High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails

• URL: http://arxiv.org/abs/2106.14343v1
• Date: Mon, 28 Jun 2021 00:17:01 GMT
• Title: High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails
• Authors: Ashok Cutkosky and Harsh Mehta
• Abstract summary: We consider non- Hilbert optimization using first-order algorithms for which the gradient estimates may have tails. We show that a combination of gradient, momentum, and normalized gradient descent convergence to critical points in high-probability with best-known iteration for smooth losses.
• Score: 55.561406656549686
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider non-convex stochastic optimization using first-order algorithms for which the gradient estimates may have heavy tails. We show that a combination of gradient clipping, momentum, and normalized gradient descent yields convergence to critical points in high-probability with best-known rates for smooth losses when the gradients only have bounded $\mathfrak{p}$th moments for some $\mathfrak{p}\in(1,2]$. We then consider the case of second-order smooth losses, which to our knowledge have not been studied in this setting, and again obtain high-probability bounds for any $\mathfrak{p}$. Moreover, our results hold for arbitrary smooth norms, in contrast to the typical SGD analysis which requires a Hilbert space norm. Further, we show that after a suitable "burn-in" period, the objective value will monotonically decrease for every iteration until a critical point is identified, which provides intuition behind the popular practice of learning rate "warm-up" and also yields a last-iterate guarantee.

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