arXiv subjects

Katya Scheinberg

Publications and source records attributed to Katya Scheinberg.

At least 37 records · Page 2Linked to original sources

arXiv2019

A Novel Smoothed Loss and Penalty Function for Noncrossing Composite Quantile Estimation via Deep Neural Networks

Uncertainty analysis in the form of probabilistic forecasting can significantly improve decision making processes in the smart power grid when integrating renewable energy sources such as wind. Whereas point forecasting provides a single expected value, probabilistic forecasts provide more information in the form of quantiles, prediction intervals, or full predictive densities. Traditionally quantile regression is applied for such forecasting and recently quantile regression neural networks have become popular for weather and renewable energy forecasting. However, one major shortcoming of composite quantile estimation in neural networks is the quantile crossover problem. This paper analyzes the effectiveness of a novel smoothed loss and penalty function for neural network architectures to prevent the quantile crossover problem. Its efficacy is examined on the wind power forecasting problem. A numerical case study is conducted using publicly available wind data from the Global Energy Forecasting Competition 2014. Multiple quantiles are estimated to form 10\%, to 90\% prediction intervals which are evaluated using a quantile score and reliability measures. Benchmark models such as the persistence and climatology distributions, multiple quantile regression, and support vector quantile regression are used for comparison where results demonstrate the proposed approach leads to improved performance while preventing the problem of overlapping quantile estimates.

eess.SP↗

arXiv2019

Optimal Generalized Decision Trees via Integer Programming

Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if allowed to grow large, they lose interpretability. In this paper, we present a mixed integer programming formulation to construct optimal decision trees of a prespecified size. We take the special structure of categorical features into account and allow combinatorial decisions (based on subsets of values of features) at each node. Our approach can also handle numerical features via thresholding. We show that very good accuracy can be achieved with small trees using moderately-sized training sets. The optimization problems we solve are tractable with modern solvers.

cs.LG↗

arXiv2019

Linear interpolation gives better gradients than Gaussian smoothing in derivative-free optimization

In this paper, we consider derivative free optimization problems, where the objective function is smooth but is computed with some amount of noise, the function evaluations are expensive and no derivative information is available. We are motivated by policy optimization problems in reinforcement learning that have recently become popular [Choromaski et al. 2018; Fazel et al. 2018; Salimans et al. 2016], and that can be formulated as derivative free optimization problems with the aforementioned characteristics. In each of these works some approximation of the gradient is constructed and a (stochastic) gradient method is applied. In [Salimans et al. 2016] the gradient information is aggregated along Gaussian directions, while in [Choromaski et al. 2018] it is computed along orthogonal direction. We provide a convergence rate analysis for a first-order line search method, similar to the ones used in the literature, and derive the conditions on the gradient approximations that ensure this convergence. We then demonstrate via rigorous analysis of the variance and by numerical comparisons on reinforcement learning tasks that the Gaussian sampling method used in [Salimans et al. 2016] is significantly inferior to the orthogonal sampling used in [Choromaski et al. 2018] as well as more general interpolation methods.

math.OC↗

arXiv2019

Novel and Efficient Approximations for Zero-One Loss of Linear Classifiers

The predictive quality of machine learning models is typically measured in terms of their (approximate) expected prediction accuracy or the so-called Area Under the Curve (AUC). Minimizing the reciprocals of these measures are the goals of supervised learning. However, when the models are constructed by the means of empirical risk minimization (ERM), surrogate functions such as the logistic loss or hinge loss are optimized instead. In this work, we show that in the case of linear predictors, the expected error and the expected ranking loss can be effectively approximated by smooth functions whose closed form expressions and those of their first (and second) order derivatives depend on the first and second moments of the data distribution, which can be precomputed. Hence, the complexity of an optimization algorithm applied to these functions does not depend on the size of the training data. These approximation functions are derived under the assumption that the output of the linear classifier for a given data set has an approximately normal distribution. We argue that this assumption is significantly weaker than the Gaussian assumption on the data itself and we support this claim by demonstrating that our new approximation is quite accurate on data sets that are not necessarily Gaussian. We present computational results that show that our proposed approximations and related optimization algorithms can produce linear classifiers with similar or better test accuracy or AUC, than those obtained using state-of-the-art approaches, in a fraction of the time.

stat.ML↗

arXiv2018

When Does Stochastic Gradient Algorithm Work Well?

