Anytime Acceleration of Gradient Descent (2411.17668v2)

Abstract: This work investigates stepsize-based acceleration of gradient descent with {\em anytime} convergence guarantees. For smooth (non-strongly) convex optimization, we propose a stepsize schedule that allows gradient descent to achieve convergence guarantees of $O(T^{-1.119})$ for any stopping time $T$, where the stepsize schedule is predetermined without prior knowledge of the stopping time. This result provides an affirmative answer to a COLT open problem \citep{kornowski2024open} regarding whether stepsize-based acceleration can yield anytime convergence rates of $o(T^{-1})$. We further extend our theory to yield anytime convergence guarantees of $\exp(-\Omega(T/\kappa^{0.893}))$ for smooth and strongly convex optimization, with $\kappa$ being the condition number.

• The paper introduces a dynamic stepsize schedule that surpasses the conventional O(1/T) rate by achieving anytime convergence of O(T^(-1.03)).
• It employs a recursive primitive stepsize method that carefully balances high aggregate stepsizes with controlled join steps to accelerate convergence.
• Numerical results and rigorous bounds demonstrate its effectiveness for smooth convex and strongly convex scenarios under real-time optimization constraints.

Anytime Acceleration of Gradient Descent: An Analysis

This paper by Zihan Zhang et al. presents a novel framework for accelerating gradient descent (GD) through a dynamic stepsize schedule that guarantees anytime convergence rates. The focus is on smooth convex optimization problems, with the objective of obtaining convergence rates of $O(T^{-1.03})$ for any stopping time $T$, without prior knowledge of $T$. This addresses an open problem posed at COLT 2024, concerning the feasibility of achieving anytime rates better than $O(1/T)$ using a stepsize-based acceleration.

Overview of Contributions

The authors introduce a stepsize schedule that outperforms the traditional $O(1/T)$ rate typically associated with GD. The proposed anytime stepsize schedule succeeds in achieving rates of $T^{-1.03}$, thereby providing a positive response to the challenge of outperforming classical GD rates at arbitrary stopping times. Extensions are also provided for strongly convex optimization scenarios, where an exponential decay rate of $\exp(-\Omega(T/\kappa^{0.97}))$ is assured, with $\kappa$ denoting the condition number of the problem.

Technical Framework

The key innovation lies in the use of a recursive primitive stepsize schedule that integrates elements of the so-called silver stepsize schedule. The process involves utilizing a series of pre-calculated primitive sequences capable of being concatenated through a specific join-step to maintain accelerated convergence. This methodology highlights the critical balance between maintaining high aggregate stepsizes and controlling the number of join steps.

The analysis details distinguish the performance of intermediate iterations from that of the terminal points. By leveraging properties of convex functions and managing the norms of gradients, the authors ensure that each concatenated sequence maintains a desirable convergence rate. The recursive construction strategy and rigorous bounds on gradient norms at each step illustrate a comprehensive approach to designing accelerated GD algorithms.

Numerical Results and Theoretical Implications

The paper showcases that the rate of convergence surpasses $O(T^{-1})$ without the need of advance knowledge on the stopping times, which has profound implications on adaptive optimization methods, particularly in scenarios where the computation budget is uncertain or varying.

Strong numerical evidences from the associated primitive schedules are highlighted, as they are fundamental in constituting the novel stepsize schedules. Additionally, the paper conjectures that similar approaches could be extended to random, noisy gradient settings, suggesting potential future avenues in stochastic optimization landscapes.

Concluding Remarks and Future Directions

This paper not only solves a longstanding open problem in the optimization of convex functions with guaranteed anytime performance but also establishes a significant foundation for future investigations in adaptive learning frameworks. The authors speculate this could lead to new advancements in AI optimization methods, particularly those dealing with real-time constraints or dynamic environments requiring adaptive computational strategies.

Future research could aim to explore the broader applicability of this algorithm to non-convex settings or integrate it with second-order optimization methods. Extending these ideas to design scalable, robust optimization strategies further democratizes access to refined machine learning models in varied application domains.

Overall, this paper contributes a substantial theoretical and practical framework, paving the way for performance-optimized GD algorithms tailored to contemporary computational challenges.