Last-Iterate Convergence of Clipped SGD with Linear Decay

Key points

• We assume that the stochastic gradients are unbiased and that the noise has a bounded central $p$-th moment, where $1<p\le2$.1
• For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order $O(LR^2\log(1/δ)/T+σR(\log(1/δ)/T)^{(p-1)/p})$.1
• We further establish last-iterate guarantees for nonsmooth composite objectives and smooth composite objectives with proximal updates.1
• In addition, when a causal adversary replaces at most $q$ gradients, the same clipped update achieves a bound with $\log(1/δ)$ replaced by $\log(1/δ)+q$.1

Sources (1)

• [1]Last-Iterate Convergence of Clipped SGD with Linear Decay
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 13, 03:29 PM

  In this paper, we study the last-iterate convergence of clipped stochastic gradient descent under heavy-tailed noise.

  We assume that the stochastic gradients are unbiased and that the noise has a bounded central $p$-th moment, where $1<p\le2$.

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