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Step-Size Stability in Stochastic Optimization: A Theoretical Perspective
Authors:
Fabian Schaipp,
Robert M. Gower,
Adrien Taylor
Abstract:
We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size becomes too large. For convex problems, we show that this quantity directly impacts the suboptimality bound of the method. Most importantly, our analysis provides d…
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We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size becomes too large. For convex problems, we show that this quantity directly impacts the suboptimality bound of the method. Most importantly, our analysis provides direct theoretical evidence that adaptive step-size methods, such as SPS or NGN, are more robust than SGD. This allows us to quantify the advantage of these adaptive methods beyond empirical evaluation. Finally, we show through experiments that our theoretical bound qualitatively mirrors the actual performance as a function of the step size, even for nonconvex problems.
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Submitted 10 February, 2026;
originally announced February 2026.
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Schedulers for Schedule-free: Theoretically inspired hyperparameters
Authors:
Yuen-Man Pun,
Matthew Buchholz,
Robert M. Gower
Abstract:
The recently proposed schedule-free method has been shown to achieve strong performance when hyperparameter tuning is limited. The current theory for schedule-free only supports a constant learning rate, where-as the implementation used in practice uses a warm-up schedule. We show how to extend the last-iterate convergence theory of schedule-free to allow for any scheduler, and how the averaging p…
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The recently proposed schedule-free method has been shown to achieve strong performance when hyperparameter tuning is limited. The current theory for schedule-free only supports a constant learning rate, where-as the implementation used in practice uses a warm-up schedule. We show how to extend the last-iterate convergence theory of schedule-free to allow for any scheduler, and how the averaging parameter has to be updated as a function of the learning rate. We then perform experiments showing how our convergence theory has some predictive power with regards to practical executions on deep neural networks, despite that this theory relies on assuming convexity. When applied to the warmup-stable-decay (wsd) schedule, our theory shows the optimal convergence rate of $\mathcal{O}(1/\sqrt{T})$. We then use convexity to design a new adaptive Polyak learning rate schedule for schedule-free. We prove an optimal anytime last-iterate convergence for our new Polyak schedule, and show that it performs well compared to a number of baselines on a black-box model distillation task.
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Submitted 10 November, 2025;
originally announced November 2025.
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Fisher meets Feynman: score-based variational inference with a product of experts
Authors:
Diana Cai,
Robert M. Gower,
David M. Blei,
Lawrence K. Saul
Abstract:
We introduce a highly expressive yet distinctly tractable family for black-box variational inference (BBVI). Each member of this family is a weighted product of experts (PoE), and each weighted expert in the product is proportional to a multivariate $t$-distribution. These products of experts can model distributions with skew, heavy tails, and multiple modes, but to use them for BBVI, we must be a…
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We introduce a highly expressive yet distinctly tractable family for black-box variational inference (BBVI). Each member of this family is a weighted product of experts (PoE), and each weighted expert in the product is proportional to a multivariate $t$-distribution. These products of experts can model distributions with skew, heavy tails, and multiple modes, but to use them for BBVI, we must be able to sample from their densities. We show how to do this by reformulating these products of experts as latent variable models with auxiliary Dirichlet random variables. These Dirichlet variables emerge from a Feynman identity, originally developed for loop integrals in quantum field theory, that expresses the product of multiple fractions (or in our case, $t$-distributions) as an integral over the simplex. We leverage this simplicial latent space to draw weighted samples from these products of experts -- samples which BBVI then uses to find the PoE that best approximates a target density. Given a collection of experts, we derive an iterative procedure to optimize the exponents that determine their geometric weighting in the PoE. At each iteration, this procedure minimizes a regularized Fisher divergence to match the scores of the variational and target densities at a batch of samples drawn from the current approximation. This minimization reduces to a convex quadratic program, and we prove under general conditions that these updates converge exponentially fast to a near-optimal weighting of experts. We conclude by evaluating this approach on a variety of synthetic and real-world target distributions.
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Submitted 24 October, 2025;
originally announced October 2025.
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An Exploration of Non-Euclidean Gradient Descent: Muon and its Many Variants
Authors:
Michael Crawshaw,
Chirag Modi,
Mingrui Liu,
Robert M. Gower
Abstract:
To define a steepest descent method over a neural network, we need to choose a norm for each layer, a way to aggregate these norms across layers, and whether to use normalization. We systematically explore different alternatives for aggregating norms across layers, both formalizing existing combinations of Adam and the recently proposed Muon as a type of non-Euclidean gradient descent, and derivin…
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To define a steepest descent method over a neural network, we need to choose a norm for each layer, a way to aggregate these norms across layers, and whether to use normalization. We systematically explore different alternatives for aggregating norms across layers, both formalizing existing combinations of Adam and the recently proposed Muon as a type of non-Euclidean gradient descent, and deriving new variants of the Muon optimizer. Through a comprehensive experimental evaluation of the optimizers within our framework, we find that Muon is sensitive to the choice of learning rate, whereas a new variant we call MuonMax is significantly more robust. We then show how to combine any non-Euclidean gradient method with model based momentum (known as Momo). The new Momo variants of Muon are significantly more robust to hyperparameter tuning, and often achieve a better validation score. Thus for new tasks, where the optimal hyperparameters are not known, we advocate for using Momo in combination with MuonMax to save on costly hyperparameter tuning.
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Submitted 10 October, 2025;
originally announced October 2025.
