[Paper Review] Adaptive Online Prediction by Following the Perturbed Leader

Marcus Hütter, Jan Poland|ArXiv.org|Apr 11, 2005

Advanced Bandit Algorithms ResearchDecision Sciences17 references113 citations

TL;DR

This paper introduces an adaptive learning rate version of the Follow the Perturbed Leader (FPL) algorithm for online prediction with expert advice, enabling simple and elegant regret analysis even with arbitrary expert weights and adaptive adversaries. It achieves optimal $√{kL}$ regret bounds for countable expert classes with general weights, matching the best-known results for finite classes, and provides the first such bounds for adaptive learning rates with arbitrary weights via a hierarchical FPL extension.

ABSTRACT

When applying aggregating strategies to Prediction with Expert Advice, the learning rate must be adaptively tuned. The natural choice of sqrt(complexity/current loss) renders the analysis of Weighted Majority derivatives quite complicated. In particular, for arbitrary weights there have been no results proven so far. The analysis of the alternative "Follow the Perturbed Leader" (FPL) algorithm from Kalai & Vempala (2003) (based on Hannan's algorithm) is easier. We derive loss bounds for adaptive learning rate and both finite expert classes with uniform weights and countable expert classes with arbitrary weights. For the former setup, our loss bounds match the best known results so far, while for the latter our results are new.

Motivation & Objective

• To address the challenge of adaptive learning rate tuning in online prediction with expert advice, especially for countable expert classes with arbitrary weights.
• To simplify the regret analysis of adaptive learning rate algorithms, which is notoriously complex for Weighted Majority-type methods.
• To extend performance guarantees to general weights and adaptive adversaries, where prior work often restricted to uniform weights or finite experts.
• To establish the first loss bounds for FPL with adaptive learning rates and arbitrary expert weights, using a hierarchical FPL construction.
• To compare FPL's performance with Bayesian prediction and other algorithms, particularly regarding leading constants in regret bounds.

Proposed method

• Proposes an adaptive learning rate strategy for FPL based on the inverse square root of current cumulative loss and expert complexity, enabling dynamic adaptation without prior knowledge.
• Introduces a hierarchical FPL variant to achieve optimal $√{kL}$ regret bounds for arbitrary weights, where standard FPL would yield a looser $k\sqrt{L}$ bound.
• Uses infeasible FPL (IFPL) as a theoretical benchmark to derive upper bounds on regret, which are then transferred to the feasible FPL algorithm.
• Employs perturbation-based prediction: at each time step, the algorithm selects the expert that minimizes the sum of past loss and a random perturbation drawn from a specific distribution (e.g., Laplace or Gumbel).
• Derives both expected and high-probability regret bounds by analyzing the difference between the feasible and infeasible FPL predictors.
• Applies the doubling trick and self-confident learning rate choices to achieve adaptive performance without prior knowledge of loss or complexity.

Experimental results

Research questions

• RQ1 Can FPL achieve optimal $√{kL}$ regret bounds with adaptive learning rates for countable expert classes with arbitrary weights?
• RQ2 How does the performance of adaptive FPL compare to Weighted Majority and Hedge algorithms in terms of leading constants and adaptivity?
• RQ3 What is the minimal regret achievable by FPL under an adaptive adversary, and can it match Bayesian prediction bounds?
• RQ4 Can the hierarchical FPL construction be used to generalize FPL to arbitrary weights while maintaining tight regret bounds?
• RQ5 Is the $√{kL}$ bound tight for FPL with adaptive learning rates, or is a larger constant unavoidable?

Key findings

• The paper establishes the first regret bound of $O(\sqrt{kL})$ for FPL with adaptive learning rates and arbitrary expert weights, achieved via a hierarchical FPL construction.
• For finite expert classes with uniform weights, the FPL algorithm achieves a regret bound of $O(\sqrt{kL})$, matching the best-known results for static learning rates.
• The hierarchical FPL variant improves the regret bound from $k\sqrt{L}$ (non-hierarchical) to $\sqrt{kL}$, demonstrating the necessity of the hierarchy for optimal performance with general weights.
• The analysis of adaptive learning rates in FPL is significantly simpler and more elegant than for Weighted Majority variants, with proofs under half a page for self-confident learning rate choices.
• The leading constant in the regret bound is $2$ for FPL, which is $\sqrt{2}$ worse than the Hedge algorithm but matches the best-known dynamic bounds for WM-type algorithms.
• The FPL bounds are comparable to Bayesian prediction bounds, with the same asymptotic order and a leading constant close to optimal, even without assuming the true sequence is generated by a known expert.