We present a simple generative framework for learning to predict previously unseen classes, based on estimating class-attribute-gated class-conditional distributions. We model each class-conditional distribution as an exponential family distribution and the parameters of the distribution of each seen/unseen class are defined as functions of the respective observed class attributes. These functions can be learned using only the seen class data and can be used to predict the parameters of the class-conditional distribution of each unseen class. Unlike most existing methods for zero-shot learning that represent classes as fixed embeddings in some vector space, our generative model naturally represents each class as a probability distribution. It is simple to implement and also allows leveraging additional unlabeled data from unseen classes to improve the estimates of their class-conditional distributions using transductive/semi-supervised learning. Moreover, it extends seamlessly to few-shot learning by easily updating these distributions when provided with a small number of additional labelled examples from unseen classes. Through a comprehensive set of experiments on several benchmark data sets, we demonstrate the efficacy of our framework.
National University of Singapore（新加坡国立大学）；California Institute of Technology（加州理工学院）
We study the regret of Thompson sampling (TS) algorithms for exponential family bandits, where the reward distribution is from a one-dimensional exponential family, which covers many common reward distributions including Bernoulli, Gaussian, Gamma, Exponential, etc. We propose a Thompson sampling algorithm, termed ExpTS, which uses a novel sampling distribution to avoid the under-estimation of the optimal arm. We provide a tight regret analysis for ExpTS, which simultaneously yields both the finite-time regret bound as well as the asymptotic regret bound. In particular, for a $K$-armed bandit …
CentraleSupélec（巴黎中央理工-高等电力学院）；Inria Saclay（法国国家信息与自动化研究所萨克莱分部）；University of Paris-Saclay（巴黎萨克雷大学）
Representing networks in a low dimensional latent space is a crucial task with many interesting applications in graph learning problems, such as link prediction and node classification. A widely applied network representation learning paradigm is based on the combination of random walks for sampling context nodes and the traditional \textit{Skip-Gram} model to capture center-context node relationships. In this paper, we emphasize on exponential family distributions to capture rich interaction patterns between nodes in random walk sequences. We introduce the generic \textit{exponential family g…
Google Brain（谷歌大脑）；Georgia Institute of Technology（佐治亚理工学院）；University College London（伦敦大学学院）；University of Illinois at Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）
We investigate penalized maximum log-likelihood estimation for exponential family distributions whose natural parameter resides in a reproducing kernel Hilbert space. Key to our approach is a novel technique, doubly dual embedding, that avoids computation of the partition function. This technique also allows the development of a flexible sampling strategy that amortizes the cost of Monte-Carlo sampling in the inference stage. The resulting estimator can be easily generalized to kernel conditional exponential families. We establish a connection between kernel exponential family estimation and M…
Thompson Sampling has been demonstrated in many complex bandit models, however the theoretical guarantees available for the parametric multi-armed bandit are still limited to the Bernoulli case. Here we extend them by proving asymptotic optimality of the algorithm using the Jeffreys prior for 1-dimensional exponential family bandits. Our proof builds on previous work, but also makes extensive use of closed forms for Kullback-Leibler divergence and Fisher information (and thus Jeffreys prior) available in an exponential family. This allow us to give a finite time exponential concentration inequ…
A nonparametric family of conditional distributions is introduced, which generalizes conditional exponential families using functional parameters in a suitable RKHS. An algorithm is provided for learning the generalized natural parameter, and consistency of the estimator is established in the well specified case. In experiments, the new method generally outperforms a competing approach with consistency guarantees, and is competitive with a deep conditional density model on datasets that exhibit abrupt transitions and heteroscedasticity.