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In Search of Adam's Secret Sauce
Authors:
Antonio Orvieto,
Robert M. Gower
Abstract:
Understanding the remarkable efficacy of Adam when training transformer-based language models has become a central research topic within the optimization community. To gain deeper insights, several simplifications of Adam have been proposed, such as the signed gradient and signed momentum methods. In this work, we conduct an extensive empirical study - training over 1500 language models across dif…
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Understanding the remarkable efficacy of Adam when training transformer-based language models has become a central research topic within the optimization community. To gain deeper insights, several simplifications of Adam have been proposed, such as the signed gradient and signed momentum methods. In this work, we conduct an extensive empirical study - training over 1500 language models across different data configurations and scales - comparing Adam to several known simplified variants. We find that signed momentum methods are faster than SGD, but consistently underperform relative to Adam, even after careful tuning of momentum, clipping setting and learning rates. However, our analysis reveals a compelling option that preserves near-optimal performance while allowing for new insightful reformulations: constraining the Adam momentum parameters to be equal, beta1 = beta2. Beyond robust performance, this choice affords new theoretical insights, highlights the "secret sauce" on top of signed momentum, and grants a precise statistical interpretation: we show that Adam in this setting implements a natural online algorithm for estimating the mean and variance of gradients-one that arises from a mean-field Gaussian variational inference perspective.
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Submitted 30 November, 2025; v1 submitted 27 May, 2025;
originally announced May 2025.
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The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm
Authors:
Noah Amsel,
David Persson,
Christopher Musco,
Robert M. Gower
Abstract:
Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon optimizer for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize h…
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Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon optimizer for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We introduce Polar Express, a new method for computing the polar decomposition. Like Newton-Schulz and other classical polynomial methods, our approach uses only matrix-matrix multiplications, making it very efficient on GPUs. Inspired by earlier work of Chen & Chow and Nakatsukasa & Freund, Polar Express adapts the update rule at each iteration by solving a minimax optimization problem. We prove that this strategy minimizes error in a worst-case sense, allowing Polar Express to converge as rapidly as possible both in the early iterations and asymptotically. We also address finite-precision issues, making it practical to use in bfloat16. When integrated into Muon, our method yields consistent improvements in validation loss for a GPT-2 model trained on one to ten billion tokens from the FineWeb dataset, outperforming recent alternatives across a range of learning rates.
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Submitted 7 April, 2026; v1 submitted 22 May, 2025;
originally announced May 2025.
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Analysis of an Idealized Stochastic Polyak Method and its Application to Black-Box Model Distillation
Authors:
Robert M. Gower,
Guillaume Garrigos,
Nicolas Loizou,
Dimitris Oikonomou,
Konstantin Mishchenko,
Fabian Schaipp
Abstract:
We provide a general convergence theorem of an idealized stochastic Polyak step size called SPS$^*$. Besides convexity, we only assume a local expected gradient bound, that includes locally smooth and locally Lipschitz losses as special cases. We refer to SPS$^*$ as idealized because it requires access to the loss for every training batch evaluated at a solution. It is also ideal, in that it achie…
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We provide a general convergence theorem of an idealized stochastic Polyak step size called SPS$^*$. Besides convexity, we only assume a local expected gradient bound, that includes locally smooth and locally Lipschitz losses as special cases. We refer to SPS$^*$ as idealized because it requires access to the loss for every training batch evaluated at a solution. It is also ideal, in that it achieves the optimal lower bound for globally Lipschitz function, and is the first Polyak step size to have an $O(1/\sqrt{t})$ anytime convergence in the smooth setting. We show how to combine SPS$^*$ with momentum to achieve the same favorable rates for the last iterate. We conclude with several experiments to validate our theory, and a more practical setting showing how we can distill a teacher GPT-2 model into a smaller student model without any hyperparameter tuning.
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Submitted 2 April, 2025;
originally announced April 2025.
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EigenVI: score-based variational inference with orthogonal function expansions
Authors:
Diana Cai,
Chirag Modi,
Charles C. Margossian,
Robert M. Gower,
David M. Blei,
Lawrence K. Saul
Abstract:
We develop EigenVI, an eigenvalue-based approach for black-box variational inference (BBVI). EigenVI constructs its variational approximations from orthogonal function expansions. For distributions over $\mathbb{R}^D$, the lowest order term in these expansions provides a Gaussian variational approximation, while higher-order terms provide a systematic way to model non-Gaussianity. These approximat…
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We develop EigenVI, an eigenvalue-based approach for black-box variational inference (BBVI). EigenVI constructs its variational approximations from orthogonal function expansions. For distributions over $\mathbb{R}^D$, the lowest order term in these expansions provides a Gaussian variational approximation, while higher-order terms provide a systematic way to model non-Gaussianity. These approximations are flexible enough to model complex distributions (multimodal, asymmetric), but they are simple enough that one can calculate their low-order moments and draw samples from them. EigenVI can also model other types of random variables (e.g., nonnegative, bounded) by constructing variational approximations from different families of orthogonal functions. Within these families, EigenVI computes the variational approximation that best matches the score function of the target distribution by minimizing a stochastic estimate of the Fisher divergence. Notably, this optimization reduces to solving a minimum eigenvalue problem, so that EigenVI effectively sidesteps the iterative gradient-based optimizations that are required for many other BBVI algorithms. (Gradient-based methods can be sensitive to learning rates, termination criteria, and other tunable hyperparameters.) We use EigenVI to approximate a variety of target distributions, including a benchmark suite of Bayesian models from posteriordb. On these distributions, we find that EigenVI is more accurate than existing methods for Gaussian BBVI.
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Submitted 31 October, 2024;
originally announced October 2024.
