arXiv subjects

Yang Kang

Publications and source records attributed to Yang Kang.

13 recordsLinked to original sources

arXiv2020

Robust Wasserstein Profile Inference and Applications to Machine Learning

We show that several machine learning estimators, including square-root LASSO (Least Absolute Shrinkage and Selection) and regularized logistic regression can be represented as solutions to distributionally robust optimization (DRO) problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (Robust Wasserstein Profile Inference), a novel inference methodology which extends the use of methods inspired by Empirical Likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence, we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings.

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arXiv2020

Machine Learning's Dropout Training is Distributionally Robust Optimal

This paper shows that dropout training in Generalized Linear Models is the minimax solution of a two-player, zero-sum game where an adversarial nature corrupts a statistician's covariates using a multiplicative nonparametric errors-in-variables model. In this game, nature's least favorable distribution is dropout noise, where nature independently deletes entries of the covariate vector with some fixed probability $\delta$. This result implies that dropout training indeed provides out-of-sample expected loss guarantees for distributions that arise from multiplicative perturbations of in-sample data. In addition to the decision-theoretic analysis, the paper makes two more contributions. First, there is a concrete recommendation on how to select the tuning parameter $\delta$ to guarantee that, as the sample size grows large, the in-sample loss after dropout training exceeds the true population loss with some pre-specified probability. Second, the paper provides a novel, parallelizable, Unbiased Multi-Level Monte Carlo algorithm to speed-up the implementation of dropout training. Our algorithm has a much smaller computational cost compared to the naive implementation of dropout, provided the number of data points is much smaller than the dimension of the covariate vector.

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arXiv2020

Sample Out-Of-Sample Inference Based on Wasserstein Distance

We present a novel inference approach that we call Sample Out-of-Sample (or SOS) inference. The approach can be used widely, ranging from semi-supervised learning to stress testing, and it is fundamental in the application of data-driven Distributionally Robust Optimization (DRO). Our method enables measuring the impact of plausible out-of-sample scenarios in a given performance measure of interest, such as a financial loss. The methodology is inspired by Empirical Likelihood (EL), but we optimize the empirical Wasserstein distance (instead of the empirical likelihood) induced by observations. From a methodological standpoint, our analysis of the asymptotic behavior of the induced Wasserstein-distance profile function shows dramatic qualitative differences relative to EL. For instance, in contrast to EL, which typically yields chi-squared weak convergence limits, our asymptotic distributions are often not chi-squared. Also, the rates of convergence that we obtain have some dependence on the dimension in a non-trivial way but remain controlled as the dimension increases.

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arXiv2019

A Distributionally Robust Boosting Algorithm

Distributionally Robust Optimization (DRO) has been shown to provide a flexible framework for decision making under uncertainty and statistical estimation. For example, recent works in DRO have shown that popular statistical estimators can be interpreted as the solutions of suitable formulated data-driven DRO problems. In turn, this connection is used to optimally select tuning parameters in terms of a principled approach informed by robustness considerations. This paper contributes to this growing literature, connecting DRO and statistics, by showing how boosting algorithms can be studied via DRO. We propose a boosting type algorithm, named DRO-Boosting, as a procedure to solve our DRO formulation. Our DRO-Boosting algorithm recovers Adaptive Boosting (AdaBoost) in particular, thus showing that AdaBoost is effectively solving a DRO problem. We apply our algorithm to a financial dataset on credit card default payment prediction. We find that our approach compares favorably to alternative boosting methods which are widely used in practice.

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arXiv2019

Data-driven Optimal Cost Selection for Distributionally Robust Optimization

Recently, (Blanchet, Kang, and Murhy 2016, and Blanchet, and Kang 2017) showed that several machine learning algorithms, such as square-root Lasso, Support Vector Machines, and regularized logistic regression, among many others, can be represented exactly as distributionally robust optimization (DRO) problems. The distributional uncertainty is defined as a neighborhood centered at the empirical distribution. We propose a methodology which learns such neighborhood in a natural data-driven way. We show rigorously that our framework encompasses adaptive regularization as a particular case. Moreover, we demonstrate empirically that our proposed methodology is able to improve upon a wide range of popular machine learning estimators.

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arXiv2019

Semi-supervised Learning based on Distributionally Robust Optimization

We propose a novel method for semi-supervised learning (SSL) based on data-driven distributionally robust optimization (DRO) using optimal transport metrics. Our proposed method enhances generalization error by using the unlabeled data to restrict the support of the worst case distribution in our DRO formulation. We enable the implementation of our DRO formulation by proposing a stochastic gradient descent algorithm which allows to easily implement the training procedure. We demonstrate that our Semi-supervised DRO method is able to improve the generalization error over natural supervised procedures and state-of-the-art SSL estimators. Finally, we include a discussion on the large sample behavior of the optimal uncertainty region in the DRO formulation. Our discussion exposes important aspects such as the role of dimension reduction in SSL.

