SA were only established under i.i.d. noise. This work instead establishes the stability and convergence of two-timescale SA under Markovian noise, a setup that is more realistic in RL. Notably, we do not need to use any projection operator and the noise does not need to live in a compact space. Our key technical novelty is to control the fast timescale parameter with the running max of the slow timescale parameter, instead of with the current slow timescale parameter, as most prior works do. As a key application, we establish the first almost sure convergence of TDC with eligibility traces under off-policy learning with linear function approximation.

Sampling from the posterior distribution poses a major computational challenge in solving inverse problems using latent diffusion models. Common methods rely on Tweedie's first-order moments, which are known to induce a quality-limiting bias. Existing second-order approximations are impractical due to prohibitive computational costs, making standard reverse diffusion processes intractable for posterior sampling. This paper introduces Second-order Tweedie sampler from Surrogate Loss (STSL), a novel sampler that offers efficiency comparable to first-order Tweedie with a tractable reverse process using second-order approximation. Our theoretical results reveal that the second-order approximation is lower bounded by our surrogate loss that only requires O(1) compute using the trace of the Hessian, and by the lower bound we derive a new drift term to make the reverse process tractable. Our method surpasses SoTA solvers PSLD and P2L, achieving 4X and 8X reduction in neural function evaluations, respectively, while notably enhancing sampling quality on FFHQ, ImageNet, and COCO benchmarks. In addition, we show STSL extends to text-guided image editing and addresses residual distortions present from corrupted images in leading text-guided image editing methods. To our best knowledge, this is the first work to offer an efficient second-order approximation in solving inverse problems using latent diffusion and editing real-world images with corruptions.

Popular machine learning approaches forgo second-order information due to the difficulty of computing curvature in high dimensions. We present FOSI, a novel meta-algorithm that improves the performance of any base first-order optimizer by efficiently incorporating second-order information during the optimization process. In each iteration, FOSI implicitly splits the function into two quadratic functions defined on orthogonal subspaces, then uses a second-order method to minimize the first, and the base optimizer to minimize the other. We formally analyze FOSI's convergence and the conditions under which it improves a base optimizer. Our empirical evaluation demonstrates that FOSI improves the convergence rate and optimization time of first-order methods such as Heavy-Ball and Adam, and outperforms second-order methods (K-FAC and L-BFGS).

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.

Recent years have witnessed the rapid progress and broad application of diffusion probabilistic models (DPMs). Sampling from DPMs can be viewed as solving an ordinary differential equation (ODE). Despite the promising performance, the generation of DPMs usually consumes much time due to the large number of function evaluations (NFE). Though recent works have accelerated the sampling to around 20 steps with high-order solvers, the sample quality with less than 10 NFE can still be improved. In this paper, we propose a unified sampling framework (USF) to study the optional strategies for solver. Under this framework, we further reveal that taking different solving strategies at different timesteps may help further decrease the truncation error, and a carefully designed solver schedule has the potential to improve the sample quality by a large margin. Therefore, we propose a new sampling framework based on the exponential integral formulation that allows free choices of solver strategy at each step and design specific decisions for the framework. Moreover, we propose S^3, a predictor-based search method that automatically optimizes the solver schedule to get a better time-quality trade-off of sampling. We demonstrate that S^3 can find outstanding solver schedules which outperform the state-of-the-art sampling methods on CIFAR-10, CelebA, ImageNet, and LSUN-Bedroom datasets. Specifically, we achieve 2.69 FID with 10 NFE and 6.86 FID with 5 NFE on CIFAR-10 dataset, outperforming the SOTA method significantly. We further apply S^3 to Stable-Diffusion model and get an acceleration ratio of 2times, showing the feasibility of sampling in very few steps without retraining the neural network.

We address the problem of optimizing a Brownian motion. We consider a (random) realization W of a Brownian motion with input space in [0,1]. Given W, our goal is to return an ε-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order log^2(1/ε). This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive---each query depends on previous values---and is an instance of the optimism-in-the-face-of-uncertainty principle.

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve mgg 1 lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling I blocks and sampling B samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of O(mepsilon^{-3I(I<m)}{II} + mepsilon^{-3}{IB}) for finding an epsilon-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schr\"{o}dinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schr\"{o}dinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.

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.

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 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 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.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

Efficiently approximating local curvature information of the loss function is a key tool for optimization and compression of deep neural networks. Yet, most existing methods to approximate second-order information have high computational or storage costs, which can limit their practicality. In this work, we investigate matrix-free, linear-time approaches for estimating Inverse-Hessian Vector Products (IHVPs) for the case when the Hessian can be approximated as a sum of rank-one matrices, as in the classic approximation of the Hessian by the empirical Fisher matrix. We propose two new algorithms as part of a framework called M-FAC: the first algorithm is tailored towards network compression and can compute the IHVP for dimension d, if the Hessian is given as a sum of m rank-one matrices, using O(dm^2) precomputation, O(dm) cost for computing the IHVP, and query cost O(m) for any single element of the inverse Hessian. The second algorithm targets an optimization setting, where we wish to compute the product between the inverse Hessian, estimated over a sliding window of optimization steps, and a given gradient direction, as required for preconditioned SGD. We give an algorithm with cost O(dm + m^2) for computing the IHVP and O(dm + m^3) for adding or removing any gradient from the sliding window. These two algorithms yield state-of-the-art results for network pruning and optimization with lower computational overhead relative to existing second-order methods. Implementations are available at [9] and [17].

