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Register a new userAn Exponential Lower Bound for Linearly-Realizable MDPs with Constant Suboptimality Gap
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Yuanhao Wang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A fundamental question in the theory of reinforcement learning is: suppose the optimal $Q$-function lies in the linear span of a given $d$ dimensional feature mapping, is sample-efficient reinforcement learning (RL) possible? The recent and remarkable result of Weisz et al. (2020) resolved this question in the negative, providing an exponential (in $d$) sample size lower bound, which holds even if the agent has access to a generative model of the environment. One may hope that this information theoretic barrier for RL can be circumvented by further supposing an even more favorable assumption: there exists a emph{constant suboptimality gap} between the optimal $Q$-value of the best action and that of the second-best action (for all states). The hope is that having a large suboptimality gap would permit easier identification of optimal actions themselves, thus making the problem tractable; indeed, provided the agent has access to a generative model, sample-efficient RL is in fact possible with the addition of this more favorable assumption. This work focuses on this question in the standard online reinforcement learning setting, where our main result resolves this question in the negative: our hardness result shows that an exponential sample complexity lower bound still holds even if a constant suboptimality gap is assumed in addition to having a linearly realizable optimal $Q$-function. Perhaps surprisingly, this implies an exponential separation between the online RL setting and the generative model setting. Complementing our negative hardness result, we give two positive results showing that provably sample-efficient RL is possible either under an additional low-variance assumption or under a novel hypercontractivity assumption (both implicitly place stronger conditions on the underlying dynamics model).
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Low-complexity models such as linear function representation play a pivotal role in enabling sample-efficient reinforcement learning (RL). The current paper pertains to a scenario with value-based linear representation, which postulates the linear realizability of the optimal Q-function (also called the linear $Q^{star}$ problem). While linear realizability alone does not allow for sample-efficient solutions in general, the presence of a large sub-optimality gap is a potential game changer, depending on the sampling mechanism in use. Informally, sample efficiency is achievable with a large sub-optimality gap when a generative model is available but is unfortunately infeasible when we turn to standard online RL settings. In this paper, we make progress towards understanding this linear $Q^{star}$ problem by investigating a new sampling protocol, which draws samples in an online/exploratory fashion but allows one to backtrack and revisit previous states in a controlled and infrequent manner. This protocol is more flexible than the standard online RL setting, while being practically relevant and far more restrictive than the generative model. We develop an algorithm tailored to this setting, achieving a sample complexity that scales polynomially with the feature dimension, the horizon, and the inverse sub-optimality gap, but not the size of the state/action space. Our findings underscore the fundamental interplay between sampling protocols and low-complexity structural representation in RL.
We derive a novel asymptotic problem-dependent lower-bound for regret minimization in finite-horizon tabular Markov Decision Processes (MDPs). While, similar to prior work (e.g., for ergodic MDPs), the lower-bound is the solution to an optimization problem, our derivation reveals the need for an additional constraint on the visitation distribution over state-action pairs that explicitly accounts for the dynamics of the MDP. We provide a characterization of our lower-bound through a series of examples illustrating how different MDPs may have significantly different complexity. 1) We first consider a difficult MDP instance, where the novel constraint based on the dynamics leads to a larger lower-bound (i.e., a larger regret) compared to the classical analysis. 2) We then show that our lower-bound recovers results previously derived for specific MDP instances. 3) Finally, we show that, in certain simple MDPs, the lower bound is considerably smaller than in the general case and it does not scale with the minimum action gap at all. We show that this last result is attainable (up to $poly(H)$ terms, where $H$ is the horizon) by providing a regret upper-bound based on policy gaps for an optimistic algorithm.
Generalization across environments is critical to the successful application of reinforcement learning algorithms to real-world challenges. In this paper, we consider the problem of learning abstractions that generalize in block MDPs, families of environments with a shared latent state space and dynamics structure over that latent space, but varying observations. We leverage tools from causal inference to propose a method of invariant prediction to learn model-irrelevance state abstractions (MISA) that generalize to novel observations in the multi-environment setting. We prove that for certain classes of environments, this approach outputs with high probability a state abstraction corresponding to the causal feature set with respect to the return. We further provide more general bounds on model error and generalization error in the multi-environment setting, in the process showing a connection between causal variable selection and the state abstraction framework for MDPs. We give empirical evidence that our methods work in both linear and nonlinear settings, attaining improved generalization over single- and multi-task baselines.
Recently, Wang et al. (2020) showed a highly intriguing hardness result for batch reinforcement learning (RL) with linearly realizable value function and good feature coverage in the finite-horizon case. In this note we show that once adapted to the discounted setting, the construction can be simplified to a 2-state MDP with 1-dimensional features, such that learning is impossible even with an infinite amount of data.
The curse of dimensionality is a widely known issue in reinforcement learning (RL). In the tabular setting where the state space $mathcal{S}$ and the action space $mathcal{A}$ are both finite, to obtain a nearly optimal policy with sampling access to a generative model, the minimax optimal sample complexity scales linearly with $|mathcal{S}|times|mathcal{A}|$, which can be prohibitively large when $mathcal{S}$ or $mathcal{A}$ is large. This paper considers a Markov decision process (MDP) that admits a set of state-action features, which can linearly express (or approximate) its probability transition kernel. We show that a model-based approach (resp.$~$Q-learning) provably learns an $varepsilon$-optimal policy (resp.$~$Q-function) with high probability as soon as the sample size exceeds the order of $frac{K}{(1-gamma)^{3}varepsilon^{2}}$ (resp.$~$$frac{K}{(1-gamma)^{4}varepsilon^{2}}$), up to some logarithmic factor. Here $K$ is the feature dimension and $gammain(0,1)$ is the discount factor of the MDP. Both sample complexity bounds are provably tight, and our result for the model-based approach matches the minimax lower bound. Our results show that for arbitrarily large-scale MDP, both the model-based approach and Q-learning are sample-efficient when $K$ is relatively small, and hence the title of this paper.
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