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 pixel-wise regression problems with sparsely annotated images. Traditional regression methods based on mean squared error emphasize pixels with labels, leading to distorted predictions in unlabeled areas. To address this limitation, we introduce Neural Point Processes, a novel approach that combines 2D Gaussian Processes with neural networks to leverage spatial correlations between sparse labels on images. This approach offers two key advantages: it imposes smoothness constraints on the model output and enables conditional predictions when sparse labels are available at inference time. Empirical results on synthetic and real-world datasets demonstrate a substantial improvement in mean-squared error and $R^2$ scores, outperforming standard regression techniques. On the real-world dataset COWC, we achieve an $R^2$ of $0.769$ with $81$ out of $40,000$ ($0.2$%) points labeled, while standard regression loss (MSE) results in an $R^2$ of $0.060$.

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 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 $(1-1/e) \approx 0.63$ using a deterministic estimation via Taylor series approximation. This drastically reduces execution time over prior art that uses sampling.