A popular approach to sample a diffusion-based generative model is to solve an ordinary differential equation (ODE). In existing samplers, the coefficients of the ODE solvers are pre-determined by the ODE formulation, the reverse discrete timesteps, and the employed ODE methods. In this paper, we consider accelerating several popular ODE-based sampling processes (including EDM, DDIM, and DPM-Solver) by optimizing certain coefficients via improved integration approximation (IIA). We propose to minimize, for each time step, a mean squared error (MSE) function with respect to the selected coefficients. The MSE is constructed by applying the original ODE solver for a set of fine-grained timesteps, which in principle provides a more accurate integration approximation in predicting the next diffusion state. The proposed IIA technique does not require any change of a pre-trained model, and only introduces a very small computational overhead for solving a number of quadratic optimization problems. Extensive experiments show that considerably better FID scores can be achieved by using IIA-EDM, IIA-DDIM, and IIA-DPM-Solver than the original counterparts when the neural function evaluation (NFE) is small (i.e., less than 25).

We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar {\alpha}, is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves O(1/{\alpha}) decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate O(1/{\alpha}^2) - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings.

Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency. For a fixed stepsize, the iterates typically admit a stationary distribution that is rarely tractable. Prior work shows that as the stepsize αdownarrow 0, the centered-and-scaled steady state converges weakly to a Gaussian random vector. However, for fixed α, this weak convergence offers no usable error bound for approximating the steady-state by its Gaussian limit. This paper provides explicit, non-asymptotic error bounds for fixed α. We first prove general-purpose theorems that bound the Wasserstein distance between the centered-scaled steady state and an appropriate Gaussian distribution, under regularity conditions for drift and moment conditions for noise. To ensure broad applicability, we cover both i.i.d. and Markovian noise models. We then instantiate these theorems for three representative SA settings: (1) stochastic gradient descent (SGD) for smooth strongly convex objectives, (2) linear SA, and (3) contractive nonlinear SA. We obtain dimension- and stepsize-dependent, explicit bounds in Wasserstein distance of order α^{1/2}log(1/α) for small α. Building on the Wasserstein approximation error, we further derive non-uniform Berry--Esseen-type tail bounds that compare the steady-state tail probability to Gaussian tails. We achieve an explicit error term that decays in both the deviation level and stepsize α. We adapt the same analysis for SGD beyond strongly convexity and study general convex objectives. We identify a non-Gaussian (Gibbs) limiting law under the correct scaling, which is validated numerically, and provide a corresponding pre-limit Wasserstein error bound.

Generating samples from limited information is a fundamental problem across scientific domains. Classical maximum entropy methods provide principled uncertainty quantification from moment constraints but require sampling via MCMC or Langevin dynamics, which typically exhibit exponential slowdown in high dimensions. In contrast, generative models based on diffusion and flow matching efficiently transport noise to data but offer limited theoretical guarantees and can overfit when data is scarce. We introduce Moment Guided Diffusion (MGD), which combines elements of both approaches. Building on the stochastic interpolant framework, MGD samples maximum entropy distributions by solving a stochastic differential equation that guides moments toward prescribed values in finite time, thereby avoiding slow mixing in equilibrium-based methods. We formally obtain, in the large-volatility limit, convergence of MGD to the maximum entropy distribution and derive a tractable estimator of the resulting entropy computed directly from the dynamics. Applications to financial time series, turbulent flows, and cosmological fields using wavelet scattering moments yield estimates of negentropy for high-dimensional multiscale processes.

Classical estimators for ARIMA parameters (MLE, CSS, OLS) assume Gaussian innovations, an assumption frequently violated in financial and economic data exhibiting asymmetric distributions with heavy tails. We develop and validate the second-order polynomial maximization method (PMM2) for estimating ARIMA(p,d,q) models with non-Gaussian innovations. PMM2 is a semiparametric technique that exploits higher-order moments and cumulants without requiring full distributional specification. Monte Carlo experiments (128,000 simulations) across sample sizes N in {100, 200, 500, 1000} and four innovation distributions demonstrate that PMM2 substantially outperforms classical methods for asymmetric innovations. For ARIMA(1,1,0) with N=500, relative efficiency reaches 1.58--1.90 for Gamma, lognormal, and χ^2(3) innovations (37--47\% variance reduction). Under Gaussian innovations PMM2 matches OLS efficiency, avoiding the precision loss typical of robust estimators. The method delivers major gains for moderate asymmetry (|γ_3| geq 0.5) and N geq 200, with computational costs comparable to MLE. PMM2 provides an effective alternative for time series with asymmetric innovations typical of financial markets, macroeconomic indicators, and industrial measurements. Future extensions include seasonal SARIMA models, GARCH integration, and automatic order selection.

