Composite Online-to-Nonconvex Conversion with Optimal Oracle Complexity

Key points

• The objective is the sum of a possibly nonsmooth nonconvex Lipschitz function and a convex regularizer, and the function is accessed through stochastic gradients or function values.1
• We extend the framework to our composite scenario by introducing new losses for the learner, which contain the regularizer itself rather than its linearization and for which a variant of online mirror descent achieves low regret.1
• We show that the resulting algorithm finds such a point with $O(δ^{-1}\varepsilon^{-3})$ stochastic gradient queries or $O(dδ^{-1}\varepsilon^{-3})$ function-value queries, where $δ$ is the Goldstein radius, $\varepsilon$ is the stationarity tolerance, and $d$ is the dimension.1
• These rates match the optimal ones for noncomposite nonsmooth nonconvex optimization, demonstrating that the additional convex regularizer does not worsen the oracle complexity.1

Sources (1)

• [1]Composite Online-to-Nonconvex Conversion with Optimal Oracle Complexity
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 8, 05:08 PM

  We consider stochastic nonsmooth nonconvex composite optimization, which includes several important problems such as constrained optimization and the regularized training of neural networks.

  The objective is the sum of a possibly nonsmooth nonconvex Lipschitz function and a convex regularizer, and the function is accessed through stochastic gradients or function values.

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