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arXiv2018

Latent Semantic Analysis Approach for Document Summarization Based on Word Embeddings

Since the amount of information on the internet is growing rapidly, it is not easy for a user to find relevant information for his/her query. To tackle this issue, much attention has been paid to Automatic Document Summarization. The key point in any successful document summarizer is a good document representation. The traditional approaches based on word overlapping mostly fail to produce that kind of representation. Word embedding, distributed representation of words, has shown an excellent performance that allows words to match on semantic level. Naively concatenating word embeddings makes the common word dominant which in turn diminish the representation quality. In this paper, we employ word embeddings to improve the weighting schemes for calculating the input matrix of Latent Semantic Analysis method. Two embedding-based weighting schemes are proposed and then combined to calculate the values of this matrix. The new weighting schemes are modified versions of the augment weight and the entropy frequency. The new schemes combine the strength of the traditional weighting schemes and word embedding. The proposed approach is experimentally evaluated on three well-known English datasets, DUC 2002, DUC 2004 and Multilingual 2015 Single-document Summarization for English. The proposed model performs comprehensively better compared to the state-of-the-art methods, by at least 1% ROUGE points, leading to a conclusion that it provides a better document representation and a better document summary as a result.

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arXiv2018

Bidirectional Attentional Encoder-Decoder Model and Bidirectional Beam Search for Abstractive Summarization

Sequence generative models with RNN variants, such as LSTM, GRU, show promising performance on abstractive document summarization. However, they still have some issues that limit their performance, especially while deal-ing with long sequences. One of the issues is that, to the best of our knowledge, all current models employ a unidirectional decoder, which reasons only about the past and still limited to retain future context while giving a prediction. This makes these models suffer on their own by generating unbalanced outputs. Moreover, unidirec-tional attention-based document summarization can only capture partial aspects of attentional regularities due to the inherited challenges in document summarization. To this end, we propose an end-to-end trainable bidirectional RNN model to tackle the aforementioned issues. The model has a bidirectional encoder-decoder architecture; in which the encoder and the decoder are bidirectional LSTMs. The forward decoder is initialized with the last hidden state of the backward encoder while the backward decoder is initialized with the last hidden state of the for-ward encoder. In addition, a bidirectional beam search mechanism is proposed as an approximate inference algo-rithm for generating the output summaries from the bidi-rectional model. This enables the model to reason about the past and future and to generate balanced outputs as a result. Experimental results on CNN / Daily Mail dataset show that the proposed model outperforms the current abstractive state-of-the-art models by a considerable mar-gin.

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arXiv2017

Doubly Robust Data-Driven Distributionally Robust Optimization

Data-driven Distributionally Robust Optimization (DD-DRO) via optimal transport has been shown to encompass a wide range of popular machine learning algorithms. The distributional uncertainty size is often shown to correspond to the regularization parameter. The type of regularization (e.g. the norm used to regularize) corresponds to the shape of the distributional uncertainty. We propose a data-driven robust optimization methodology to inform the transportation cost underlying the definition of the distributional uncertainty. We show empirically that this additional layer of robustification, which produces a method we called doubly robust data-driven distributionally robust optimization (DD-R-DRO), allows to enhance the generalization properties of regularized estimators while reducing testing error relative to state-of-the-art classifiers in a wide range of data sets.

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arXiv2017

Distributionally Robust Groupwise Regularization Estimator

Regularized estimators in the context of group variables have been applied successfully in model and feature selection in order to preserve interpretability. We formulate a Distributionally Robust Optimization (DRO) problem which recovers popular estimators, such as Group Square Root Lasso (GSRL). Our DRO formulation allows us to interpret GSRL as a game, in which we learn a regression parameter while an adversary chooses a perturbation of the data. We wish to pick the parameter to minimize the expected loss under any plausible model chosen by the adversary - who, on the other hand, wishes to increase the expected loss. The regularization parameter turns out to be precisely determined by the amount of perturbation on the training data allowed by the adversary. In this paper, we introduce a data-driven (statistical) criterion for the optimal choice of regularization, which we evaluate asymptotically, in closed form, as the size of the training set increases. Our easy-to-evaluate regularization formula is compared against cross-validation, showing good (sometimes superior) performance.

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arXiv2010

Diffusive propagation of wave packets in a fluctuating periodic potential

We consider the evolution of a tight binding wave packet propagating in a fluctuating periodic potential. If the fluctuations stem from a stationary Markov process satisfying certain technical criteria, we show that the square amplitude of the wave packet after diffusive rescaling converges to a superposition of solutions of a heat equation.

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arXiv2009

The Liouville equation for singular ergodic magnetic Schrödinger operators

We study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schrödinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation, and solve a Liouville equation in an appropriate Hilbert space.

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arXiv2009

Diffusion of wave packets in a Markov random potential

We consider the evolution of a tight binding wave packet propagating in a time dependent potential. If the potential evolves according to a stationary Markov process, we show that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.

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