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A General Reduction for High-Probability Analysis with General Light-Tailed Distributions

Amit Attia, Tomer Koren

TL;DR

The paper introduces a black-box reduction that enables high-probability analyses for algorithms using light-tailed randomness by reducing to a bounded-noise variant while preserving expectations. It formalizes a main theorem showing a $(\gamma,\sigma,c)$-tailed oracle can be simulated by a $B$-bounded oracle with $B(x)=4\sigma(x)(\max\{\log(2c(x)n/\delta),2/\gamma(x)\})^{1/\gamma(x)}$, guaranteeing identical outputs w.h.p. The framework extends classical results such as Azuma's inequality, stochastic gradient descent, and UCB bandits to general light-tailed noise without bespoke concentration tools, at the cost of polylogarithmic factors. The authors prove the key truncation/rejection sampling lemma to construct bounded surrogates that preserve expectations and then demonstrate the tightness of the bound in the maximum-of-RVs setting. Overall, the work provides a unified, black-box approach to generalize high-probability guarantees to a broad class of light-tailed distributions, with practical implications for robust analyses in stochastic optimization and bandits.

Abstract

We describe a general reduction technique for analyzing learning algorithms that are subject to light-tailed (but not necessarily bounded) randomness, a scenario that is often the focus of theoretical analysis. We show that the analysis of such an algorithm can be reduced, in a black-box manner and with only a small loss in logarithmic factors, to an analysis of a simpler variant of the same algorithm that uses bounded random variables and is often easier to analyze. This approach simultaneously applies to any light-tailed randomization, including exponential, sub-Gaussian, and more general fast-decaying distributions, without needing to appeal to specialized concentration inequalities. Derivations of a generalized Azuma inequality, convergence bounds in stochastic optimization, and regret analysis in multi-armed bandits with general light-tailed randomization are provided to illustrate the technique.

A General Reduction for High-Probability Analysis with General Light-Tailed Distributions

TL;DR

The paper introduces a black-box reduction that enables high-probability analyses for algorithms using light-tailed randomness by reducing to a bounded-noise variant while preserving expectations. It formalizes a main theorem showing a -tailed oracle can be simulated by a -bounded oracle with , guaranteeing identical outputs w.h.p. The framework extends classical results such as Azuma's inequality, stochastic gradient descent, and UCB bandits to general light-tailed noise without bespoke concentration tools, at the cost of polylogarithmic factors. The authors prove the key truncation/rejection sampling lemma to construct bounded surrogates that preserve expectations and then demonstrate the tightness of the bound in the maximum-of-RVs setting. Overall, the work provides a unified, black-box approach to generalize high-probability guarantees to a broad class of light-tailed distributions, with practical implications for robust analyses in stochastic optimization and bandits.

Abstract

We describe a general reduction technique for analyzing learning algorithms that are subject to light-tailed (but not necessarily bounded) randomness, a scenario that is often the focus of theoretical analysis. We show that the analysis of such an algorithm can be reduced, in a black-box manner and with only a small loss in logarithmic factors, to an analysis of a simpler variant of the same algorithm that uses bounded random variables and is often easier to analyze. This approach simultaneously applies to any light-tailed randomization, including exponential, sub-Gaussian, and more general fast-decaying distributions, without needing to appeal to specialized concentration inequalities. Derivations of a generalized Azuma inequality, convergence bounds in stochastic optimization, and regret analysis in multi-armed bandits with general light-tailed randomization are provided to illustrate the technique.
Paper Structure (16 sections, 12 theorems, 55 equations)

This paper contains 16 sections, 12 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Generalizing concentration results.
  4. Different choices of noise assumption in applications.
  5. Main result
  6. Main theorem
  7. Discussion and proof ideas
  8. Examples
  9. A prelude: Azuma's inequality
  10. Stochastic first-order optimization
  11. Stochastic multi-armed bandits
  12. Bounding the maximum of RVs
  13. Proof of main theorem
  14. Proof of Lemma 2
  15. Tightness of Lemma 1
  16. ...and 1 more sections

Key Result

Theorem 1

Given an algorithm $\mathcal{A}$, number of rounds $n$ and a $(\gamma,\sigma,c)$-tailed sampling oracle $\mathcal{O}$, for any $\delta > 0$ there exists a $B$-bounded sampling oracle ${\widetilde{\mathcal{O}}}$ with for all $x \in \mathcal{X}$, such that $\mathbb{E}[\mathcal{O}(x)]=\mathbb{E}[{\widetilde{\mathcal{O}}}(x)]$ for all queries $x \in \mathcal{X}$, and with probability at least $1-\del

Theorems & Definitions (21)

  • Definition 1: light-tailed RV / sampling oracle
  • Theorem 1
  • Theorem 2: Azuma's Inequality
  • Theorem 3: Azuma's Inequality for light-tailed RVs
  • proof : Proof of \ref{['thm:azuma-tailed']}
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • proof : Proof sketch
  • Theorem 7
  • ...and 11 more