The problem of estimating the divergence between 2 high dimensional distributions with limited samples is an important problem in various fields such as machine learning. Although previous methods perform well with moderate dimensional data, their accuracy starts to degrade in situations with 100s of binary variables. Therefore, we propose the use of decomposable models for estimating divergences in high dimensional data. These allow us to factorize the estimated density of the high-dimensional distribution into a product of lower dimensional functions. We conduct formal and experimental analyses to explore the properties of using decomposable models in the context of divergence estimation. To this end, we show empirically that estimating the Kullback-Leibler divergence using decomposable models from a maximum likelihood estimator outperforms existing methods for divergence estimation in situations where dimensionality is high and useful decomposable models can be learnt from the available data.
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Learning data representations under uncertainty is an important task that emerges in numerous machine learning applications. However, uncertainty quantification (UQ) techniques are computationally intensive and become prohibitively expensive for high-dimensional data. In this paper, we present a novel surrogate model for representation learning and uncertainty quantification, which aims to deal with data of moderate to high dimensions. The proposed model combines a neural network approach for dimensionality reduction of the (potentially high-dimensional) data, with a surrogate model method for learning the data distribution. We first employ a variational autoencoder (VAE) to learn a low-dimensional representation of the data distribution. We then propose to harness polynomial chaos expansion (PCE) formulation to map this distribution to the output target. The coefficients of PCE are learned from the distribution representation of the training data using a maximum mean discrepancy (MMD) approach. Our model enables us to (a) learn a representation of the data, (b) estimate uncertainty in the high-dimensional data system, and (c) match high order moments of the output distribution; without any prior statistical assumptions on the data. Numerical experimental results are presented to illustrate the performance of the proposed method.
The Wasserstein distance is a distance between two probability distributions and has recently gained increasing popularity in statistics and machine learning, owing to its attractive properties. One important approach to extending this distance is using low-dimensional projections of distributions to avoid a high computational cost and the curse of dimensionality in empirical estimation, such as the sliced Wasserstein or max-sliced Wasserstein distances. Despite their practical success in machine learning tasks, the availability of statistical inferences for projection-based Wasserstein distances is limited owing to the lack of distributional limit results. In this paper, we consider distances defined by integrating or maximizing Wasserstein distances between low-dimensional projections of two probability distributions. Then we derive limit distributions regarding these distances when the two distributions are supported on finite points. We also propose a bootstrap procedure to estimate quantiles of limit distributions from data. This facilitates asymptotically exact interval estimation and hypothesis testing for these distances. Our theoretical results are based on the arguments of Sommerfeld and Munk (2018) for deriving distributional limits regarding the original Wasserstein distance on finite spaces and the theory of sensitivity analysis in nonlinear programming. Finally, we conduct numerical experiments to illustrate the theoretical results and demonstrate the applicability of our inferential methods to real data analysis.
Many modern applications require detecting change points in complex sequential data. Most existing methods for change point detection are unsupervised and, as a consequence, lack any information regarding what kind of changes we want to detect or if some kinds of changes are safe to ignore. This often results in poor change detection performance. We present a novel change point detection framework that uses true change point instances as supervision for learning a ground metric such that Sinkhorn divergences can be then used in two-sample tests on sliding windows to detect change points in an online manner. Our method can be used to learn a sparse metric which can be useful for both feature selection and interpretation in high-dimensional change point detection settings. Experiments on simulated as well as real world sequences show that our proposed method can substantially improve change point detection performance over existing unsupervised change point detection methods using only few labeled change point instances.
A central problem in business concerns the optimal allocation of limited resources to a set of available tasks, where the payoff of these tasks is inherently uncertain. In credit card fraud detection, for instance, a bank can only assign a small subset of transactions to their fraud investigations team. Typically, such problems are solved using a classification framework, where the focus is on predicting task outcomes given a set of characteristics. Resources are then allocated to the tasks that are predicted to be the most likely to succeed. However, we argue that using classification to address task uncertainty is inherently suboptimal as it does not take into account the available capacity. Therefore, we first frame the problem as a type of assignment problem. Then, we present a novel solution using learning to rank by directly optimizing the assignment's expected profit given limited, stochastic capacity. This is achieved by optimizing a specific instance of the net discounted cumulative gain, a commonly used class of metrics in learning to rank. Empirically, we demonstrate that our new method achieves higher expected profit and expected precision compared to a classification approach for a wide variety of application areas and data sets. This illustrates the benefit of an integrated approach and of explicitly considering the available resources when learning a predictive model.
