In many real-world hybrid (mixed discrete continuous) planning problems such as Reservoir Control, Heating, Ventilation and Air Conditioning (HVAC), and Navigation, it is difficult to obtain a model of the complex nonlinear dynamics that govern state evolution. However, the ubiquity of modern sensors allow us to collect large quantities of data from each of these complex systems and build accurate, nonlinear deep network models of their state transitions. But there remains one major problem for the task of control -how can we plan with deep network learned transition models without resorting to Monte Carlo Tree Search and other black-box transition model techniques that ignore model structure and do not easily extend to mixed discrete and continuous domains? In this paper, we make the critical observation that the popular Rectified Linear Unit (ReLU) transfer function for deep networks not only allows accurate nonlinear deep net model learning, but also permits a direct compilation of the deep network transition model to a MixedInteger Linear Program (MILP) encoding in a planner we call Hybrid Deep MILP Planning (HD-MILP-PLAN). We identify deep net specific optimizations and a simple sparsification method for HD-MILP-PLAN that improve performance over a naïve encoding, and show that we are able to plan optimally with respect to the learned deep network.
Previous highly scalable one-class collaborative filtering methods such as Projected Linear Recommendation (PLRec) have advocated using fast randomized SVD to embed items into a latent space, followed by linear regression methods to learn personalized recommendation models per user. Unfortunately, naive SVD embedding methods often exhibit a popularity bias that skews the ability to accurately embed niche items. To address this, we leverage insights from Noise Contrastive Estimation (NCE) to derive a closed-form, efficiently computable "depopularized" embedding. While this method is not ideal for direct recommendation using methods like PureSVD since popularity still plays an important role in recommendation, we find that embedding followed by linear regression to learn personalized user models in a novel method we call NCE-PLRec leverages the improved item embedding of NCE while correcting for its popularity unbiasing in final recommendations. An analysis of the recommendation popularity distribution demonstrates that NCE-PLRec uniformly distributes its recommendations over the popularity spectrum while other methods exhibit distinct biases towards specific popularity subranges, thus artificially restricting their recommendations. Empirically, NCE-PLRec outperforms state-ofthe-art methods as well as various ablations of itself on a variety of large-scale recommendation datasets.
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