Weak Proximal Newton Oracles for Composite Convex Optimization

Second-order methods are of great importance for composite convex optimization problems due to their local super-linear convergence rates (under appropriate assumptions). However, the presence of even a simple nonsmooth function in the model most often renders the subproblems in proximal Newton methods computationally-difficult to solve in high-dimension. We introduce a novel approach based on a weak proximal Newton oracle (WPNO), which only requires to solve such subproblems to accuracy that is comparable to that of the optimal solution of the global problem, while maintaining local super-linear convergence under standard assumptions. We show that when the optimal solution of the global problem admits some sparse structure, a WPNO could be implemented very efficiently using specialized first-order methods.