This paper builds upon the work of Pfau (2013), which generalized the bias variance tradeoff to any Bregman divergence loss function. Pfau (2013) showed that for Bregman divergences, the bias and variances are defined with respect to a central label, defined as the expected mean of the label, and a central prediction, of a more complex form. We show that, similarly to the label, the central prediction can be interpreted as the mean of a random variable, where the mean operates in a dual space defined by the loss function itself. Viewing the bias-variance tradeoff through operations taken in dual space, we subsequently derive several results of interest. In particular, (a) the variance terms satisfy a generalized law of total variance; (b) if a source of randomness cannot be controlled, its contribution to the bias and variance has a closed form; (c) there exist natural ensembling operations in the label and prediction spaces which reduce the variance and do not affect the bias.
This paper serves as a survey of recent advances in large margin training and its theoretical foundations, mostly for (nonlinear) deep neural networks (DNNs) that are probably the most prominent machine learning models for large-scale data in the community over the past decade. We generalize the formulation of classification margins from classical research to latest DNNs, summarize theoretical connections between the margin, network generalization, and robustness, and introduce recent efforts in enlarging the margins for DNNs comprehensively. Since the viewpoint of different methods is discrepant, we categorize them into groups for ease of comparison and discussion in the paper. Hopefully, our discussions and overview inspire new research work in the community that aim to improve the performance of DNNs, and we also point to directions where the large margin principle can be verified to provide theoretical evidence why certain regularizations for DNNs function well in practice. We managed to shorten the paper such that the crucial spirit of large margin learning and related methods are better emphasized.
This paper proposes a recommender system to alleviate the cold-start problem that can estimate user preferences based on only a small number of items. To identify a user's preference in the cold state, existing recommender systems, such as Netflix, initially provide items to a user; we call those items evidence candidates. Recommendations are then made based on the items selected by the user. Previous recommendation studies have two limitations: (1) the users who consumed a few items have poor recommendations and (2) inadequate evidence candidates are used to identify user preferences. We propose a meta-learning-based recommender system called MeLU to overcome these two limitations. From meta-learning, which can rapidly adopt new task with a few examples, MeLU can estimate new user's preferences with a few consumed items. In addition, we provide an evidence candidate selection strategy that determines distinguishing items for customized preference estimation. We validate MeLU with two benchmark datasets, and the proposed model reduces at least 5.92% mean absolute error than two comparative models on the datasets. We also conduct a user study experiment to verify the evidence selection strategy.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
While Generative Adversarial Networks (GANs) have empirically produced impressive results on learning complex real-world distributions, recent work has shown that they suffer from lack of diversity or mode collapse. The theoretical work of Arora et al.~\cite{AroraGeLiMaZh17} suggests a dilemma about GANs' statistical properties: powerful discriminators cause overfitting, whereas weak discriminators cannot detect mode collapse. In contrast, we show in this paper that GANs can in principle learn distributions in Wasserstein distance (or KL-divergence in many cases) with polynomial sample complexity, if the discriminator class has strong distinguishing power against the particular generator class (instead of against all possible generators). For various generator classes such as mixture of Gaussians, exponential families, and invertible neural networks generators, we design corresponding discriminators (which are often neural nets of specific architectures) such that the Integral Probability Metric (IPM) induced by the discriminators can provably approximate the Wasserstein distance and/or KL-divergence. This implies that if the training is successful, then the learned distribution is close to the true distribution in Wasserstein distance or KL divergence, and thus cannot drop modes. Our preliminary experiments show that on synthetic datasets the test IPM is well correlated with KL divergence, indicating that the lack of diversity may be caused by the sub-optimality in optimization instead of statistical inefficiency.
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.
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