Columbia University（哥伦比亚大学）；Univ. of Cambridge（剑桥大学）
Word embeddings are a powerful approach for capturing semantic similarity among terms in a vocabulary. In this paper, we develop exponential family embeddings, a class of methods that extends the idea of word embeddings to other types of high-dimensional data. As examples, we studied neural data with real-valued observations, count data from a market basket analysis, and ratings data from a movie recommendation system. The main idea is to model each observation conditioned on a set of other observations. This set is called the context, and the way the context is defined is a modeling choice th…
Aalto University（阿尔托大学）；School of Science and Engineering（科学与工程学院）；Helsinki Institute for Information Technology（赫尔辛基信息技术研究所）
Exponential family extensions of principal component analysis (EPCA) have received a considerable amount of attention in recent years, demonstrating the growing need for basic modeling tools that do not assume the squared loss or Gaussian distribution. We extend the EPCA model toolbox by presenting the first exponential family multi-view learning methods of the partial least squares and canonical correlation analysis, based on a unified representation of EPCA as matrix factorization of the natural parameters of exponential family. The models are based on a new family of priors that are general…
Linköping University（林雪平大学）；Middle East Technical University（中东技术大学）
In this paper, a Bayesian inference technique based on Taylor series approximation of the logarithm of the likelihood function is presented. The proposed approximation is devised for the case, where the prior distribution belongs to the exponential family of distributions. The logarithm of the likelihood function is linearized with respect to the sufficient statistic of the prior distribution in exponential family such that the posterior obtains the same exponential family form as the prior. Similarities between the proposed method and the extended Kalman filter for nonlinear filtering are ill…
We present a general method for deriving collapsed variational inference algo- rithms for probabilistic models in the conjugate exponential family. Our method unifies many existing approaches to collapsed variational inference. Our collapsed variational inference leads to a new lower bound on the marginal likelihood. We exploit the information geometry of the bound to derive much faster optimization methods based on conjugate gradients for these models. Our approach is very general and is easily applied to any model where the mean field update equations have been derived. Empirically we show s…
UC Santa Cruz（加州大学圣克鲁兹分校）；Northwestern University（西北大学）；Yale University（耶鲁大学）
We propose an optimistic model-based algorithm, dubbed SMRL, for finite-horizon episodic reinforcement learning (RL) when the transition model is specified by exponential family distributions with $d$ parameters and the reward is bounded and known. SMRL uses score matching, an unnormalized density estimation technique that enables efficient estimation of the model parameter by ridge regression. Under standard regularity assumptions, SMRL achieves $\tilde O(d\sqrt{H^3T})$ online regret, where $H$ is the length of each episode and $T$ is the total number of interactions (ignoring polynomial depe…
Dept. of Neuroscience Columbia University（哥伦比亚大学神经科学系）；Dept. of Statistics Columbia University（哥伦比亚大学统计系）；Columbia University（哥伦比亚大学）
Recently much attention has been paid to deep generative models, since they have been used to great success for variational inference, generation of complex data types, and more. In most all of these settings, the goal has been to find a particular member of that model family: optimized parameters index a distribution that is close (via a divergence or classification metric) to a target distribution. Much less attention, however, has been paid to the problem of learning a model itself. Here we introduce a two-network architecture and optimization procedure for learning intractable exponential …
Information Sciences Institute, University of Southern California（南加州大学信息科学研究所）；University of British Columbia（不列颠哥伦比亚大学）
The recently proposed Thermodynamic Variational Objective (TVO) leverages thermodynamic integration to provide a family of variational inference objectives, which both tighten and generalize the ubiquitous Evidence Lower Bound (ELBO). However, the tightness of TVO bounds was not previously known, an expensive grid search was used to choose a "schedule" of intermediate distributions, and model learning suffered with ostensibly tighter bounds. In this work, we propose an exponential family interpretation of the geometric mixture curve underlying the TVO and various path sampling methods, which a…
School of Computing（计算学院）；University of Utah（犹他大学）
We present an approach to semi-supervised learning based on an exponential family characterization. Our approach generalizes previous work on coupled priors for hybrid generative/discriminative models. Our model is more flexible and natural than previous approaches. Experimental results on several data sets show that our approach also performs better in practice.
