Learning Zero-Sum Linear Quadratic Games with Improved Sample Complexity

Zero-sum Linear Quadratic (LQ) games are fundamental in optimal control and can be used (i) as a dynamic game formulation for risk-sensitive or robust control, or (ii) as a benchmark setting for multi-agent reinforcement learning with two competing agents in continuous state-control spaces. In contrast to the well-studied single-agent linear quadratic regulator problem, zero-sum LQ games entail solving a challenging nonconvex-nonconcave min-max problem with an objective function that lacks coercivity. Recently, Zhang et al. discovered an implicit regularization property of natural policy gradient methods which is crucial for safety-critical control systems since it preserves the robustness of the controller during learning. Moreover, in the model-free setting where the knowledge of model parameters is not available, Zhang et al. proposed the first polynomial sample complexity algorithm to reach an $\epsilon$-neighborhood of the Nash equilibrium while maintaining the desirable implicit regularization property. In this work, we propose a simpler nested Zeroth-Order (ZO) algorithm improving sample complexity by several orders of magnitude. Our main result guarantees a $\widetilde{\mathcal{O}}(\epsilon^{-3})$ sample complexity under the same assumptions using a single-point ZO estimator. Furthermore, when the estimator is replaced by a two-point estimator, our method enjoys a better $\widetilde{\mathcal{O}}(\epsilon^{-2})$ sample complexity. Our key improvements rely on a more sample-efficient nested algorithm design and finer control of the ZO natural gradient estimation error

GNN-encoder: Learning a Dual-encoder Architecture via Graph Neural Networks for Passage Retrieval

Recently, retrieval models based on dense representations are dominant in passage retrieval tasks, due to their outstanding ability in terms of capturing semantics of input text compared to the traditional sparse vector space models. A common practice of dense retrieval models is to exploit a dual-encoder architecture to represent a query and a passage independently. Though efficient, such a structure loses interaction between the query-passage pair, resulting in inferior accuracy. To enhance the performance of dense retrieval models without loss of efficiency, we propose a GNN-encoder model in which query (passage) information is fused into passage (query) representations via graph neural networks that are constructed by queries and their top retrieved passages. By this means, we maintain a dual-encoder structure, and retain some interaction information between query-passage pairs in their representations, which enables us to achieve both efficiency and efficacy in passage retrieval. Evaluation results indicate that our method significantly outperforms the existing models on MSMARCO, Natural Questions and TriviaQA datasets, and achieves the new state-of-the-art on these datasets.Comment: 11 pages, 6 figure

Learning Zero-Sum Linear Quadratic Games with Improved Sample Complexity

ISSN:0743-154