Recursive Decomposition for Nonconvex Optimization

TL;DR

The paper introduces RDIS, a recursive decomposition algorithm for nonconvex optimization that exploits combinatorial structure to efficiently find solutions, outperforming traditional methods in complex AI problems.

Contribution

It presents a novel recursive decomposition approach, RDIS, that leverages problem structure to solve nonconvex optimization problems more efficiently than existing techniques.

Findings

RDIS solves certain nonconvex problems exponentially faster than gradient descent.

RDIS outperforms standard techniques on structure from motion and protein folding tasks.

The method effectively decomposes complex problems into simpler subproblems.

Abstract

Continuous optimization is an important problem in many areas of AI, including vision, robotics, probabilistic inference, and machine learning. Unfortunately, most real-world optimization problems are nonconvex, causing standard convex techniques to find only local optima, even with extensions like random restarts and simulated annealing. We observe that, in many cases, the local modes of the objective function have combinatorial structure, and thus ideas from combinatorial optimization can be brought to bear. Based on this, we propose a problem-decomposition approach to nonconvex optimization. Similarly to DPLL-style SAT solvers and recursive conditioning in probabilistic inference, our algorithm, RDIS, recursively sets variables so as to simplify and decompose the objective function into approximately independent sub-functions, until the remaining functions are simple enough to be optimized by standard techniques like gradient descent. The variables to set are chosen by graph partitioning, ensuring decomposition whenever possible. We show analytically that RDIS can solve a broad class of nonconvex optimization problems exponentially faster than gradient descent with random restarts. Experimentally, RDIS outperforms standard techniques on problems like structure from motion and protein folding.