We consider on-line density estimation with a parameterized density from the exponential family. The on-line algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss which is the negative log-likelihood of the example w.r.t. the past parameter of the algorithm. An off-line algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the on-line algorithm over the total loss of the off-line algorithm. These relative loss boun…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室公司）；CSIRO Data61（澳大利亚联邦科学与工业研究组织Data61）；Australian National University（澳大利亚国立大学）
It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler, $α$-divergences, and Jeffreys' divergences between densities belonging to a same exponential family have generic closed-form formulas relying on the strictly convex and real-analytic cumulant function characterizing the exponential family. In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function and highlight the role of quasi-arithmetic means and their multivariate mean operator extensions. In practice, these cumulant-free formulas are handy when implementing these (dis)similar…
We propose a novel approach for density estimation with exponential families for the case when the true density may not fall within the chosen family. Our approach augments the sufficient statistics with features designed to accumulate probability mass in the neighborhood of the observed points, resulting in a non-parametric model similar to kernel density estimators. We show that under mild conditions, the resulting model uses only the sufficient statistics if the density is within the chosen exponential family, and asymptotically, it approximates densities outside of the chosen exponential f…
University of British Columbia（不列颠哥伦比亚大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
Natural-gradient methods enable fast and simple algorithms for variational inference, but due to computational difficulties, their use is mostly limited to \emph{minimal} exponential-family (EF) approximations. In this paper, we extend their application to estimate \emph{structured} approximations such as mixtures of EF distributions. Such approximations can fit complex, multimodal posterior distributions and are generally more accurate than unimodal EF approximations. By using a \emph{minimal conditional-EF} representation of such approximations, we derive simple natural-gradient updates. Our…
Motivated by the needs of online large-scale recommender systems, we specialize the decoupled extended Kalman filter (DEKF) to factorization models, including factorization machines, matrix and tensor factorization, and illustrate the effectiveness of the approach through numerical experiments on synthetic and on real-world data. Online learning of model parameters through the DEKF makes factorization models more broadly useful by (i) allowing for more flexible observations through the entire exponential family, (ii) modeling parameter drift, and (iii) producing parameter uncertainty estimates…
Munich Data Science Institute（慕尼黑数据科学研究所）；Technical University of Munich（慕尼黑工业大学）
Uncertainty awareness is crucial to develop reliable machine learning models. In this work, we propose the Natural Posterior Network (NatPN) for fast and high-quality uncertainty estimation for any task where the target distribution belongs to the exponential family. Thus, NatPN finds application for both classification and general regression settings. Unlike many previous approaches, NatPN does not require out-of-distribution (OOD) data at training time. Instead, it leverages Normalizing Flows to fit a single density on a learned low-dimensional and task-dependent latent space. For any input …
We investigate a class of hierarchical mixtures-of-experts (HME) models where exponential family regression models with generalized linear mean functions of the form psi(ga+fx^Tfgb) are mixed. Here psi(...) is the inverse link function. Suppose the true response y follows an exponential family regression model with mean function belonging to a class of smooth functions of the form psi(h(fx)) where h(...)in W_2^infty (a Sobolev class over [0,1]^{s}). It is shown that the HME probability density functions can approximate the true density, at a rate of O(m^{-2/s}) in L_p norm, and at a rate of O(…
We consider the question of learning the natural parameters of a $k$ parameter minimal exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a computationally efficient estimator that is consistent as well as asymptotically normal unde…
The Ohio State University（俄亥俄州立大学）；Boston University（波士顿大学）
Conjugate pairs of distributions over infinite dimensional spaces are prominent in statistical learning theory, particularly due to the widespread adoption of Bayesian nonparametric methodologies for a host of models and applications. Much of the existing literature in the learning community focuses on processes possessing some form of computationally tractable conjugacy as is the case for the beta and gamma processes (and, via normalization, the Dirichlet process). For these processes, proofs of conjugacy and requisite derivation of explicit computational formulae for posterior density parame…
Guiding Time-Varying Generative Models with Natural Gradients on Exponential Family Manifold