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Enhancing Policy Gradient with the Polyak Step-Size Adaption
Authors:
Yunxiang Li,
Rui Yuan,
Chen Fan,
Mark Schmidt,
Samuel Horváth,
Robert M. Gower,
Martin Takáč
Abstract:
Policy gradient is a widely utilized and foundational algorithm in the field of reinforcement learning (RL). Renowned for its convergence guarantees and stability compared to other RL algorithms, its practical application is often hindered by sensitivity to hyper-parameters, particularly the step-size. In this paper, we introduce the integration of the Polyak step-size in RL, which automatically a…
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Policy gradient is a widely utilized and foundational algorithm in the field of reinforcement learning (RL). Renowned for its convergence guarantees and stability compared to other RL algorithms, its practical application is often hindered by sensitivity to hyper-parameters, particularly the step-size. In this paper, we introduce the integration of the Polyak step-size in RL, which automatically adjusts the step-size without prior knowledge. To adapt this method to RL settings, we address several issues, including unknown f* in the Polyak step-size. Additionally, we showcase the performance of the Polyak step-size in RL through experiments, demonstrating faster convergence and the attainment of more stable policies.
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Submitted 11 April, 2024;
originally announced April 2024.
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Directional Smoothness and Gradient Methods: Convergence and Adaptivity
Authors:
Aaron Mishkin,
Ahmed Khaled,
Yuanhao Wang,
Aaron Defazio,
Robert M. Gower
Abstract:
We develop new sub-optimality bounds for gradient descent (GD) that depend on the conditioning of the objective along the path of optimization rather than on global, worst-case constants. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. Minimizing these upper-bounds requires solving implicit equations to obtain a seq…
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We develop new sub-optimality bounds for gradient descent (GD) that depend on the conditioning of the objective along the path of optimization rather than on global, worst-case constants. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. Minimizing these upper-bounds requires solving implicit equations to obtain a sequence of strongly adapted step-sizes; we show that these equations are straightforward to solve for convex quadratics and lead to new guarantees for two classical step-sizes. For general functions, we prove that the Polyak step-size and normalized GD obtain fast, path-dependent rates despite using no knowledge of the directional smoothness. Experiments on logistic regression show our convergence guarantees are tighter than the classical theory based on $L$-smoothness.
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Submitted 13 January, 2025; v1 submitted 6 March, 2024;
originally announced March 2024.
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Level Set Teleportation: An Optimization Perspective
Authors:
Aaron Mishkin,
Alberto Bietti,
Robert M. Gower
Abstract:
We study level set teleportation, an optimization routine which tries to accelerate gradient descent (GD) by maximizing the gradient norm over a level set of the objective. While teleportation intuitively speeds-up GD via bigger steps, current work lacks convergence theory for convex functions, guarantees for solving the teleportation operator, and even clear empirical evidence showing this accele…
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We study level set teleportation, an optimization routine which tries to accelerate gradient descent (GD) by maximizing the gradient norm over a level set of the objective. While teleportation intuitively speeds-up GD via bigger steps, current work lacks convergence theory for convex functions, guarantees for solving the teleportation operator, and even clear empirical evidence showing this acceleration. We resolve these open questions. For convex functions satisfying Hessian stability, we prove that GD with teleportation obtains a combined sub-linear/linear convergence rate which is strictly faster than GD when the optimality gap is small. This is in sharp contrast to the standard (strongly) convex setting, where teleportation neither improves nor worsens convergence. To evaluate teleportation in practice, we develop a projected-gradient method requiring only Hessian-vector products. We use this to show that gradient methods with access to a teleportation oracle out-perform their standard versions on a variety of problems. We also find that GD with teleportation is faster than truncated Newton methods, particularly for non-convex optimization.
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Submitted 18 March, 2025; v1 submitted 5 March, 2024;
originally announced March 2024.
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Batch and match: black-box variational inference with a score-based divergence
Authors:
Diana Cai,
Chirag Modi,
Loucas Pillaud-Vivien,
Charles C. Margossian,
Robert M. Gower,
David M. Blei,
Lawrence K. Saul
Abstract:
Most leading implementations of black-box variational inference (BBVI) are based on optimizing a stochastic evidence lower bound (ELBO). But such approaches to BBVI often converge slowly due to the high variance of their gradient estimates and their sensitivity to hyperparameters. In this work, we propose batch and match (BaM), an alternative approach to BBVI based on a score-based divergence. Not…
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Most leading implementations of black-box variational inference (BBVI) are based on optimizing a stochastic evidence lower bound (ELBO). But such approaches to BBVI often converge slowly due to the high variance of their gradient estimates and their sensitivity to hyperparameters. In this work, we propose batch and match (BaM), an alternative approach to BBVI based on a score-based divergence. Notably, this score-based divergence can be optimized by a closed-form proximal update for Gaussian variational families with full covariance matrices. We analyze the convergence of BaM when the target distribution is Gaussian, and we prove that in the limit of infinite batch size the variational parameter updates converge exponentially quickly to the target mean and covariance. We also evaluate the performance of BaM on Gaussian and non-Gaussian target distributions that arise from posterior inference in hierarchical and deep generative models. In these experiments, we find that BaM typically converges in fewer (and sometimes significantly fewer) gradient evaluations than leading implementations of BBVI based on ELBO maximization.
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Submitted 12 June, 2024; v1 submitted 22 February, 2024;
originally announced February 2024.
