Convex Optimization with Unbounded Nonconvex Oracles using Simulated Annealing

TL;DR

This paper introduces a polynomial-time algorithm combining simulated annealing with Langevin dynamics to minimize convex functions under complex, unbounded nonconvex noise conditions, extending prior work beyond bounded noise assumptions.

Contribution

It proposes a novel optimization method that handles unbounded, multiplicative noise in convex minimization, surpassing previous approaches limited to bounded noise scenarios.

Findings

The algorithm achieves polynomial-time convergence under the new noise model.

It effectively finds approximate minimizers despite the noise complexity.

The method extends to functions close to convex, even with deteriorating closeness away from the optimum.

Abstract

We consider the problem of minimizing a convex objective function $F$ when one can only evaluate its noisy approximation $\hat{F}$. Unless one assumes some structure on the noise, $\hat{F}$ may be an arbitrary nonconvex function, making the task of minimizing $F$ intractable. To overcome this, prior work has often focused on the case when $F(x)-\hat{F}(x)$ is uniformly-bounded. In this paper we study the more general case when the noise has magnitude $\alpha F(x) + \beta$ for some $\alpha, \beta > 0$, and present a polynomial time algorithm that finds an approximate minimizer of $F$ for this noise model. Previously, Markov chains, such as the stochastic gradient Langevin dynamics, have been used to arrive at approximate solutions to these optimization problems. However, for the noise model considered in this paper, no single temperature allows such a Markov chain to both mix quickly and concentrate near the global minimizer. We bypass this by combining "simulated annealing" with the stochastic gradient Langevin dynamics, and gradually decreasing the temperature of the chain in order to approach the global minimizer. As a corollary one can approximately minimize a nonconvex function that is close to a convex function; however, the closeness can deteriorate as one moves away from the optimum.