Song Liu School of Mathematics University of Bristol United Kingdom（索恩·刘 布里斯托大学数学学院）；Leyang Wang Department of Computer Science University College London United Kingdom（雷杨·王 伦敦大学学院计算机科学系）；Yakun Wang School of Mathematics University of Bristol United Kingdom（亚坤·王 布里斯托大学数学学院）
Coresets are efficient representations of data sets such that models trained on the coreset are provably competitive with models trained on the original data set. As such, they have been successfully used to scale up clustering models such as K-Means and Gaussian mixture models to massive data sets. However, until now, the algorithms and the corresponding theory were usually specific to each clustering problem. We propose a single, practical algorithm to construct strong coresets for a large class of hard and soft clustering problems based on Bregman divergences. This class includes hard clust…
Mila；Université de Montreal（蒙特利尔大学）；University of British Columbia（不列颠哥伦比亚大学）；Samsung（三星）；SAIT AI Lab（SAIT AI实验室）
We consider the problem of upper bounding the expected log-likelihood sub-optimality of the maximum likelihood estimate (MLE), or a conjugate maximum a posteriori (MAP) for an exponential family, in a non-asymptotic way. Surprisingly, we found no general solution to this problem in the literature. In particular, current theories do not hold for a Gaussian or in the interesting few samples regime. After exhibiting various facets of the problem, we show we can interpret the MAP as running stochastic mirror descent (SMD) on the log-likelihood. However, modern convergence results do not apply for …
University of California at Berkeley（加州大学伯克利分校）；Queensland University of Technology（昆士兰科技大学）；CWI（荷兰数学与计算机科学研究院）；Copenhagen Business College（哥本哈根商学院）；Poznań University of Technology（波兹南理工大学）
We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the optimal strategy can be calculated without knowing the time horizon in advance. They put forward the question what families have exchangeable SNML strategies. This paper fully answers this open problem for one-dimensional exponential families. The exchangeability can happen only for …
What if there is a teacher who knows the learning goal and wants to design good training data for a machine learner? We propose an optimal teaching framework aimed at learners who employ Bayesian models. Our framework is expressed as an optimization problem over teaching examples that balance the future loss of the learner and the effort of the teacher. This optimization problem is in general hard. In the case where the learner employs conjugate exponential family models, we present an approximate algorithm for finding the optimal teaching set. Our algorithm optimizes the aggregate sufficient …
The kernel exponential family is a rich class of distributions, which can be fit efficiently and with statistical guarantees by score matching. Being required to choose a priori a simple kernel such as the Gaussian, however, limits its practical applicability. We provide a scheme for learning a kernel parameterized by a deep network, which can find complex location-dependent local features of the data geometry. This gives a very rich class of density models, capable of fitting complex structures on moderate-dimensional problems. Compared to deep density models fit via maximum likelihood, our a…
We study the problem of episodic reinforcement learning in continuous state-action spaces with unknown rewards and transitions. Specifically, we consider the setting where the rewards and transitions are modeled using parametric bilinear exponential families. We propose an algorithm, BEF-RLSVI, that a) uses penalized maximum likelihood estimators to learn the unknown parameters, b) injects a calibrated Gaussian noise in the parameter of rewards to ensure exploration, and c) leverages linearity of the exponential family with respect to an underlying RKHS to perform tractable planning. We furthe…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Keio University（庆应义塾大学）；The University of Tokyo（东京大学）
Exponential family plays an important role in information geometry. In arXiv:1811.01394, we introduced a method to construct an exponential family $\mathcal{P}=\{p_θ\}_{θ\inΘ}$ on a homogeneous space $G/H$ from a pair $(V,v_0)$. Here $V$ is a representation of $G$ and $v_0$ is an $H$-fixed vector in $V$. Then the following questions naturally arise: (Q1) when is the correspondence $θ\mapsto p_θ$ injective? (Q2) when do distinct pairs $(V,v_0)$ and $(V',v_0')$ generate the same family? In this paper, we answer these two questions (Theorems 1 and 2). Moreover, in Section 3, we consider the case …
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
We proposea graphical model for multi-view feature extraction that automatically adapts its structure to achieve better representation of data distribution. The proposed model, structure-adapting multi-view harmonium (SA-MVH) has switch parameters that control the connection between hidden nodes and input views, and learn the switch parameter while training. Numerical experiments on synthetic and a real-world dataset demonstrate the useful behavior of the SA-MVH, compared to existing multi-view feature extraction methods.
