skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Provable Policy Gradient Methods for Average-Reward Markov Potential Games
We study Markov potential games under the infinite horizon average reward criterion. Most previous studies have been for discounted rewards. We prove that both algorithms based on independent policy gradient and independent natural policy gradient converge globally to a Nash equilibrium for the average reward criterion. To set the stage for gradient-based methods, we first establish that the average reward is a smooth function of policies and provide sensitivity bounds for the differential value functions, under certain conditions on ergodicity and the second largest eigenvalue of the underlying Markov decision process (MDP). We prove that three algorithms, policy gradient, proximal-Q, and natural policy gradient (NPG), converge to an ϵ-Nash equilibrium with time complexity O(1ϵ2), given a gradient/differential Q function oracle. When policy gradients have to be estimated, we propose an algorithm with ~O(1mins,aπ(a|s)δ) sample complexity to achieve δ approximation error w.r.t~the ℓ2 norm. Equipped with the estimator, we derive the first sample complexity analysis for a policy gradient ascent algorithm, featuring a sample complexity of ~O(1/ϵ5). Simulation studies are presented.  more » « less
Award ID(s):
2038625
PAR ID:
10563429
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
Proceedings of the 27th International Conference on Artificial Intelligence and Statistics (AISTATS) 2024
Date Published:
Volume:
38
Format(s):
Medium: X
Location:
Valencia, Spain
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR)
    Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)
    Average reward reinforcement learning (RL) provides a suitable framework for capturing the objective (i.e. long-run average reward) for continuing tasks, where there is often no natural way to identify a discount factor. However, existing average reward RL algorithms with sample complexity guarantees are not feasible, as they take as input the (unknown) mixing time of the Markov decision process (MDP). In this paper, we make initial progress towards addressing this open problem. We design a feasible average-reward $$Q$$-learning framework that requires no knowledge of any problem parameter as input. Our framework is based on discounted $$Q$$-learning, while we dynamically adapt the discount factor (and hence the effective horizon) to progressively approximate the average reward. In the synchronous setting, we solve three tasks: (i) learn a policy that is $$\epsilon$$-close to optimal, (ii) estimate optimal average reward with $$\epsilon$$-accuracy, and (iii) estimate the bias function (similar to $$Q$$-function in discounted case) with $$\epsilon$$-accuracy. We show that with carefully designed adaptation schemes, (i) can be achieved with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^{8}}{\epsilon^{8}})$$ samples, (ii) with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^5}{\epsilon^5})$$ samples, and (iii) with $$\tilde{O}(\frac{SA B}{\epsilon^9})$$ samples, where $$t_\mathrm{mix}$$ is the mixing time, and $B > 0$ is an MDP-dependent constant. To our knowledge, we provide the first finite-sample guarantees that are polynomial in $$S, A, t_{\mathrm{mix}}, \epsilon$$ for a feasible variant of $$Q$$-learning. That said, the sample complexity bounds have tremendous room for improvement, which we leave for the community’s best minds. Preliminary simulations verify that our framework is effective without prior knowledge of parameters as input. 
    more » « less
  2. ; ; ; (, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR)
    Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)
    Average reward reinforcement learning (RL) provides a suitable framework for capturing the objective (i.e. long-run average reward) for continuing tasks, where there is often no natural way to identify a discount factor. However, existing average reward RL algorithms with sample complexity guarantees are not feasible, as they take as input the (unknown) mixing time of the Markov decision process (MDP). In this paper, we make initial progress towards addressing this open problem. We design a feasible average-reward $$Q$$-learning framework that requires no knowledge of any problem parameter as input. Our framework is based on discounted $$Q$$-learning, while we dynamically adapt the discount factor (and hence the effective horizon) to progressively approximate the average reward. In the synchronous setting, we solve three tasks: (i) learn a policy that is $$\epsilon$$-close to optimal, (ii) estimate optimal average reward with $$\epsilon$$-accuracy, and (iii) estimate the bias function (similar to $$Q$$-function in discounted case) with $$\epsilon$$-accuracy. We show that with carefully designed adaptation schemes, (i) can be achieved with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^{8}}{\epsilon^{8}})$$ samples, (ii) with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^5}{\epsilon^5})$$ samples, and (iii) with $$\tilde{O}(\frac{SA B}{\epsilon^9})$$ samples, where $$t_\mathrm{mix}$$ is the mixing time, and $B > 0$ is an MDP-dependent constant. To our knowledge, we provide the first finite-sample guarantees that are polynomial in $$S, A, t_{\mathrm{mix}}, \epsilon$$ for a feasible variant of $$Q$$-learning. That said, the sample complexity bounds have tremendous room for improvement, which we leave for the community’s best minds. Preliminary simulations verify that our framework is effective without prior knowledge of parameters as input. 
