We present a novel method for frequentist statistical inference in $M$-estimation problems, based on stochastic gradient descent (SGD) with a fixed step size: we demonstrate that the average of such SGD sequences can be used for statistical inference, after proper scaling. An intuitive analysis using the Ornstein-Uhlenbeck process suggests that such averages are asymptotically normal. From a practical perspective, our SGD-based inference procedure is a first order method, and is well-suited for large scale problems. To show its merits, we apply it to both synthetic and real datasets, and demonstrate that its accuracy is comparable to classical statistical methods, while requiring potentially far less computation.

Using conditional probability gives Bayes Nets strong analytical advantages over traditional regression-based models. This adds to several advantages we discussed in an earlier article. But what is conditional probability and what makes it different? In short, conditional probability means that the effects of one variable depend on, of flow from, the distribution of another variable (or others). The complete state of one variable determines how another acts.

We present a robust generalization of the synthetic control method for comparative case studies. Like the classical method, we present an algorithm to estimate the unobservable counterfactual of a treatment unit. A distinguishing feature of our algorithm is that of de-noising the data matrix via singular value thresholding, which renders our approach robust in multiple facets: it automatically identifies a good subset of donors, overcomes the challenges of missing data, and continues to work well in settings where covariate information may not be provided. To begin, we establish the condition under which the fundamental assumption in synthetic control-like approaches holds, i.e. when the linear relationship between the treatment unit and the donor pool prevails in both the pre- and post-intervention periods. We provide the first finite sample analysis for a broader class of models, the Latent Variable Model, in contrast to Factor Models previously considered in the literature. Further, we show that our de-noising procedure accurately imputes missing entries, producing a consistent estimator of the underlying signal matrix provided $p = \Omega( T^{-1 + \zeta})$ for some $\zeta > 0$; here, $p$ is the fraction of observed data and $T$ is the time interval of interest. Under the same setting, we prove that the mean-squared-error (MSE) in our prediction estimation scales as $O(\sigma^2/p + 1/\sqrt{T})$, where $\sigma^2$ is the noise variance. Using a data aggregation method, we show that the MSE can be made as small as $O(T^{-1/2+\gamma})$ for any $\gamma \in (0, 1/2)$, leading to a consistent estimator. We also introduce a Bayesian framework to quantify the model uncertainty through posterior probabilities. Our experiments, using both real-world and synthetic datasets, demonstrate that our robust generalization yields an improvement over the classical synthetic control method.

We introduce the Givens Transform, a novel transform between the space of orthonormal matrices and $\mathbb{R}^D$. The Givens Transform allows for the application of any general Bayesian inference algorithm to probabilistic models containing constrained unit-vectors or orthonormal matrix parameters. This includes a variety of matrix factorizations and dimensionality reduction models such as Probabilistic PCA (PPCA), Exponential Family PPCA (BXPCA), and Canonical Correlation Analysis (CCA). While previous Bayesian approaches to these models relied on separate sampling update rules for constrained and unconstrained parameters, the Givens Transform enables the treatment of unit-vectors and orthonormal matrices agnostically as unconstrained parameters. Thus any Bayesian inference algorithm can be used on these models without modification. This opens the door to not just sampling algorithms, but Variational Inference (VI) as well. We illustrate with several examples and supplied code, how the Givens Transform allows end-users to easily build complex models in their favorite Bayesian modeling framework such as Stan, Edward, or PyMC3, a task that was previously intractable due to technical constraints.

We present an information-theoretic framework for bounding the number of labeled samples needed to train a classifier in a parametric Bayesian setting. We derive bounds on the average $L_p$ distance between the learned classifier and the true maximum a posteriori classifier, which are well-established surrogates for the excess classification error due to imperfect learning. We provide lower and upper bounds on the rate-distortion function, using $L_p$ loss as the distortion measure, of a maximum a priori classifier in terms of the differential entropy of the posterior distribution and a quantity called the interpolation dimension, which characterizes the complexity of the parametric distribution family. In addition to expressing the information content of a classifier in terms of lossy compression, the rate-distortion function also expresses the minimum number of bits a learning machine needs to extract from training data to learn a classifier to within a specified $L_p$ tolerance. We use results from universal source coding to express the information content in the training data in terms of the Fisher information of the parametric family and the number of training samples available. The result is a framework for computing lower bounds on the Bayes $L_p$ risk. This framework complements the well-known probably approximately correct (PAC) framework, which provides minimax risk bounds involving the Vapnik-Chervonenkis dimension or Rademacher complexity. Whereas the PAC framework provides upper bounds the risk for the worst-case data distribution, the proposed rate-distortion framework lower bounds the risk averaged over the data distribution. We evaluate the bounds for a variety of data models, including categorical, multinomial, and Gaussian models. In each case the bounds are provably tight orderwise, and in two cases we prove that the bounds are tight up to multiplicative constants.

