Given enough data, Deep Neural Networks (DNNs) are capable of learning complex input-output relations with high accuracy. In several domains, however, data is scarce or expensive to retrieve, while a substantial amount of expert knowledge is available. It seems reasonable that if we can inject this additional information in the DNN, we could ease the learning process. One such case is that of Constraint Problems, for which declarative approaches exists and pure ML solutions have obtained mixed success. Using a classical constrained problem as a case study, we perform controlled experiments to probe the impact of progressively adding domain and empirical knowledge in the DNN. Our results are very encouraging, showing that (at least in our setup) embedding domain knowledge at training time can have a considerable effect and that a small amount of empirical knowledge is sufficient to obtain practically useful results. 4

Given enough data, Deep Neural Networks (DNNs) are capable of learning complex input-output relations with high accuracy. In several domains, however, data is scarce or expensive to retrieve, while a substantial amount of expert knowledge is available. It seems reasonable that if we can inject this additional information in the DNN, we could ease the learning process. One such case is that of Constraint Problems, for which declarative approaches exists and pure ML solutions have obtained mixed success. Using a classical constrained problem as a case study, we perform controlled experiments to probe the impact of progressively adding domain and empirical knowledge in the DNN. Our results are very encouraging, showing that (at least in our setup) embedding domain knowledge at training time can have a considerable effect and that a small amount of empirical knowledge is sufficient to obtain practically useful results.

Constrained decision problems in the real world are subject to uncertainty. If predictive information about the stochastic elements is available offline, recent works have shown that it is possible to rely on an (expensive) parameter tuning phase to improve the behavior of a simple online solver so that it roughly matches the solution quality of an anticipative approach but maintains its original efficiency. Here, we start from a state-of-the-art offline/online optimization method that relies on optimality conditions to inject knowledge of a (convex) online approach into an offline solver used for parameter tuning. We then propose to replace the offline step with (Deep) Reinforcement Learning (RL) approaches, which results in a simpler integration scheme with a higher potential for generalization. We introduce two hybrid methods that combine both learning and optimization: the őrst optimizes all the parameters at once, whereas the second exploits the sequential nature of the online problem via the Markov Decision Process framework. In a case study in energy management, we show the effectiveness of our hybrid approaches, w.r.t. the state-of-the-art and pure RL methods. The combination proves capable of faster convergence and naturally handles constraint satisfaction.

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