# Adaptive Learning Rates with Surrogate Probability for Follow-the-Perturbed-Leader

arXiv: https://arxiv.org/abs/2606.06043 · HTML: https://mlsanity.com/paper/2606.06043/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 20 of 20)

- [Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1807.07623/) (2018)
- [More Adaptive Algorithms for Adversarial Bandits](https://mlsanity.com/paper/1801.03265/) (2018)
- [A Second-order Bound with Excess Losses](https://mlsanity.com/paper/1402.2044/) (2014)
- [Beating Stochastic and Adversarial Semi-bandits Optimally and Simultaneously](https://mlsanity.com/paper/1901.08779/) (2019)
- [An Improved Parametrization and Analysis of the EXP3++ Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1702.06103/) (2017)
- [A Blackbox Approach to Best of Both Worlds in Bandits and Beyond](https://mlsanity.com/paper/2302.09739/) (2023)
- [Adaptive Best-of-Both-Worlds Algorithm for Heavy-Tailed Multi-Armed Bandits](https://mlsanity.com/paper/2201.11921/) (2022)
- [Improved Best-of-Both-Worlds Guarantees for Multi-Armed Bandits: FTRL with General Regularizers and Multiple Optimal Arms](https://mlsanity.com/paper/2302.13534/) (2023)
- [Importance weighting without importance weights: An efficient algorithm for combinatorial semi-bandits](https://mlsanity.com/paper/1503.05087/) (2015)
- [Stability-penalty-adaptive follow-the-regularized-leader: Sparsity, game-dependency, and best-of-both-worlds](https://mlsanity.com/paper/2305.17301/) (2023)
- [On the Optimality of Perturbations in Stochastic and Adversarial Multi-armed Bandit Problems](https://mlsanity.com/paper/1902.00610/) (2019)
- [Adaptive Learning Rate for Follow-the-Regularized-Leader: Competitive Analysis and Best-of-Both-Worlds](https://mlsanity.com/paper/2403.00715/) (2024)
- [Follow-the-Perturbed-Leader with Fr'echet-type Tail Distributions: Optimality in Adversarial Bandits and Best-of-Both-Worlds](https://mlsanity.com/paper/2403.05134/) (2024)
- [Heavy-tailed Linear Bandits: Adversarial Robustness, Best-of-both-worlds, and Beyond](https://mlsanity.com/paper/2508.13679/) (2025)
- [A Simple and Adaptive Learning Rate for FTRL in Online Learning with Minimax Regret of Theta(T2/3) and its Application to Best-of-Both-Worlds](https://mlsanity.com/paper/2405.20028/) (2024)
- [Optimism in the Face of Ambiguity Principle for Multi-Armed Bandits](https://mlsanity.com/paper/2409.20440/) (2024)
- [Revisiting Follow-the-Perturbed-Leader with Unbounded Perturbations in Bandit Problems](https://mlsanity.com/paper/2508.18604/) (2025)
- [A Further Efficient Algorithm with Best-of-Both-Worlds Guarantees for m-Set Semi-Bandit Problem](https://mlsanity.com/paper/2603.11764/) (2026)
- [Data-dependent Bounds with T-Optimal Best-of-Both-Worlds Guarantees in Multi-Armed Bandits using Stability-Penalty Matching](https://mlsanity.com/paper/2502.08143/) (2025)
- [Follow-the-Perturbed-Leader for Decoupled Bandits: Best-of-Both-Worlds and Practicality](https://mlsanity.com/paper/2510.12152/) (2025)

## Built on by (most cited first, 1 of 1)

- [Follow the Perturbed Leader for Linear Bandits: √T Regret over General Convex Sets](https://mlsanity.com/paper/2610.17818/) (2026)

## Abstract (arXiv)

Follow-the-regularized-leader framework has shown effectiveness and flexibility in online learning problems, where the choice of learning rates are known to be crucial. Recently, adaptive learning rates defined in terms of the arm-selection probabilities, obtained by solving convex optimization, have achieved improved best-of-both-worlds (BOBW) guarantees in various bandit problems. In contrast, BOBW guarantees for its computationally efficient alternative, follow-the-perturbed-leader (FTPL), remain relatively limited since its optimization-free nature ironically makes the design of adaptive, probability-dependent learning rates non-trivial. To address this challenge, we propose an adaptive learning rate for FTPL by introducing surrogate probability functions that can be computed only from the available quantities, without requiring the exact probabilities. Based on these learning rates with surrogate functions, we provide the BOBW guarantee for FTPL with Pareto perturbations for any shape parameter α >1, generalizing prior results restricted to specific choices of α=2. We further show the BOBW guarantees for FTPL with adaptive learning rates in the bandit problem with expert advices. Our approach preserves the computational simplicity of FTPL while enabling probability-dependent adaptivity, and the surrogate-based methodology may be of independent interest in other algorithmic frameworks beyond FTPL and learning rate designs.