arXiv subjects

Robert M. Gower

Publications and source records attributed to Robert M. Gower.

At least 37 records · Page 2Linked to original sources

arXiv2023

Function Value Learning: Adaptive Learning Rates Based on the Polyak Stepsize and Function Splitting in ERM

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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arXiv2023

A Model-Based Method for Minimizing CVaR and Beyond

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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arXiv2023

A Stochastic Proximal Polyak Step Size

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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arXiv2023

The Power of Factorial Powers: New Parameter settings for (Stochastic) Optimization

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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arXiv2023

Linear Convergence of Natural Policy Gradient Methods with Log-Linear Policies

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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arXiv2022

A general sample complexity analysis of vanilla policy gradient

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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arXiv2022

SP2: A Second Order Stochastic Polyak Method

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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arXiv2022

Cutting Some Slack for SGD with Adaptive Polyak Stepsizes

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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arXiv2022

Sketched Newton-Raphson

We propose a new globally convergent stochastic second order method. Our starting point is the development of a new Sketched Newton-Raphson (SNR) method for solving large scale nonlinear equations of the form $F(x)=0$ with $F:\mathbb{R}^p \rightarrow \mathbb{R}^m$. We then show how to design several stochastic second order optimization methods by re-writing the optimization problem of interest as a system of nonlinear equations and applying SNR. For instance, by applying SNR to find a stationary point of a generalized linear model (GLM), we derive completely new and scalable stochastic second order methods. We show that the resulting method is very competitive as compared to state-of-the-art variance reduced methods. Furthermore, using a variable splitting trick, we also show that the Stochastic Newton method (SNM) is a special case of SNR, and use this connection to establish the first global convergence theory of SNM. We establish the global convergence of SNR by showing that it is a variant of the stochastic gradient descent (SGD) method, and then leveraging proof techniques of SGD. As a special case, our theory also provides a new global convergence theory for the original Newton-Raphson method under strictly weaker assumptions as compared to the classic monotone convergence theory.

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arXiv2022

SAN: Stochastic Average Newton Algorithm for Minimizing Finite Sums

We present a principled approach for designing stochastic Newton methods for solving finite sum optimization problems. Our approach has two steps. First, we re-write the stationarity conditions as a system of nonlinear equations that associates each data point to a new row. Second, we apply a Subsampled Newton Raphson method to solve this system of nonlinear equations. Using our approach, we develop a new Stochastic Average Newton (SAN) method, which is incremental by design, in that it requires only a single data point per iteration. It is also cheap to implement when solving regularized generalized linear models, with a cost per iteration of the order of the number of the parameters. We show through extensive numerical experiments that SAN requires no knowledge about the problem, neither parameter tuning, while remaining competitive as compared to classical variance reduced gradient methods (e.g. SAG and SVRG), incremental Newton and quasi-Newton methods (e.g. SNM, IQN).

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arXiv2021

Stochastic Polyak Stepsize with a Moving Target

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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arXiv2021

Towards closing the gap between the theory and practice of SVRG

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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arXiv2021

$\texttt{RidgeSketch}$: A Fast sketching based solver for large scale ridge regression

We propose new variants of the sketch-and-project method for solving large scale ridge regression problems. Firstly, we propose a new momentum alternative and provide a theorem showing it can speed up the convergence of sketch-and-project, through a fast $\textit{sublinear}$ convergence rate. We carefully delimit under what settings this new sublinear rate is faster than the previously known linear rate of convergence of sketch-and-project without momentum. Secondly, we consider combining the sketch-and-project method with new modern sketching methods such as the count sketch, subcount sketch (a new method we propose), and subsampled Hadamard transforms. We show experimentally that when combined with the sketch-and-project method, the (sub)count sketch is very effective on sparse data and the standard subsample sketch is effective on dense data. Indeed, we show that these sketching methods, combined with our new momentum scheme, result in methods that are competitive even when compared to the Conjugate Gradient method on real large scale data. On the contrary, we show the subsampled Hadamard transform does not perform well in this setting, despite the use of fast Hadamard transforms, and nor do recently proposed acceleration schemes work well in practice. To support all of our experimental findings, and invite the community to validate and extend our results, with this paper we are also releasing an open source software package: $\texttt{RidgeSketch}$. We designed this object-oriented package in Python for testing sketch-and-project methods and benchmarking ridge regression solvers. $\texttt{RidgeSketch}$ is highly modular, and new sketching methods can easily be added as subclasses. We provide code snippets of our package in the appendix.

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arXiv2021

SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation

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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arXiv2021

Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball

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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arXiv2021

Fast Linear Convergence of Randomized BFGS

Since the late 1950's when quasi-Newton methods first appeared, they have become one of the most widely used and efficient algorithmic paradigms for unconstrained optimization. Despite their immense practical success, there is little theory that shows why these methods are so efficient. We provide a semi-local rate of convergence for the randomized BFGS method which can be significantly better than that of gradient descent, finally giving theoretical evidence supporting the superior empirical performance of the method.

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arXiv2020

Variance-Reduced Methods for Machine Learning

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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arXiv2020

A new framework for the computation of Hessians

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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