Linear Convergence of SVRG in Statistical Estimation
This addresses the optimization bottleneck for non-convex and non-strongly convex statistical models, providing a theoretical guarantee for faster convergence in machine learning applications.
The paper proves that SVRG achieves linear convergence for statistical M-estimators like Lasso and logistic regression, even without strong convexity, by using restricted strong convexity to converge to the statistical precision of the model.
SVRG and its variants are among the state of art optimization algorithms for large scale machine learning problems. It is well known that SVRG converges linearly when the objective function is strongly convex. However this setup can be restrictive, and does not include several important formulations such as Lasso, group Lasso, logistic regression, and some non-convex models including corrected Lasso and SCAD. In this paper, we prove that, for a class of statistical M-estimators covering examples mentioned above, SVRG solves the formulation with {\em a linear convergence rate} without strong convexity or even convexity. Our analysis makes use of {\em restricted strong convexity}, under which we show that SVRG converges linearly to the fundamental statistical precision of the model, i.e., the difference between true unknown parameter $θ^*$ and the optimal solution $\hatθ$ of the model.