Optimizing Optimizers: Regret-optimal gradient descent algorithms

Philippe Casgrain , Anastasis Kratsios

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Session: Optimization(A)

Session Chair: Simon Du

Poster: Poster Session 1

Abstract: This paper treats the task of designing optimization algorithms as an optimal control problem. Using regret as a metric for an algorithm's performance, we study the existence, uniqueness and consistency of regret-optimal algorithms. By providing first-order optimality conditions for the control problem, we show that regret-optimal algorithms must satisfy a specific structure in their dynamics which we show is equivalent to performing \emph{dual-preconditioned gradient descent} on the value function generated by its regret. Using these optimal dynamics, we provide bounds on their rates of convergence to solutions of convex optimization problems. Though closed-form optimal dynamics cannot be obtained in general, we present fast numerical methods for approximating them, generating optimization algorithms which directly optimize their long-term regret. These are benchmarked against commonly used optimization algorithms to demonstrate their effectiveness.

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