Composing Optimized Stepsize Schedules for Gradient Descent (2410.16249v1)

Abstract: Recent works by Altschuler and Parrilo and the authors have shown that it is possible to accelerate the convergence of gradient descent on smooth convex functions, even without momentum, just by picking special stepsizes. In this paper, we provide a general theory for composing stepsize schedules capturing all recent advances in this area and more. We propose three notions of ``composable'' stepsize schedules with elementary associated composition operations for combining them. From these operations, in addition to recovering recent works, we construct three highly optimized sequences of stepsize schedules. We first construct optimized stepsize schedules of every length generalizing the exponentially spaced silver stepsizes. We then construct highly optimized stepsizes schedules for minimizing final objective gap or gradient norm, improving on prior rates by constants and, more importantly, matching or beating the numerically computed minimax optimal schedules. We conjecture these schedules are in fact minimax (information theoretic) optimal. Several novel tertiary results follow from our theory including recovery of the recent dynamic gradient norm minimizing short stepsizes and extending them to objective gap minimization.

• The paper introduces three notions of composable stepsize schedules ($f$, $g$, $s$) optimized for different gradient descent performance criteria, formalized with tight convergence guarantees.
• It defines composition operations for combining these composable schedules, creating a theoretical framework that recovers and extends previous state-of-the-art stepsize results.
• Utilizing the PEP framework, the authors propose methods to derive Optimized Basic Schedules (OBS) that minimize convergence rates and aim for minimax optimality without requiring momentum.

Composing Optimized Stepsize Schedules for Gradient Descent

The paper "Composing Optimized Stepsize Schedules for Gradient Descent" by Grimmer, Shu, and Wang presents a comprehensive theory for constructing optimized stepsize schedules that enhance the convergence efficiency of gradient descent algorithms when applied to smooth convex functions. This research builds on recent findings that demonstrate the potential for accelerated convergence via specific stepsize strategies, eschewing the traditional reliance on momentum.

Summary of Contributions

1. Composability Concepts: The authors introduce three notions of composable stepsize schedules: $f$-composable, $g$-composable, and $s$-composable. Each type of composability pertains to optimizing different performance objectives: reducing final objective gap, reducing gradient norm, and simultaneously balancing both, respectively. These definitions are formalized with tight convergence guarantees.
2. Composition Operations: The paper introduces composition operations (denoted as $\triangleright$, $\triangleleft$, and $\Join$) that allow for the combination of stepsize schedules while maintaining their composability characteristics. These operations provide theoretical underpinnings for constructing new schedules that are guaranteed to achieve certain convergence rates.
3. Theoretical Framework and Proofs: Utilizing the performance estimation problem (PEP) framework, the authors prove the equivalence of their $f$-composability, $g$-composability, and $s$-composability definitions with corresponding real-world implications on gradient descent performance. The proofs rely on intricate algebraic and analytical techniques, embedding robust mathematical structures within the newly proposed schedules.
4. Recovery of Previous Results and Extensions: The proposed framework encapsulates and extends several noted recent advances in stepsize optimization. The paper recovers state-of-the-art stepsize schedules such as the Silver Stepsizes, the numerically derived minimax optimal stepsizes, and dynamic short stepsize schedules. The framework not only subsumes these results but also provides improved convergence guarantees in certain cases.
5. Optimized Basic Schedules (OBS): The researchers propose methods for deriving Optimized Basic Stepsize Schedules (OBS) that minimize rates among basic stepsize schedules. This involves systematic approaches to crafting $f$-composable, $g$-composable, and $s$-composable schedules that yield tight convergence bounds, posited to approach minimax optimality.

Implications and Future Directions

The implications of this work are significant for both theoretical insights and practical applications in optimization. By constructing stepsize schedules that maximize efficiency without requiring momentum, the work offers a pathway towards more streamlined and potentially more intuitive optimization methods. Practically, these results could improve the runtimes and efficiencies of machine learning and data analysis algorithms that rely heavily on gradient-based optimization.

Future research could explore the following avenues:

• Validation and Extension: Incorporating empirical evaluations over diverse datasets and problem instances to validate the theoretical results and potentially extending the principles to non-convex or stochastic settings.
• Automated Composition Tools: Developing algorithms capable of automatically discovering optimal compositions for given problem classes, potentially leveraging reinforcement learning or genetic algorithm frameworks.
• Exploration of Duality and Symmetry in Gradient Descent: Further exploration of the symmetries between $f$-composable and $g$-composable schedules, exploiting these relationships could yield novel dualistic algorithms that share structural and performance benefits.

In conclusion, the paper establishes a systematic theory for constructing and composing stepsize schedules for gradient descent, offering a unifying framework that not only synthesizes but also advances current understanding in convex optimization. Through rigorous mathematical development and potential real-world application insights, this work represents a substantial contribution to the optimization community.