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.
In the parametric optimization setting, @cite_16 develop an optimization model that encodes KKT optimality conditions for imputing objective function coefficients of a convex optimization problem. @cite_28 focus on the same problem under the assumption of noisy measurements, developing a bilevel problem and two algorithms which are shown to maintain statistical consistency. Saez-Gallego and Morales @cite_20 address the case of learning @math and @math jointly in a parametric setting where the @math vector is assumed to be an affine function of a regressor. The general case of learning the weights of a parametric linear optimization problem where @math , @math and @math are functions of @math (Figure (iii)) has not been addressed in the literature.
@cite_28 @cite_16 @cite_20
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We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . This paper addresses the problem of minimizing a convex, Lipschitz function @math over a convex, compact set @math under a stochastic bandit (i.e., noisy zeroth-order) feedback model. In this model, the algorithm is allowed to observe noisy realizations of the function value @math at any query point @math . The quantity of interest is the regret of the algorithm, which is the sum of the function values at algorithm's query points minus the optimal function value. We demonstrate a generalization of the ellipsoid algorithm that incurs @math regret. Since any algorithm has regret at least @math on this problem, our algorithm is optimal in terms of the scaling with @math . The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter @math . However, in many settings the non-differentiable convex function @math is not uniformly Lipschitz continuous -- for example (i) the classical support vector machine (SVM) problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with @math . Herein we develop a notion of "relative continuity" that is determined relative to a user-specified "reference function" @math (that should be computationally tractable for algorithms), and we show that many non-differentiable convex functions are relatively continuous with respect to a correspondingly fairly-simple reference function @math . We also similarly develop a notion of "relative stochastic continuity" for the stochastic setting. We analysis two standard algorithms -- the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm -- for solving optimization problems in these two new settings, and we develop for the first time computational guarantees for instances where the objective function is not uniformly Lipschitz continuous. This paper is a companion paper for non-differentiable convex optimization to the recent paper by Lu, Freund, and Nesterov, which developed similar sorts of results for differentiable convex optimization. We derive the mean squared error convergence rates of kernel density-based plug-in estimators of mutual information measures between two multidimensional random variables @math and @math for two cases: 1) @math and @math are both continuous; 2) @math is continuous and @math is discrete. Using the derived rates, we propose an ensemble estimator of these information measures for the second case by taking a weighted sum of the plug-in estimators with varied bandwidths. The resulting ensemble estimator achieves the @math parametric convergence rate when the conditional densities of the continuous variables are sufficiently smooth. To the best of our knowledge, this is the first nonparametric mutual information estimator known to achieve the parametric convergence rate for this case, which frequently arises in applications (e.g. variable selection in classification). The estimator is simple to implement as it uses the solution to an offline convex optimization problem and simple plug-in estimators. A central limit theorem is also derived for the ensemble estimator. Ensemble estimators that achieve the parametric rate are also derived for the first case ( @math and @math are both continuous) and another case 3) @math and @math may have any mixture of discrete and continuous components.