Given a set of observations generated by an optimization process, the goal of inverse optimization is to determine likely parameters of that process. We cast inverse optimization as a form of deep learning. Our method, called deep inverse optimization, is to unroll an iterative optimization process and then use backpropagation to learn parameters that generate the observations. We demonstrate that by backpropagating through the interior point algorithm we can learn the coefficients determining the cost vector and the constraints, independently or jointly, for both non-parametric and parametric linear programs, starting from one or multiple observations. With this approach, inverse optimization can leverage concepts and algorithms from deep learning.
Recent work in machine learning @cite_27 @cite_23 @cite_33 views inverse optimization through the lens of online learning, where new observations appear over time rather than as one batch. Our approach may be applicable in online settings, but we focus on generality in the batch setting and do not investigate real-time cases.
@cite_27 @cite_33 @cite_23
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Inverse optimization is a powerful paradigm for learning preferences and restrictions that explain the behavior of a decision maker, based on a set of external signal and the corresponding decision pairs. However, most inverse optimization algorithms are designed specifically in batch setting, where all the data is available in advance. As a consequence, there has been rare use of these methods in an online setting suitable for real-time applications. In this paper, we propose a general framework for inverse optimization through online learning. Specifically, we develop an online learning algorithm that uses an implicit update rule which can handle noisy data. Moreover, under additional regularity assumptions in terms of the data and the model, we prove that our algorithm converges at a rate of @math and is statistically consistent. In our experiments, we show the online learning approach can learn the parameters with great accuracy and is very robust to noises, and achieves a dramatic improvement in computational efficacy over the batch learning approach. While deep learning methods have achieved state-of-theart performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach, each inverse problem requires its own dedicated network. In scenarios where we need to solve a wide variety of problems, e.g., on a mobile camera, it is inefficient and expensive to use these problem-specific networks. On the other hand, traditional methods using analytic signal priors can be used to solve any linear inverse problem; this often comes with a performance that is worse than learning-based methods. In this work, we provide a middle ground between the two kinds of methods — we propose a general framework to train a single deep neural network that solves arbitrary linear inverse problems. We achieve this by training a network that acts as a quasi-projection operator for the set of natural images and show that any linear inverse problem involving natural images can be solved using iterative methods. We empirically show that the proposed framework demonstrates superior performance over traditional methods using wavelet sparsity prior while achieving performance comparable to specially-trained networks on tasks including compressive sensing and pixel-wise inpainting. While deep learning methods have achieved state-of-the-art performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach, different problems require different networks. In scenarios where we need to solve a wide variety of problems, e.g., on a mobile camera, it is inefficient and costly to use these specially-trained networks. On the other hand, traditional methods using signal priors can be used in all linear inverse problems but often have worse performance on challenging tasks. In this work, we provide a middle ground between the two kinds of methods --- we propose a general framework to train a single deep neural network that solves arbitrary linear inverse problems. The proposed network acts as a proximal operator for an optimization algorithm and projects non-image signals onto the set of natural images defined by the decision boundary of a classifier. In our experiments, the proposed framework demonstrates superior performance over traditional methods using a wavelet sparsity prior and achieves comparable performance of specially-trained networks on tasks including compressive sensing and pixel-wise inpainting. While classic machine learning paradigms assume training and test data are generated from the same process, domain adaptation addresses the more realistic setting in which the learner has large quantities of labeled data from some source task but limited or no labeled data from the target task it is attempting to learn. In this work, we give the first formal analysis showing that using active learning for domain adaptation yields a way to address the statistical challenges inherent in this setting. We propose a novel nonparametric algorithm, ANDA, that combines an active nearest neighbor querying strategy with nearest neighbor prediction. We provide analyses of its querying behavior and of finite sample convergence rates of the resulting classifier under covariate shift. Our experiments show that ANDA successfully corrects for dataset bias in multiclass image categorization.