Partial monitoring games are repeated games where the learner receives feedback that might be different from adversary's move or even the reward gained by the learner. Recently, a general model of combinatorial partial monitoring (CPM) games was proposed \cite{lincombinatorial2014}, where the learner's action space can be exponentially large and adversary samples its moves from a bounded, continuous space, according to a fixed distribution. The paper gave a confidence bound based algorithm (GCB) that achieves $O(T^{2/3}\log T)$ distribution independent and $O(\log T)$ distribution dependent regret bounds. The implementation of their algorithm depends on two separate offline oracles and the distribution dependent regret additionally requires existence of a unique optimal action for the learner. Adopting their CPM model, our first contribution is a Phased Exploration with Greedy Exploitation (PEGE) algorithmic framework for the problem.

Policy gradient methods are widely used for control in reinforcement learning, particularly for the continuous action setting. There have been a host of theoretically sound algorithms proposed for the on-policy setting, due to the existence of the policy gradient theorem which provides a simplified form for the gradient. In off-policy learning, however, where the behaviour policy is not necessarily attempting to learn and follow the optimal policy for the given task, the existence of such a theorem has been elusive. In this work, we solve this open problem by providing the first off-policy policy gradient theorem. The key to the derivation is the use of emphatic weightings. We develop a new actor-critic algorithm--called Actor Critic with Emphatic weightings (ACE)--that approximates the simplified gradients provided by the theorem. We demonstrate in a simple counterexample that previous off-policy policy gradient methods--particularly OffPAC and DPG--converge to the wrong solution whereas ACE finds the optimal solution.

In contrast, we design a discrete-time algorithm and derive its convergence rate solely under weakly monotone conditions.

The optimization phase in deep learning consists in minimizing an objective function w.r.t. the set of

However, these results can be too general to be applicable to natural data distributions. Indeed, humans are quite robust for tasks involving vision.

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We thank the reviewers for their comments. We then address reviewer's comments individually (due to space limits please zoom in the tiny figures). For [18] we used Alg. 2 We thank the reviewer for the additional reference, which we will add to the paper. Gradient Descent) applied in parallel to multiple starting points. We thank R2 for the reference "Entropic regularization of continuous optimal transport problems".

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