In training neural networks, it is common practice to use partial gradients computed over batches, mostly very small subsets of the training set. This approach is motivated by the argument that such a partial gradient is close to the true one, with precision growing only with the square root of the batch size. A theoretical justification is with the help of stochastic approximation theory. However, the conditions for the validity of this theory are not satisfied in the usual learning rate schedules. Batch processing is also difficult to combine with efficient second-order optimization methods. This proposal is based on another hypothesis: the loss minimum of the training set can be expected to be well-approximated by the minima of its subsets. Such subset minima can be computed in a fraction of the time necessary for optimizing over the whole training set. This hypothesis has been tested with the help of the MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks, optionally extended by training data augmentation. The experiments have confirmed that results equivalent to conventional training can be reached. In summary, even small subsets are representative if the overdetermination ratio for the given model parameter set sufficiently exceeds unity. The computing expense can be reduced to a tenth or less.

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called Walk-on-Spheres Neural Operator (WoS-NO) which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75times improvement in L_2-error compared to standard physics-informed training schemes, up to 6.31times improvement in training speed, and reductions of up to 2.97times in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs, as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in 100,000 effective dimensions in 12 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. Its continuous-time formulation can be interpreted as an interacting particle system driven by a Kalman-type preconditioned descent direction. A well-known limitation of this dynamics is the possible premature collapse of the covariance of the ensemble, which makes the method sensitive to the initial ensemble. We introduce a second-order particle system in which the particles evolve according to an inertial dynamics. The model combines a Kalman-type relaxation force with damping, attraction towards the ensemble mean, and a short-range repulsive interaction designed to counteract ensemble collapse. The resulting dynamics can be interpreted as a heavy-ball reformulation of continuous-time EKI enriched by competing attractive and repulsive mechanisms. For linear inverse problems, we analyze the induced mean and fluctuation dynamics and identify a parameter regime in which fully collapsed configurations are linearly unstable. We further characterize asymptotic equilibria through a constrained optimality condition on the subspace retained by the limiting ensemble covariance and derive an exponential decay estimate. Numerical experiments illustrate the effect of inertia and repulsion on the ensemble dynamics and compare the proposed second-order method with first-order EKI-type

Training learned image compression (LIC) models entails navigating a challenging optimization landscape defined by the fundamental trade-off between rate and distortion. Standard first-order optimizers, such as SGD and Adam, struggle with gradient conflicts arising from competing objectives, leading to slow convergence and suboptimal rate-distortion performance. In this work, we demonstrate that a simple utilization of a second-order quasi-Newton optimizer, SOAP, dramatically improves both training efficiency and final performance across diverse LICs. Our theoretical and empirical analyses reveal that Newton preconditioning inherently resolves the intra-step and inter-step update conflicts intrinsic to the R-D objective, facilitating faster, more stable convergence. Beyond acceleration, we uncover a critical deployability benefit: second-order trained models exhibit significantly fewer activation and latent outliers. This substantially enhances robustness to post-training quantization. Together, these results establish second-order optimization, achievable as a seamless drop-in replacement of the imported optimizer, as a powerful, practical tool for advancing the efficiency and real-world readiness of LICs.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

Approximating Stochastic Gradient Descent (SGD) as a Stochastic Differential Equation (SDE) has allowed researchers to enjoy the benefits of studying a continuous optimization trajectory while carefully preserving the stochasticity of SGD. Analogous study of adaptive gradient methods, such as RMSprop and Adam, has been challenging because there were no rigorously proven SDE approximations for these methods. This paper derives the SDE approximations for RMSprop and Adam, giving theoretical guarantees of their correctness as well as experimental validation of their applicability to common large-scaling vision and language settings. A key practical result is the derivation of a square root scaling rule to adjust the optimization hyperparameters of RMSprop and Adam when changing batch size, and its empirical validation in deep learning settings.

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.

Iterative methods are ubiquitous in large-scale scientific computing applications, and a number of approaches based on meta-learning have been recently proposed to accelerate them. However, a systematic study of these approaches and how they differ from meta-learning is lacking. In this paper, we propose a framework to analyze such learning-based acceleration approaches, where one can immediately identify a departure from classical meta-learning. We show that this departure may lead to arbitrary deterioration of model performance. Based on our analysis, we introduce a novel training method for learning-based acceleration of iterative methods. Furthermore, we theoretically prove that the proposed method improves upon the existing methods, and demonstrate its significant advantage and versatility through various numerical applications.

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.

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.

We analyze the Accelerated Noisy Power Method, an algorithm for Principal Component Analysis in the setting where only inexact matrix-vector products are available, which can arise for instance in decentralized PCA. While previous works have established that acceleration can improve convergence rates compared to the standard Noisy Power Method, these guarantees require overly restrictive upper bounds on the magnitude of the perturbations, limiting their practical applicability. We provide an improved analysis of this algorithm, which preserves the accelerated convergence rate under much milder conditions on the perturbations. We show that our new analysis is worst-case optimal, in the sense that the convergence rate cannot be improved, and that the noise conditions we derive cannot be relaxed without sacrificing convergence guarantees. We demonstrate the practical relevance of our results by deriving an accelerated algorithm for decentralized PCA, which has similar communication costs to non-accelerated methods. To our knowledge, this is the first decentralized algorithm for PCA with provably accelerated convergence.

This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an O(d/epsilon) mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known O(d/epsilon) result and is optimal (in terms of order) in both dimension d and accuracy tolerance epsilon for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.

