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arXiv · 2009.13016

Escaping Saddle-Points Faster under Interpolation-like Conditions

Abhishek Roy·Krishnakumar Balasubramanian·Saeed Ghadimi·Prasant Mohapatra
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Abstract

In this paper, we show that under over-parametrization several standard stochastic optimization algorithms escape saddle-points and converge to local-minimizers much faster. One of the fundamental aspects of over-parametrized models is that they are capable of interpolating the training data. We show that, under interpolation-like assumptions satisfied by the stochastic gradients in an over-parametrization setting, the first-order oracle complexity of Perturbed Stochastic Gradient Descent (PSGD) algorithm to reach an $\epsilon$-local-minimizer, matches the corresponding deterministic rate of $\tilde{\mathcal{O}}(1/\epsilon^{2})$. We next analyze Stochastic Cubic-Regularized Newton (SCRN) algorithm under interpolation-like conditions, and show that the oracle complexity to reach an $\epsilon$-local-minimizer under interpolation-like conditions, is $\tilde{\mathcal{O}}(1/\epsilon^{2.5})$. While this obtained complexity is better than the corresponding complexity of either PSGD, or SCRN without interpolation-like assumptions, it does not match the rate of $\tilde{\mathcal{O}}(1/\epsilon^{1.5})$ corresponding to deterministic Cubic-Regularized Newton method. It seems further Hessian-based interpolation-like assumptions are necessary to bridge this gap. We also discuss the corresponding improved complexities in the zeroth-order settings.

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BibTeXRIS

Abhishek Roy, Krishnakumar Balasubramanian, Saeed Ghadimi, Prasant Mohapatra. 2020-09-28. Escaping Saddle-Points Faster under Interpolation-like Conditions. https://arxiv.org/abs/2009.13016

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