[Paper Review] The Physical Systems Behind Optimization Algorithms

Lin F. Yang, Raman Arora+2 more|arXiv (Cornell University)|Jan 1, 2018

Stochastic Gradient Optimization Techniques7 citations

TL;DR

This paper introduces a unified physical systems framework using differential equations to analyze optimization algorithms such as gradient descent, Newton's method, and their Nesterov-accelerated variants. By modeling these algorithms as dynamical systems governed by physical laws, the authors provide new insights into their convergence behavior under general conditions like Polyak-Öjasiewicz and error bound, extending beyond convexity.

ABSTRACT

We use differential equations based approaches to provide some {\it extbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a natural connection of optimization algorithms to physical systems. Our analysis is applicable to more general algorithms and optimization problems {\it extbf{beyond}} convexity and strong convexity, e.g. Polyak-\L ojasiewicz and error bound conditions (possibly nonconvex).

Motivation & Objective

• To unify the analysis of popular optimization algorithms through a physical systems lens, revealing deeper dynamical insights.
• To extend convergence analysis beyond convex and strongly convex settings to more general conditions such as Polyak-Öjasiewicz and error bound.
• To provide a framework that captures the behavior of both standard and accelerated variants (e.g., Nesterov's) in a coherent, physically motivated way.
• To model optimization dynamics using continuous-time differential equations that mirror physical motion, enabling stability and convergence analysis.

Proposed method

• Model optimization algorithms as continuous-time dynamical systems using second-order ordinary differential equations (ODEs), inspired by Newtonian mechanics.
• Formulate gradient descent and its variants as systems with mass, damping, and potential energy, where the objective function defines the potential energy landscape.
• Use the concept of mechanical energy (kinetic + potential) to analyze convergence, with energy decay indicating algorithm progress.
• Introduce a generalized framework that accommodates nonconvex objectives by leveraging conditions like the Polyak-Öjasiewicz inequality and error bound.
• Apply asymptotic stability and Lyapunov analysis to prove convergence under weak assumptions, avoiding strong convexity.
• Derive continuous-time analogues of Nesterov's acceleration by incorporating momentum terms with specific damping and mass scaling.

Experimental results

Research questions

• RQ1 How can optimization algorithms be systematically interpreted as physical dynamical systems governed by differential equations?
• RQ2 What physical principles underlie the convergence of standard and accelerated optimization methods like Nesterov's?
• RQ3 To what extent can the framework analyze nonconvex optimization problems under weak conditions such as the Polyak-Öjasiewicz inequality?
• RQ4 How does the energy decay in the physical system relate to the convergence rate of the corresponding optimization algorithm?
• RQ5 Can the framework unify the analysis of diverse algorithms including coordinate descent, proximal methods, and Newton-type methods?

Key findings

• The framework successfully models gradient descent and its accelerated variants as second-order ODEs with physical analogs of mass, damping, and force, enabling a unified dynamical interpretation.
• Convergence is established under the Polyak-Öjasiewicz condition and error bound assumptions, extending results beyond strong convexity.
• The energy decay rate in the physical system corresponds to the convergence rate of the optimization algorithm, providing a direct link between physical and algorithmic behavior.
• Nesterov's acceleration is naturally explained as a form of over-damped oscillation with optimal damping, derived from the physical model.
• The approach reveals that proximal and coordinate descent methods also fit within the same physical framework, suggesting a common dynamical origin.
• The analysis provides a systematic way to derive and understand new variants of optimization algorithms through physical intuition and ODE stability analysis.