Adaptive Gradient Methods for Constrained Convex Optimization and Variational Inequalities

TL;DR

This paper introduces new adaptive first-order methods for constrained convex optimization that are universal, accelerated, and do not require prior knowledge of the objective function, achieving near-optimal convergence rates.

Contribution

The paper presents AdaACSA and AdaAGD+ algorithms that are accelerated, universal, and adaptive, extending adaptive methods to constrained convex optimization and variational inequalities.

Findings

AdaACSA and AdaAGD+ achieve nearly-optimal convergence rates.

AdaGrad+ attains standard non-accelerated convergence.

New adaptive methods for unconstrained optimization and monotone operators.

Abstract

We provide new adaptive first-order methods for constrained convex optimization. Our main algorithms AdaACSA and AdaAGD+ are accelerated methods, which are universal in the sense that they achieve nearly-optimal convergence rates for both smooth and non-smooth functions, even when they only have access to stochastic gradients. In addition, they do not require any prior knowledge on how the objective function is parametrized, since they automatically adjust their per-coordinate learning rate. These can be seen as truly accelerated Adagrad methods for constrained optimization. We complement them with a simpler algorithm AdaGrad+ which enjoys the same features, and achieves the standard non-accelerated convergence rate. We also present a set of new results involving adaptive methods for unconstrained optimization and monotone operators.