  It is well known that the historical logs are used for evaluating and
learning policies in interactive systems, e.g. recommendation, search, and
online advertising. Since direct online policy learning usually harms user
experiences, it is more crucial to apply off-policy learning in real-world
applications instead. Though there have been some existing works, most are
focusing on learning with one single historical policy. However, in practice,
usually a number of parallel experiments, e.g. multiple AB tests, are performed
simultaneously. To make full use of such historical data, learning policies
from multiple loggers becomes necessary. Motivated by this, in this paper, we
investigate off-policy learning when the training data coming from multiple
historical policies. Specifically, policies, e.g. neural networks, can be
learned directly from multi-logger data, with counterfactual estimators. In
order to understand the generalization ability of such estimator better, we
conduct generalization error analysis for the empirical risk minimization
problem. We then introduce the generalization error bound as the new risk
function, which can be reduced to a constrained optimization problem. Finally,
we give the corresponding learning algorithm for the new constrained problem,
where we can appeal to the minimax problems to control the constraints.
Extensive experiments on benchmark datasets demonstrate that the proposed
methods achieve better performances than the state-of-the-arts.