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Function Value Learning: Adaptive Learning Rates Based on the Polyak Stepsize and Function Splitting in ERM
Authors:
Guillaume Garrigos,
Robert M. Gower,
Fabian Schaipp
Abstract:
Here we develop variants of SGD (stochastic gradient descent) with an adaptive step size that make use of the sampled loss values. In particular, we focus on solving a finite sum-of-terms problem, also known as empirical risk minimization. We first detail an idealized adaptive method called $\texttt{SPS}_+$ that makes use of the sampled loss values and assumes knowledge of the sampled loss at opti…
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Here we develop variants of SGD (stochastic gradient descent) with an adaptive step size that make use of the sampled loss values. In particular, we focus on solving a finite sum-of-terms problem, also known as empirical risk minimization. We first detail an idealized adaptive method called $\texttt{SPS}_+$ that makes use of the sampled loss values and assumes knowledge of the sampled loss at optimality. This $\texttt{SPS}_+$ is a minor modification of the SPS (Stochastic Polyak Stepsize) method, where the step size is enforced to be positive. We then show that $\texttt{SPS}_+$ achieves the best known rates of convergence for SGD in the Lipschitz non-smooth. We then move onto to develop $\texttt{FUVAL}$, a variant of $\texttt{SPS}_+$ where the loss values at optimality are gradually learned, as opposed to being given. We give three viewpoints of $\texttt{FUVAL}$, as a projection based method, as a variant of the prox-linear method, and then as a particular online SGD method. We then present a convergence analysis of $\texttt{FUVAL}$ and experimental results. The shortcomings of our work is that the convergence analysis of $\texttt{FUVAL}$ shows no advantage over SGD. Another shortcomming is that currently only the full batch version of $\texttt{FUVAL}$ shows a minor advantages of GD (Gradient Descent) in terms of sensitivity to the step size. The stochastic version shows no clear advantage over SGD. We conjecture that large mini-batches are required to make $\texttt{FUVAL}$ competitive.
Currently the new $\texttt{FUVAL}$ method studied in this paper does not offer any clear theoretical or practical advantage. We have chosen to make this draft available online nonetheless because of some of the analysis techniques we use, such as the non-smooth analysis of $\texttt{SPS}_+$, and also to show an apparently interesting approach that currently does not work.
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Submitted 26 July, 2023;
originally announced July 2023.
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A Model-Based Method for Minimizing CVaR and Beyond
Authors:
Si Yi Meng,
Robert M. Gower
Abstract:
We develop a variant of the stochastic prox-linear method for minimizing the Conditional Value-at-Risk (CVaR) objective. CVaR is a risk measure focused on minimizing worst-case performance, defined as the average of the top quantile of the losses. In machine learning, such a risk measure is useful to train more robust models. Although the stochastic subgradient method (SGM) is a natural choice for…
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We develop a variant of the stochastic prox-linear method for minimizing the Conditional Value-at-Risk (CVaR) objective. CVaR is a risk measure focused on minimizing worst-case performance, defined as the average of the top quantile of the losses. In machine learning, such a risk measure is useful to train more robust models. Although the stochastic subgradient method (SGM) is a natural choice for minimizing the CVaR objective, we show that our stochastic prox-linear (SPL+) algorithm can better exploit the structure of the objective, while still providing a convenient closed form update. Our SPL+ method also adapts to the scaling of the loss function, which allows for easier tuning. We then specialize a general convergence theorem for SPL+ to our setting, and show that it allows for a wider selection of step sizes compared to SGM. We support this theoretical finding experimentally.
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Submitted 27 May, 2023;
originally announced May 2023.
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Improving Convergence and Generalization Using Parameter Symmetries
Authors:
Bo Zhao,
Robert M. Gower,
Robin Walters,
Rose Yu
Abstract:
In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only sp…
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In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.
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Submitted 13 April, 2024; v1 submitted 22 May, 2023;
originally announced May 2023.
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MoMo: Momentum Models for Adaptive Learning Rates
Authors:
Fabian Schaipp,
Ruben Ohana,
Michael Eickenberg,
Aaron Defazio,
Robert M. Gower
Abstract:
Training a modern machine learning architecture on a new task requires extensive learning-rate tuning, which comes at a high computational cost. Here we develop new Polyak-type adaptive learning rates that can be used on top of any momentum method, and require less tuning to perform well. We first develop MoMo, a Momentum Model based adaptive learning rate for SGD-M (stochastic gradient descent wi…
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Training a modern machine learning architecture on a new task requires extensive learning-rate tuning, which comes at a high computational cost. Here we develop new Polyak-type adaptive learning rates that can be used on top of any momentum method, and require less tuning to perform well. We first develop MoMo, a Momentum Model based adaptive learning rate for SGD-M (stochastic gradient descent with momentum). MoMo uses momentum estimates of the losses and gradients sampled at each iteration to build a model of the loss function. Our model makes use of any known lower bound of the loss function by using truncation, e.g. most losses are lower-bounded by zero. The model is then approximately minimized at each iteration to compute the next step. We show how MoMo can be used in combination with any momentum-based method, and showcase this by developing MoMo-Adam, which is Adam with our new model-based adaptive learning rate. We show that MoMo attains a $\mathcal{O}(1/\sqrt{K})$ convergence rate for convex problems with interpolation, needing knowledge of no problem-specific quantities other than the optimal value. Additionally, for losses with unknown lower bounds, we develop on-the-fly estimates of a lower bound, that are incorporated in our model. We show that MoMo and MoMo-Adam improve over SGD-M and Adam in terms of robustness to hyperparameter tuning for training image classifiers on MNIST, CIFAR, and Imagenet, for recommender systems on Criteo, for a transformer model on the translation task IWSLT14, and for a diffusion model.
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Submitted 5 June, 2024; v1 submitted 12 May, 2023;
originally announced May 2023.
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A Stochastic Proximal Polyak Step Size
Authors:
Fabian Schaipp,
Robert M. Gower,
Michael Ulbrich
Abstract:
Recently, the stochastic Polyak step size (SPS) has emerged as a competitive adaptive step size scheme for stochastic gradient descent. Here we develop ProxSPS, a proximal variant of SPS that can handle regularization terms. Developing a proximal variant of SPS is particularly important, since SPS requires a lower bound of the objective function to work well. When the objective function is the sum…
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Recently, the stochastic Polyak step size (SPS) has emerged as a competitive adaptive step size scheme for stochastic gradient descent. Here we develop ProxSPS, a proximal variant of SPS that can handle regularization terms. Developing a proximal variant of SPS is particularly important, since SPS requires a lower bound of the objective function to work well. When the objective function is the sum of a loss and a regularizer, available estimates of a lower bound of the sum can be loose. In contrast, ProxSPS only requires a lower bound for the loss which is often readily available. As a consequence, we show that ProxSPS is easier to tune and more stable in the presence of regularization. Furthermore for image classification tasks, ProxSPS performs as well as AdamW with little to no tuning, and results in a network with smaller weight parameters. We also provide an extensive convergence analysis for ProxSPS that includes the non-smooth, smooth, weakly convex and strongly convex setting.
