We study offline reinforcement learning under $Q^\star$-approximation and partial coverage, a setting that motivates practical algorithms such as Conservative $Q$-Learning (CQL; Kumar et al., 2020) but has received limited theoretical attention. Our work is inspired by the following open question: "Are $Q^\star$-realizability and Bellman completeness sufficient for sample-efficient offline RL under partial coverage?" We answer in the negative by establishing an information-theoretic lower bound. Going substantially beyond this, we introduce a general framework that characterizes the intrinsic complexity of a given $Q^\star$ function class, inspired by model-free decision-estimation coefficients (DEC) for online RL (Foster et al., 2023b; Liu et al., 2025b). This complexity recovers and improves the quantities underlying the guarantees of Chen and Jiang (2022) and Uehara et al. (2023), and extends to broader settings. Our decision-estimation decomposition can be combined with a wide range of $Q^\star$ estimation procedures, modularizing and generalizing existing approaches. Beyond the general framework, we make further contributions: By developing a novel second-order performance difference lemma, we obtain the first $ε^{-2}$ sample complexity under partial coverage for soft $Q$-learning, improving the $ε^{-4}$ bound of Uehara et al. (2023). We remove Chen and Jiang's (2022) need for additional online interaction when the value gap of $Q^\star$ is unknown. We also give the first characterization of offline learnability for general low-Bellman-rank MDPs without Bellman completeness (Jiang et al., 2017; Du et al., 2021; Jin et al., 2021), a canonical setting in online RL that remains unexplored in offline RL except for special cases. Finally, we provide the first analysis for CQL under $Q^\star$-realizability and Bellman completeness beyond the tabular case.

Furthermore, whencontrollingcomplex high-dimensional morphologies, such as humanoid bodies, tasks often require coordination of multiple skills simultaneously.

OfflineRLisdeemed to be promising [16, 14] as online learning requires the agent to continuously interact with the environment, which howevermaybecostly,time-consuming, orevendangerous.

Deep RL(DRL) tackles the curse of dimensionality due to large state spaces by utilizing a Deep Neural Network (DNN) to approximate the value function and/or the policy.

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This approach is often preferable when using "standard" parametric5 regression algorithms, as the weighted p-norm can be directly minimized at learning time (e.g., in least-squares6 regression,the`2-normisminimized). This is not surprising as in the worst case the inherent Bellman error may12 be unbounded and standard AVI tends to diverge. Recent work (Jinglin Chen, Nan Jiang,Information-Theoretic13 Considerations in Batch Reinforcement Learning, ICML 2019, Conjecture 8) has even conjectured an exponential14 lower bound in case of unbounded Bellman error. In the paper we propose afirst heuristic algorithm to automatically construct aset22 ofanchor points (see beginning ofpage 6). Akeybenefit ofourapproach isthat itdoes notmodify theunderlying (linear) feature51 representation, allowing theuser tousethelinear representation with, forexample, approximate value iteration, and52 should this fail, the user can switch toour algorithm and progressively increase the number of support points while53 keeping the same feature representation.

Delusionalbias ariseswhenthe approximation architecture limits the class ofexpressible greedy policies.

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