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Parameter-free Stochastic Optimization of Variationally Coherent Functions

Francesco Orabona, Dávid Pál

TL;DR

This work introduces a parameter-free stochastic optimization algorithm for differentiable functions on $\mathbb{R}^d$ that simultaneously achieves almost-sure convergence to the global minimizer for variationally coherent objectives and near-optimal convex-rate guarantees without averaging. Central to the method is Follow The Regularized Leader with gradient rescaling and a novel linearithmic regularizer, yielding robust last-iterate convergence in non-convex settings and favorable $O\left(T^{-(1-\alpha)}\right)$ rates in convex scenarios. The authors develop explicit regularizers $\phi_t$ via the functions $\psi$ and $\psi^*$, prove key regret bounds, and establish convergence through Bregman-divergence analysis. The work also discusses limitations, adaptivity, and potential extensions, highlighting the framework’s applicability to broader online-to-batch contexts and parameter-free stochastic optimization problems.

Abstract

We design and analyze an algorithm for first-order stochastic optimization of a large class of functions on $\mathbb{R}^d$. In particular, we consider the \emph{variationally coherent} functions which can be convex or non-convex. The iterates of our algorithm on variationally coherent functions converge almost surely to the global minimizer $\boldsymbol{x}^*$. Additionally, the very same algorithm with the same hyperparameters, after $T$ iterations guarantees on convex functions that the expected suboptimality gap is bounded by $\widetilde{O}(\|\boldsymbol{x}^* - \boldsymbol{x}_0\| T^{-1/2+ε})$ for any $ε>0$. It is the first algorithm to achieve both these properties at the same time. Also, the rate for convex functions essentially matches the performance of parameter-free algorithms. Our algorithm is an instance of the Follow The Regularized Leader algorithm with the added twist of using \emph{rescaled gradients} and time-varying linearithmic regularizers.

Parameter-free Stochastic Optimization of Variationally Coherent Functions

TL;DR

This work introduces a parameter-free stochastic optimization algorithm for differentiable functions on that simultaneously achieves almost-sure convergence to the global minimizer for variationally coherent objectives and near-optimal convex-rate guarantees without averaging. Central to the method is Follow The Regularized Leader with gradient rescaling and a novel linearithmic regularizer, yielding robust last-iterate convergence in non-convex settings and favorable rates in convex scenarios. The authors develop explicit regularizers via the functions and , prove key regret bounds, and establish convergence through Bregman-divergence analysis. The work also discusses limitations, adaptivity, and potential extensions, highlighting the framework’s applicability to broader online-to-batch contexts and parameter-free stochastic optimization problems.

Abstract

We design and analyze an algorithm for first-order stochastic optimization of a large class of functions on . In particular, we consider the \emph{variationally coherent} functions which can be convex or non-convex. The iterates of our algorithm on variationally coherent functions converge almost surely to the global minimizer . Additionally, the very same algorithm with the same hyperparameters, after iterations guarantees on convex functions that the expected suboptimality gap is bounded by for any . It is the first algorithm to achieve both these properties at the same time. Also, the rate for convex functions essentially matches the performance of parameter-free algorithms. Our algorithm is an instance of the Follow The Regularized Leader algorithm with the added twist of using \emph{rescaled gradients} and time-varying linearithmic regularizers.

Paper Structure

This paper contains 26 sections, 26 theorems, 169 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setup and Notation
  4. Problem Setup
  5. Notation
  6. Main Results
  7. FTRL with Rescaled Gradients
  8. Linearithmic Regularizer
  9. Definition of $\phi_t$ and its properties
  10. Explicit formulas
  11. Proofs of the Main Results
  12. Proof of Theorem \ref{['theorem:variationally-coherent-convergence']}
  13. Proof of Theorem \ref{['theorem:convex-average-non-adaptive']}
  14. Proof of Theorem \ref{['theorem:convex-last-iterate']}
  15. Discussions on Limitations and Future Work
  16. ...and 11 more sections

Key Result

Theorem 2

Let $F:\mathbb{R}^d \to \mathbb{R}$ be a variationally coherent function with minimizer $\boldsymbol{x}^*$. Assume the stochastic gradients satisfy equation:stochastic-gradient and equation:gradient-bound. Assume that learning rate $\eta_t$ is a non-negative $\mathcal{F}_t$-measurable random variabl Then, the sequence $\boldsymbol{x}_1, \boldsymbol{x}_2, \dots$ generated by Algorithm algorithm:ftr

Theorems & Definitions (53)

  • Definition 1: Variatonally coherent function
  • Definition 1: Variatonally coherent function
  • Theorem 2: $\boldsymbol{x}_t$ converges to $\boldsymbol{x}^*$
  • Theorem 3: Convergence rate of running average for convex functions
  • Theorem 4: Convergence rate of last iterate for convex functions
  • Lemma 5: FTRL regret equality
  • Lemma 6: FTRL for stochastic optimization
  • proof
  • Lemma 7: Convergence of Bregman divergences
  • Lemma 8: $\left\|{\boldsymbol{x}_t-\boldsymbol{x}^*}\right\|^2$ is squeezed
  • ...and 43 more