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. We consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded-rank hypergraphs with multiple edge cost functions. For a fixed number of edge cost functions, we show that the total number of supported non-dominated (SND) cuts is bounded by a polynomial in the numbers of nodes and edges, i.e., is strongly polynomial. This bound also applies to the combinatorial facet complexity of the problem, i.e., the maximum number of facets (linear pieces) of the parametric curve for the parametrized (linear combination) objective, over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive. We sharpen this bound in the case of two objectives (the bicriteria problem), for which we also derive a strongly polynomial upper bound on the total number of non-dominated (Pareto optimal) cuts. In particular, the bicriteria global minimum cut problem in an n-node graph admits @math O(n3logn) SND cuts and @math O(n5logn) Pareto optimal cuts. These results significantly improve on earlier graph cut results by Mulmuley (SIAM J Comput 28(4):1460---1509, 1999) and Armon and Zwick (Algorithmica 46(1):15---26, 2006). They also imply that the parametric curve and all SND cuts, and, for the bicriteria problems, all Pareto optimal cuts, can be computed in strongly polynomial time when the number of objectives is fixed. In this paper, we present a new conditional gradient type method for convex optimization by calling a linear optimization ( @math ) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time and, as a result, can achieve the optimal complexity bounds in terms of not only the number of calls to the @math oracle but also the number of gradient evaluations. More specifically, we show that the CGS method requires @math and @math gradient evaluations, respectively, for solving smooth and strongly convex problems, while still maintaining the optimal @math bound on the number of calls to the @math oracle. We also develop variants of the CGS method which can achieve the optimal complexity bounds for solving stochastic optimization problems and an important class... Consider a set of @math agents seeking to solve distributively the minimization problem @math where the convex functions @math are local to the agents. The popular Alternating Direction Method of Multipliers has the potential to handle distributed optimization problems of this kind. We provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures. The rate of convergence of our method is considered. It is assumed that the infimum of the problem is reached at a point @math , the functions @math are twice differentiable at this point and @math in the positive definite ordering of symmetric matrices. With these assumptions, it is shown that the convergence to the consensus @math is linear and the exact rate is provided. Application examples where this rate can be optimized with respect to the ADMM free parameter @math are also given. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . We provide new theoretical insights on why over-parametrization is effective in learning neural networks. For a @math hidden node shallow network with quadratic activation and @math training data points, we show as long as @math , over-parametrization enables local search algorithms to find a optimal solution for general smooth and convex loss functions. Further, despite that the number of parameters may exceed the sample size, using theory of Rademacher complexity, we show with weight decay, the solution also generalizes well if the data is sampled from a regular distribution such as Gaussian. To prove when @math , the loss function has benign landscape properties, we adopt an idea from smoothed analysis, which may have other applications in studying loss surfaces of neural networks. We propose a new distributed algorithm based on alternating direction method of multipliers (ADMM) to minimize sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. Our algorithm allows for a general choice of the communication weight matrix, which is used to combine the iterates at different nodes. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate @math , where @math is the number of iterations. We then show that when functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an @math -optimal solution can be computed with @math iterations, where @math is the condition number of the problem. Our analysis highlights the effect of network and communication weights on the convergence rate through degrees of the nodes, the smallest nonzero eigenvalue, and operator norm of the communication matrix. The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant @math . However, in many settings the differentiable convex function @math is not uniformly smooth---for example, in @math -optimal design where @math and @math , or even the univariate setting with @math . In this paper we develop a notion of “relative smoothness” and relative strong convexity that is determined relative to a user-specified “reference function” @math (that should be computationally tractable for algorithms), and we show that many differentiable convex functions are relatively smooth with respect to a correspondingly fairly simple reference function @math . We extend two standard algorithms---the primal gradient scheme and the dual averaging scheme---to our new setting, with associated computational guarantees. We apply our new a... Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., the problem @math minxEvfv(Ew[gw(x)]). In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of @math fv,gw and use an auxiliary variable to track the unknown quantity @math Ewgw(x). We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieves a convergence rate of @math O(k-1 4) in the general case and @math O(k-2 3) in the strongly convex case, after taking k samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of @math O(k-2 7) in the general case and @math O(k-4 5) in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc. Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone @math K, a norm @math ?·? and a smooth convex function @math f, we want either (1) to minimize the norm over the intersection of the cone and a level set of @math f, or (2) to minimize over the cone the sum of @math f and a multiple of the norm. We focus on the case where (a) the dimension of the problem is too large to allow for interior point algorithms, (b) @math ?·? is "too complicated" to allow for computationally cheap Bregman projections required in the first-order proximal gradient algorithms. On the other hand, we assume that it is relatively easy to minimize linear forms over the intersection of @math K and the unit @math ?·?