Improved Oracle Complexity of Variance Reduced Methods for Nonsmooth Convex Stochastic Composition Optimization
This work addresses optimization challenges in machine learning for problems like sparse mean-variance optimization, though it is incremental as it extends existing SCVRG methods to nonsmooth convex cases.
The paper tackles nonsmooth convex composition optimization problems, which include important examples like Lasso and deep neural nets, by analyzing stochastic compositional variance reduced gradient (SCVRG) methods and proving they achieve a total incremental first-order oracle complexity of O((m+n)log(1/ε)+1/ε^3), improving over prior methods such as SCGD and accelerated gradient descent.
We consider the nonsmooth convex composition optimization problem where the objective is a composition of two finite-sum functions and analyze stochastic compositional variance reduced gradient (SCVRG) methods for them. SCVRG and its variants have recently drawn much attention given their edge over stochastic compositional gradient descent (SCGD); but the theoretical analysis exclusively assumes strong convexity of the objective, which excludes several important examples such as Lasso, logistic regression, principle component analysis and deep neural nets. In contrast, we prove non-asymptotic incremental first-order oracle (IFO) complexity of SCVRG or its novel variants for nonsmooth convex composition optimization and show that they are provably faster than SCGD and gradient descent. More specifically, our method achieves the total IFO complexity of $O\left((m+n)\log\left(1/ε\right)+1/ε^3\right)$ which improves that of $O\left(1/ε^{3.5}\right)$ and $O\left((m+n)/\sqrtε\right)$ obtained by SCGD and accelerated gradient descent (AGD) respectively. Experimental results confirm that our methods outperform several existing methods, e.g., SCGD and AGD, on sparse mean-variance optimization problem.