Constants of Motion: The Antidote to Chaos in Optimization and Game Dynamics

TL;DR

This paper demonstrates that identifying invariant functions, or constants of motion, can prevent chaos in optimization and game dynamics, ensuring regularity and stability in these systems.

Contribution

It introduces the concept of invariant functions as a tool to guarantee regularity in optimization and game dynamics, with explicit forms and proofs for various algorithms.

Findings

Invariant functions exist for several algorithms like gradient descent and multiplicative weights.

Explicit closed-form invariants are provided for some cases.

Non-constructive proofs using dynamical systems tools are also presented.

Abstract

Several recent works in online optimization and game dynamics have established strong negative complexity results including the formal emergence of instability and chaos even in small such settings, e.g., $2\times 2$ games. These results motivate the following question: Which methodological tools can guarantee the regularity of such dynamics and how can we apply them in standard settings of interest such as discrete-time first-order optimization dynamics? We show how proving the existence of invariant functions, i.e., constant of motions, is a fundamental contribution in this direction and establish a plethora of such positive results (e.g. gradient descent, multiplicative weights update, alternating gradient descent and manifold gradient descent) both in optimization as well as in game settings. At a technical level, for some conservation laws we provide an explicit and concise closed form, whereas for other ones we present non-constructive proofs using tools from dynamical systems.