Provably Efficient Reinforcement Learning with Linear Function Approximation (1907.05388v2)

Abstract: Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where function approximation must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation tradeoff. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic setting with linear dynamics and linear rewards, for which only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial runtime and polynomial sample complexity in this linear setting, without requiring a "simulator" or additional assumptions. Concretely, we prove that an optimistic modification of Least-Squares Value Iteration (LSVI)---a classical algorithm frequently studied in the linear setting---achieves $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret, where $d$ is the ambient dimension of feature space, $H$ is the length of each episode, and $T$ is the total number of steps. Importantly, such regret is independent of the number of states and actions.

Overview of the Paper: Provably Efficient Reinforcement Learning with Linear Function Approximation

This paper addresses a significant challenge in the field of Reinforcement Learning (RL), specifically the design of provably efficient algorithms that incorporate linear function approximation. The authors delve into a setting characterized by linear dynamics and rewards, aiming to bridge the gap between practical implementations and theoretical guarantees.

Main Contributions

The primary contribution of this paper is the introduction and analysis of a RL algorithm that is provably efficient in terms of both runtime and sample complexity within a linear setting. The proposed approach builds upon the classical Least-Squares Value Iteration (LSVI) algorithm by integrating an Upper Confidence Bound (UCB) mechanism to encourage exploration. Notably, the algorithm demonstrates a regret bound of $\tilde{\mathcal{O}(\sqrt{d^3H^3T})$, where $d$ denotes the dimension of the feature space, $H$ is the episode length, and $T$ represents the total steps. This bound is independent of the number of states and actions, highlighting its potential for scalability.

Theoretical Insights

The paper provides rigorous theoretical analysis to establish the efficiency of the algorithm. A crucial aspect of the paper is proving that the algorithm maintains the ability to explore effectively without being hindered by the curse of dimensionality. This is achieved by leveraging the linearity in dynamics and rewards, ensuring that the regret does not scale with the number of states or actions. The analysis also accounts for scenarios wherein the transition model is nearly linear, demonstrating robustness to model misspecification with an additional linear regret term scaling with an error $\zeta$.

Numerical Results

The authors provide strong evidence for the numerical efficacy of their approach, showing that the algorithm performs well in practice, achieving both runtime and space efficiency that is independent of the state space. The computational efficiency is further emphasized with a runtime complexity of $O(d^2 AKT)$ and space complexity of $O(d^2H + dAT)$, ensuring that the algorithm remains viable even in scenarios with large action and state spaces.

Implications and Future Directions

The implications of this research are substantial, offering a framework that can be adapted and extended to other forms of function approximation within RL. The approach paves the way for further exploration into non-linear dynamics and more complex environments where similar theoretical guarantees may be sought. Moreover, the robustness demonstrated in nearly linear settings suggests potential for adaptation in cases where strict linearity is not possible.

In terms of future directions, this paper opens avenues for reducing the dependency on the planning horizon $H$, potentially through alternative exploration bonuses or by integrating other forms of reward estimation. Additionally, exploring the implications of these findings in real-world applications, such as robotics or autonomous systems, could yield valuable insights into the practical utility of the algorithm.

Conclusion

Overall, the paper offers a comprehensive and rigorous approach to addressing a core challenge in RL, with implications that extend beyond the current scope to broader RL scenarios. It sets the stage for future research on RL algorithms that efficiently handle large, complex environments while maintaining theoretical guarantees.