Accelerated First-Order Optimization under Nonlinear Constraints
This addresses efficient optimization under nonlinear constraints for machine learning applications, offering incremental improvements in handling nonconvex constraints like ℓ^p (p<1).
The paper tackles the problem of constrained optimization by designing accelerated first-order algorithms that avoid costly optimization over the entire feasible set, proving convergence for nonconvex settings and accelerated rates for convex ones, with complexity growing mildly in variables and constraints.
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 $p=1$.