High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails
This addresses robust optimization in machine learning for scenarios with noisy or heavy-tailed data, offering theoretical guarantees that improve upon standard SGD analyses, though it is incremental in extending existing techniques to new conditions.
The paper tackles non-convex stochastic optimization with heavy-tailed gradients by proposing a method combining gradient clipping, momentum, and normalized gradient descent, achieving high-probability convergence to critical points with best-known rates for smooth losses under bounded p-th moments, and extends this to second-order smooth losses with arbitrary norms.
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.