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Submitted 4 May, 2023; v1 submitted 12 January, 2023;
originally announced January 2023.
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Linear Convergence of Natural Policy Gradient Methods with Log-Linear Policies
Authors:
Rui Yuan,
Simon S. Du,
Robert M. Gower,
Alessandro Lazaric,
Lin Xiao
Abstract:
We consider infinite-horizon discounted Markov decision processes and study the convergence rates of the natural policy gradient (NPG) and the Q-NPG methods with the log-linear policy class. Using the compatible function approximation framework, both methods with log-linear policies can be written as inexact versions of the policy mirror descent (PMD) method. We show that both methods attain linea…
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We consider infinite-horizon discounted Markov decision processes and study the convergence rates of the natural policy gradient (NPG) and the Q-NPG methods with the log-linear policy class. Using the compatible function approximation framework, both methods with log-linear policies can be written as inexact versions of the policy mirror descent (PMD) method. We show that both methods attain linear convergence rates and $\tilde{\mathcal{O}}(1/ε^2)$ sample complexities using a simple, non-adaptive geometrically increasing step size, without resorting to entropy or other strongly convex regularization. Lastly, as a byproduct, we obtain sublinear convergence rates for both methods with arbitrary constant step size.
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Submitted 21 February, 2023; v1 submitted 4 October, 2022;
originally announced October 2022.
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SP2: A Second Order Stochastic Polyak Method
Authors:
Shuang Li,
William J. Swartworth,
Martin Takáč,
Deanna Needell,
Robert M. Gower
Abstract:
Recently the "SP" (Stochastic Polyak step size) method has emerged as a competitive adaptive method for setting the step sizes of SGD. SP can be interpreted as a method specialized to interpolated models, since it solves the interpolation equations. SP solves these equation by using local linearizations of the model. We take a step further and develop a method for solving the interpolation equatio…
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Recently the "SP" (Stochastic Polyak step size) method has emerged as a competitive adaptive method for setting the step sizes of SGD. SP can be interpreted as a method specialized to interpolated models, since it solves the interpolation equations. SP solves these equation by using local linearizations of the model. We take a step further and develop a method for solving the interpolation equations that uses the local second-order approximation of the model. Our resulting method SP2 uses Hessian-vector products to speed-up the convergence of SP. Furthermore, and rather uniquely among second-order methods, the design of SP2 in no way relies on positive definite Hessian matrices or convexity of the objective function. We show SP2 is very competitive on matrix completion, non-convex test problems and logistic regression. We also provide a convergence theory on sums-of-quadratics.
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Submitted 17 July, 2022;
originally announced July 2022.
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Cutting Some Slack for SGD with Adaptive Polyak Stepsizes
Authors:
Robert M. Gower,
Mathieu Blondel,
Nidham Gazagnadou,
Fabian Pedregosa
Abstract:
Tuning the step size of stochastic gradient descent is tedious and error prone. This has motivated the development of methods that automatically adapt the step size using readily available information. In this paper, we consider the family of SPS (Stochastic gradient with a Polyak Stepsize) adaptive methods. These are methods that make use of gradient and loss value at the sampled points to adapti…
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Tuning the step size of stochastic gradient descent is tedious and error prone. This has motivated the development of methods that automatically adapt the step size using readily available information. In this paper, we consider the family of SPS (Stochastic gradient with a Polyak Stepsize) adaptive methods. These are methods that make use of gradient and loss value at the sampled points to adaptively adjust the step size. We first show that SPS and its recent variants can all be seen as extensions of the Passive-Aggressive methods applied to nonlinear problems. We use this insight to develop new variants of the SPS method that are better suited to nonlinear models. Our new variants are based on introducing a slack variable into the interpolation equations. This single slack variable tracks the loss function across iterations and is used in setting a stable step size. We provide extensive numerical results supporting our new methods and a convergence theory.
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Submitted 20 May, 2022; v1 submitted 24 February, 2022;
originally announced February 2022.
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A general sample complexity analysis of vanilla policy gradient
Authors:
Rui Yuan,
Robert M. Gower,
Alessandro Lazaric
Abstract:
We adapt recent tools developed for the analysis of Stochastic Gradient Descent (SGD) in non-convex optimization to obtain convergence and sample complexity guarantees for the vanilla policy gradient (PG). Our only assumptions are that the expected return is smooth w.r.t. the policy parameters, that its $H$-step truncated gradient is close to the exact gradient, and a certain ABC assumption. This…
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We adapt recent tools developed for the analysis of Stochastic Gradient Descent (SGD) in non-convex optimization to obtain convergence and sample complexity guarantees for the vanilla policy gradient (PG). Our only assumptions are that the expected return is smooth w.r.t. the policy parameters, that its $H$-step truncated gradient is close to the exact gradient, and a certain ABC assumption. This assumption requires the second moment of the estimated gradient to be bounded by $A\geq 0$ times the suboptimality gap, $B \geq 0$ times the norm of the full batch gradient and an additive constant $C \geq 0$, or any combination of aforementioned. We show that the ABC assumption is more general than the commonly used assumptions on the policy space to prove convergence to a stationary point. We provide a single convergence theorem that recovers the $\widetilde{\mathcal{O}}(ε^{-4})$ sample complexity of PG to a stationary point. Our results also affords greater flexibility in the choice of hyper parameters such as the step size and the batch size $m$, including the single trajectory case (i.e., $m=1$). When an additional relaxed weak gradient domination assumption is available, we establish a novel global optimum convergence theory of PG with $\widetilde{\mathcal{O}}(ε^{-3})$ sample complexity. We then instantiate our theorems in different settings, where we both recover existing results and obtain improved sample complexity, e.g., $\widetilde{\mathcal{O}}(ε^{-3})$ sample complexity for the convergence to the global optimum for Fisher-non-degenerated parametrized policies.