We propose and analyze a general-purpose dataset-distance-based utility function family, Duff, for differential privacy's exponential mechanism. Given a particular dataset and a statistic (e.g., median, mode), this function family assigns utility to a possible output o based on the number of individuals whose data would have to be added to or removed from the dataset in order for the statistic to take on value o. We show that the exponential mechanism based on Duff often offers provably higher fidelity to the statistic's true value compared to existing differential privacy mechanisms based on …
The Gaussian-Head OFL Family: One-Shot Federated Learning from Client Global Statistics
Department of Sciences and Methods for Engineering（工程科学与方法系）；Artificial Intelligence Research and Innovation Center（人工智能研究与创新中心）；University of Modena and Reggio Emilia（摩德纳和雷吉奥艾米利亚大学）
Neuromorphic hardware platforms, such as Intel's Loihi chip, support the implementation of Spiking Neural Networks (SNNs) as an energy-efficient alternative to Artificial Neural Networks (ANNs). SNNs are networks of neurons with internal analogue dynamics that communicate by means of binary time series. In this work, a probabilistic model is introduced for a generalized set-up in which the synaptic time series can take values in an arbitrary alphabet and are characterized by both causal and instantaneous statistical dependencies. The model, which can be considered as an extension of exponentia…
ETH Zürich（苏黎世联邦理工学院）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；Caltech（加州理工学院）
Fitting a graphical model to a collection of random variables given sample observations is a challenging task if the observed variables are influenced by latent variables, which can induce significant confounding statistical dependencies among the observed variables. We present a new convex relaxation framework based on regularized conditional likelihood for latent-variable graphical modeling in which the conditional distribution of the observed variables conditioned on the latent variables is given by an exponential family graphical model. In comparison to previously proposed tractable method…
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；Neural Magic Inc.（Neural Magic公司）
We show for the first time that large-scale generative pretrained transformer (GPT) family models can be pruned to at least 50% sparsity in one-shot, without any retraining, at minimal loss of accuracy. This is achieved via a new pruning method called SparseGPT, specifically designed to work efficiently and accurately on massive GPT-family models. We can execute SparseGPT on the largest available open-source models, OPT-175B and BLOOM-176B, in under 4.5 hours, and can reach 60% unstructured sparsity with negligible increase in perplexity: remarkably, more than 100 billion weights from these mo…
Duke University（杜克大学）；University of California, Berkeley（加州大学伯克利分校）
Recent progress in center-based clustering algorithms combats poor local minima by implicit annealing, using a family of generalized means. These methods are variations of Lloyd's celebrated $k$-means algorithm, and are most appropriate for spherical clusters such as those arising from Gaussian data. In this paper, we bridge these algorithmic advances to classical work on hard clustering under Bregman divergences, which enjoy a bijection to exponential family distributions and are thus well-suited for clustering objects arising from a breadth of data generating mechanisms. The elegant properti…
This document describes concisely the ubiquitous class of exponential family distributions met in statistics. The first part recalls definitions and summarizes main properties and duality with Bregman divergences (all proofs are skipped). The second part lists decompositions and related formula of common exponential family distributions. We recall the Fisher-Rao-Riemannian geometries and the dual affine connection information geometries of statistical manifolds. It is intended to maintain and update this document and catalog by adding new distribution items.
NEC Laboratories Europe（NEC欧洲实验室）；University College London（伦敦大学学院）；Istituto Italiano di Tecnologia（意大利理工学院）
Combining discrete probability distributions and combinatorial optimization problems with neural network components has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable as it only requires the ability to compute the most probable states and does not rely on smooth relaxations. The framework encompasses several approaches such as perturbation-based implicit differentiation …
