An Online Newton's Method for Time-varying Linear Equality Constraints

TL;DR

This paper introduces OPEN-M, an online Newton's method for time-varying linear equality constraints, achieving sublinear regret and constraint violation bounds in adversarial settings without requiring convexity.

Contribution

The paper presents a novel online Newton's method tailored for time-varying linear equality constraints with theoretical guarantees and superior empirical performance.

Findings

Achieves sublinear dynamic regret and constraint violation bounds.

Outperforms existing algorithms in network flow applications.

Operates under mild conditions without convexity requirements.

Abstract

We consider online optimization problems with time-varying linear equality constraints. In this framework, an agent makes sequential decisions using only prior information. At every round, the agent suffers an environment-determined loss and must satisfy time-varying constraints. Both the loss functions and the constraints can be chosen adversarially. We propose the Online Projected Equality-constrained Newton Method (OPEN-M) to tackle this family of problems. We obtain sublinear dynamic regret and constraint violation bounds for OPEN-M under mild conditions. Namely, smoothness of the loss function and boundedness of the inverse Hessian at the optimum are required, but not convexity. Finally, we show OPEN-M outperforms state-of-the-art online constrained optimization algorithms in a numerical network flow application.