In this paper, we consider a general stochastic optimization problem which is often at the core of supervised learning, such as deep learning and linear classification. We consider a standard stochastic gradient descent (SGD) method with a fixed, large step size and propose a novel assumption on the objective function, under which this method has the improved convergence rates (to a neighborhood of the optimal solutions). We then empirically demonstrate that these assumptions hold for logistic regression and standard deep neural networks on classical data sets. Thus our analysis helps to explain when efficient behavior can be expected from the SGD method in training classification models and deep neural networks.

stat.ML↗

arXiv2018

Convergence Rate Analysis of a Stochastic Trust Region Method via Submartingales

We propose a novel framework for analyzing convergence rates of stochastic optimization algorithms with adaptive step sizes. This framework is based on analyzing properties of an underlying generic stochastic process, in particular by deriving a bound on the expected stopping time of this process. We utilize this framework to analyze the bounds on expected global convergence rates of a stochastic variant of a traditional trust region method, introduced in \cite{ChenMenickellyScheinberg2014}. While traditional trust region methods rely on exact computations of the gradient, Hessian and values of the objective function, this method assumes that these values are available up to some dynamically adjusted accuracy. Moreover, this accuracy is assumed to hold only with some sufficiently large, but fixed, probability, without any additional restrictions on the variance of the errors. This setting applies, for example, to standard stochastic optimization and machine learning formulations. Improving upon the analysis in \cite{ChenMenickellyScheinberg2014}, we show that the stochastic process defined by the algorithm satisfies the assumptions of our proposed general framework, with the stopping time defined as reaching accuracy $\|\nabla f(x)\|\leq ε$. The resulting bound for this stopping time is $O(ε^{-2})$, under the assumption of sufficiently accurate stochastic gradient, and is the first global complexity bound for a stochastic trust-region method. Finally, we apply the same framework to derive second order complexity bound under some additional assumptions.

math.OC↗

arXiv2018

A Stochastic Line Search Method with Convergence Rate Analysis

For deterministic optimization, line-search methods augment algorithms by providing stability and improved efficiency. We adapt a classical backtracking Armijo line-search to the stochastic optimization setting. While traditional line-search relies on exact computations of the gradient and values of the objective function, our method assumes that these values are available up to some dynamically adjusted accuracy which holds with some sufficiently large, but fixed, probability. We show the expected number of iterations to reach a near stationary point matches the worst-case efficiency of typical first-order methods, while for convex and strongly convex objective, it achieves rates of deterministic gradient descent in function values.

math.OC↗

arXiv2018

A Stochastic Trust Region Algorithm Based on Careful Step Normalization

An algorithm is proposed for solving stochastic and finite sum minimization problems. Based on a trust region methodology, the algorithm employs normalized steps, at least as long as the norms of the stochastic gradient estimates are within a specified interval. The complete algorithm---which dynamically chooses whether or not to employ normalized steps---is proved to have convergence guarantees that are similar to those possessed by a traditional stochastic gradient approach under various sets of conditions related to the accuracy of the stochastic gradient estimates and choice of stepsize sequence. The results of numerical experiments are presented when the method is employed to minimize convex and nonconvex machine learning test problems. These results illustrate that the method can outperform a traditional stochastic gradient approach.

math.OC↗

arXiv2018

SGD and Hogwild! Convergence Without the Bounded Gradients Assumption

Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime, which results in more relaxed conditions than those in (Bottou et al.,2016). We then move on the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate.

math.OC↗

arXiv2018

An Empirical Analysis of Constrained Support Vector Quantile Regression for Nonparametric Probabilistic Forecasting of Wind Power

Uncertainty analysis in the form of probabilistic forecasting can provide significant improvements in decision-making processes in the smart power grid for better integrating renewable energies such as wind. Whereas point forecasting provides a single expected value, probabilistic forecasts provide more information in the form of quantiles, prediction intervals, or full predictive densities. This paper analyzes the effectiveness of an approach for nonparametric probabilistic forecasting of wind power that combines support vector machines and nonlinear quantile regression with non-crossing constraints. A numerical case study is conducted using publicly available wind data from the Global Energy Forecasting Competition 2014. Multiple quantiles are estimated to form 20%, 40%, 60% and 80% prediction intervals which are evaluated using the pinball loss function and reliability measures. Three benchmark models are used for comparison where results demonstrate the proposed approach leads to significantly better performance while preventing the problem of overlapping quantile estimates.

stat.ML↗

arXiv2018

Directly and Efficiently Optimizing Prediction Error and AUC of Linear Classifiers

The predictive quality of machine learning models is typically measured in terms of their (approximate) expected prediction error or the so-called Area Under the Curve (AUC) for a particular data distribution. However, when the models are constructed by the means of empirical risk minimization, surrogate functions such as the logistic loss are optimized instead. This is done because the empirical approximations of the expected error and AUC functions are nonconvex and nonsmooth, and more importantly have zero derivative almost everywhere. In this work, we show that in the case of linear predictors, and under the assumption that the data has normal distribution, the expected error and the expected AUC are not only smooth, but have closed form expressions, which depend on the first and second moments of the normal distribution. Hence, we derive derivatives of these two functions and use these derivatives in an optimization algorithm to directly optimize the expected error and the AUC. In the case of real data sets, the derivatives can be approximated using empirical moments. We show that even when data is not normally distributed, computed derivatives are sufficiently useful to render an efficient optimization method and high quality solutions. Thus, we propose a gradient-based optimization method for direct optimization of the prediction error and AUC. Moreover, the per-iteration complexity of the proposed algorithm has no dependence on the size of the data set, unlike those for optimizing logistic regression and all other well known empirical risk minimization problems.

