# Uniform Last-Iterate Guarantee for Bandits and Reinforcement Learning

arXiv: https://arxiv.org/abs/2402.12711 · HTML: https://mlsanity.com/paper/2402.12711/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 20 Feb 2024.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 8 of 8)

- [Achieving Near Instance-Optimality and Minimax-Optimality in Stochastic and Adversarial Linear Bandits Simultaneously](https://mlsanity.com/paper/2102.05858/) (2021)
- [Contextual Bandits with Large Action Spaces: Made Practical](https://mlsanity.com/paper/2207.05836/) (2022)
- [Last-Iterate Convergent Policy Gradient Primal-Dual Methods for Constrained MDPs](https://mlsanity.com/paper/2306.11700/) (2023)
- [ReLOAD: Reinforcement Learning with Optimistic Ascent-Descent for Last-Iterate Convergence in Constrained MDPs](https://mlsanity.com/paper/2302.01275/) (2023)
- [Instance-optimal PAC Algorithms for Contextual Bandits](https://mlsanity.com/paper/2207.02357/) (2022)
- [Geometric Exploration for Online Control](https://mlsanity.com/paper/2010.13178/) (2020)
- [Generalized Policy Elimination: an efficient algorithm for Nonparametric Contextual Bandits](https://mlsanity.com/paper/2003.02873/) (2020)
- [Uniform-PAC Guarantees for Model-Based RL with Bounded Eluder Dimension](https://mlsanity.com/paper/2305.08350/) (2023)

## Built on by (most cited first, 5 of 5)

- [The Computational Limits of State-Space Models and Mamba via the Lens of Circuit Complexity](https://mlsanity.com/paper/2412.06148/) (2024)
- [Breaking the Frozen Subspace: Importance Sampling for Low-Rank Optimization in LLM Pretraining](https://mlsanity.com/paper/2502.05790/) (2025)
- [Scaling Law Phenomena Across Regression Paradigms: Multiple and Kernel Approaches](https://mlsanity.com/paper/2503.01314/) (2025)
- [Near-Optimal Last-Iterate Convergence for Zero-Sum Games with Bandit Feedback and Opponent Actions](https://mlsanity.com/paper/2605.09363/) (2026)
- [Time and Memory Trade-off of KV-Cache Compression in Tensor Transformer Decoding](https://mlsanity.com/paper/2503.11108/) (2025)

## Abstract (arXiv)

Existing metrics for reinforcement learning (RL) such as regret, PAC bounds, or uniform-PAC (Dann et al., 2017), typically evaluate the cumulative performance, while allowing the agent to play an arbitrarily bad policy at any finite time t. Such a behavior can be highly detrimental in high-stakes applications. This paper introduces a stronger metric, uniform last-iterate (ULI) guarantee, capturing both cumulative and instantaneous performance of RL algorithms. Specifically, ULI characterizes the instantaneous performance by ensuring that the per-round suboptimality of the played policy is bounded by a function, monotonically decreasing w.r.t. round t, preventing revisiting bad policies when sufficient samples are available. We demonstrate that a near-optimal ULI guarantee directly implies near-optimal cumulative performance across aforementioned metrics, but not the other way around. To examine the achievability of ULI, we first provide two positive results for bandit problems with finite arms, showing that elimination-based algorithms and high-probability adversarial algorithms with stronger analysis or additional designs, can attain near-optimal ULI guarantees. We also provide a negative result, indicating that optimistic algorithms cannot achieve near-optimal ULI guarantee. Furthermore, we propose an efficient algorithm for linear bandits with infinitely many arms, which achieves the ULI guarantee, given access to an optimization oracle. Finally, we propose an algorithm that achieves near-optimal ULI guarantee for the online reinforcement learning setting.