Stochastic second-order optimization for over-parameterized machine learning models

We consider stochastic second-order methods for minimizing smooth and strongly-convex functions under an interpolation condition, which can be satisfied by over-parameterized machine learning models. Under this condition, we show that the regularized subsampled Newton’s method (R-SSN) achieves global linear convergence with an adaptive step-size and a constant batch-size. By growing the batch size for both the subsampled gradient and Hessian, we show that R-SSN can converge at a quadratic rate in a local neighbourhood of the solution. We also show that R-SSN attains local linear convergence for the family of self-concordant functions. Furthermore, we analyze stochastic BFGS algorithms in the interpolation setting and prove their global linear convergence. We empirically evaluate stochastic limited-memory BFGS (L-BFGS) and a “Hessian-free” implementation of R-SSN for binary classification on synthetic, linearly-separable datasets and real datasets under a kernel mapping. Our experimental results demonstrate the fast convergence of these methods, both in terms of the number of iterations and wall-clock time.Science, Faculty ofComputer Science, Department ofGraduat