Generative models have demonstrated remarkable success in domains such as text, image, and video synthesis. In this work, we explore the application of generative models to fluid dynamics, specifically for turbulence simulation, where classical numerical solvers are computationally expensive. We propose a novel stochastic generative model based on stochastic interpolants, which enables probabilistic forecasting while incorporating physical constraints such as energy stability and divergence-freeness. Unlike conventional stochastic generative models, which are often agnostic to underlying physical laws, our approach embeds energy consistency by making the parameters of the stochastic interpolant learnable coefficients. We evaluate our method on a benchmark turbulence problem - Kolmogorov flow - demonstrating superior accuracy and stability over state-of-the-art alternatives such as autoregressive conditional diffusion models (ACDMs) and PDE-Refiner. Furthermore, we achieve stable results for significantly longer roll-outs than standard stochastic interpolants. Our results highlight the potential of physics-aware generative models in accelerating and enhancing turbulence simulations while preserving fundamental conservation properties.

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 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.

Accurate Zeroth-Order (ZO) Hessian estimation is a cornerstone of derivative-free methods, essential for tasks such as bilevel optimization, Bayesian inference, and uncertainty quantification. However, obtaining a complete suite of low-variance estimators for the Hessian and its inverse in high-dimensional settings remains a significant challenge. To address this, we propose a unified framework that reinterprets ZO Hessian approximation through the lens of single-step Policy Optimization (PO). This perspective establishes a theoretical equivalence between general ZO Hessian estimators and the Hessian of a smoothed PO objective, unifying distinct classical randomized estimators as specific instances of baseline selection. Building on this foundation, we introduce ZoVH, a comprehensive suite of variance-reduced estimators for the full Hessian matrix, its regularized inverse, and the bias-corrected inverse Hessian-gradient product. ZoVH leverages two key techniques: (1) a unique optimal baseline derived to provably minimize variance, and (2) a query reuse strategy that incorporates historical function queries to enhance sample efficiency without inflating costs. Our rigorous theoretical analysis confirms the unbiasedness of the Hessian estimator, validates the variance optimality of our baseline, provides error bounds for the entire ZoVH suite, and establishes convergence guarantees for the resulting curvature-aware ZO algorithm. Extensive empirical results validate our theoretical findings, demonstrating that ZoVH achieves superior estimation accuracy and convergence performance in real-world applications. Code is available at https://github.com/Qjbtiger/ZoVH

Diffusion probabilistic models (DPMs) have achieved impressive success in visual generation. While, they suffer from slow inference speed due to iterative sampling. Employing fewer sampling steps is an intuitive solution, but this will also introduces discretization error. Existing fast samplers make inspiring efforts to reduce discretization error through the adoption of high-order solvers, potentially reaching a plateau in terms of optimization. This raises the question: can the sampling process be accelerated further? In this paper, we re-examine the nature of sampling errors, discerning that they comprise two distinct elements: the widely recognized discretization error and the less explored approximation error. Our research elucidates the dynamics between these errors and the step by implementing a dual-error disentanglement strategy. Building on these foundations, we introduce an unified and training-free acceleration framework, DualFast, designed to enhance the speed of DPM sampling by concurrently accounting for both error types, thereby minimizing the total sampling error. DualFast is seamlessly compatible with existing samplers and significantly boost their sampling quality and speed, particularly in extremely few sampling steps. We substantiate the effectiveness of our framework through comprehensive experiments, spanning both unconditional and conditional sampling domains, across both pixel-space and latent-space DPMs.

We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. Our main result is a novel analysis for the effect of mini-batches through the lens of differential operator splitting, revising previous literature results. The stochastic component of a Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This leads to the identification of a convergence bottleneck: when considering mini-batches, the best achievable error rate is O(eta^2), with eta being the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.

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.

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.

Diffusion models (DMs) have been adopted across diverse fields with its remarkable abilities in capturing intricate data distributions. In this paper, we propose a Fast Diffusion Model (FDM) to significantly speed up DMs from a stochastic optimization perspective for both faster training and sampling. We first find that the diffusion process of DMs accords with the stochastic optimization process of stochastic gradient descent (SGD) on a stochastic time-variant problem. Then, inspired by momentum SGD that uses both gradient and an extra momentum to achieve faster and more stable convergence than SGD, we integrate momentum into the diffusion process of DMs. This comes with a unique challenge of deriving the noise perturbation kernel from the momentum-based diffusion process. To this end, we frame the process as a Damped Oscillation system whose critically damped state -- the kernel solution -- avoids oscillation and yields a faster convergence speed of the diffusion process. Empirical results show that our FDM can be applied to several popular DM frameworks, e.g., VP, VE, and EDM, and reduces their training cost by about 50% with comparable image synthesis performance on CIFAR-10, FFHQ, and AFHQv2 datasets. Moreover, FDM decreases their sampling steps by about 3x to achieve similar performance under the same samplers. The code is available at https://github.com/sail-sg/FDM.

