A Note on High-Probability Analysis of Algorithms with Exponential, Sub-Gaussian, and General Light Tails

2024-03-05

arXiv.org

AI Summary

High-probability analysis of learning algorithms involving light-tailed (e.g., sub-exponential, sub-Gaussian) but possibly unbounded random variables poses significant technical challenges due to the lack of uniform concentration tools across distribution families. Method: We propose a generic black-box reduction that systematically transforms high-probability analysis of any algorithm relying on light-tailed randomness into the corresponding analysis under bounded-variable assumptions, incurring only controllable logarithmic-factor overheads. Contribution/Results: This is the first unified framework handling diverse light-tailed distributions without ad hoc concentration inequalities—greatly simplifying theoretical analysis. As applications, we reconstruct a generalized Azuma’s inequality and derive tight high-probability convergence bounds for stochastic optimization algorithms under light-tailed noise, demonstrating both the method’s effectiveness and broad applicability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This short note describes a simple technique for analyzing probabilistic algorithms that rely on a light-tailed (but not necessarily bounded) source of randomization. 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 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. Analyses of a generalized Azuma inequality and stochastic optimization with general light-tailed noise are provided to illustrate the technique.

Problem

• Reducing analysis of light-tailed randomized algorithms to bounded cases
• Generalizing high-probability bounds for exponential and sub-Gaussian distributions
• Simplifying proofs for stochastic optimization and bandit problems

Innovation

• Reduces analysis to bounded random variables
• Applies to various light-tailed distributions
• Uses black-box technique with minimal loss