Pre-print articles on Adaptive Optimization

1. A simple uniformly optimal method without line search for convex optimization #

Authors: Tianjiao Li, Guanghui Lan

Abstract: Line search (or backtracking) procedures have been widely employed into first-order methods for solving convex optimization problems, especially those with unknown problem parameters (e.g., Lipschitz constant). In this paper, we show that line search is superfluous in attaining the optimal rate of convergence for solving a convex optimization problem whose parameters are not given a priori. In particular, we present a novel accelerated gradient descent type algorithm called auto-conditioned fast gradient method (AC-FGM) that can achieve an optimal $\mathcal{O}(1/k^2)$ rate of convergence for smooth convex optimization without requiring the estimate of a global Lipschitz constant or the employment of line search procedures. We then extend AC-FGM to solve convex optimization problems with Hölder continuous gradients and show that it automatically achieves the optimal rates of convergence uniformly for all problem classes with the desired accuracy of the solution as the only input. Finally, we report some encouraging numerical results that demonstrate the advantages of AC-FGM over the previously developed parameter-free methods for convex optimization.

2. Adaptive Proximal Gradient Method for Convex Optimization #

Authors: Yura Malitsky, Konstantin Mishchenko

Abstract: In this paper, we explore two fundamental first-order algorithms in convex optimization, namely, gradient descent (GD) and proximal gradient method (ProxGD). Our focus is on making these algorithms entirely adaptive by leveraging local curvature information of smooth functions. We propose adaptive versions of GD and ProxGD that are based on observed gradient differences and, thus, have no added computational costs. Moreover, we prove convergence of our methods assuming only local Lipschitzness of the gradient. In addition, the proposed versions allow for even larger stepsizes than those initially suggested in [MM20].

3. An Adaptive Stochastic Gradient Method with Non-negative Gauss-Newton Stepsizes #

Authors: Antonio Orvieto, Lin Xiao

Abstract: We consider the problem of minimizing the average of a large number of smooth but possibly non-convex functions. In the context of most machine learning applications, each loss function is non-negative and thus can be expressed as the composition of a square and its real-valued square root. This reformulation allows us to apply the Gauss-Newton method, or the Levenberg-Marquardt method when adding a quadratic regularization. The resulting algorithm, while being computationally as efficient as the vanilla stochastic gradient method, is highly adaptive and can automatically warmup and decay the effective stepsize while tracking the non-negative loss landscape. We provide a tight convergence analysis, leveraging new techniques, in the stochastic convex and non-convex settings. In particular, in the convex case, the method does not require access to the gradient Lipshitz constant for convergence, and is guaranteed to never diverge. The convergence rates and empirical evaluations compare favorably to the classical (stochastic) gradient method as well as to several other adaptive methods.

4. Stochastic Polyak Step-sizes and Momentum: Convergence Guarantees and Practical Performance #

Authors: Antonio Orvieto, Lin Xiao

Abstract: Stochastic gradient descent with momentum, also known as Stochastic Heavy Ball method (SHB), is one of the most popular algorithms for solving large-scale stochastic optimization problems in various machine learning tasks. In practical scenarios, tuning the step-size and momentum parameters of the method is a prohibitively expensive and time-consuming process. In this work, inspired by the recent advantages of stochastic Polyak step-size in the performance of stochastic gradient descent (SGD), we propose and explore new Polyak-type variants suitable for the update rule of the SHB method. In particular, using the Iterate Moving Average (IMA) viewpoint of SHB, we propose and analyze three novel step-size selections: $\text{MomSPS} _{\max}$, $\text{MomDecSPS}$, and $\text{MomAdaSPS}$. For $\text{MomSPS} _{\max}$, we provide convergence guarantees for SHB to a neighborhood of the solution for convex and smooth problems (without assuming interpolation). If interpolation is also satisfied, then using $\text{MomSPS} _{\max}$, SHB converges to the true solution at a fast rate matching the deterministic HB. The other two variants, MomDecSPS and MomAdaSPS, are the first adaptive step-size for SHB that guarantee convergence to the exact minimizer - without a priori knowledge of the problem parameters and without assuming interpolation. Our convergence analysis of SHB is tight and obtains the convergence guarantees of stochastic Polyak step-size for SGD as a special case. We supplement our analysis with experiments validating our theory and demonstrating the effectiveness and robustness of our algorithms.

Where: 13th International Conference on Learning Representations (ICLR 2025)