Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization

Creating impact in real-world settings requires artificial intelligence techniques to span the full pipeline from data, to predictive models, to decisions. These components are typically approached separately: a machine learning model is first trained via a measure of predictive accuracy, and then its predictions are used as input into an optimization algorithm which produces a decision. However, the loss function used to train the model may easily be misaligned with the end goal, which is to make the best decisions possible. Hand-tuning the loss function to align with optimization is a difficult and error-prone process (which is often skipped entirely). We focus on combinatorial optimization problems and introduce a general framework for decision-focused learning, where the machine learning model is directly trained in conjunction with the optimization algorithm to produce high-quality decisions. Technically, our contribution is a means of integrating common classes of discrete optimization problems into deep learning or other predictive models, which are typically trained via gradient descent. The main idea is to use a continuous relaxation of the discrete problem to propagate gradients through the optimization procedure. We instantiate this framework for two broad classes of combinatorial problems: linear programs and submodular maximization. Experimental results across a variety of domains show that decision-focused learning often leads to improved optimization performance compared to traditional methods. We find that standard measures of accuracy are not a reliable proxy for a predictive model's utility in optimization, and our method's ability to specify the true goal as the model's training objective yields substantial dividends across a range of decision problems.Comment: Full version of paper accepted at AAAI 201

A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching

We propose a combinatorial solution for the problem of non-rigidly matching a 3D shape to 3D image data. To this end, we model the shape as a triangular mesh and allow each triangle of this mesh to be rigidly transformed to achieve a suitable matching to the image. By penalising the distance and the relative rotation between neighbouring triangles our matching compromises between image and shape information. In this paper, we resolve two major challenges: Firstly, we address the resulting large and NP-hard combinatorial problem with a suitable graph-theoretic approach. Secondly, we propose an efficient discretisation of the unbounded 6-dimensional Lie group SE(3). To our knowledge this is the first combinatorial formulation for non-rigid 3D shape-to-image matching. In contrast to existing local (gradient descent) optimisation methods, we obtain solutions that do not require a good initialisation and that are within a bound of the optimal solution. We evaluate the proposed method on the two problems of non-rigid 3D shape-to-shape and non-rigid 3D shape-to-image registration and demonstrate that it provides promising results.Comment: 10 pages, 7 figure

An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure

We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common theory, leading to novel methods for working with several important models of interest in statistics, machine learning and computer vision. In Part 1, we review the concepts of network flows and submodular function optimization theory foundational to our results. We then examine the connections between network flows and the minimum-norm algorithm from submodular optimization, extending and improving several current results. This leads to a concise representation of the structure of a large class of pairwise regularized models important in machine learning, statistics and computer vision. In Part 2, we describe the full regularization path of a class of penalized regression problems with dependent variables that includes the graph-guided LASSO and total variation constrained models. This description also motivates a practical algorithm. This allows us to efficiently find the regularization path of the discretized version of TV penalized models. Ultimately, our new algorithms scale up to high-dimensional problems with millions of variables