Understanding the Acceleration Phenomenon via High-Resolution Differential Equations (1810.08907v3)

Abstract: Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms---Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Polyak's heavy-ball method---we study an alternative limiting process that yields high-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG-SC and Polyak's heavy-ball method, but they allow the identification of a term that we refer to as "gradient correction" that is present in NAG-SC but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov's accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result---that NAG-C minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG-C for smooth convex functions.

• The paper introduces high-resolution ODEs to differentiate between NAG-SC and heavy-ball methods and clarify their dynamic behaviors.
• It develops a Lyapunov function framework for analyzing convergence in both continuous and discrete systems, offering deeper theoretical insight.
• The study reveals that modifying NAG-C's high-resolution ODE leads to inverse cubic gradient norm minimization and accelerated convergence.

Understanding High-Resolution Differential Equations in Optimization

The paper presents a detailed exploration of gradient-based optimization algorithms through the lens of high-resolution ordinary differential equations (ODEs), offering a new perspective on well-established acceleration techniques. These techniques have significant implications for large-scale machine learning problems, where understanding the dynamics of optimization algorithms can lead to improved performance and efficiency.

Key Contributions

1. High-Resolution ODEs: The authors introduce high-resolution ODEs as a tool for distinguishing between Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Polyak's heavy-ball method. While both share similarities in traditional ODE frameworks, these high-resolution models incorporate a term referred to as "gradient correction," offering finer insight into their differing behaviors.
2. Lyapunov Function Framework: Through these ODEs, a Lyapunov function framework is developed for analyzing convergence in both continuous and discrete-time settings. This methodology not only captures algorithm dynamics but also illuminates the qualitative differences in convergence between NAG-SC and the heavy-ball method.
3. Gradient Norm Minimization: The paper uncovers that Nesterov's accelerated gradient method for convex functions (NAG-C) achieves an inverse cubic rate for minimizing the squared gradient norm, a hitherto unrecognized result. This is facilitated by the introduction of high-resolution ODEs which capture second-order information even in first-order methods.
4. Family of New Methods: By altering the high-resolution ODE of NAG-C, the authors propose a family of new optimization methods that maintain NAG-C's accelerated convergence rates for smooth convex functions.

Implications and Future Directions

The implications of these findings are twofold. Practically, understanding the finer dynamics of these optimization methods can lead to algorithms with better convergence properties and stability, essential for large-scale machine learning applications. Theoretically, this approach provides a framework for potentially uncovering new optimization methods that could outperform current best practices.

The introduction of high-resolution ODEs is particularly intriguing as it bridges a gap in the quantitative understanding of momentum-based methods—a critical component in optimization. The gradient correction term, identified and formalized in this high-resolution setting, could lead to new insights into why certain methods succeed where others do not.

Looking forward, the high-resolution ODE framework could serve as a foundation for further exploration into non-Euclidean settings and extensions to non-convex optimization. Moreover, the potential to apply this framework to other momentum-based methods, such as Adam or RMSProp, offers a rich avenue for research, particularly in adapting these methods to different learning environments and constraints.

The development of high-resolution ODEs represents a significant step forward in the analysis of optimization algorithms, setting the stage for both refinements to existing methods and the creation of novel approaches that harness the power of these insights for practical and theoretical advancements.