Democratization of machine learning requires architectures that automatically adapt to new problems. Neural Differential Equations (NDEs) have emerged as a popular modeling framework by removing the need for ML practitioners to choose the number of layers in a recurrent model. While we can control the computational cost by choosing the number of layers in standard architectures, in NDEs the number of neural network evaluations for a forward pass can depend on the number of steps of the adaptive ODE solver. But, can we force the NDE to learn the version with the least steps while not increasing the training cost? Current strategies to overcome slow prediction require high order automatic differentiation, leading to significantly higher training time. We describe a novel regularization method that uses the internal cost heuristics of adaptive differential equation solvers combined with discrete adjoint sensitivities to guide the training process towards learning NDEs that are easier to solve. This approach opens up the blackbox numerical analysis behind the differential equation solver's algorithm and directly uses its local error estimates and stiffness heuristics as cheap and accurate cost estimates. We incorporate our method without any change in the underlying NDE framework and show that our method extends beyond Ordinary Differential Equations to accommodate Neural Stochastic Differential Equations. We demonstrate how our approach can halve the prediction time and, unlike other methods which can increase the training time by an order of magnitude, we demonstrate similar reduction in training times. Together this showcases how the knowledge embedded within state-of-the-art equation solvers can be used to enhance machine learning.

Flow and diffusion models generate high-quality samples in many modalities; however, many network evaluations are required during inference due to numerical integration of an underlying differential equation. Flow maps alleviate this problem by learning the solution map of the differential equation directly, enabling few-step sampling. Yet, current methods are restricted to approximating the solution map of ODEs. These methods can be used to learn the transition kernel of an SDE, thereby obtaining a solution map that recovers the marginal distributions of the process (weak convergence) rather than the solution path (strong convergence). We propose Strong Stochastic Flow Maps (SSFMs) as a novel framework for learning the strong solution map of additive-noise SDEs, directly generalizing deterministic flow maps to the stochastic setting. Further, a polynomial approximation to Brownian motion is introduced and shown to converge pathwise. These results enable a simulation-free training objective for the solution map of diffusion models. We demonstrate that SSFMs outperform previous stochastic flow map methods on image generation and enable few-step sampling of molecular systems.

This work proposes a simple yet effective sampling framework for combinatorial optimization (CO). Our method builds on discrete Langevin dynamics (LD), an efficient gradient-guided generative paradigm. However, we observe that directly applying LD often leads to limited exploration. To overcome this limitation, we propose the Regularized Langevin Dynamics (RLD), which enforces an expected distance between the sampled and current solutions, effectively avoiding local minima. We develop two CO solvers on top of RLD, one based on simulated annealing (SA), and the other one based on neural network (NN). Empirical results on three classic CO problems demonstrate that both of our methods can achieve comparable or better performance against the previous state-of-the-art (SOTA) SA- and NN-based solvers. In particular, our SA algorithm reduces the runtime of the previous SOTA SA method by up to 80\%, while achieving equal or superior performance. In summary, RLD offers a promising framework for enhancing both traditional heuristics and NN models to solve CO problems. Our code is available at https://github.com/Shengyu-Feng/RLD4CO.

Diffusion Probabilistic Models (DPMs) have achieved considerable success in generation tasks. As sampling from DPMs is equivalent to solving diffusion SDE or ODE which is time-consuming, numerous fast sampling methods built upon improved differential equation solvers are proposed. The majority of such techniques consider solving the diffusion ODE due to its superior efficiency. However, stochastic sampling could offer additional advantages in generating diverse and high-quality data. In this work, we engage in a comprehensive analysis of stochastic sampling from two aspects: variance-controlled diffusion SDE and linear multi-step SDE solver. Based on our analysis, we propose SA-Solver, which is an improved efficient stochastic Adams method for solving diffusion SDE to generate data with high quality. Our experiments show that SA-Solver achieves: 1) improved or comparable performance compared with the existing state-of-the-art sampling methods for few-step sampling; 2) SOTA FID scores on substantial benchmark datasets under a suitable number of function evaluations (NFEs).

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.

Diffusion models suffer from slow sample generation at inference time. Therefore, developing a principled framework for fast deterministic/stochastic sampling for a broader class of diffusion models is a promising direction. We propose two complementary frameworks for accelerating sample generation in pre-trained models: Conjugate Integrators and Splitting Integrators. Conjugate integrators generalize DDIM, mapping the reverse diffusion dynamics to a more amenable space for sampling. In contrast, splitting-based integrators, commonly used in molecular dynamics, reduce the numerical simulation error by cleverly alternating between numerical updates involving the data and auxiliary variables. After extensively studying these methods empirically and theoretically, we present a hybrid method that leads to the best-reported performance for diffusion models in augmented spaces. Applied to Phase Space Langevin Diffusion [Pandey & Mandt, 2023] on CIFAR-10, our deterministic and stochastic samplers achieve FID scores of 2.11 and 2.36 in only 100 network function evaluations (NFE) as compared to 2.57 and 2.63 for the best-performing baselines, respectively. Our code and model checkpoints will be made publicly available at https://github.com/mandt-lab/PSLD.

Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically. Training a neural differential equation is effectively a search over a space of plausible dynamical systems. However, controlling the computational cost for these models is difficult since it relies on the number of steps the adaptive solver takes. Most prior works have used higher-order methods to reduce prediction timings while greatly increasing training time or reducing both training and prediction timings by relying on specific training algorithms, which are harder to use as a drop-in replacement due to strict requirements on automatic differentiation. In this manuscript, we use internal cost heuristics of adaptive differential equation solvers at stochastic time points to guide the training toward learning a dynamical system that is easier to integrate. We "close the black-box" and allow the use of our method with any adjoint technique for gradient calculations of the differential equation solution. We perform experimental studies to compare our method to global regularization to show that we attain similar performance numbers without compromising the flexibility of implementation on ordinary differential equations (ODEs) and stochastic differential equations (SDEs). We develop two sampling strategies to trade off between performance and training time. Our method reduces the number of function evaluations to 0.556-0.733x and accelerates predictions by 1.3-2x.

