We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n x n, n >= 3, positive-definite symmetric matrix of random variables and f(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W_{11}-W_{12}W_{22}^{-1}W_{12}' is independent of {W_{12}, W_{22}} for every block partitioning W_{11}, W_{12}, W_{12}', W_{22} of W. Similar characterizations of the normal and normal-Wishart distributions are provided as well. We also show how to construct a prior for every DAG model over X from the prior of a single regression model.

We describe two techniques that significantly improve the running time of several standard machine-learning algorithms when data is sparse. The first technique is an algorithm that efficiently extracts one-way and two-way counts--either real or expected-- from discrete data. Extracting such counts is a fundamental step in learning algorithms for constructing a variety of models including decision trees, decision graphs, Bayesian networks, and naive-Bayes clustering models. The second technique is an algorithm that efficiently performs the E-step of the EM algorithm (i.e., inference) when applied to a naive-Bayes clustering model. Using realworld data sets, we demonstrate a dramatic decrease in running time for algorithms that incorporate these techniques.

The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions, such as when there is some sparsity pattern of the optimal parameter. This work characterizes a certain strong convexity property of general exponential families, which allow their generalization ability to be quantified. In particular, we show how this property can be used to analyze generic exponential families under L_1 regularization.

Interesting theoretical associations have been established by recent papers between the fields of active learning and stochastic convex optimization due to the common role of feedback in sequential querying mechanisms. In this paper, we continue this thread in two parts by exploiting these relations for the first time to yield novel algorithms in both fields, further motivating the study of their intersection. First, inspired by a recent optimization algorithm that was adaptive to unknown uniform convexity parameters, we present a new active learning algorithm for one-dimensional thresholds that can yield minimax rates by adapting to unknown noise parameters. Next, we show that one can perform $d$-dimensional stochastic minimization of smooth uniformly convex functions when only granted oracle access to noisy gradient signs along any coordinate instead of real-valued gradients, by using a simple randomized coordinate descent procedure where each line search can be solved by $1$-dimensional active learning, provably achieving the same error convergence rate as having the entire real-valued gradient. Combining these two parts yields an algorithm that solves stochastic convex optimization of uniformly convex and smooth functions using only noisy gradient signs by repeatedly performing active learning, achieves optimal rates and is adaptive to all unknown convexity and smoothness parameters.

In active learning, the user sequentially chooses values for feature $X$ and an oracle returns the corresponding label $Y$. In this paper, we consider the effect of feature noise in active learning, which could arise either because $X$ itself is being measured, or it is corrupted in transmission to the oracle, or the oracle returns the label of a noisy version of the query point. In statistics, feature noise is known as "errors in variables" and has been studied extensively in non-active settings. However, the effect of feature noise in active learning has not been studied before. We consider the well-known Berkson errors-in-variables model with additive uniform noise of width $\sigma$. Our simple but revealing setting is that of one-dimensional binary classification setting where the goal is to learn a threshold (point where the probability of a $+$ label crosses half). We deal with regression functions that are antisymmetric in a region of size $\sigma$ around the threshold and also satisfy Tsybakov's margin condition around the threshold. We prove minimax lower and upper bounds which demonstrate that when $\sigma$ is smaller than the minimiax active/passive noiseless error derived in \cite{CN07}, then noise has no effect on the rates and one achieves the same noiseless rates. For larger $\sigma$, the \textit{unflattening} of the regression function on convolution with uniform noise, along with its local antisymmetry around the threshold, together yield a behaviour where noise \textit{appears} to be beneficial. Our key result is that active learning can buy significant improvement over a passive strategy even in the presence of feature noise.

Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This motivates us to consider the problem of low rank tensor recovery from a class of linear measurements called separable measurements. As specific examples, we focus on two distinct types of separable measurement mechanisms (a) Random projections, where each measurement corresponds to an inner product of the tensor with a suitable random tensor, and (b) the completion problem where measurements constitute revelation of a random set of entries. We present a computationally efficient algorithm, with rigorous and order-optimal sample complexity results (upto logarithmic factors) for tensor recovery. Our method is based on reduction to matrix completion sub-problems and adaptation of Leurgans' method for tensor decomposition. We extend the methodology and sample complexity results to higher order tensors, and experimentally validate our theoretical results.