cs.LG↗

arXiv2017

Proximal Quasi-Newton Methods for Regularized Convex Optimization with Linear and Accelerated Sublinear Convergence Rates

In [19], a general, inexact, efficient proximal quasi-Newton algorithm for composite optimization problems has been proposed and a sublinear global convergence rate has been established. In this paper, we analyze the convergence properties of this method, both in the exact and inexact setting, in the case when the objective function is strongly convex. We also investigate a practical variant of this method by establishing a simple stopping criterion for the subproblem optimization. Furthermore, we consider an accelerated variant, based on FISTA [1], to the proximal quasi-Newton algorithm. A similar accelerated method has been considered in [7], where the convergence rate analysis relies on very strong impractical assumptions. We present a modified analysis while relaxing these assumptions and perform a practical comparison of the accelerated proximal quasi- Newton algorithm and the regular one. Our analysis and computational results show that acceleration may not bring any benefit in the quasi-Newton setting.

math.NA↗

arXiv2017

Smooth Pinball Neural Network for Probabilistic Forecasting of Wind Power

Uncertainty analysis in the form of probabilistic forecasting can significantly improve decision making processes in the smart power grid for better integrating renewable energy sources such as wind. Whereas point forecasting provides a single expected value, probabilistic forecasts provide more information in the form of quantiles, prediction intervals, or full predictive densities. This paper analyzes the effectiveness of a novel approach for nonparametric probabilistic forecasting of wind power that combines a smooth approximation of the pinball loss function with a neural network architecture and a weighting initialization scheme to prevent the quantile cross over problem. A numerical case study is conducted using publicly available wind data from the Global Energy Forecasting Competition 2014. Multiple quantiles are estimated to form 10%, to 90% prediction intervals which are evaluated using a quantile score and reliability measures. Benchmark models such as the persistence and climatology distributions, multiple quantile regression, and support vector quantile regression are used for comparison where results demonstrate the proposed approach leads to improved performance while preventing the problem of overlapping quantile estimates.

stat.ML↗

arXiv2017

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.

stat.ML↗

arXiv2017

SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient

In this paper, we propose a StochAstic Recursive grAdient algoritHm (SARAH), as well as its practical variant SARAH+, as a novel approach to the finite-sum minimization problems. Different from the vanilla SGD and other modern stochastic methods such as SVRG, S2GD, SAG and SAGA, SARAH admits a simple recursive framework for updating stochastic gradient estimates; when comparing to SAG/SAGA, SARAH does not require a storage of past gradients. The linear convergence rate of SARAH is proven under strong convexity assumption. We also prove a linear convergence rate (in the strongly convex case) for an inner loop of SARAH, the property that SVRG does not possess. Numerical experiments demonstrate the efficiency of our algorithm.

stat.ML↗

arXiv2017

Stochastic Recursive Gradient Algorithm for Nonconvex Optimization

In this paper, we study and analyze the mini-batch version of StochAstic Recursive grAdient algoritHm (SARAH), a method employing the stochastic recursive gradient, for solving empirical loss minimization for the case of nonconvex losses. We provide a sublinear convergence rate (to stationary points) for general nonconvex functions and a linear convergence rate for gradient dominated functions, both of which have some advantages compared to other modern stochastic gradient algorithms for nonconvex losses.

stat.ML↗

arXiv2017

Black-Box Optimization in Machine Learning with Trust Region Based Derivative Free Algorithm

In this work, we utilize a Trust Region based Derivative Free Optimization (DFO-TR) method to directly maximize the Area Under Receiver Operating Characteristic Curve (AUC), which is a nonsmooth, noisy function. We show that AUC is a smooth function, in expectation, if the distributions of the positive and negative data points obey a jointly normal distribution. The practical performance of this algorithm is compared to three prominent Bayesian optimization methods and random search. The presented numerical results show that DFO-TR surpasses Bayesian optimization and random search on various black-box optimization problem, such as maximizing AUC and hyperparameter tuning.

cs.LG↗

arXiv2017

Global convergence rate analysis of unconstrained optimization methods based on probabilistic models

We present global convergence rates for a line-search method which is based on random first-order models and directions whose quality is ensured only with certain probability. We show that in terms of the order of the accuracy, the evaluation complexity of such a method is the same as its counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. We particularize and improve these results in the convex and strongly convex case. We also analyze a probabilistic cubic regularization variant that allows approximate probabilistic second-order models and show improved complexity bounds compared to probabilistic first-order methods; again, as a function of the accuracy, the probabilistic cubic regularization bounds are of the same (optimal) order as for the deterministic case.

math.OC↗