Despite the remarkable success of diffusion models in image generation, slow sampling remains a persistent issue. To accelerate the sampling process, prior studies have reformulated diffusion sampling as an ODE/SDE and introduced higher-order numerical methods. However, these methods often produce divergence artifacts, especially with a low number of sampling steps, which limits the achievable acceleration. In this paper, we investigate the potential causes of these artifacts and suggest that the small stability regions of these methods could be the principal cause. To address this issue, we propose two novel techniques. The first technique involves the incorporation of Heavy Ball (HB) momentum, a well-known technique for improving optimization, into existing diffusion numerical methods to expand their stability regions. We also prove that the resulting methods have first-order convergence. The second technique, called Generalized Heavy Ball (GHVB), constructs a new high-order method that offers a variable trade-off between accuracy and artifact suppression. Experimental results show that our techniques are highly effective in reducing artifacts and improving image quality, surpassing state-of-the-art diffusion solvers on both pixel-based and latent-based diffusion models for low-step sampling. Our research provides novel insights into the design of numerical methods for future diffusion work.

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.

We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative function distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression to determine optimal approximation coefficients. Furthermore, we propose the use of neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction,-an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.

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.

The LogSumExp function, dual to the Kullback-Leibler (KL) divergence, plays a central role in many important optimization problems, including entropy-regularized optimal transport (OT) and distributionally robust optimization (DRO). In practice, when the number of exponential terms inside the logarithm is large or infinite, optimization becomes challenging since computing the gradient requires differentiating every term. We propose a novel convexity- and smoothness-preserving approximation to LogSumExp that can be efficiently optimized using stochastic gradient methods. This approximation is rooted in a sound modification of the KL divergence in the dual, resulting in a new f-divergence called the Safe KL divergence. Our experiments and theoretical analysis of the LogSumExp-based stochastic optimization, arising in DRO and continuous OT, demonstrate the advantages of our approach over existing baselines.

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models.

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only O(t) time and space per iteration, and, when the only information on the losses is their convexity, achieve a minimax optimal O(T) regret. Nonetheless, many common losses in kernel problems, such as squared loss, logistic loss, and squared hinge loss posses stronger curvature that can be exploited. In this case, second-order KOCO methods achieve O(log(Det(K))) regret, which we show scales as O(d_{eff}log T), where d_{eff} is the effective dimension of the problem and is usually much smaller than O(T). The main drawback of second-order methods is their much higher O(t^2) space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new second-order KOCO method that also achieves O(d_{eff}log T) regret. To address the computational complexity of second-order methods, we introduce a new matrix sketching algorithm for the kernel matrix K_t, and show that for a chosen parameter γleq 1 our Sketched-KONS reduces the space and time complexity by a factor of γ^2 to O(t^2γ^2) space and time per iteration, while incurring only 1/γ times more regret.

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.

The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among existing approaches, the Newton-Schulz iteration has emerged as a particularly effective solution, as it relies solely on matrix multiplications and thus achieves high computational efficiency on GPU hardware. Despite its efficiency, the method has inherent limitations - its coefficients are fixed and thus not optimized for a given matrix. In this paper we address this issue by proposing a Chebyshev-optimized version of Newton-Schulz (CANS). Based on the Chebyshev's alternance theorem, we theoretically derive optimal coefficients for the 3-rd order Newton-Schulz iteration and apply a Remez algorithm to compute optimal higher-degree polynomials. We leverage these polynomials to construct controlled approximate orthogonalization schemes, which is of interest in deep learning applications. Practically, we demonstrate the method's effectiveness in two key applications: orthogonalization in the Muon optimizer, and providing an efficient retraction alternative for Riemannian optimization on the Stiefel manifold.

In parallel simulation, convergence and parallelism are often seen as inherently conflicting objectives. Improved parallelism typically entails lighter local computation and weaker coupling, which unavoidably slow the global convergence. This paper presents a novel GPU algorithm that achieves convergence rates comparable to fullspace Newton's method while maintaining good parallelizability just like the Jacobi method. Our approach is built on a key insight into the phenomenon of overshoot. Overshoot occurs when a local solver aggressively minimizes its local energy without accounting for the global context, resulting in a local update that undermines global convergence. To address this, we derive a theoretically second-order optimal solution to mitigate overshoot. Furthermore, we adapt this solution into a pre-computable form. Leveraging Cubature sampling, our runtime cost is only marginally higher than the Jacobi method, yet our algorithm converges nearly quadratically as Newton's method. We also introduce a novel full-coordinate formulation for more efficient pre-computation. Our method integrates seamlessly with the incremental potential contact method and achieves second-order convergence for both stiff and soft materials. Experimental results demonstrate that our approach delivers high-quality simulations and outperforms state-of-the-art GPU methods with 50 to 100 times better convergence.