    more » « less
  3. ; ; ; (, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR)
    Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)
    Average reward reinforcement learning (RL) provides a suitable framework for capturing the objective (i.e. long-run average reward) for continuing tasks, where there is often no natural way to identify a discount factor. However, existing average reward RL algorithms with sample complexity guarantees are not feasible, as they take as input the (unknown) mixing time of the Markov decision process (MDP). In this paper, we make initial progress towards addressing this open problem. We design a feasible average-reward $$Q$$-learning framework that requires no knowledge of any problem parameter as input. Our framework is based on discounted $$Q$$-learning, while we dynamically adapt the discount factor (and hence the effective horizon) to progressively approximate the average reward. In the synchronous setting, we solve three tasks: (i) learn a policy that is $$\epsilon$$-close to optimal, (ii) estimate optimal average reward with $$\epsilon$$-accuracy, and (iii) estimate the bias function (similar to $$Q$$-function in discounted case) with $$\epsilon$$-accuracy. We show that with carefully designed adaptation schemes, (i) can be achieved with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^{8}}{\epsilon^{8}})$$ samples, (ii) with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^5}{\epsilon^5})$$ samples, and (iii) with $$\tilde{O}(\frac{SA B}{\epsilon^9})$$ samples, where $$t_\mathrm{mix}$$ is the mixing time, and $B > 0$ is an MDP-dependent constant. To our knowledge, we provide the first finite-sample guarantees that are polynomial in $$S, A, t_{\mathrm{mix}}, \epsilon$$ for a feasible variant of $$Q$$-learning. That said, the sample complexity bounds have tremendous room for improvement, which we leave for the community’s best minds. Preliminary simulations verify that our framework is effective without prior knowledge of parameters as input. 
    more » « less
  4. ; ; ; ; (, Proceedings of Machine Learning Research)
    The problem of two-player zero-sum Markov games has recently attracted increasing interests in theoretical studies of multi-agent reinforcement learning (RL). In particular, for finite-horizon episodic Markov decision processes (MDPs), it has been shown that model-based algorithms can find an ϵ-optimal Nash Equilibrium (NE) with the sample complexity of O(H3SAB/ϵ2), which is optimal in the dependence of the horizon H and the number of states S (where A and B denote the number of actions of the two players, respectively). However, none of the existing model-free algorithms can achieve such an optimality. In this work, we propose a model-free stage-based Q-learning algorithm and show that it achieves the same sample complexity as the best model-based algorithm, and hence for the first time demonstrate that model-free algorithms can enjoy the same optimality in the H dependence as model-based algorithms. The main improvement of the dependency on H arises by leveraging the popular variance reduction technique based on the reference-advantage decomposition previously used only for single-agent RL. However, such a technique relies on a critical monotonicity property of the value function, which does not hold in Markov games due to the update of the policy via the coarse correlated equilibrium (CCE) oracle. Thus, to extend such a technique to Markov games, our algorithm features a key novel design of updating the reference value functions as the pair of optimistic and pessimistic value functions whose value difference is the smallest in the history in order to achieve the desired improvement in the sample efficiency. 
    more » « less
  5. ; ; ; ; (, Journal of Artificial Intelligence Research)
    Robust Markov decision processes (MDPs) aim to find a policy that optimizes the worst-case performance over an uncertainty set of MDPs. Existing studies mostly have focused on the robust MDPs under the discounted reward criterion, leaving the ones under the average-reward criterion largely unexplored. In this paper, we develop the first comprehensive and systematic study of robust average-reward MDPs, where the goal is to optimize the long-term average performance under the worst case. Our contributions are four-folds: (1) we prove the uniform convergence of the robust discounted value function to the robust average-reward function as the discount factor γ goes to 1; (2) we derive the robust average-reward Bellman equation, characterize the structure of its solution set, and prove the equivalence between solving the robust Bellman equation and finding the optimal robust policy; (3) we design robust dynamic programming algorithms, and theoretically characterize their convergence to the optimal policy; and (4) we design two model-free algorithms unitizing the multi-level Monte-Carlo approach, and prove their asymptotic convergence 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email