Accelerated First-Order Optimization under Nonlinear Constraints

TL;DR

This paper introduces a novel class of accelerated first-order algorithms for constrained optimization that leverage velocity-based constraints, enabling efficient handling of nonconvex constraints and demonstrating strong empirical results in machine learning tasks.

Contribution

The paper presents a new velocity-based approach to constrained optimization that avoids full feasible set optimization and achieves accelerated convergence in both convex and nonconvex settings.

Findings

Algorithms converge to stationary points in nonconvex settings.

Achieve accelerated rates in convex problems in continuous and discrete time.

Effective in sparse regression and compressed sensing with nonconvex constraints.

Abstract

We exploit analogies between first-order algorithms for constrained optimization and non-smooth dynamical systems to design a new class of accelerated first-order algorithms for constrained optimization. Unlike Frank-Wolfe or projected gradients, these algorithms avoid optimization over the entire feasible set at each iteration. We prove convergence to stationary points even in a nonconvex setting and we derive accelerated rates for the convex setting both in continuous time, as well as in discrete time. An important property of these algorithms is that constraints are expressed in terms of velocities instead of positions, which naturally leads to sparse, local and convex approximations of the feasible set (even if the feasible set is nonconvex). Thus, the complexity tends to grow mildly in the number of decision variables and in the number of constraints, which makes the algorithms suitable for machine learning applications. We apply our algorithms to a compressed sensing and a sparse regression problem, showing that we can treat nonconvex $\ell^p$ constraints ($p<1$) efficiently, while recovering state-of-the-art performance for $p=1$.