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arXiv · 1910.04920

Fast and Furious Convergence: Stochastic Second Order Methods under Interpolation

Si Yi Meng·Sharan Vaswani·Issam Laradji·Mark Schmidt·Simon Lacoste-Julien
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Abstract

We consider stochastic second-order methods for minimizing smooth and strongly-convex functions under an interpolation condition satisfied by over-parameterized models. Under this condition, we show that the regularized subsampled Newton 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 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.

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Si Yi Meng, Sharan Vaswani, Issam Laradji, Mark Schmidt, Simon Lacoste-Julien. 2019-10-11. Fast and Furious Convergence: Stochastic Second Order Methods under Interpolation. https://arxiv.org/abs/1910.04920

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