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Fully Unconstrained Online Learning

Ashok Cutkosky, Zakaria Mhammedi

TL;DR

The paper addresses the challenge of fully unconstrained online learning for convex losses, aiming to achieve sublinear regret without prior knowledge of the loss sequence magnitude or the comparator norm. It introduces a parameter-free algorithm that achieves a near-optimal regret bound by combining magnitude hints with a small regularization term cast through an epigraph constraint, and then reducing the general problem to a 1D setting via standard dimensionality reductions. The approach balances a tractable, data-driven regularization schedule with careful reductions to obtain regret that matches the $G\|w_\star\|\sqrt{T}$ rate up to logarithmic factors in all practically interesting cases. The work also explores generalizations to other regularizers, provides lower bounds showing near tightness, and discusses extensions to stochastic optimization and adaptivity, highlighting both the theoretical significance and potential for robust, parameter-free online learning in practice.

Abstract

We provide an online learning algorithm that obtains regret $G\|w_\star\|\sqrt{T\log(\|w_\star\|G\sqrt{T})} + \|w_\star\|^2 + G^2$ on $G$-Lipschitz convex losses for any comparison point $w_\star$ without knowing either $G$ or $\|w_\star\|$. Importantly, this matches the optimal bound $G\|w_\star\|\sqrt{T}$ available with such knowledge (up to logarithmic factors), unless either $\|w_\star\|$ or $G$ is so large that even $G\|w_\star\|\sqrt{T}$ is roughly linear in $T$. Thus, it matches the optimal bound in all cases in which one can achieve sublinear regret, which arguably most "interesting" scenarios.

Fully Unconstrained Online Learning

TL;DR

The paper addresses the challenge of fully unconstrained online learning for convex losses, aiming to achieve sublinear regret without prior knowledge of the loss sequence magnitude or the comparator norm. It introduces a parameter-free algorithm that achieves a near-optimal regret bound by combining magnitude hints with a small regularization term cast through an epigraph constraint, and then reducing the general problem to a 1D setting via standard dimensionality reductions. The approach balances a tractable, data-driven regularization schedule with careful reductions to obtain regret that matches the rate up to logarithmic factors in all practically interesting cases. The work also explores generalizations to other regularizers, provides lower bounds showing near tightness, and discusses extensions to stochastic optimization and adaptivity, highlighting both the theoretical significance and potential for robust, parameter-free online learning in practice.

Abstract

We provide an online learning algorithm that obtains regret on -Lipschitz convex losses for any comparison point without knowing either or . Importantly, this matches the optimal bound available with such knowledge (up to logarithmic factors), unless either or is so large that even is roughly linear in . Thus, it matches the optimal bound in all cases in which one can achieve sublinear regret, which arguably most "interesting" scenarios.
Paper Structure (32 sections, 30 theorems, 187 equations, 6 algorithms)

This paper contains 32 sections, 30 theorems, 187 equations, 6 algorithms.

Table of Contents

  1. Unconstrained Online Learning
  2. Our new upper bound.
  3. Comparison with previous bounds
  4. Notation
  5. Overview of Approach
  6. Hints and Regularization
  7. Challenge of achieving (\ref{['eqn:regularizedgoal']}).
  8. Re-parametrizing to achieve (\ref{['eqn:regularizedgoal']}).
  9. Proof Sketch of Theorem \ref{['thm:finalphalfqone']}
  10. Regularized Online Learning via Epigraph Constraints
  11. Generalizations
  12. Lower Bounds
  13. Discussion
  14. Other forms of Unconstrained Online Learning
  15. Parameter-free Algorithms and Stochastic Convex Optimization
  16. ...and 17 more sections

Key Result

Theorem 1

There exists an online learning algorithm that requires $O(d)$ space and takes $O(d)$ time per update, takes as input scalar values $\gamma$, $h_1$, and $\epsilon$ and ensures that for any sequence $g_1,g_2,\dots\subset \mathbb{R}^d$, the outputs $w_1,w_1,\dots\subset \mathbb{R}^d$ satisfy for all $ where $G=\max(h_1, \max_{t\in[T]} \|g_t\|)$ and $V=G^2 + \sum_{t=1}^T \|g_t\|^2$.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 4
  • proof
  • Theorem 5: cutkosky2018black
  • Corollary 5
  • proof
  • Corollary 5
  • ...and 46 more