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Submitted 18 November, 2022; v1 submitted 23 July, 2021;
originally announced July 2021.
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Stochastic Polyak Stepsize with a Moving Target
Authors:
Robert M. Gower,
Aaron Defazio,
Michael Rabbat
Abstract:
We propose a new stochastic gradient method called MOTAPS (Moving Targetted Polyak Stepsize) that uses recorded past loss values to compute adaptive stepsizes. MOTAPS can be seen as a variant of the Stochastic Polyak (SP) which is also a method that also uses loss values to adjust the stepsize. The downside to the SP method is that it only converges when the interpolation condition holds. MOTAPS i…
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We propose a new stochastic gradient method called MOTAPS (Moving Targetted Polyak Stepsize) that uses recorded past loss values to compute adaptive stepsizes. MOTAPS can be seen as a variant of the Stochastic Polyak (SP) which is also a method that also uses loss values to adjust the stepsize. The downside to the SP method is that it only converges when the interpolation condition holds. MOTAPS is an extension of SP that does not rely on the interpolation condition. The MOTAPS method uses $n$ auxiliary variables, one for each data point, that track the loss value for each data point. We provide a global convergence theory for SP, an intermediary method TAPS, and MOTAPS by showing that they all can be interpreted as a special variant of online SGD. We also perform several numerical experiments on convex learning problems, and deep learning models for image classification and language translation. In all of our tasks we show that MOTAPS is competitive with the relevant baseline method.
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Submitted 23 September, 2021; v1 submitted 22 June, 2021;
originally announced June 2021.
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Variance-Reduced Methods for Machine Learning
Authors:
Robert M. Gower,
Mark Schmidt,
Francis Bach,
Peter Richtarik
Abstract:
Stochastic optimization lies at the heart of machine learning, and its cornerstone is stochastic gradient descent (SGD), a method introduced over 60 years ago. The last 8 years have seen an exciting new development: variance reduction (VR) for stochastic optimization methods. These VR methods excel in settings where more than one pass through the training data is allowed, achieving a faster conver…
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Stochastic optimization lies at the heart of machine learning, and its cornerstone is stochastic gradient descent (SGD), a method introduced over 60 years ago. The last 8 years have seen an exciting new development: variance reduction (VR) for stochastic optimization methods. These VR methods excel in settings where more than one pass through the training data is allowed, achieving a faster convergence than SGD in theory as well as practice. These speedups underline the surge of interest in VR methods and the fast-growing body of work on this topic. This review covers the key principles and main developments behind VR methods for optimization with finite data sets and is aimed at non-expert readers. We focus mainly on the convex setting, and leave pointers to readers interested in extensions for minimizing non-convex functions.
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Submitted 2 October, 2020;
originally announced October 2020.
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A new framework for the computation of Hessians
Authors:
Robert M. Gower,
Margarida P. Mello
Abstract:
We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on Griewank and Walther's state transformations, synthesizes the calculation of the Hessian as a formula. These dual points of view, graphical and algebraic, lead…
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We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on Griewank and Walther's state transformations, synthesizes the calculation of the Hessian as a formula. These dual points of view, graphical and algebraic, lead to a new framework for Hessian computation. This is illustrated by developing edge_pushing, a new truly reverse Hessian computation algorithm that fully exploits the Hessian's symmetry. Computational experiments compare the performance of edge_pushing on sixteen functions from the CUTE collection against two algorithms available as drivers of the software ADOL-C, and the results are very promising.
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Submitted 29 July, 2020;
originally announced July 2020.
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Unified Analysis of Stochastic Gradient Methods for Composite Convex and Smooth Optimization
Authors:
Ahmed Khaled,
Othmane Sebbouh,
Nicolas Loizou,
Robert M. Gower,
Peter Richtárik
Abstract:
We present a unified theorem for the convergence analysis of stochastic gradient algorithms for minimizing a smooth and convex loss plus a convex regularizer. We do this by extending the unified analysis of Gorbunov, Hanzely \& Richtárik (2020) and dropping the requirement that the loss function be strongly convex. Instead, we only rely on convexity of the loss function. Our unified analysis appli…
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We present a unified theorem for the convergence analysis of stochastic gradient algorithms for minimizing a smooth and convex loss plus a convex regularizer. We do this by extending the unified analysis of Gorbunov, Hanzely \& Richtárik (2020) and dropping the requirement that the loss function be strongly convex. Instead, we only rely on convexity of the loss function. Our unified analysis applies to a host of existing algorithms such as proximal SGD, variance reduced methods, quantization and some coordinate descent type methods. For the variance reduced methods, we recover the best known convergence rates as special cases. For proximal SGD, the quantization and coordinate type methods, we uncover new state-of-the-art convergence rates. Our analysis also includes any form of sampling and minibatching. As such, we are able to determine the minibatch size that optimizes the total complexity of variance reduced methods. We showcase this by obtaining a simple formula for the optimal minibatch size of two variance reduced methods (\textit{L-SVRG} and \textit{SAGA}). This optimal minibatch size not only improves the theoretical total complexity of the methods but also improves their convergence in practice, as we show in several experiments.
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Submitted 20 June, 2020;
originally announced June 2020.
