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A recent work introduces the problem of learning set functions from data generated by a so-called optimal subset oracle. Their approach approximates the underlying utility function with an energy-based model, whose parameters are estimated via mean-field variational inference. This approximation reduces to fixed point iterations; however, as the number of iterations increases, automatic differentiation quickly becomes computationally prohibitive due to the size of the Jacobians that are stacked during backpropagation. We address this challenge with implicit differentiation and examine the convergence conditions for the fixed-point iterations. We empirically demonstrate the efficiency of our method on synthetic and real-world subset selection applications including product recommendation, set anomaly detection and compound selection tasks. 

We study monotone submodular maximization under general matroid constraints in the online setting. We prove that online optimization of a large class of submodular functions, namely, threshold potential functions, reduces to online convex optimization (OCO). This is precisely because functions in this class admit a concave relaxation; as a result, OCO policies, coupled with an appropriate rounding scheme, can be used to achieve sublinear regret in the combinatorial setting. We also show that our reduction extends to many different versions of the online learning problem, including the dynamic regret, bandit, and optimistic-learning settings. 

We introduce the problem of optimal congestion control in cache networks, whereby both rate allocations and content placements are optimized jointly. We formulate this as a maximization problem with non-convex constraints, and propose solving this problem via (a) a Lagrangian barrier algorithm and (b) a convex relaxation. We prove different optimality guarantees for each of these two algorithms; our proofs exploit the fact that the non-convex constraints of our problem involve DR-submodular functions. 

We study submodular maximization problems with matroid constraints, in particular, problems where the objective can be expressed via compositions of analytic and multilinear functions. We show that for functions of this form, the so-called continuous greedy algorithm attains a ratio arbitrarily close to (1 – 1/e) ≈ 0.63 using a deterministic estimation via Taylor series approximation. This drastically reduces execution time over prior art that uses sampling. 

Graph embeddings have been tremendously successful at producing node representations that are discriminative for downstream tasks. In this paper, we study the problem of graph transfer learning: given two graphs and labels in the nodes of the first graph, we wish to predict the labels on the second graph. We propose a tractable, noncombinatorial method for solving the graph transfer learning problem by combining classification and embedding losses with a continuous, convex penalty motivated by tractable graph distances. We demonstrate that our method successfully predicts labels across graphs with almost perfect accuracy; in the same scenarios, training embeddings through standard methods leads to predictions that are no better than random. 

The mean squared error loss is widely used in many applications, including auto-encoders, multi-target regression, and matrix factorization, to name a few. Despite computational advantages due to its differentiability, it is not robust to outliers. In contrast, ℓ𝑝 norms are known to be robust, but cannot be optimized via, e.g., stochastic gradient descent, as they are non-differentiable. We propose an algorithm inspired by so-called model-based optimization (MBO), which replaces a non-convex objective with a convex model function and alternates between optimizing the model function and updating the solution. We apply this to robust regression, proposing SADM, a stochastic variant of the Online Alternating Direction Method of Multipliers (OADM) to solve the inner optimization in MBO. We show that SADM converges with the rate 𝑂(log𝑇/𝑇) . Finally, we demonstrate experimentally (a) the robustness of ℓ𝑝 norms to outliers and (b) the efficiency of our proposed model-based algorithms in comparison with gradient methods on autoencoders and multi-target regression.