Learning with Good Feature Representations in Bandits and in RL with a Generative Model (1911.07676v2)

Abstract: The construction by Du et al. (2019) implies that even if a learner is given linear features in $\mathbb R^d$ that approximate the rewards in a bandit with a uniform error of $\epsilon$, then searching for an action that is optimal up to $O(\epsilon)$ requires examining essentially all actions. We use the Kiefer-Wolfowitz theorem to prove a positive result that by checking only a few actions, a learner can always find an action that is suboptimal with an error of at most $O(\epsilon \sqrt{d})$. Thus, features are useful when the approximation error is small relative to the dimensionality of the features. The idea is applied to stochastic bandits and reinforcement learning with a generative model where the learner has access to $d$-dimensional linear features that approximate the action-value functions for all policies to an accuracy of $\epsilon$. For linear bandits, we prove a bound on the regret of order $\sqrt{dn \log(k)} + \epsilon n \sqrt{d} \log(n)$ with $k$ the number of actions and $n$ the horizon. For RL we show that approximate policy iteration can learn a policy that is optimal up to an additive error of order $\epsilon \sqrt{d}/(1 - \gamma)^2$ and using $d/(\epsilon^2(1 - \gamma)^4)$ samples from a generative model. These bounds are independent of the finer details of the features. We also investigate how the structure of the feature set impacts the tradeoff between sample complexity and estimation error.

• The paper shows that even with linear features in bandits, finding a near-optimal action may require evaluating almost all actions if approximation error is not small relative to feature count.
• It demonstrates using the Kiefer-Wolfowitz theorem that finding suboptimal actions efficiently is possible with few evaluations when approximation errors are sufficiently minimal.
• Strong regret bounds are derived for stochastic bandits, and the paper confirms approximate policy iteration works well in RL with generative models within bounded errors and sample complexities.

Learning with Good Feature Representations in Bandits and Reinforcement Learning

The paper "Learning with Good Feature Representations in Bandits and in RL with a Generative Model" tackles the problem of whether having good feature representations is sufficient to ensure efficient learning in the settings of bandits and reinforcement learning (RL) equipped with a generative model. This work builds upon the findings of Du et al. (2019), investigating the limits and potential of feature representations in these learning contexts.

Key Results and Theoretical Insights

1. Negative Results from Prior Construction: Inspired by the work of Du et al. (2019), the paper initially addresses the negative result derived from their feature construction. Specifically, the claim is that even with linear features in a bandit case, a near-optimal action cannot be found without evaluating nearly every action when the error in approximation is not explicitly small relative to the number of features.
2. Positive Outcomes Using Kiefer-Wolfowitz Theorem: To counterbalance the negative results, the authors employ the Kiefer-Wolfowitz theorem to demonstrate a positive result. They prove that it is indeed possible to efficiently find suboptimal actions with bounded error using only a few evaluations. The theorem supports that good feature representations are indeed useful when approximation errors are sufficiently minimal in relation to feature dimensionality.
3. Regret Bounds and Efficiency: For stochastic bandits, strong bounds on regret are derived, showing that the regret is of order $\sqrt{dn \log(k)} + n \sqrt{d} \log(n)$, where $k$ is the number of actions and $n$ is the time horizon. In reinforcement learning scenarios with generative models, the paper validates that approximate policy iteration can achieve a policy with near-optimal performance within bounded additive errors and sample complexities.
4. Independent of Feature Detail: Notably, the bounds provided are independent of the specific feature details, showcasing a flexibility and robustness in approach that aligns strongly with theoretical correctness.

Implications and Speculation on Future Directions

The implications of these findings are multifold:

• Practical Application: They suggest strategic avenues for designing algorithms where feature representations can be leveraged efficiently, even in cases of model misspecification. This could extend to real-world applications where action space or state space exploration is costly.
• Structural Insights: There is a clear indication that the choice and structure of feature sets significantly impact sample complexity and estimation error, highlighting the importance of feature engineering and selection in practical scenarios.
• Future Developments in AI: The paper prompt further investigation into the potential need for hybrid models that adopt generative approaches to sample generation, thus maximizing learning efficiency where the exploration is a bottleneck.

Speculation on the future developments might involve delving deeper into nonlinear models or sparse feature distributions to further improve the efficiency in action or state evaluations. Additionally, exploring the established link between feature dimensionality and learning efficacy could foster advancements that inform the architectural design of machine learning systems, potentially improving adaptability and scalability of AI systems in dynamic environments.

Conclusion

This paper offers substantial theoretical grounding and practical insights into the use of linear feature representations within bandit and reinforcement learning frameworks. While certain negative results underscore the complexity, the positive results derived through sophisticated theoretical constructs provide valuable guidance for effectively utilizing feature representations. As research in AI progresses, integrating these findings could significantly enhance the ability to realize efficient learning outcomes in diverse settings.