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SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation
Authors:
Robert M. Gower,
Othmane Sebbouh,
Nicolas Loizou
Abstract:
Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumption…
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Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumptions. In particular, we focus on two large classes of structured non-convex functions: (i) Quasar (Strongly) Convex functions (a generalization of convex functions) and (ii) functions satisfying the Polyak-Lojasiewicz condition (a generalization of strongly-convex functions). Our analysis relies on an Expected Residual condition which we show is a strictly weaker assumption than previously used growth conditions, expected smoothness or bounded variance assumptions. We provide theoretical guarantees for the convergence of SGD for different step-size selections including constant, decreasing and the recently proposed stochastic Polyak step-size. In addition, all of our analysis holds for the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching and determine an optimal minibatch size. Finally, we show that for models that interpolate the training data, we can dispense of our Expected Residual condition and give state-of-the-art results in this setting.
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Submitted 22 March, 2021; v1 submitted 18 June, 2020;
originally announced June 2020.
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Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball
Authors:
Othmane Sebbouh,
Robert M. Gower,
Aaron Defazio
Abstract:
We study stochastic gradient descent (SGD) and the stochastic heavy ball method (SHB, otherwise known as the momentum method) for the general stochastic approximation problem.
For SGD, in the convex and smooth setting, we provide the first \emph{almost sure} asymptotic convergence \emph{rates} for a weighted average of the iterates . More precisely, we show that the convergence rate of the funct…
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We study stochastic gradient descent (SGD) and the stochastic heavy ball method (SHB, otherwise known as the momentum method) for the general stochastic approximation problem.
For SGD, in the convex and smooth setting, we provide the first \emph{almost sure} asymptotic convergence \emph{rates} for a weighted average of the iterates . More precisely, we show that the convergence rate of the function values is arbitrarily close to $o(1/\sqrt{k})$, and is exactly $o(1/k)$ in the so-called overparametrized case. We show that these results still hold when using stochastic line search and stochastic Polyak stepsizes, thereby giving the first proof of convergence of these methods in the non-overparametrized regime.
Using a substantially different analysis, we show that these rates hold for SHB as well, but at the last iterate. This distinction is important because it is the last iterate of SGD and SHB which is used in practice. We also show that the last iterate of SHB converges to a minimizer \emph{almost surely}. Additionally, we prove that the function values of the deterministic HB converge at a $o(1/k)$ rate, which is faster than the previously known $O(1/k)$.
Finally, in the nonconvex setting, we prove similar rates on the lowest gradient norm along the trajectory of SGD.
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Submitted 5 February, 2021; v1 submitted 14 June, 2020;
originally announced June 2020.
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The Power of Factorial Powers: New Parameter settings for (Stochastic) Optimization
Authors:
Aaron Defazio,
Robert M. Gower
Abstract:
The convergence rates for convex and non-convex optimization methods depend on the choice of a host of constants, including step sizes, Lyapunov function constants and momentum constants. In this work we propose the use of factorial powers as a flexible tool for defining constants that appear in convergence proofs. We list a number of remarkable properties that these sequences enjoy, and show how…
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The convergence rates for convex and non-convex optimization methods depend on the choice of a host of constants, including step sizes, Lyapunov function constants and momentum constants. In this work we propose the use of factorial powers as a flexible tool for defining constants that appear in convergence proofs. We list a number of remarkable properties that these sequences enjoy, and show how they can be applied to convergence proofs to simplify or improve the convergence rates of the momentum method, accelerated gradient and the stochastic variance reduced method (SVRG).
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Submitted 11 April, 2023; v1 submitted 1 June, 2020;
originally announced June 2020.
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Towards closing the gap between the theory and practice of SVRG
Authors:
Othmane Sebbouh,
Nidham Gazagnadou,
Samy Jelassi,
Francis Bach,
Robert M. Gower
Abstract:
Among the very first variance reduced stochastic methods for solving the empirical risk minimization problem was the SVRG method (Johnson & Zhang 2013). SVRG is an inner-outer loop based method, where in the outer loop a reference full gradient is evaluated, after which $m \in \mathbb{N}$ steps of an inner loop are executed where the reference gradient is used to build a variance reduced estimate…
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Among the very first variance reduced stochastic methods for solving the empirical risk minimization problem was the SVRG method (Johnson & Zhang 2013). SVRG is an inner-outer loop based method, where in the outer loop a reference full gradient is evaluated, after which $m \in \mathbb{N}$ steps of an inner loop are executed where the reference gradient is used to build a variance reduced estimate of the current gradient. The simplicity of the SVRG method and its analysis have led to multiple extensions and variants for even non-convex optimization. We provide a more general analysis of SVRG than had been previously done by using arbitrary sampling, which allows us to analyse virtually all forms of mini-batching through a single theorem. Furthermore, our analysis is focused on more practical variants of SVRG including a new variant of the loopless SVRG (Hofman et al 2015, Kovalev et al 2019, Kulunchakov and Mairal 2019) and a variant of k-SVRG (Raj and Stich 2018) where $m=n$ and where $n$ is the number of data points. Since our setup and analysis reflect what is done in practice, we are able to set the parameters such as the mini-batch size and step size using our theory in such a way that produces a more efficient algorithm in practice, as we show in extensive numerical experiments.
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Submitted 2 July, 2021; v1 submitted 31 July, 2019;
originally announced August 2019.
