Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems
This addresses a bottleneck in optimization for scenarios with unbounded noise, offering improved performance in machine learning and statistical applications.
The paper tackles stochastic optimization with heavy-tailed noise by introducing a method using smoothed medians of means to stabilize gradients, achieving faster convergence rates than previous bounds, specifically improving from O(K^{-2(α-1)/α}) for α in (1,2].
We consider stochastic optimization problems with heavy-tailed noise with structured density. For such problems, we show that it is possible to get faster rates of convergence than $\mathcal{O}(K^{-2(α- 1)/α})$, when the stochastic gradients have finite moments of order $α\in (1, 2]$. In particular, our analysis allows the noise norm to have an unbounded expectation. To achieve these results, we stabilize stochastic gradients, using smoothed medians of means. We prove that the resulting estimates have negligible bias and controllable variance. This allows us to carefully incorporate them into clipped-SGD and clipped-SSTM and derive new high-probability complexity bounds in the considered setup.