-ball. Motivating examples are given by the nuclear norm with @math K being the entire space of matrices, or the positive semidefinite cone in the space of symmetric matrices, and the Total Variation norm on the space of 2D images. We discuss versions of the Conditional Gradient algorithm capable to handle our problems of interest, provide the related theoretical efficiency estimates and outline some applications. The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates @math given by @math where @math is a collection of maximal monotone operators on a separable Hilbert space, @math is an independent identically distributed sequence of random variables on @math and @math is a positive sequence in @math . The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation @math under the assumption that the latter is maximal. We consider applications to stochastic optimization problems of the form @math , where @math is a normal convex integrand and @math is a collection of closed convex sets. In this case, the iterations are closely related to a stochastic proximal algorithm recently proposed by Wang and Bertsekas [Increment... The alternating direction method of multipliers (ADMM) has been widely used for solving structured convex optimization problems. In particular, the ADMM can solve convex programs that minimize the sum of @math convex functions whose variables are linked by some linear constraints. While the convergence of the ADMM for @math was well established in the literature, it remained an open problem for a long time whether the ADMM for @math is still convergent. Recently, it was shown in [, Math. Program. (2014), DOI 10.1007 s10107-014-0826-5.] that without additional conditions the ADMM for @math generally fails to converge. In this paper, we show that under some easily verifiable and reasonable conditions the global linear convergence of the ADMM when @math can still be ensured, which is important since the ADMM is a popular method for solving large-scale multiblock optimization models and is known to perform very well in practice even when @math . Our study aims to offer an explanation for... In this paper we study the problem of learning Rectified Linear Units (ReLUs) which are functions of the form @math with @math denoting the weight vector. We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to a planted weight vector. We show that projected gradient descent, when initialization at 0, converges at a linear rate to the planted model with a number of samples that is optimal up to numerical constants. Our results on the dynamics of convergence of these very shallow neural nets may provide some insights towards understanding the dynamics of deeper architectures. Nonconvex optimization is becoming more and more important in machine learning and operations research. In spite of recent progresses, the development of provably efficient algorithm for optimization with nonconvex functional constraints remains open. Such problems have potential applications in risk-averse machine learning, semisupervised learning and robust optimization among others. In this paper, we introduce a new proximal point type method for solving this important class of nonconvex problems by transforming them into a sequence of convex constrained subproblems. We establish the convergence and rate of convergence of this algorithm to the KKT point under different types of constraint qualifications. In particular, we prove that our algorithm will converge to an @math -KKT point in @math iterations under a properly defined condition. For practical use, we present inexact variants of this approach, in which approximate solutions of the subproblems are computed by either primal or primal-dual type algorithms, and establish their associated rate of convergence. To the best of our knowledge, this is the first time that proximal point type method is developed for nonlinear programing with nonconvex functional constraints, and most of the convergence and complexity results seem to be new in the literature. This paper presents a lower bound for optimizing a finite sum of @math functions, where each function is @math -smooth and the sum is @math -strongly convex. We show that no algorithm can reach an error @math in minimizing all functions from this class in fewer than @math iterations, where @math is a surrogate condition number. We then compare this lower bound to upper bounds for recently developed methods specializing to this setting. When the functions involved in this sum are not arbitrary, but based on i.i.d. random data, then we further contrast these complexity results with those for optimal first-order methods to directly optimize the sum. The conclusion we draw is that a lot of caution is necessary for an accurate comparison, and identify machine learning scenarios where the new methods help computationally. We consider @math -median clustering in finite metric spaces and @math -means clustering in Euclidean spaces, in the setting where @math is part of the input (not a constant). For the @math -means problem, show that if the optimal @math -means clustering of the input is more expensive than the optimal @math -means clustering by a factor of @math , then one can achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math by using a variant of Lloyd's algorithm. In this work we substantially improve this approximation guarantee. We show that given only the condition that the @math -means optimal is more expensive than the @math -means optimal by a factor @math for some constant @math , we can obtain a PTAS. In particular, under this assumption, for any @math we achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math , and exponential in @math and @math . We thus decouple the strength of the assumption from the quality of the approximation ratio. We also give a PTAS for the @math -median problem in finite metrics under the analogous assumption as well. For @math -means, we in addition give a randomized algorithm with improved running time of @math . Our technique also obtains a PTAS under the assumption of that all @math approximations are @math -close to a desired target clustering, in the case that all target clusters have size greater than @math and @math is constant. Note that the motivation of is that for many clustering problems, the objective function is only a proxy for the true goal of getting close to the target. From this perspective, our improvement is that for @math -means in Euclidean spaces we reduce the distance of the clustering found to the target from @math to @math when all target clusters are large, and for @math -median we improve the largeness'' condition needed in the work of to get exactly @math -close from @math to @math . Our results are based on a new notion of clustering stability. We study Byzantine fault-tolerant distributed optimization of a sum of convex (cost) functions with real-valued scalar input ouput. In particular, the goal is to optimize a global cost function @math , where @math is the set of non-faulty agents, and @math is agent @math 's local cost function, which is initially known only to agent @math . In general, when some of the agents may be Byzantine faulty, the above goal is unachievable, because the identity of the faulty agents is not necessarily known to the non-faulty agents, and the faulty agents may behave arbitrarily. Since the above global cost function cannot be optimized exactly in presence of Byzantine agents, we define a weaker version of the problem. The goal for the weaker problem is to generate an output that is an optimum of a function formed as a convex combination of local cost functions of the non-faulty agents. More precisely, for some choice of weights @math for @math such that @math and @math , the output must be an optimum of the cost function @math . Ideally, we would like @math for all @math -- however, this cannot be guaranteed due to the presence of faulty agents. In fact, we show that the maximum achievable number of nonzero weights ( @math 's) is @math , where @math is the upper bound on the number of Byzantine agents. In addition, we present algorithms that ensure that at least @math agents have weights that are bounded away from 0. We also propose a low-complexity suboptimal algorithm, which ensures that at least @math agents have weights that are bounded away from 0, where @math is the total number of agents, and @math ( @math ) is the actual number of Byzantine agents.