Modern proximal and stochastic gradient descent (SGD) methods are believed to efficiently minimize large composite objective functions, but such methods have two algorithmic challenges: (1) a lack of fast or justified stop conditions, and (2) sensitivity to the objective function's conditioning. In response to the first challenge, modern proximal and SGD methods guarantee convergence only after multiple epochs, but such a guarantee renders proximal and SGD methods infeasible when the number of component functions is very large or infinite. In response to the second challenge, second order SGD methods have been developed, but they are marred by the complexity of their analysis. In this work, we address these challenges on the limited, but important, linear regression problem by introducing and analyzing a second order proximal/SGD method based on Kalman Filtering (kSGD). Through our analysis, we show kSGD is asymptotically optimal, develop a fast algorithm for very large, infinite or streaming data sources with a justified stop condition, prove that kSGD is insensitive to the problem's conditioning, and develop a unique approach for analyzing the complex second order dynamics. Our theoretical results are supported by numerical experiments on three regression problems (linear, nonparametric wavelet, and logistic) using three large publicly available datasets. Moreover, our analysis and experiments lay a foundation for embedding kSGD in multiple epoch algorithms, extending kSGD to other problem classes, and developing parallel and low memory kSGD implementations.

Bayesian inference has great promise for the privacy-preserving analysis of sensitive data, as posterior sampling automatically preserves differential privacy, an algorithmic notion of data privacy, under certain conditions (Dimitrakakis et al., 2014; Wang et al., 2015b). While this one posterior sample (OPS) approach elegantly provides privacy "for free," it is data inefficient in the sense of asymptotic relative efficiency (ARE). We show that a simple alternative based on the Laplace mechanism, the workhorse of differential privacy, is as asymptotically efficient as non-private posterior inference, under general assumptions. This technique also has practical advantages including efficient use of the privacy budget for MCMC. We demonstrate the practicality of our approach on a time-series analysis of sensitive military records from the Afghanistan and Iraq wars disclosed by the Wikileaks organization.

Completely random measures (CRM) represent the key building block of a wide variety of popular stochastic models and play a pivotal role in modern Bayesian Nonparametrics. A popular representation of CRMs as a random series with decreasing jumps is due to Ferguson and Klass (1972). This can immediately be turned into an algorithm for sampling realizations of CRMs or more elaborate models involving transformed CRMs. However, concrete implementation requires to truncate the random series at some threshold resulting in an approximation error. The goal of this paper is to quantify the quality of the approximation by a moment-matching criterion, which consists in evaluating a measure of discrepancy between actual moments and moments based on the simulation output. Seen as a function of the truncation level, the methodology can be used to determine the truncation level needed to reach a certain level of precision. The resulting moment-matching \FK algorithm is then implemented and illustrated on several popular Bayesian nonparametric models.

The most common question asked by prospective data scientists is – "What is the best programming language for Machine Learning?" The answer to this question always results in a debate whether to choose R, Python or MATLAB for Machine Learning. Nobody can, in reality, answer the question as to whether Python or R is best language for Machine Learning. However, the programming language one should choose for machine learning directly depends on the requirements of a given data problem, the likes and preferences of the data scientist and the context of machine learning activities they want to perform. According to a survey on Kaggler's Favourite Tools, the open source R programming language turned out to be the favourite among 543 Kagglers of the 1714 Kaggler's listing their data science tools.

A graphical model is a statistical model that is associated to a graph whose nodes correspond to variables of interest. The edges of the graph reflect allowed conditional dependencies among the variables. Graphical models admit computationally convenient factorization properties and have long been a valuable tool for tractable modeling of multivariate distributions. More recently, applications such as reconstructing gene regulatory networks from gene expression data have driven major advances in structure learning, that is, estimating the graph underlying a model. We review some of these advances and discuss methods such as the graphical lasso and neighborhood selection for undirected graphical models (or Markov random fields), and the PC algorithm and score-based search methods for directed graphical models (or Bayesian networks).

We propose a novel classification model for weak signal data, building upon a recent model for Bayesian multi-view learning, Group Factor Analysis (GFA). Instead of assuming all data to come from a single GFA model, we allow latent clusters, each having a different GFA model and producing a different class distribution. We show that sharing information across the clusters, by sharing factors, increases the classification accuracy considerably; the shared factors essentially form a flexible noise model that explains away the part of data not related to classification. Motivation for the setting comes from single-trial functional brain imaging data, having a very low signal-to-noise ratio and a natural multi-view setting, with the different sensors, measurement modalities (EEG, MEG, fMRI) and possible auxiliary information as views. We demonstrate our model on a MEG dataset.

We consider the problem of learning the preferences of a heterogeneous population by observing choices from an assortment of products, ads, or other offerings. Our observation model takes a form common in assortment planning applications: each arriving customer is offered an assortment consisting of a subset of all possible offerings; we observe only the assortment and the customer's single choice. In this paper we propose a mixture choice model with a natural underlying low-dimensional structure, and show how to estimate its parameters. In our model, the preferences of each customer or segment follow a separate parametric choice model, but the underlying structure of these parameters over all the models has low dimension. We show that a nuclear-norm regularized maximum likelihood estimator can learn the preferences of all customers using a number of observations much smaller than the number of item-customer combinations. This result shows the potential for structural assumptions to speed up learning and improve revenues in assortment planning and customization. We provide a specialized factored gradient descent algorithm and study the success of the approach empirically.

The EM algorithm is a novel numerical method to obtain maximum likelihood estimates and is often used for practical calculations. However, many of maximum likelihood estimation problems are nonconvex, and it is known that the EM algorithm fails to give the optimal estimate by being trapped by local optima. In order to deal with this difficulty, we propose a deterministic quantum annealing EM algorithm by introducing the mathematical mechanism of quantum fluctuations into the conventional EM algorithm because quantum fluctuations induce the tunnel effect and are expected to relax the difficulty of nonconvex optimization problems in the maximum likelihood estimation problems. We show a theorem that guarantees its convergence and give numerical experiments to verify its efficiency.

In this series we've been using the empirical Bayes method to estimate batting averages of baseball players. Empirical Bayes is useful here because when we don't have a lot of information about a batter, they're "shrunken" towards the average across all players, as a natural consequence of the beta prior. When players are better, they are given more chances to bat! (Hat tip to Hadley Wickham to pointing this complication out to me). That means there's a relationship between the number of at-bats (AB) and the true batting average. For reasons I explain below, this makes our estimates systematically inaccurate.

Recent advances in neural variational inference have spawned a renaissance in deep latent variable models. In this paper we introduce a generic variational inference framework for generative and conditional models of text. While traditional variational methods derive an analytic approximation for the intractable distributions over latent variables, here we construct an inference network conditioned on the discrete text input to provide the variational distribution. We validate this framework on two very different text modelling applications, generative document modelling and supervised question answering. Our neural variational document model combines a continuous stochastic document representation with a bag-of-words generative model and achieves the lowest reported perplexities on two standard test corpora. The neural answer selection model employs a stochastic representation layer within an attention mechanism to extract the semantics between a question and answer pair. On two question answering benchmarks this model exceeds all previous published benchmarks.