Self-Supervised Learning (SSL) for Combinatorial Optimization (CO) is an emerging paradigm for solving combinatorial problems using neural networks. In this paper, we address a central challenge of SSL for CO: solving problems with discrete constraints. We design an end-to-end differentiable framework that enables us to solve discrete constrained optimization problems with neural networks. Concretely, we leverage algorithmic techniques from the literature on convex geometry and Carathéodory's theorem to decompose neural network outputs into convex combinations of polytope corners that correspond to feasible sets. This decomposition-based approach enables self-supervised training but also ensures efficient quality-preserving rounding of the neural net output into feasible solutions. Extensive experiments in cardinality-constrained optimization show that our approach can consistently outperform neural baselines. We further provide workedout examples of how our method can be applied beyond cardinality-constrained problems to a diverse set of combinatorial optimization tasks, including finding independent sets in graphs, and solving matroid-constrained problems.

The famous theorem of Fritz John states that any convex body has a unique maximal volume inscribed ellipsoid, known as the John Ellipsoid. Computing the John Ellipsoid is a fundamental problem in convex optimization. In this paper, we focus on approximating the John Ellipsoid inscribed in a convex and centrally symmetric polytope defined by P:= {x Rd: 1n Ax 1n}, where A Rn d is a rank-d matrix and 1n Rn is the all-ones vector. We develop two efficient algorithms for approximating the John Ellipsoid. The first is a sketchingbased algorithm that runs in nearly input-sparsity time eO(nnz(A)+dω), where nnz(A)denotes the number of nonzero entries in the matrix Aand ω 2.37is the current matrix multiplication exponent. The second is a treewidth-based algorithm that runs in time eO(nτ2), where τ is the treewidth of the dual graph of the matrix A. Our algorithms significantly improve upon the state-of-the-art running time of eO(nd2)achieved by [Cohen, Cousins, Lee, and Yang, COLT 2019].

In strategic classification, the standard supervised learning setting is extended to support the notion of strategic user behavior in the form of costly feature manipulations made in response to a classifier. While standard learning supports a broad range of model classes, the study of strategic classification has, so far, been dedicated mostly to linear classifiers. This work aims to expand the horizon by exploring how strategic behavior manifests under non-linear classifiers and what this implies for learning. We take a bottom-up approach showing how non-linearity affects decision boundary points, classifier expressivity, and model class complexity. Our results show how, unlike the linear case, strategic behavior may either increase or decrease effective class complexity, and that the complexity decrease may be arbitrarily large. Another key finding is that universal approximators (e.g., neural nets) are no longer universal once the environment is strategic. We demonstrate empirically how this can create performance gaps even on an unrestricted model class.

We develop a Frank-Wolfe algorithm with corrective steps, generalizing previous algorithms including Blended Conditional Gradients, Blended Pairwise Conditional Gradients, and Fully-Corrective Frank-Wolfe. For this, we prove tight convergence guarantees together with an optimal face identification property. Furthermore, we propose two highly efficient corrective steps for convex quadratic objectives based on linear optimization or linear system solving, akin to Wolfe's MinimumNorm Point algorithm, and prove finite-time convergence under suitable conditions. Beyond optimization problems that are directly quadratic, we revisit two algorithms, Split Conditional Gradient and Second-Order Conditional Gradient Sliding, which can leverage quadratic corrections to accelerate the solution of their quadratic subproblems. We show improved convergence rates for the first and prove broader applicability for the second. Finally, we demonstrate substantial computational speedups for Frank-Wolfe-based algorithms with quadratic corrections across the considered problem classes.

Programmatic Weak Supervision (PWS) aims to reduce the cost of constructing large high quality labeled datasets often used in training modern machine learning models. A major component of the PWS pipeline is the label model, which amalgamates predictions from multiple noisy weak supervision sources, i.e. labeling functions (LFs), to label datapoints. While most label models are either probabilistic or adversarial, a recently proposed label model achieves strong empirical performance without falling into either camp. That label model constructs a polytope of plausible labelings using the LF predictions and outputs the center of that polytope as its proposed labeling. In this paper, we attempt to theoretically study that strategy by proposing Bayesian Balsubramani-Freund (BBF), a label model that implicitly constructs a polytope of plausible labelings and selects a labeling in its interior. We show an assortment of statistical results for BBF: log-concavity of its posterior, its form of solution, consistency, and rates of convergence. Extensive experiments compare our proposed method against twelve baseline label models over eleven datasets. BBF compares favorably to other Bayesian label models and label models that don't use datapoint features -- matching or exceeding their performance on eight out of eleven datasets.

Neural Information Processing Systems http://nips.cc/

This paper presents new projection-free algorithms for Online Convex Optimization (OCO) over a convex domain K Rd. Classical OCO algorithms (such as Online Gradient Descent) typically need to perform Euclidean projections onto the convex set K to ensure feasibility of their iterates. Alternative algorithms, such as those based on the Frank-Wolfe method, swap potentially-expensive Euclidean projections onto Kfor linear optimization over K. However, such algorithms have a sub-optimal regret in OCO compared to projection-based algorithms. In this paper, we look at a third type of algorithms that output approximate Newton iterates using a self-concordant barrier for the set of interest. The use of a self-concordant barrier automatically ensures feasibility without the need of projections. However, the computation of the Newton iterates requires a matrix inverse, which can still be expensive. As our main contribution, we show how the stability of the Newton iterates can be leveraged to only compute the inverse Hessian a vanishing fractions of the rounds, leading to a new efficient projection-free OCO algorithm with a state-of-the-art regret bound.

Neural Information Processing Systems http://nips.cc/

Ensuring that a DNN satisfies a desired property is critical when deploying DNNs in safety-critical applications. There are efficient methods that can verify whether a DNN satisfies a property, as seen in the annual DNN verification competition (VNN-COMP). However, the problem of provably editing a DNN to satisfy a property remains challenging.