Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems

• URL: http://arxiv.org/abs/2311.04161v2
• Date: Wed, 17 Apr 2024 10:12:59 GMT
• Title: Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems
• Authors: Nikita Puchkin, Eduard Gorbunov, Nikolay Kutuzov, Alexander Gasnikov,
• Abstract summary: We consider clipped optimization problems with heavy-tailed noise with structured density. We show that it is possible to get faster rates of convergence than $mathcalO(K-(alpha - 1)/alpha)$, when the gradients have finite moments of order. We prove that the resulting estimates have negligible bias and controllable variance.
• Score: 56.86067111855056
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\mathcal{O}(K^{-2(\alpha - 1)/\alpha})$, when the stochastic gradients have finite moments of order $\alpha \in (1, 2]$. 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.

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