Combining abstract, symbolic reasoning with continuous neural reasoning is a grand challenge of representation learning. As a step in this direction, we propose a new architecture, called neural equivalence networks, for the problem of learning continuous semantic representations of algebraic and logical expressions. These networks are trained to represent semantic equivalence, even of expressions that are syntactically very different. The challenge is that semantic representations must be computed in a syntax-directed manner, because semantics is compositional, but at the same time, small changes in syntax can lead to very large changes in semantics, which can be difficult for continuous neural architectures. We perform an exhaustive evaluation on the task of checking equivalence on a highly diverse class of symbolic algebraic and boolean expression types, showing that our model significantly outperforms existing architectures.

State-level minimum Bayes risk (sMBR) training has become the de facto standard for sequence-level training of speech recognition acoustic models. It has an elegant formulation using the expectation semiring, and gives large improvements in word error rate (WER) over models trained solely using cross-entropy (CE) or connectionist temporal classification (CTC). sMBR training optimizes the expected number of frames at which the reference and hypothesized acoustic states differ. It may be preferable to optimize the expected WER, but WER does not interact well with the expectation semiring, and previous approaches based on computing expected WER exactly involve expanding the lattices used during training. In this paper we show how to perform optimization of the expected WER by sampling paths from the lattices used during conventional sMBR training. The gradient of the expected WER is itself an expectation, and so may be approximated using Monte Carlo sampling. We show experimentally that optimizing WER during acoustic model training gives 5% relative improvement in WER over a well-tuned sMBR baseline on a 2-channel query recognition task (Google Home).

Generalized cross validation (GCV) is one of the most important approaches used to estimate parameters in the context of inverse problems and regularization techniques. A notable example is the determination of the smoothness parameter in splines. When the data are generated by a state space model, like in the spline case, efficient algorithms are available to evaluate the GCV score with complexity that scales linearly in the data set size. However, these methods are not amenable to on-line applications since they rely on forward and backward recursions. Hence, if the objective has been evaluated at time $t-1$ and new data arrive at time t, then O(t) operations are needed to update the GCV score. In this paper we instead show that the update cost is $O(1)$, thus paving the way to the on-line use of GCV. This result is obtained by deriving the novel GCV filter which extends the classical Kalman filter equations to efficiently propagate the GCV score over time. We also illustrate applications of the new filter in the context of state estimation and on-line regularized linear system identification.

We present a Distributionally Robust Optimization (DRO) approach to outlier detection in a linear regression setting, where the closeness of probability distributions is measured using the Wasserstein metric. Training samples contaminated with outliers skew the regression plane computed by least squares and thus impede outlier detection. Classical approaches, such as robust regression, remedy this problem by downweighting the contribution of atypical data points. In contrast, our Wasserstein DRO approach hedges against a family of distributions that are close to the empirical distribution. We show that the resulting formulation encompasses a class of models, which include the regularized Least Absolute Deviation (LAD) as a special case. We provide new insights into the regularization term and give guidance on the selection of the regularization coefficient from the standpoint of a confidence region. We establish two types of performance guarantees for the solution to our formulation under mild conditions. One is related to its out-of-sample behavior, and the other concerns the discrepancy between the estimated and true regression planes. Extensive numerical results demonstrate the superiority of our approach to both robust regression and the regularized LAD in terms of estimation accuracy and outlier detection rates.

The widespread use of machine learning to make consequential decisions about individual citizens (including in domains such as credit, employment, education and criminal sentencing [3, 4, 26, 29]) has been accompanied by increased reports of instances in which the algorithms and models employed can be unfair or discriminatory in a variety of ways [2, 30]. As a result, research on fairness in machine learning and statistics has seen rapid growth in recent years [1, 5-7, 9-11, 13, 14, 18-21, 25, 27], and several mathematical formulations have been proposed as metrics of (un)fairness for a number of different learning frameworks. While much of the attention to date has focused on (binary) classification settings, where standard fairness notions include equal false positive or negative rates across different populations, less attention has been paid to fairness in (linear and logistic) regression settings, where the target and/or predicted values are continuous, and the same value may not occur even twice in the training data. In this work, we introduce a rich family of fairness metrics for regression models that take the form of a fairness regularizer and apply them to the standard loss functions for linear and logistic regression. Since these loss functions and our fairness regularizer are convex, the combined objective functions obtained from our framework are also convex, and thus permit efficient optimization. Furthermore, our family of fairness metrics covers the spectrum from the type of group fairness that is common in classification formulations (where e.g.

