TLDR: This paper introduces “partial qπ-realizability,” a new framework for reinforcement learning with linear function approximation that sits between existing strong and weak assumptions. It proves that learning an optimal policy in this setting is computationally hard: NP-hard for greedy policies and exponentially hard for softmax policies. These findings suggest that computational difficulties in RL persist even when the set of policies considered is expanded beyond just the optimal one.
Reinforcement Learning (RL) has driven incredible advancements in artificial intelligence, from mastering complex games like Go to controlling robots. However, a fundamental challenge in RL, especially when dealing with vast state spaces, is computational complexity. This refers to the sheer amount of computing power and time an algorithm needs to find a near-optimal solution.
To tackle this, researchers often employ ‘function approximation’ techniques. These methods help estimate the value of different actions and states without needing to store every single possibility, making RL scalable for real-world applications. Traditionally, two main settings for linear function approximation have been studied: ‘q*-realizability’ and ‘qπ-realizability’.
Q*-realizability is a relatively weak assumption, only requiring that the optimal value function can be approximated linearly. It makes no assumptions about non-optimal policies. On the other hand, qπ-realizability is a much stronger condition, assuming that the value functions for *all* possible policies are linearly realizable.
A new research paper, “Computational Hardness of Reinforcement Learning with Partial qπ-Realizability”, introduces a novel framework called ‘partial qπ-realizability’. This new concept aims to bridge the gap between the two extreme cases. In this setting, the assumption is that value functions for a *specific subset* of policies (denoted as Π) are linearly realizable. This offers a more practical model for many RL scenarios where function approximation is naturally used, as it’s less restrictive than assuming all policies are linearly realizable, but more structured than only considering the optimal one.
Despite this more flexible and practical approach, the paper delivers a significant finding: learning an optimal policy in this newly defined partial qπ-realizability setting is computationally hard. The researchers established two key results:
NP-Hardness for Greedy Policies
For scenarios where the policy set Π consists of ‘deterministic greedy policies’ (meaning actions are chosen by simply picking the one that maximizes an estimated value), the problem of learning a near-optimal policy is proven to be NP-hard. This implies that, unless P=NP (a major unsolved problem in computer science), there is no polynomial-time algorithm that can efficiently solve this problem for certain levels of accuracy.
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Exponential Hardness for Softmax Policies
Moving beyond deterministic greedy policies, the paper also investigated ‘stochastic softmax policies’ (where actions are chosen probabilistically based on their estimated values, often used for exploration). Under the Randomized Exponential Time Hypothesis (rETH), which is a stronger assumption than P≠NP, the researchers showed that learning a near-optimal policy in this setting exhibits ‘exponential computational hardness’. This means the time required to solve the problem grows exponentially with the feature vector dimension.
These hardness results are crucial because they mirror those found in the q*-realizability settings. They strongly suggest that computational difficulty persists even when the class of policies considered (Π) is expanded beyond just the single optimal policy. This reinforces the idea that fundamental computational challenges remain in RL, even with more accommodating assumptions about value function realizability.
The technical core of these findings involves reducing complex problems, specifically variants of the δ-MAX-3SAT problem, to instances of the defined RL problems (GLINEAR-κ-RL for greedy policies and SLINEAR-κ-RL for softmax policies). This reduction demonstrates that if you could efficiently solve these RL problems, you could also efficiently solve these known hard problems, leading to a contradiction unless the hardness results hold.
In conclusion, this research indicates that achieving positive computational results in partial qπ-realizability is generally unattainable without additional, stronger assumptions. While these are ‘worst-case’ complexity results, they highlight the critical need for developing efficient algorithms under specific structural constraints on policies or features in practical RL applications.
