Domain adaptation refers to the problem of leveraging labeled data in a source domain to learn an accurate model in a target domain where labels are scarce or unavailable. A recent approach for finding a common representation of the two domains is via domain adversarial training (Ganin & Lempitsky, 2015), which attempts to induce a feature extractor that matches the source and target feature distributions in some feature space. However, domain adversarial training faces two critical limitations: 1) if the feature extraction function has high-capacity, then feature distribution matching is a weak constraint, 2) in non-conservative domain adaptation (where no single classifier can perform well in both the source and target domains), training the model to do well on the source domain hurts performance on the target domain. In this paper, we address these issues through the lens of the cluster assumption, i.e., decision boundaries should not cross high-density data regions. We propose two novel and related models: 1) the Virtual Adversarial Domain Adaptation (VADA) model, which combines domain adversarial training with a penalty term that punishes the violation the cluster assumption; 2) the Decision-boundary Iterative Refinement Training with a Teacher (DIRT-T) model, which takes the VADA model as initialization and employs natural gradient steps to further minimize the cluster assumption violation. Extensive empirical results demonstrate that the combination of these two models significantly improve the state-of-the-art performance on the digit, traffic sign, and Wi-Fi recognition domain adaptation benchmarks.

Over-parameterized models are crucial in deep learning, but their workings are far from understood. Over-parameterization -- the technique of using more parameters than statistically necessary -- apparently improves the training: theoretical and empirical results have suggested that it can enhance the geometric properties of the optimization landscape in simplified settings [24, 18, 17, 34] and thus make it easier to train over-parameterized models. On the other hand, over-parameterization often doesn't hurt the test performance, even if the number of parameters is much larger than the number of examples. Large neural networks used in practice have enough expressiveness to fit any labels of the training datasets [42, 17]. The training objective function may have multiple global minima with almost zero training error, some of which generalize better than the others [21, 11]. However, local improvement algorithms such as stochastic gradient descent, starting with proper initialization, may prefer some generalizable local minima to the others and thus provide an implicit effect of regularization [38, 28, 19, 27, 41]. Such regularization seems to depend on the algorithmic choice, the initialization scheme, and certain intrinsic properties of the data. The phenomenon and intuition above can be theoretically fleshed out in the context of linear models [35], whereas less is known for nonlinear models whose training objectives are usually nonconvex. The very important work of Gunasekar et al. [16] initiates the study of low-rank matrix factorization models with over-parameterization and conjectures that gradient descent prefers small trace norm solution in over-parameterized models with thorough empirical evidences.

Exponential families and mixture families are parametric probability models that can be geometrically studied as smooth statistical manifolds with respect to any statistical divergence like the Kullback-Leibler (KL) divergence or the Hellinger divergence. When equipping a statistical manifold with the KL divergence, the induced manifold structure is dually flat, and the KL divergence between distributions amounts to an equivalent Bregman divergence on their corresponding parameters. In practice, the corresponding Bregman generators of mixture/exponential families require to perform definite integral calculus that can either be too time-consuming (for exponentially large discrete support case) or even do not admit closed-form formula (for continuous support case). In these cases, the dually flat construction remains theoretical and cannot be used by information-geometric algorithms. To bypass this problem, we consider performing stochastic Monte Carlo (MC) estimation of those integral-based mixture/exponential family Bregman generators. We show that, under natural assumptions, these MC generators are almost surely Bregman generators. We define a series of dually flat information geometries, termed Monte Carlo Information Geometries, that increasingly-finely approximate the untractable geometry. The advantage of this MCIG is that it allows a practical use of the Bregman algorithmic toolbox on a wide range of probability distribution families. We demonstrate our approach with a clustering task on a mixture family manifold.

A Gaussian process (GP) [1] is a prior distribution over functions with a support that includes a wide class of phenomena via the design of its mean and covariance functions, the parameters of which provide meaningful interpretation of the process at hand. Beyond regression [2], GPs have been extensively used in the last two decades for classification [3], density estimation [4], filter design [5], model identification [6] and optimisation [7]. In general terms, all these generative models have two stages: The latent process is modelled as a GP and the observation is modelled (conditional to the latent process) as a non-Gaussian variable. This class of models is referred to as GP with non-Gaussian likelihood, or as Generalised GPs. These usually consider likelihood functions from the exponential family such as the Laplace, Poisson, beta and gamma distributions [8]. A well-known example is the GP classification model, where the classes are represented by the output of an activation neuron into which a latent GP is fed. A slightly different approach to non-Gaussian models, which is not constrained to the exponential family, is the warped GP (WGP, [9]). The WGP models non-Gaussian data by assuming that there is a transformation φ such that the observations can be passed through φ to yield a GP, therefore, the likelihood function of this model is not designed directly but, rather, induced by the transformation (a.k.a.

