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Parameter-free Mirror Descent

Andrew Jacobsen, Ashok Cutkosky

TL;DR

This work develops a modified online mirror descent framework that is suitable for building adaptive and parameter-free algorithms in unbounded domains and develops the first unconstrained online linear optimization algorithm achieving an optimal dynamic regret bound.

Abstract

We develop a modified online mirror descent framework that is suitable for building adaptive and parameter-free algorithms in unbounded domains. We leverage this technique to develop the first unconstrained online linear optimization algorithm achieving an optimal dynamic regret bound, and we further demonstrate that natural strategies based on Follow-the-Regularized-Leader are unable to achieve similar results. We also apply our mirror descent framework to build new parameter-free implicit updates, as well as a simplified and improved unconstrained scale-free algorithm.

Parameter-free Mirror Descent

TL;DR

This work develops a modified online mirror descent framework that is suitable for building adaptive and parameter-free algorithms in unbounded domains and develops the first unconstrained online linear optimization algorithm achieving an optimal dynamic regret bound.

Abstract

We develop a modified online mirror descent framework that is suitable for building adaptive and parameter-free algorithms in unbounded domains. We leverage this technique to develop the first unconstrained online linear optimization algorithm achieving an optimal dynamic regret bound, and we further demonstrate that natural strategies based on Follow-the-Regularized-Leader are unable to achieve similar results. We also apply our mirror descent framework to build new parameter-free implicit updates, as well as a simplified and improved unconstrained scale-free algorithm.
Paper Structure (32 sections, 35 theorems, 162 equations, 4 figures, 7 algorithms)

This paper contains 32 sections, 35 theorems, 162 equations, 4 figures, 7 algorithms.

Table of Contents

  1. Online Learning
  2. Centered Mirror Descent
  3. Parameter-free Learning
  4. Dynamic Regret
  5. Lower Bounds
  6. Dynamic Regret Algorithm
  7. Adapting to Gradient Variability
  8. Lipschitz Adaptivity and Scale-free Learning
  9. Trade-offs in the Horizon Dependence
  10. Conclusion
  11. A Strong Mirror Descent Lemma
  12. Proofs for Section \ref{['sec:centered-md']} (Centered Mirror Descent)
  13. Proof of Lemma \ref{['lemma:centered-md']}
  14. Proof of Lemma \ref{['lemma:md-stability']}
  15. Proofs for Section \ref{['sec:pf']} (Parameter-free Learning)
  16. ...and 17 more sections

Key Result

Lemma 1

(Centered Mirror Descent Lemma) Let $\psi_{t}(\cdot)$ be an arbitrary sequence of differentiable non-negative convex functions, and assume that $w_{1}\in\operatorname{arg\,min}_{w\in\mathbb{R}^{d}}\psi_{t}(w)$ for all $t$. Let $\varphi_{t}(\cdot)$ be an arbitrary sequence of sub-differentiable non-n where $g_{t}\in\partial\ell_{t}(w_t)$ and $\phi_{t}(w)=\Delta_{t}(w)+\varphi_{t}(w)$.

Figures (4)

  • Figure : Centered Mirror Descent
  • Figure : Dynamic Regret Algorithm
  • Figure : Implicit-Optimistic Centered Mirror Descent
  • Figure : Lazy Reduction for Amortized Computation

Theorems & Definitions (56)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 3
  • Theorem 3
  • Proposition 1
  • Theorem 4
  • Theorem 5
  • ...and 46 more