# Bilinear Classes: A Structural Framework for Provable Generalization in RL

@inproceedings{Du2021BilinearCA, title={Bilinear Classes: A Structural Framework for Provable Generalization in RL}, author={Simon Shaolei Du and Sham M. Kakade and Jason D. Lee and Shachar Lovett and Gaurav Mahajan and Wen Sun and Ruosong Wang}, booktitle={ICML}, year={2021} }

This work introduces Bilinear Classes, a new structural framework, which permit generalization in reinforcement learning in a wide variety of settings through the use of function approximation. The framework incorporates nearly all existing models in which a polynomial sample complexity is achievable, and, notably, also includes new models, such as the Linear Q∗/V ∗ model in which both the optimal Q-function and the optimal V -function are linear in some known feature space. Our main result…

## 66 Citations

Sample-Efficient Reinforcement Learning for POMDPs with Linear Function Approximations

- Mathematics, Computer ScienceArXiv
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An RL algorithm is proposed that constructs optimistic estimators of undercomplete POMDPs with linear function approximations via reproducing kernel Hilbert space (RKHS) embedding and it is theoretically proved that the proposed algorithm has an ε -optimal policy with e O (1 /ε 2 ) episodes of exploration.

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A new algorithm is provided along with statistical guarantees that efficiently exploits low rank structure given access to a generative model, achieving a sample complexity of Õ ( d5(|S|+ |A |)poly(H)/ǫ2 ) for a rank d setting, which is minimax optimal with respect to the scaling of |S|, |A|, and ǫ.

TensorPlan and the Few Actions Lower Bound for Planning in MDPs under Linear Realizability of Optimal Value Functions

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The minimax query complexity of online planning with a generative model in fixedhorizon Markov decision processes (MDPs) with linear function approximation is considered and an exponentially large lower bound holds when A = Ω(min(d1/4, H1/2), under either (i), (ii) or (iii).

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This work proposes two new algorithms for discounted Markov decision processes with linear function approximation and a simulator that have polynomial query and computational cost in the dimension of the features, the effective planning horizon and the targeted sub-optimality, while the cost remains independent of the size of the state space.

Representation Learning for Online and Offline RL in Low-rank MDPs

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An algorithm REP-UCB—Upper Confidence Bound driven REPresentation learning for RL, which significantly improves the sample complexity and is simpler than FLAMBE, as it directly balances the interplay between representation learning, exploration, and exploitation.

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A new offline actor-critic algorithm is proposed that naturally incorporates the pessimism principle, leading to several key advantages compared to the state of the art, and an upper bound on the suboptimality gap of the policy returned by the procedure is proved.

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- Computer ScienceNeurIPS
- 2021

A new complexity measure—Bellman Eluder (BE) dimension is introduced and it is proved that both algorithms learn the near-optimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space.

Bellman Eluder Dimension: New Rich Classes of RL Problems, and Sample-Efﬁcient Algorithms

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It is proved that both algorithms learn the near-optimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space.

Sample-Efficient Reinforcement Learning Is Feasible for Linearly Realizable MDPs with Limited Revisiting

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A new sampling protocol is investigated, which draws samples in an online/exploratory fashion but allows one to backtrack and revisit previous states but not the size of the state/action space, and an algorithm is developed that achieves a sample complexity that scales polynomially with the feature dimension, the horizon, and the inverse sub-optimality gap.

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