Intelligent control using adaptive/approximate dynamic programming (ADP), sometimes referred to by reinforcement learning (RL) or neuro-dynamic programming (NDP), is a set of powerful tools for obtaining approximate solutions to difficult and mathematically intractable problems which seek optimum while sometimes even no knowledge of the system model/dynamics is available. The dramatic potential of the tools in practice has attracted many researchers within the last few decades, [1]- [13]. The multitude of appeared papers and success stories on applications of ADP to different problems, however, has intensified the need for firm mathematical analyses for guaranteeing the convergence of the learning processes and the stability of the results. Besides the classifications of heuristic dynamic programming (HDP), dual heuristic programming (DHP), etc. [7], which are in terms of the variables subject to approximation and their dependencies, the learning algorithms are typically based on either value iteration (VI) or policy iteration (PI), [3], [14]. These algorithms are well investigated both by computer scientists for machine learning [3] and by control scientists for feedback control of dynamical systems [14].

In image classification, visual separability between different object categories is highly uneven, and some categories are more difficult to distinguish than others. Such difficult categories demand more dedicated classifiers. However, existing deep convolutional neural networks (CNN) are trained as flat N-way classifiers, and few efforts have been made to leverage the hierarchical structure of categories. In this paper, we introduce hierarchical deep CNNs (HD-CNNs) by embedding deep CNNs into a category hierarchy. An HD-CNN separates easy classes using a coarse category classifier while distinguishing difficult classes using fine category classifiers. During HD-CNN training, component-wise pretraining is followed by global finetuning with a multinomial logistic loss regularized by a coarse category consistency term. In addition, conditional executions of fine category classifiers and layer parameter compression make HD-CNNs scalable for large-scale visual recognition. We achieve state-of-the-art results on both CIFAR100 and large-scale ImageNet 1000-class benchmark datasets. In our experiments, we build up three different HD-CNNs and they lower the top-1 error of the standard CNNs by 2.65%, 3.1% and 1.1%, respectively.

We consider the problem of $n$-class classification ($n\geq 2$), where the classifier can choose to abstain from making predictions at a given cost, say, a factor $\alpha$ of the cost of misclassification. Designing consistent algorithms for such $n$-class classification problems with a `reject option' is the main goal of this paper, thereby extending and generalizing previously known results for $n=2$. We show that the Crammer-Singer surrogate and the one vs all hinge loss, albeit with a different predictor than the standard argmax, yield consistent algorithms for this problem when $\alpha=\frac{1}{2}$. More interestingly, we design a new convex surrogate that is also consistent for this problem when $\alpha=\frac{1}{2}$ and operates on a much lower dimensional space ($\log(n)$ as opposed to $n$). We also generalize all three surrogates to be consistent for any $\alpha\in[0, \frac{1}{2}]$.

Outlier detection aims to identify unusual data instances that deviate from expected patterns. The outlier detection is particularly challenging when outliers are context dependent and when they are defined by unusual combinations of multiple outcome variable values. In this paper, we develop and study a new conditional outlier detection approach for multivariate outcome spaces that works by (1) transforming the conditional detection to the outlier detection problem in a new (unconditional) space and (2) defining outlier scores by analyzing the data in the new space. Our approach relies on the classifier chain decomposition of the multi-dimensional classification problem that lets us transform the output space into a probability vector, one probability for each dimension of the output space. Outlier scores applied to these transformed vectors are then used to detect the outliers. Experiments on multiple multi-dimensional classification problems with the different outlier injection rates show that our methodology is robust and able to successfully identify outliers when outliers are either sparse (manifested in one or very few dimensions) or dense (affecting multiple dimensions).