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A simple and improved algorithm for noisy, convex, zeroth-order optimisation

Alexandra Carpentier

TL;DR

This work provides a conceptually simple method inspired by the textbook centre of gravity method, but adapted to the noisy and zeroth-order setting, and proves that this method is such that the f(\hat x) - f(x) is of smaller order than d^{2}/\sqrt{n} up to poly-logarithmic terms.

Abstract

In this paper, we study the problem of noisy, convex, zeroth order optimisation of a function $f$ over a bounded convex set $\bar{\mathcal X}\subset \mathbb{R}^d$. Given a budget $n$ of noisy queries to the function $f$ that can be allocated sequentially and adaptively, our aim is to construct an algorithm that returns a point $\hat x\in \bar{\mathcal X}$ such that $f(\hat x)$ is as small as possible. We provide a conceptually simple method inspired by the textbook center of gravity method, but adapted to the noisy and zeroth order setting. We prove that this method is such that the $f(\hat x) - \min_{x\in \bar{\mathcal X}} f(x)$ is of smaller order than $d^2/\sqrt{n}$ up to poly-logarithmic terms. We slightly improve upon existing literature, where to the best of our knowledge the best known rate is in [Lattimore, 2024] is of order $d^{2.5}/\sqrt{n}$, albeit for a more challenging problem. Our main contribution is however conceptual, as we believe that our algorithm and its analysis bring novel ideas and are significantly simpler than existing approaches.

A simple and improved algorithm for noisy, convex, zeroth-order optimisation

TL;DR

This work provides a conceptually simple method inspired by the textbook centre of gravity method, but adapted to the noisy and zeroth-order setting, and proves that this method is such that the f(\hat x) - f(x) is of smaller order than d^{2}/\sqrt{n} up to poly-logarithmic terms.

Abstract

In this paper, we study the problem of noisy, convex, zeroth order optimisation of a function over a bounded convex set . Given a budget of noisy queries to the function that can be allocated sequentially and adaptively, our aim is to construct an algorithm that returns a point such that is as small as possible. We provide a conceptually simple method inspired by the textbook center of gravity method, but adapted to the noisy and zeroth order setting. We prove that this method is such that the is of smaller order than up to poly-logarithmic terms. We slightly improve upon existing literature, where to the best of our knowledge the best known rate is in [Lattimore, 2024] is of order , albeit for a more challenging problem. Our main contribution is however conceptual, as we believe that our algorithm and its analysis bring novel ideas and are significantly simpler than existing approaches.
Paper Structure (23 sections, 11 theorems, 74 equations, 3 algorithms)

This paper contains 23 sections, 11 theorems, 74 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary results and notations
  3. Smoothed functional notations and results on smoothed convex functions
  4. Estimators of the function and of the gradient of smoothed convex functions
  5. Algorithm
  6. Part 1: finding a cutting point
  7. Part 2: routine for effectively cutting the space
  8. Part 3: final algorithm
  9. Acknowledgements.
  10. Proofs of the results in this paper
  11. Proof of Theorem \ref{['thm:cut']}
  12. Step 1: Definition of a near-optimal set and lower bound on its volume.
  13. Step 2: Results on $\mathbf{FCP}$.
  14. Step 3: Results on a single run of $\mathbf{Cut}$.
  15. Step 4: Induction on several runs of $\mathbf{Cut}$.
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $c>0$ and $z,\tilde{z}\in \mathbb R^d$. If $g_{2c} (z) - g_c(z) \leq 2^{-2}[g_c(z)) - g(\tilde{z})]$

Theorems & Definitions (16)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Proposition 4.5: KLS Lemma
  • Proposition 4.6: Approximate barycentric cutting of an isotropic convex
  • Corollary 4.7: Approximate barycentric cutting of a convex
  • ...and 6 more