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We study model-free reinforcement learning (RL) algorithms for infinite-horizon average-reward Markov decision process (MDP), which is more appropriate for applications that involve continuing operations not divided into episodes. In contrast to episodic/discounted MDPs, theoretical understanding of model-free RL algorithms is relatively inadequate for the average-reward setting. In this paper, we consider both the online setting and the setting with access to a simulator. We develop computationally efficient model-free algorithms that achieve sharper guarantees on regret/sample complexity compared with existing results. In the online setting, we design an algorithm, UCB-AVG, based on an optimistic variant of variance-reduced Q-learning. We show that UCB-AVG achieves a regret bound $$\widetilde{O}(S^5A^2sp(h^*)\sqrt{T})$$ after $$T$$ steps, where $$S\times A$$ is the size of state-action space, and $sp(h^*)$ the span of the optimal bias function. Our result provides the first computationally efficient model-free algorithm that achieves the optimal dependence in $$T$$ (up to log factors) for weakly communicating MDPs, which is necessary for low regret. In contrast, prior results either are suboptimal in $$T$$ or require strong assumptions of ergodicity or uniformly mixing of MDPs. In the simulator setting, we adapt the idea of UCB-AVG to develop a model-free algorithm that finds an $$\epsilon$$-optimal policy with sample complexity $$\widetilde{O}(SAsp^2(h^*)\epsilon^{-2} + S^2Asp(h^*)\epsilon^{-1}).$$ This sample complexity is near-optimal for weakly communicating MDPs, in view of the minimax lower bound $$\Omega(SAsp(^*)\epsilon^{-2})$$. Existing work mainly focuses on ergodic MDPs and the results typically depend on $$t_{mix},$$ the worst-case mixing time induced by a policy. We remark that the diameter $$D$$ and mixing time $$t_{mix}$$ are both lower bounded by $sp(h^*)$, and $$t_{mix}$$ can be arbitrarily large for certain MDPs. On the technical side, our approach integrates two key ideas: learning an $$\gamma$$-discounted MDP as an approximation, and leveraging reference-advantage decomposition for variance in optimistic Q-learning. As recognized in prior work, a naive approximation by discounted MDPs results in suboptimal guarantees. A distinguishing feature of our method is maintaining estimates of value-difference between state pairs to provide a sharper bound on the variance of reference advantage. We also crucially use a careful choice of the discounted factor $$\gamma$$ to balance approximation error due to discounting and the statistical learning error, and we are able to maintain a good-quality reference value function with $O(SA)$ space complexity.
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No-Regret Linear Bandits beyond Realizability
We study linear bandits when the underlying reward function is not linear. Existing work relies on a uniform misspecification parameter $$\epsilon$$ that measures the sup-norm error of the best linear approximation. This results in an unavoidable linear regret whenever $$\epsilon > 0$$. We describe a more natural model of misspecification which only requires the approximation error at each input $$x$$ to be proportional to the suboptimality gap at $$x$$. It captures the intuition that, for optimization problems, near-optimal regions should matter more and we can tolerate larger approximation errors in suboptimal regions. Quite surprisingly, we show that the classical LinUCB algorithm — designed for the realizable case — is automatically robust against such gap-adjusted misspecification. It achieves a near-optimal $$\sqrt{T}$$ regret for problems that the best-known regret is almost linear in time horizon $$T$$. Technically, our proof relies on a novel self-bounding argument that bounds the part of the regret due to misspecification by the regret itself.
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- PAR ID:
- 10466938
- Editor(s):
- Evans, Robin J.; Shpitser, Ilya
- Publisher / Repository:
- UAI 2023
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 216
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 1294--1303
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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