Recently, crowdsourcing has emerged as an effective paradigm for human-powered large scale problem solving in various domains. However, task requester usually has a limited amount of budget, thus it is desirable to have a policy to wisely allocate the budget to achieve better quality. In this paper, we study the principle of information maximization for active sampling strategies in the framework of HodgeRank, an approach based on Hodge Decomposition of pairwise ranking data with multiple workers. The principle exhibits two scenarios of active sampling: Fisher information maximization that leads to unsupervised sampling based on a sequential maximization of graph algebraic connectivity without considering labels; and Bayesian information maximization that selects samples with the largest information gain from prior to posterior, which gives a supervised sampling involving the labels collected. Experiments show that the proposed methods boost the sampling efficiency as compared to traditional sampling schemes and are thus valuable to practical crowdsourcing experiments.

Many modern unsupervised or semi-supervised machine learning algorithms rely on Bayesian probabilistic models. These models are usually intractable and thus require approximate inference. Variational inference (VI) lets us approximate a high-dimensional Bayesian posterior with a simpler variational distribution by solving an optimization problem. This approach has been successfully used in various models and large-scale applications. In this review, we give an overview of recent trends in variational inference. We first introduce standard mean field variational inference, then review recent advances focusing on the following aspects: (a) scalable VI, which includes stochastic approximations, (b) generic VI, which extends the applicability of VI to a large class of otherwise intractable models, such as non-conjugate models, (c) accurate VI, which includes variational models beyond the mean field approximation or with atypical divergences, and (d) amortized VI, which implements the inference over local latent variables with inference networks. Finally, we provide a summary of promising future research directions.

We present the Variational Adaptive Newton (VAN) method which is a black-box optimization method especially suitable for explorative-learning tasks such as active learning and reinforcement learning. Similar to Bayesian methods, VAN estimates a distribution that can be used for exploration, but requires computations that are similar to continuous optimization methods. Our theoretical contribution reveals that VAN is a second-order method that unifies existing methods in distinct fields of continuous optimization, variational inference, and evolution strategies. Our experimental results show that VAN performs well on a wide-variety of learning tasks. This work presents a general-purpose explorative-learning method that has the potential to improve learning in areas such as active learning and reinforcement learning.

Sequential hypothesis testing is a desirable decision making strategy in any time sensitive scenario. Compared with fixed sample-size testing, sequential testing is capable of achieving identical probability of error requirements using less samples in average. For a binary detection problem, it is well known that for known density functions accumulating the likelihood ratio statistics is time optimal under a fixed error rate constraint. This paper considers the problem of learning a binary sequential detector from training samples when density functions are unavailable. We formulate the problem as a constrained likelihood ratio estimation which can be solved efficiently through convex optimization by imposing Reproducing Kernel Hilbert Space (RKHS) structure on the log-likelihood ratio function. In addition, we provide a computationally efficient approximated solution for large scale data set. The proposed algorithm, namely Wald-Kernel, is tested on a synthetic data set and two real world data sets, together with previous approaches for likelihood ratio estimation. Our empirical results show that the classifier trained through the proposed technique achieves smaller average sampling cost than previous approaches proposed in the literature for the same error rate.

Bayes Nets (or Bayesian Networks) give remarkable results in determining the effects of many variables on an outcome. They typically perform strongly even in cases when other methods falter or fail. These networks have had relatively little use with business-related problems, although they have worked successfully for years in fields such as scientific research, public safety, aircraft guidance systems and national defense. Importantly, they often outperform regression, particularly in determining variables' effects. Regression is one of the most august multivariate methods, and among the most studied and applied.