Deep reinforcement learning (RL) has shown remarkable success in specific offline decisionmaking scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a nonasymptotic estimation error of pessimistic offline RL using general neural network approximation with Cmixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for Cmixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.
This content will become publicly available on April 17, 2025
The learning speed of feedforward neural networks is notoriously slow and has presented a bottleneck in deep learning applications for several decades. For instance, gradientbased learning algorithms, which are used extensively to train neural networks, tend to work slowly when all of the network parameters must be iteratively tuned. To counter this, both researchers and practitioners have tried introducing randomness to reduce the learning requirement. Based on the original construction of Igelnik and Pao, single layer neuralnetworks with random inputtohidden layer weights and biases have seen success in practice, but the necessary theoretical justification is lacking. In this study, we begin to fill this theoretical gap. We then extend this result to the nonasymptotic setting using a concentration inequality for MonteCarlo integral approximations. We provide a (corrected) rigorous proof that the Igelnik and Pao construction is a universal approximator for continuous functions on compact domains, with approximation error squared decaying asymptotically like
 Award ID(s):
 1928930
 NSFPAR ID:
 10529332
 Publisher / Repository:
 Frontiers in Applied Mathematics and Statistics
 Date Published:
 Journal Name:
 Frontiers in Applied Mathematics and Statistics
 Volume:
 10
 ISSN:
 22974687
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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