On the Differentiability of the Solution to Convex Optimization Problems

In this paper, we provide conditions under which one can take derivatives of the solution to convex optimization problems with respect to problem data. These conditions are (roughly) that Slater's condition holds, the functions involved are twice differentiable, and that a certain Jacobian matrix is non-singular. The derivation involves applying the implicit function theorem to the necessary and sufficient KKT system for optimality

InterpNET: Neural Introspection for Interpretable Deep Learning

Humans are able to explain their reasoning. On the contrary, deep neural networks are not. This paper attempts to bridge this gap by introducing a new way to design interpretable neural networks for classification, inspired by physiological evidence of the human visual system's inner-workings. This paper proposes a neural network design paradigm, termed InterpNET, which can be combined with any existing classification architecture to generate natural language explanations of the classifications. The success of the module relies on the assumption that the network's computation and reasoning is represented in its internal layer activations. While in principle InterpNET could be applied to any existing classification architecture, it is evaluated via an image classification and explanation task. Experiments on a CUB bird classification and explanation dataset show qualitatively and quantitatively that the model is able to generate high-quality explanations. While the current state-of-the-art METEOR score on this dataset is 29.2, InterpNET achieves a much higher METEOR score of 37.9.Comment: Presented at NIPS 2017 Symposium on Interpretable Machine Learnin

Active Robotic Mapping through Deep Reinforcement Learning

We propose an approach to learning agents for active robotic mapping, where the goal is to map the environment as quickly as possible. The agent learns to map efficiently in simulated environments by receiving rewards corresponding to how fast it constructs an accurate map. In contrast to prior work, this approach learns an exploration policy based on a user-specified prior over environment configurations and sensor model, allowing it to specialize to the specifications. We evaluate the approach through a simulated Disaster Mapping scenario and find that it achieves performance slightly better than a near-optimal myopic exploration scheme, suggesting that it could be useful in more complicated problem scenarios

A Matrix Gaussian Distribution

In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian distribution. We also derive useful results regarding the expected value of certain quadratic forms based solely on covariances between matrices. Previous definitions of matrix normal distributions are severely under-parameterized, assuming unrealistic structure on the covariance (see Section 2). We believe that our generalization is better equipped for use in practice.Comment: 4 page

Optimizing for Generalization in Machine Learning with Cross-Validation Gradients

Cross-validation is the workhorse of modern applied statistics and machine learning, as it provides a principled framework for selecting the model that maximizes generalization performance. In this paper, we show that the cross-validation risk is differentiable with respect to the hyperparameters and training data for many common machine learning algorithms, including logistic regression, elastic-net regression, and support vector machines. Leveraging this property of differentiability, we propose a cross-validation gradient method (CVGM) for hyperparameter optimization. Our method enables efficient optimization in high-dimensional hyperparameter spaces of the cross-validation risk, the best surrogate of the true generalization ability of our learning algorithm.Comment: 11 page

Stochastic Control with Affine Dynamics and Extended Quadratic Costs

An extended quadratic function is a quadratic function plus the indicator function of an affine set, that is, a quadratic function with embedded linear equality constraints. We show that, under some technical conditions, random convex extended quadratic functions are closed under addition, composition with an affine function, expectation, and partial minimization, that is, minimizing over some of its arguments. These properties imply that dynamic programming can be tractably carried out for stochastic control problems with random affine dynamics and extended quadratic cost functions. While the equations for the dynamic programming iterations are much more complicated than for traditional linear quadratic control, they are well suited to an object-oriented implementation, which we describe. We also describe a number of known and new applications.Comment: 46 pages, 16 figure

A Distributed Method for Fitting Laplacian Regularized Stratified Models

Stratified models are models that depend in an arbitrary way on a set of selected categorical features, and depend linearly on the other features. In a basic and traditional formulation a separate model is fit for each value of the categorical feature, using only the data that has the specific categorical value. To this formulation we add Laplacian regularization, which encourages the model parameters for neighboring categorical values to be similar. Laplacian regularization allows us to specify one or more weighted graphs on the stratification feature values. For example, stratifying over the days of the week, we can specify that the Sunday model parameter should be close to the Saturday and Monday model parameters. The regularization improves the performance of the model over the traditional stratified model, since the model for each value of the categorical `borrows strength' from its neighbors. In particular, it produces a model even for categorical values that did not appear in the training data set. We propose an efficient distributed method for fitting stratified models, based on the alternating direction method of multipliers (ADMM). When the fitting loss functions are convex, the stratified model fitting problem is convex, and our method computes the global minimizer of the loss plus regularization; in other cases it computes a local minimizer. The method is very efficient, and naturally scales to large data sets or numbers of stratified feature values. We illustrate our method with a variety of examples.Comment: 37 pages, 6 figure

Learning Probabilistic Trajectory Models of Aircraft in Terminal Airspace from Position Data

Models for predicting aircraft motion are an important component of modern aeronautical systems. These models help aircraft plan collision avoidance maneuvers and help conduct offline performance and safety analyses. In this article, we develop a method for learning a probabilistic generative model of aircraft motion in terminal airspace, the controlled airspace surrounding a given airport. The method fits the model based on a historical dataset of radar-based position measurements of aircraft landings and takeoffs at that airport. We find that the model generates realistic trajectories, provides accurate predictions, and captures the statistical properties of aircraft trajectories. Furthermore, the model trains quickly, is compact, and allows for efficient real-time inference.Comment: IEEE Transactions on Intelligent Transportation System

Optimal Representative Sample Weighting

We consider the problem of assigning weights to a set of samples or data records, with the goal of achieving a representative weighting, which happens when certain sample averages of the data are close to prescribed values. We frame the problem of finding representative sample weights as an optimization problem, which in many cases is convex and can be efficiently solved. Our formulation includes as a special case the selection of a fixed number of the samples, with equal weights, i.e., the problem of selecting a smaller representative subset of the samples. While this problem is combinatorial and not convex, heuristic methods based on convex optimization seem to perform very well. We describe rsw, an open-source implementation of the ideas described in this paper, and apply it to a skewed sample of the CDC BRFSS dataset

Multi-Period Liability Clearing via Convex Optimal Control

We consider the problem of determining a sequence of payments among a set of entities that clear (if possible) the liabilities among them. We formulate this as an optimal control problem, which is convex when the objective function is, and therefore readily solved. For this optimal control problem, we give a number of useful and interesting convex costs and constraints that can be combined in any way for different applications. We describe a number of extensions, for example to handle unknown changes in cash and liabilities, to allow bailouts, to find the minimum time to clear the liabilities, or to minimize the number of non-cleared liabilities, when fully clearing the liabilities is impossible