Partial Information as Full: Reward Imputation with Sketching in Bandits

Xiao Zhang, Ninglu Shao, Zihua Si, Jun Xu, Wenhan Wang, hanjing su, Ji-Rong Wen

Keywords: reward imputation, bandit, sketching, regret analysis

Abstract: We focus on the setting of contextual batched bandit (CBB), where a batch of rewards is observed from the environment in each episode. But these rewards are partial-information feedbacks where the rewards of the non-executed actions are unobserved. Existing approaches for CBB usually ignore the potential rewards of the non-executed actions, resulting in feedback information being underutilized. In this paper, we propose an efficient reward imputation approach using sketching in CBB, which completes the unobserved rewards with the imputed rewards approximating the full-information feedbacks. Specifically, we formulate the reward imputation as a problem of imputation regularized ridge regression, which captures the feedback mechanisms of both the non-executed and executed actions. To reduce the time complexity of reward imputation on a large batch of data, we use randomized sketching for solving the regression problem of imputation. We prove that the proposed reward imputation approach obtains a relative-error bound for sketching approximation, achieves an instantaneous regret with an exponentially-decaying bias and a smaller variance than that without reward imputation, and enjoys a sublinear regret bound against the optimal policy. Moreover, we present two extensions of our approach, including the rate-scheduled version and the version for nonlinear rewards, which makes our approach more feasible. Experimental results demonstrated that our approach can outperform the state-of-the-art baselines on a synthetic dataset, the Criteo dataset, and a dataset from a commercial app.