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Applications such as virtual reality and online gaming require low delays for acceptable user experience. A key task for over-the-top (OTT) service providers who provide these applications is sending traffic through the networks to minimize delays. OTT traffic is typically generated from multiple data centers which are multi-homed to several network ingresses. However, information about the path characteristics of the underlying network from the ingresses to destinations is not explicitly available to OTT services. These can only be inferred from external probing. In this paper, we combine network tomography with machine learning to minimize delays. We consider this problem in a general setting where traffic sources can choose a set of ingresses through which their traffic enter a black box network. The problem in this setting can be viewed as a reinforcement learning problem with constraints on a continuous action space, which to the best of our knowledge have not been investigated by the machine learning community. Key technical challenges to solving this problem include the high dimensionality of the problem and handling constraints that are intrinsic to networks. Evaluation results show that our methods achieve up to 60% delay reductions in comparison to standard heuristics. Moreover, the methods we develop can be used in a centralized manner or in a distributed manner by multiple independent agents.