We study the $A$-optimal design problem where we are given vectors $v_1,\ldots,v_n\in\mathbb{R}^d$, an integer $k\geq d$, and the goal is to select a set $S$ of $k$ vectors that minimizes the trace of $(\sum_{i\in S}v_iv_i^\top)^{-1}$. Traditionally, the problem is an instance of optimal design of experiments in statistics where each vector corresponds to a linear measurement of an unknown vector and the goal is to pick $k$ of them that minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks, sparse least squares regression, feature selection for $k$-means clustering, and matrix approximation. In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for $A$-optimal design. Given a matrix, proportional volume sampling picks a set of columns $S$ of size $k$ with probability proportional to $\mu(S)$ times $\det(\sum_{i\in S}v_iv_i^\top)$ for some measure $\mu$. Our main result is to show the approximability of the $A$-optimal design problem can be reduced to approximate independence properties of the measure $\mu$. We appeal to hard-core distributions as candidate distributions $\mu$ that allow us to obtain improved approximation algorithms for the $A$-optimal design. Our results include a $d$-approximation when $k=d$, an $(1+\epsilon)$-approximation when $k=\Omega\left(\frac{d}{\epsilon}+\frac{1}{\epsilon^2}\log\frac{1}{\epsilon}\right)$ and $\frac{k}{k-d+1}$-approximation when repetitions of vectors are allowed in the solution. We consider generalization of the problem for $k\leq d$ and obtain a $k$-approximation. The last result implies a restricted invertibility principle for the harmonic mean of singular values. We also show that the problem is $\mathsf{NP}$-hard to approximate within a fixed constant when $k=d$.

In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamental computational challenges often associated with problems involving higher order tensors, yet very little is known about their statistical performance. To fill in this void, in this article, we characterize the fundamental statistical limits of noisy tensor completion by establishing minimax optimal rates of convergence for estimating a $k$th order low rank tensor under the general $\ell_p$ ($1\le p\le 2$) norm which suggest significant room for improvement over the existing approaches. Furthermore, we propose a polynomial-time computable estimating procedure based upon power iteration and a second-order spectral initialization that achieves the optimal rates of convergence. Our method is fairly easy to implement and numerical experiments are presented to further demonstrate the practical merits of our estimator.

International Centre for Theoretical Physics, Trieste, Italy Stochasticity and limited precision of synaptic weights in neural network models are key aspects of both biological and hardware modeling of learning processes. Here we show that a neural network model with stochastic binary weights naturally gives prominence to exponentially rare dense regions of solutions with a number of desirable properties such as robustness and good generalization performance, while typical solutions are isolated and hard to find. Binary solutions of the standard perceptron problem are obtained from a simple gradient descent procedure on a set of real values parametrizing a probability distribution over the binary synapses. Both analytical and numerical results are presented. An algorithmic extension aimed at training discrete deep neural networks is also investigated. Learning can be regarded as an optimization process over the connection weights of a neural network. In nature, synaptic weights are known to be plastic, low precision and unreliable, and it is an interesting issue to understand if this stochasticity can help learning or if it is an obstacle.

Ensemble methods -- particularly those based on decision trees -- have recently demonstrated superior performance in a variety of machine learning settings. Specifically, Random Forest (RF) was found to outperform >100 other methods in several manuscripts, and gradient boosting trees have been a crucial component of several recent Kaggle competition victories. Building off these successes and recent advances in sparse learning and random matrix theory, we propose a novel ensemble tree method called "Randomer Forest" (RerF). The key intuition behind RerF is that we can use sparse linear combinations at each decision node rather than just one feature (as in RF) or all of them (as in Rotation Forests). RerF significantly outperforms other methods on a standard benchmark suite containing 105 problems with varying dimension, sample size, and number of classes. Moreover, we provide an implementation that scales as or more efficiently than other available packages. Via a combination of basic principles, theory, and extensive numerical experiments, we demonstrate why, when, and how RerF achieves its performance properties.

A recent survey of over 16,000 data professionals showed that the most common challenges to data science included dirty data (36%), lack of data science talent (30%) and lack of management support (27%). Also, data professionals reported experiencing around three challenges in the previous year. A principal component analysis of the 20 challenges studied showed that challenges can be grouped into five categories. Data science is about finding useful insights and putting them to use. Data science, however, doesn't occur in a vacuum.

Real-valued word representations have transformed NLP applications; popular examples are word2vec and GloVe, recognized for their ability to capture linguistic regularities. In this paper, we demonstrate a {\em very simple}, and yet counter-intuitive, postprocessing technique -- eliminate the common mean vector and a few top dominating directions from the word vectors -- that renders off-the-shelf representations {\em even stronger}. The postprocessing is empirically validated on a variety of lexical-level intrinsic tasks (word similarity, concept categorization, word analogy) and sentence-level tasks (semantic textural similarity and { text classification}) on multiple datasets and with a variety of representation methods and hyperparameter choices in multiple languages; in each case, the processed representations are consistently better than the original ones.