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Nonsmooth Optimization with Zeroth Order Comparison Feedback

Taha El Bakkali, El Mahdi Chayti, Omar Saadi

TL;DR

By leveraging a Russian-roulette truncation on the Bernoulli-product expansion of the inverse link, this work constructs an exactly unbiased estimator for directional differences, and plugs this into the smoothed gradient identity enables a standard nonconvex SGD analysis.

Abstract

We study unconstrained optimization problems of nonsmooth, nonconvex Lipschitz functions, using only noisy pairwise comparisons governed by a known link function. Our goal is to compute a $(δ,\varepsilon)$-Goldstein stationary point. We combine randomized smoothing with a novel unbiased reduction from comparisons to local value differences. By leveraging a Russian-roulette truncation on the Bernoulli-product expansion of the inverse link, we construct an exactly unbiased estimator for directional differences. This estimator has finite expected cost and variance scaling quadratically with the function gap, $\mathcal{O}(B^2)$, under mild conditions. Plugging this into the smoothed gradient identity enables a standard nonconvex SGD analysis, yielding explicit comparison-complexity bounds for common symmetric links such as logistic, probit, and cauchit.

Nonsmooth Optimization with Zeroth Order Comparison Feedback

TL;DR

By leveraging a Russian-roulette truncation on the Bernoulli-product expansion of the inverse link, this work constructs an exactly unbiased estimator for directional differences, and plugs this into the smoothed gradient identity enables a standard nonconvex SGD analysis.

Abstract

We study unconstrained optimization problems of nonsmooth, nonconvex Lipschitz functions, using only noisy pairwise comparisons governed by a known link function. Our goal is to compute a -Goldstein stationary point. We combine randomized smoothing with a novel unbiased reduction from comparisons to local value differences. By leveraging a Russian-roulette truncation on the Bernoulli-product expansion of the inverse link, we construct an exactly unbiased estimator for directional differences. This estimator has finite expected cost and variance scaling quadratically with the function gap, , under mild conditions. Plugging this into the smoothed gradient identity enables a standard nonconvex SGD analysis, yielding explicit comparison-complexity bounds for common symmetric links such as logistic, probit, and cauchit.
Paper Structure (37 sections, 12 theorems, 39 equations)

This paper contains 37 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Comparison model and link function.
  3. Main contributions.
  4. $(\delta,\varepsilon)$-Goldstein Stationarity Through Pairwise Comparisons
  5. Goldstein stationarity and random smoothing
  6. Goldstein $\delta$-subdifferential.
  7. Random smoothing.
  8. A standard smoothing-based certificate.
  9. How comparisons enter.
  10. Comparison model and link function
  11. Comparison model.
  12. Operated probability range.
  13. Logistic link and unbiased gap estimation from comparisons
  14. Comparison oracle (logistic link).
  15. Bounded gaps via symmetric queries.
  16. ...and 22 more sections

Key Result

Lemma 2.1

Assume $f$ is locally Lipschitz. Then $f_\delta$ is differentiable and, for every $x\in\mathbb{R}^d$, $\nabla f_\delta(x)\in \partial_\delta f(x).$ Consequently, In particular, any point $x$ satisfying $\|\nabla f_\delta(x)\|\le \varepsilon$ is $(\delta,\varepsilon)$-Goldstein stationary.

Theorems & Definitions (22)

  • Lemma 2.1: Smoothed gradients certify Goldstein stationarity
  • Lemma 3.1: Logit series
  • proof
  • Lemma 3.2: Unbiasedness
  • proof
  • Lemma 3.3: Expected comparison cost
  • proof
  • Lemma 3.4: Finite second moment
  • proof
  • Corollary 3.5: Second moment scales as $O(B^2)$
  • ...and 12 more