A Stochastic Semismooth Newton Method for Nonsmooth Nonconvex Optimization
This addresses optimization challenges in machine learning for problems with nonsmooth nonconvex objectives, offering a hybrid method that is incremental in combining existing techniques.
The paper tackles stochastic optimization problems with smooth nonconvex and nonsmooth convex terms by proposing a globalized stochastic semismooth Newton method, achieving global convergence to stationary points in expectation and local r-superlinear convergence with high probability, as demonstrated in numerical tests on tasks like ℓ1-regularized logistic regression.
In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via calling stochastic first and second order oracles. The proposed method can be seen as a hybrid approach combining stochastic semismooth Newton steps and stochastic proximal gradient steps. Two inexact growth conditions are incorporated to monitor the convergence and the acceptance of the semismooth Newton steps and it is shown that the algorithm converges globally to stationary points in expectation. Moreover, under standard assumptions and utilizing random matrix concentration inequalities, we prove that the proposed approach locally turns into a pure stochastic semismooth Newton method and converges r-superlinearly with high probability. We present numerical results and comparisons on $\ell_1$-regularized logistic regression and nonconvex binary classification that demonstrate the efficiency of our algorithm.