Learning the structure of dependencies among multiple random variables is a problem of considerable theoretical and practical interest. In practice, score optimisation with multiple restarts provides a practical and surprisingly successful solution, yet the conditions under which this may be a well founded strategy are poorly understood. In this paper, we prove that the problem of identifying the structure of a Bayesian Network via regularised score optimisation can be recast, in expectation, as a submodular optimisation problem, thus guaranteeing optimality with high probability. This result both explains the practical success of optimisation heuristics, and suggests a way to improve on such algorithms by artificially simulating multiple data sets via a bootstrap procedure. We show on several synthetic data sets that the resulting algorithm yields better recovery performance than the state of the art, and illustrate in a real cancer genomic study how such an approach can lead to valuable practical insights.

In 2013, a man named Eric L. Loomis was sentenced for eluding police and driving a car without the owner's consent. When the judge weighed Loomis' sentence, he considered an array of evidence, including the results of an automated risk assessment tool called COMPAS. Loomis' COMPAS score indicated he was at a "high risk" of committing new crimes. Considering this prediction, the judge sentenced him to seven years. Loomis challenged his sentence, arguing it was unfair to use the data-driven score against him.

Explain what regularization is and why it is useful. What are the benefits and drawbacks of specific methods, such as ridge regression and LASSO? Explain what a local optimum is and why it is important in a specific context, such as k-means clustering. What are specific ways for determining if you have a local optimum problem? What can be done to avoid local optima?

The Expectation-Maximization (EM) algorithm is a widely used method for maximum likelihood estimation in models with latent variables. For estimating mixtures of Gaussians, its iteration can be viewed as a soft version of the k-means clustering algorithm. Despite its wide use and applications, there are essentially no known convergence guarantees for this method. We provide global convergence guarantees for mixtures of two Gaussians with known covariance matrices. We show that the population version of EM, where the algorithm is given access to infinitely many samples from the mixture, converges geometrically to the correct mean vectors, and provide simple, closed-form expressions for the convergence rate. As a simple illustration, we show that, in one dimension, ten steps of the EM algorithm initialized at infinity result in less than 1\% error estimation of the means. In the finite sample regime, we show that, under a random initialization, $\tilde{O}(d/\epsilon^2)$ samples suffice to compute the unknown vectors to within $\epsilon$ in Mahalanobis distance, where $d$ is the dimension. In particular, the error rate of the EM based estimator is $\tilde{O}\left(\sqrt{d \over n}\right)$ where $n$ is the number of samples, which is optimal up to logarithmic factors.

Large-scale Hierarchical Classification (HC) involves datasets consisting of thousands of classes and millions of training instances with high-dimensional features posing several big data challenges. Feature selection that aims to select the subset of discriminant features is an effective strategy to deal with large-scale HC problem. It speeds up the training process, reduces the prediction time and minimizes the memory requirements by compressing the total size of learned model weight vectors. Majority of the studies have also shown feature selection to be competent and successful in improving the classification accuracy by removing irrelevant features. In this work, we investigate various filter-based feature selection methods for dimensionality reduction to solve the large-scale HC problem. Our experimental evaluation on text and image datasets with varying distribution of features, classes and instances shows upto 3x order of speed-up on massive datasets and upto 45% less memory requirements for storing the weight vectors of learned model without any significant loss (improvement for some datasets) in the classification accuracy. Source Code: https://cs.gmu.edu/~mlbio/featureselection.