Second-order optimizers, maintaining a matrix termed a preconditioner, are superior to first-order optimizers in both theory and practice. The states forming the preconditioner and its inverse root restrict the maximum size of models trained by second-order optimizers. To address this, compressing 32-bit optimizer states to lower bitwidths has shown promise in reducing memory usage. However, current approaches only pertain to first-order optimizers. In this paper, we propose the first 4-bit second-order optimizers, exemplified by 4-bit Shampoo, maintaining performance similar to that of 32-bit ones. We show that quantizing the eigenvector matrix of the preconditioner in 4-bit Shampoo is remarkably better than quantizing the preconditioner itself both theoretically and experimentally. By rectifying the orthogonality of the quantized eigenvector matrix, we enhance the approximation of the preconditioner's eigenvector matrix, which also benefits the computation of its inverse 4-th root. Besides, we find that linear square quantization slightly outperforms dynamic tree quantization when quantizing second-order optimizer states. Evaluation on various networks for image classification demonstrates that our 4-bit Shampoo achieves comparable test accuracy to its 32-bit counterpart while being more memory-efficient. The source code will be made available.

Learning-based methods have gained attention as general-purpose solvers due to their ability to automatically learn problem-specific heuristics, reducing the need for manually crafted heuristics. However, these methods often face scalability challenges. To address these issues, the improved Sampling algorithm for Combinatorial Optimization (iSCO), using discrete Langevin dynamics, has been proposed, demonstrating better performance than several learning-based solvers. This study proposes a different approach that integrates gradient-based update through continuous relaxation, combined with Quasi-Quantum Annealing (QQA). QQA smoothly transitions the objective function, starting from a simple convex function, minimized at half-integral values, to the original objective function, where the relaxed variables are minimized only in the discrete space. Furthermore, we incorporate parallel run communication leveraging GPUs to enhance exploration capabilities and accelerate convergence. Numerical experiments demonstrate that our method is a competitive general-purpose solver, achieving performance comparable to iSCO and learning-based solvers across various benchmark problems. Notably, our method exhibits superior speed-quality trade-offs for large-scale instances compared to iSCO, learning-based solvers, commercial solvers, and specialized algorithms.

Deep generative models based on neural differential equations have quickly become the state-of-the-art for numerous generation tasks across many different applications. These models rely on ODE/SDE solvers which integrate from a prior distribution to the data distribution. In many applications it is highly desirable to then integrate in the other direction. The standard solvers, however, accumulate discretization errors which don't align with the forward trajectory, thereby prohibiting an exact inversion. In applications where the precision of the generative model is paramount this inaccuracy in inversion is often unacceptable. Current approaches to solving the inversion of these models results in significant downstream issues with poor stability and low-order of convergence; moreover, they are strictly limited to the ODE domain. In this work, we propose a new family of reversible exponential (stochastic) Runge-Kutta solvers which we refer to as Rex developed by an application of Lawson methods to convert any explicit (stochastic) Runge-Kutta scheme into a reversible one. In addition to a rigorous theoretical analysis of the proposed solvers, we also empirically demonstrate the utility of Rex on improving the sampling of Boltzmann distributions with flow models, and improving image generation and editing capabilities with diffusion models.

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a preconditioning procedure that accelerates Newton-Schulz convergence and reduces its computational cost. We evaluate its impact and show that the overhead of our preconditioning can be made negligible. Furthermore, the faster convergence it enables allows us to remove one iteration out of the usual five without degrading approximation quality. Our publicly available implementation achieves up to a 2.8x speedup in the Newton-Schulz approximation. We also show that this has a direct impact on end-to-end training runtime with 5-10% improvement in realistic training scenarios across two efficiency-focused tasks. On challenging language or vision tasks, we validate that our method maintains equal or superior model performance while improving runtime. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Our code is publicly available on github.

Lagrangian Relaxation (LR) is a powerful technique for solving large-scale Mixed Integer Linear Programming (MILP), particularly those with decomposable structures, such as vehicle routing or unit commitment problems. By relaxing the coupling constraints, LR enables parallel subproblem solving and often yields tighter dual bounds than standard linear programming relaxations, which is crucial for efficient branch-and-bound pruning. While recent empirical work has shown promising results using machine learning to predict these multipliers, a theoretical understanding of such methods remains an open question. In this work, we bridge this gap by analyzing the problem of learning LR through the lens of Data-driven Algorithm Design, i.e., a statistical learning problem over a distribution of problem instances. Our contributions are as follows: first, we derive a generalization bound of O(s^{1.5}/N) for the learned multipliers, where s is the number of coupling constraints and N is the sample size. Second, we provide a minimax lower-bound of Ω(s/N), proving that a linear dependency is unavoidable. Third, we constructively close this theoretical gap by proving that Stochastic Gradient Ascent (SGA) with averaging achieves the minimax optimal rate Θ(s/N). Finally, we extend our framework to the learning-to-warm-start setting, proving that it achieves a fast, minimax-optimal rate of Θ(s/N) and establishing a theoretical advantage over direct multiplier prediction.

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

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.

