Revisiting Smoothed Online Learning: Competitive Ratio, Dynamic Regret with Switching Cost, and Best of Both Worlds

Key points

• Surprisingly, minimizing the hitting cost alone is $\max(1,\frac{2}α)$-competitive for $α$-polyhedral functions under the $\ell2$-norm switching cost and $1+\frac{4}λ$-competitive for $λ$-quadratic growth functions under the squared $\ell2$-norm switching cost, significantly improving existing guarantees.1
• The polyhedral bound is tight for this naive algorithm, and we establish a matching lower bound for $α\geq1$.1
• For dynamic regret with the $\ell2$-norm switching cost, we develop Smoothed Ader (SAder), which achieves the optimal $O(\sqrt{T(1+PT)})$ rate in the standard online convex optimization setting, where $PT$ is the comparator's path-length.1
• Finally, for convex hitting costs on a bounded convex domain, we develop UDNP to aggregate any two experts in the lookahead setting under both switching costs.1

Sources (1)

• [1]Revisiting Smoothed Online Learning: Competitive Ratio, Dynamic Regret with Switching Cost, and Best of Both Worlds
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 17, 01:50 AM

  We revisit smoothed online learning and develop algorithms with improved competitive ratios, optimal dynamic regret with switching cost, and best-of-both-worlds guarantees.

  Surprisingly, minimizing the hitting cost alone is $\max(1,\frac{2}α)$-competitive for $α$-polyhedral functions under the $\ell_2$-norm switching cost and $1+\frac{4}λ$-competitive for $λ$-quadratic growth functions under the squared $\ell_2$-norm switching cost, significantly improving existing guarantees.

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