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Optimal mini-batch and step sizes for SAGA
Authors:
Nidham Gazagnadou,
Robert M. Gower,
Joseph Salmon
Abstract:
Recently it has been shown that the step sizes of a family of variance reduced gradient methods called the JacSketch methods depend on the expected smoothness constant. In particular, if this expected smoothness constant could be calculated a priori, then one could safely set much larger step sizes which would result in a much faster convergence rate. We fill in this gap, and provide simple closed…
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Recently it has been shown that the step sizes of a family of variance reduced gradient methods called the JacSketch methods depend on the expected smoothness constant. In particular, if this expected smoothness constant could be calculated a priori, then one could safely set much larger step sizes which would result in a much faster convergence rate. We fill in this gap, and provide simple closed form expressions for the expected smoothness constant and careful numerical experiments verifying these bounds. Using these bounds, and since the SAGA algorithm is part of this JacSketch family, we suggest a new standard practice for setting the step sizes and mini-batch size for SAGA that are competitive with a numerical grid search. Furthermore, we can now show that the total complexity of the SAGA algorithm decreases linearly in the mini-batch size up to a pre-defined value: the optimal mini-batch size. This is a rare result in the stochastic variance reduced literature, only previously shown for the Katyusha algorithm. Finally we conjecture that this is the case for many other stochastic variance reduced methods and that our bounds and analysis of the expected smoothness constant is key to extending these results.
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Submitted 18 September, 2019; v1 submitted 31 January, 2019;
originally announced February 2019.
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SGD: General Analysis and Improved Rates
Authors:
Robert Mansel Gower,
Nicolas Loizou,
Xun Qian,
Alibek Sailanbayev,
Egor Shulgin,
Peter Richtarik
Abstract:
We propose a general yet simple theorem describing the convergence of SGD under the arbitrary sampling paradigm. Our theorem describes the convergence of an infinite array of variants of SGD, each of which is associated with a specific probability law governing the data selection rule used to form mini-batches. This is the first time such an analysis is performed, and most of our variants of SGD w…
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We propose a general yet simple theorem describing the convergence of SGD under the arbitrary sampling paradigm. Our theorem describes the convergence of an infinite array of variants of SGD, each of which is associated with a specific probability law governing the data selection rule used to form mini-batches. This is the first time such an analysis is performed, and most of our variants of SGD were never explicitly considered in the literature before. Our analysis relies on the recently introduced notion of expected smoothness and does not rely on a uniform bound on the variance of the stochastic gradients. By specializing our theorem to different mini-batching strategies, such as sampling with replacement and independent sampling, we derive exact expressions for the stepsize as a function of the mini-batch size. With this we can also determine the mini-batch size that optimizes the total complexity, and show explicitly that as the variance of the stochastic gradient evaluated at the minimum grows, so does the optimal mini-batch size. For zero variance, the optimal mini-batch size is one. Moreover, we prove insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime.
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Submitted 1 May, 2019; v1 submitted 27 January, 2019;
originally announced January 2019.
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Greedy stochastic algorithms for entropy-regularized optimal transport problems
Authors:
Brahim Khalil Abid,
Robert M. Gower
Abstract:
Optimal transport (OT) distances are finding evermore applications in machine learning and computer vision, but their wide spread use in larger-scale problems is impeded by their high computational cost. In this work we develop a family of fast and practical stochastic algorithms for solving the optimal transport problem with an entropic penalization. This work extends the recently developed Green…
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Optimal transport (OT) distances are finding evermore applications in machine learning and computer vision, but their wide spread use in larger-scale problems is impeded by their high computational cost. In this work we develop a family of fast and practical stochastic algorithms for solving the optimal transport problem with an entropic penalization. This work extends the recently developed Greenkhorn algorithm, in the sense that, the Greenkhorn algorithm is a limiting case of this family. We also provide a simple and general convergence theorem for all algorithms in the class, with rates that match the best known rates of Greenkorn and the Sinkhorn algorithm, and conclude with numerical experiments that show under what regime of penalization the new stochastic methods are faster than the aforementioned methods.
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Submitted 4 March, 2018;
originally announced March 2018.
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Tracking the gradients using the Hessian: A new look at variance reducing stochastic methods
Authors:
Robert M. Gower,
Nicolas Le Roux,
Francis Bach
Abstract:
Our goal is to improve variance reducing stochastic methods through better control variates. We first propose a modification of SVRG which uses the Hessian to track gradients over time, rather than to recondition, increasing the correlation of the control variates and leading to faster theoretical convergence close to the optimum. We then propose accurate and computationally efficient approximatio…
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Our goal is to improve variance reducing stochastic methods through better control variates. We first propose a modification of SVRG which uses the Hessian to track gradients over time, rather than to recondition, increasing the correlation of the control variates and leading to faster theoretical convergence close to the optimum. We then propose accurate and computationally efficient approximations to the Hessian, both using a diagonal and a low-rank matrix. Finally, we demonstrate the effectiveness of our method on a wide range of problems.
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Submitted 31 March, 2018; v1 submitted 20 October, 2017;
originally announced October 2017.
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Higher-order Reverse Automatic Differentiation with emphasis on the third-order
Authors:
Robert M. Gower,
Artur L. Gower
Abstract:
It is commonly assumed that calculating third order information is too expensive for most applications. But we show that the directional derivative of the Hessian ($D^3f(x)\cdot d$) can be calculated at a cost proportional to that of a state-of-the-art method for calculating the Hessian matrix. We do this by first presenting a simple procedure for designing high order reverse methods and applying…
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It is commonly assumed that calculating third order information is too expensive for most applications. But we show that the directional derivative of the Hessian ($D^3f(x)\cdot d$) can be calculated at a cost proportional to that of a state-of-the-art method for calculating the Hessian matrix. We do this by first presenting a simple procedure for designing high order reverse methods and applying it to deduce several methods including a reverse method that calculates $D^3f(x)\cdot d$. We have implemented this method taking into account symmetry and sparsity, and successfully calculated this derivative for functions with a million variables. These results indicate that the use of third order information in a general nonlinear solver, such as Halley-Chebyshev methods, could be a practical alternative to Newton's method.
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Submitted 21 September, 2013;
originally announced September 2013.