Modeling Lagrangian turbulence remains a fundamental challenge due to its multiscale, intermittent, and non-Gaussian nature. Recent advances in data-driven diffusion models have enabled the generation of realistic Lagrangian velocity trajectories that accurately reproduce statistical properties across scales and capture rare extreme events. This study investigates three key aspects of diffusion-based modeling for Lagrangian turbulence. First, we assess architectural robustness by comparing a U-Net backbone with a transformer-based alternative, finding strong consistency in generated trajectories, with only minor discrepancies at small scales. Second, leveraging a deterministic variant of diffusion model formulation, namely the deterministic denoising diffusion implicit model (DDIM), we identify structured features in the initial latent noise that align consistently with extreme acceleration events. Third, we explore accelerated generation by reducing the number of diffusion steps, and find that DDIM enables substantial speedups with minimal loss of statistical fidelity. These findings highlight the robustness of diffusion models and their potential for interpretable, scalable modeling of complex turbulent systems.

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.

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.

We consider the problem of global optimization with noisy zeroth order oracles - a well-motivated problem useful for various applications ranging from hyper-parameter tuning for deep learning to new material design. Existing work relies on Gaussian processes or other non-parametric family, which suffers from the curse of dimensionality. In this paper, we propose a new algorithm GO-UCB that leverages a parametric family of functions (e.g., neural networks) instead. Under a realizable assumption and a few other mild geometric conditions, we show that GO-UCB achieves a cumulative regret of O(T) where T is the time horizon. At the core of GO-UCB is a carefully designed uncertainty set over parameters based on gradients that allows optimistic exploration. Synthetic and real-world experiments illustrate GO-UCB works better than Bayesian optimization approaches in high dimensional cases, even if the model is misspecified.

Motivated by variational inference methods, we propose a zeroth-order algorithm for solving optimization problems in the space of Gaussian probability measures. The algorithm is based on an interacting system of Gaussian particles that stochastically explore the search space and self-organize around global minima via a consensus-based optimization (CBO) mechanism. Its construction relies on the Linearized Bures-Wasserstein (LBW) space, a novel parametrization of Gaussian measures we introduce for efficient computations. LBW is inspired by linearized optimal transport and preserves key geometric features while enabling computational tractability. We establish well-posedness and study the convergence properties of the particle dynamics via a mean-field approximation. Numerical experiments on variational inference tasks demonstrate the algorithm's robustness and superior performance with respect to deterministic gradient-based method in presence of low-dimensional non log-concave targets.

Diffusion probabilistic models (DPMs) are emerging powerful generative models. Despite their high-quality generation performance, DPMs still suffer from their slow sampling as they generally need hundreds or thousands of sequential function evaluations (steps) of large neural networks to draw a sample. Sampling from DPMs can be viewed alternatively as solving the corresponding diffusion ordinary differential equations (ODEs). In this work, we propose an exact formulation of the solution of diffusion ODEs. The formulation analytically computes the linear part of the solution, rather than leaving all terms to black-box ODE solvers as adopted in previous works. By applying change-of-variable, the solution can be equivalently simplified to an exponentially weighted integral of the neural network. Based on our formulation, we propose DPM-Solver, a fast dedicated high-order solver for diffusion ODEs with the convergence order guarantee. DPM-Solver is suitable for both discrete-time and continuous-time DPMs without any further training. Experimental results show that DPM-Solver can generate high-quality samples in only 10 to 20 function evaluations on various datasets. We achieve 4.70 FID in 10 function evaluations and 2.87 FID in 20 function evaluations on the CIFAR10 dataset, and a 4sim 16times speedup compared with previous state-of-the-art training-free samplers on various datasets.

In machine learning applications, it is well known that carefully designed learning rate (step size) schedules can significantly improve the convergence of commonly used first-order optimization algorithms. Therefore how to set step size adaptively becomes an important research question. A popular and effective method is the Polyak step size, which sets step size adaptively for gradient descent or stochastic gradient descent without the need to estimate the smoothness parameter of the objective function. However, there has not been a principled way to generalize the Polyak step size for algorithms with momentum accelerations. This paper presents a general framework to set the learning rate adaptively for first-order optimization methods with momentum, motivated by the derivation of Polyak step size. It is shown that the resulting methods are much less sensitive to the choice of momentum parameter and may avoid the oscillation of the heavy-ball method on ill-conditioned problems. These adaptive step sizes are further extended to the stochastic settings, which are attractive choices for stochastic gradient descent with momentum. Our methods are demonstrated to be more effective for stochastic gradient methods than prior adaptive step size algorithms in large-scale machine learning tasks.

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.

Discrete diffusion models have emerged as a powerful generative modeling framework for discrete data with successful applications spanning from text generation to image synthesis. However, their deployment faces challenges due to the high dimensionality of the state space, necessitating the development of efficient inference algorithms. Current inference approaches mainly fall into two categories: exact simulation and approximate methods such as tau-leaping. While exact methods suffer from unpredictable inference time and redundant function evaluations, tau-leaping is limited by its first-order accuracy. In this work, we advance the latter category by tailoring the first extension of high-order numerical inference schemes to discrete diffusion models, enabling larger step sizes while reducing error. We rigorously analyze the proposed schemes and establish the second-order accuracy of the theta-trapezoidal method in KL divergence. Empirical evaluations on GPT-2 level text and ImageNet-level image generation tasks demonstrate that our method achieves superior sample quality compared to existing approaches under equivalent computational constraints.