We give an efficient algorithm for learning a binary function in a given class C of bounded VC dimension, with training data distributed according to P and test data according to Q, where P and Q may be arbitrary distributions over X. This is the generic form of what is called covariate shift, which is impossible in general as arbitrary P and Q may not even overlap. However, recently guarantees were given in a model called PQ-learning (Goldwasser et al., 2020) where the learner has: (a) access to unlabeled test examples from Q (in addition to labeled samples from P, i.e., semi-supervised learning); and (b) the option to reject any example and abstain from classifying it (i.e., selective classification). The algorithm of Goldwasser et al. (2020) requires an (agnostic) noise tolerant learner for C. The present work gives a polynomial-time PQ-learning algorithm that uses an oracle to a "reliable" learner for C, where reliable learning (Kalai et al., 2012) is a model of learning with one-sided noise. Furthermore, our reduction is optimal in the sense that we show the equivalence of reliable and PQ learning.

The use of target networks is a common practice in deep reinforcement learning for stabilizing the training; however, theoretical understanding of this technique is still limited. In this paper, we study the so-called periodic Q-learning algorithm (PQ-learning for short), which resembles the technique used in deep Q-learning for solving infinite-horizon discounted Markov decision processes (DMDP) in the tabular setting. PQ-learning maintains two separate Q-value estimates - the online estimate and target estimate. The online estimate follows the standard Q-learning update, while the target estimate is updated periodically. In contrast to the standard Q-learning, PQ-learning enjoys a simple finite time analysis and achieves better sample complexity for finding an epsilon-optimal policy. Our result provides a preliminary justification of the effectiveness of utilizing target estimates or networks in Q-learning algorithms.