Task-based End-to-end Model Learning in Stochastic Optimization

With the increasing popularity of machine learning techniques, it has become common to see prediction algorithms operating within some larger process. However, the criteria by which we train these algorithms often differ from the ultimate criteria on which we evaluate them. This paper proposes an end-to-end approach for learning probabilistic machine learning models in a manner that directly captures the ultimate task-based objective for which they will be used, within the context of stochastic programming. We present three experimental evaluations of the proposed approach: a classical inventory stock problem, a real-world electrical grid scheduling task, and a real-world energy storage arbitrage task. We show that the proposed approach can outperform both traditional modeling and purely black-box policy optimization approaches in these applications.Comment: In NIPS 2017. Code available at https://github.com/locuslab/e2e-model-learnin

Fast L1-Minimization Algorithm for Sparse Approximation Based on an Improved LPNN-LCA framework

The aim of sparse approximation is to estimate a sparse signal according to the measurement matrix and an observation vector. It is widely used in data analytics, image processing, and communication, etc. Up to now, a lot of research has been done in this area, and many off-the-shelf algorithms have been proposed. However, most of them cannot offer a real-time solution. To some extent, this shortcoming limits its application prospects. To address this issue, we devise a novel sparse approximation algorithm based on Lagrange programming neural network (LPNN), locally competitive algorithm (LCA), and projection theorem. LPNN and LCA are both analog neural network which can help us get a real-time solution. The non-differentiable objective function can be solved by the concept of LCA. Utilizing the projection theorem, we further modify the dynamics and proposed a new system with global asymptotic stability. Simulation results show that the proposed sparse approximation method has the real-time solutions with satisfactory MSEs

Solving the L1 regularized least square problem via a box-constrained smooth minimization

In this paper, an equivalent smooth minimization for the L1 regularized least square problem is proposed. The proposed problem is a convex box-constrained smooth minimization which allows applying fast optimization methods to find its solution. Further, it is investigated that the property "the dual of dual is primal" holds for the L1 regularized least square problem. A solver for the smooth problem is proposed, and its affinity to the proximal gradient is shown. Finally, the experiments on L1 and total variation regularized problems are performed, and the corresponding results are reported.Comment: 5 page

A Dual Approach to Scalable Verification of Deep Networks

This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks

Optimization under Uncertainty in the Era of Big Data and Deep Learning: When Machine Learning Meets Mathematical Programming

This paper reviews recent advances in the field of optimization under uncertainty via a modern data lens, highlights key research challenges and promise of data-driven optimization that organically integrates machine learning and mathematical programming for decision-making under uncertainty, and identifies potential research opportunities. A brief review of classical mathematical programming techniques for hedging against uncertainty is first presented, along with their wide spectrum of applications in Process Systems Engineering. A comprehensive review and classification of the relevant publications on data-driven distributionally robust optimization, data-driven chance constrained program, data-driven robust optimization, and data-driven scenario-based optimization is then presented. This paper also identifies fertile avenues for future research that focuses on a closed-loop data-driven optimization framework, which allows the feedback from mathematical programming to machine learning, as well as scenario-based optimization leveraging the power of deep learning techniques. Perspectives on online learning-based data-driven multistage optimization with a learning-while-optimizing scheme is presented

Machine learning approach to chance-constrained problems: An algorithm based on the stochastic gradient descent

We consider chance-constrained problems with discrete random distribution. We aim for problems with a large number of scenarios. We propose a novel method based on the stochastic gradient descent method which performs updates of the decision variable based only on considering a few scenarios. We modify it to handle the non-separable objective. Complexity analysis and a comparison with the standard (batch) gradient descent method is provided. We give three examples with non-convex data and show that our method provides a good solution fast even when the number of scenarios is large

Differentiable MPC for End-to-end Planning and Control

We present foundations for using Model Predictive Control (MPC) as a differentiable policy class for reinforcement learning in continuous state and action spaces. This provides one way of leveraging and combining the advantages of model-free and model-based approaches. Specifically, we differentiate through MPC by using the KKT conditions of the convex approximation at a fixed point of the controller. Using this strategy, we are able to learn the cost and dynamics of a controller via end-to-end learning. Our experiments focus on imitation learning in the pendulum and cartpole domains, where we learn the cost and dynamics terms of an MPC policy class. We show that our MPC policies are significantly more data-efficient than a generic neural network and that our method is superior to traditional system identification in a setting where the expert is unrealizable.Comment: NeurIPS 201

OptNet: Differentiable Optimization as a Layer in Neural Networks

This paper presents OptNet, a network architecture that integrates optimization problems (here, specifically in the form of quadratic programs) as individual layers in larger end-to-end trainable deep networks. These layers encode constraints and complex dependencies between the hidden states that traditional convolutional and fully-connected layers often cannot capture. In this paper, we explore the foundations for such an architecture: we show how techniques from sensitivity analysis, bilevel optimization, and implicit differentiation can be used to exactly differentiate through these layers and with respect to layer parameters; we develop a highly efficient solver for these layers that exploits fast GPU-based batch solves within a primal-dual interior point method, and which provides backpropagation gradients with virtually no additional cost on top of the solve; and we highlight the application of these approaches in several problems. In one notable example, we show that the method is capable of learning to play mini-Sudoku (4x4) given just input and output games, with no a priori information about the rules of the game; this highlights the ability of our architecture to learn hard constraints better than other neural architectures.Comment: ICML 201

The Ontology of Knowledge Based Optimization

Optimization has been becoming a central of studies in mathematic and has many areas with different applications. However, many themes of optimization came from different area have not ties closing to origin concepts. This paper is to address some variants of optimization problems using ontology in order to building basic of knowledge about optimization, and then using it to enhance strategy to achieve knowledge based optimization.Comment: 20 pages, Proceedings of International/National Seminar Matematika dan Terapan (SiManTap), FMIPA Universitas Sumatera Utara, Medan: 11-3

Optimization Methods for Supervised Machine Learning: From Linear Models to Deep Learning

The goal of this tutorial is to introduce key models, algorithms, and open questions related to the use of optimization methods for solving problems arising in machine learning. It is written with an INFORMS audience in mind, specifically those readers who are familiar with the basics of optimization algorithms, but less familiar with machine learning. We begin by deriving a formulation of a supervised learning problem and show how it leads to various optimization problems, depending on the context and underlying assumptions. We then discuss some of the distinctive features of these optimization problems, focusing on the examples of logistic regression and the training of deep neural networks. The latter half of the tutorial focuses on optimization algorithms, first for convex logistic regression, for which we discuss the use of first-order methods, the stochastic gradient method, variance reducing stochastic methods, and second-order methods. Finally, we discuss how these approaches can be employed to the training of deep neural networks, emphasizing the difficulties that arise from the complex, nonconvex structure of these models