Several scalable sample-based methods to compute the Kullback Leibler (KL) divergence between two distributions have been proposed and applied in large-scale machine learning models. While they have been found to be unstable, the theoretical root cause of the problem is not clear. In this paper, we study a generative adversarial network based approach that uses a neural network discriminator to estimate KL divergence. We argue that, in such case, high fluctuations in the estimates are a consequence of not controlling the complexity of the discriminator function space. We provide a theoretical underpinning and remedy for this problem by first constructing a discriminator in the Reproducing Kernel Hilbert Space (RKHS). This enables us to leverage sample complexity and mean embedding to theoretically relate the error probability bound of the KL estimates to the complexity of the discriminator in RKHS. Based on this theory, we then present a scalable way to control the complexity of the discriminator for a reliable estimation of KL divergence. We support both our proposed theory and method to control the complexity of the RKHS discriminator through controlled experiments.
Recently, a method called the Mutual Information Neural Estimator (MINE) that uses neural networks has been proposed to estimate mutual information and more generally the Kullback-Leibler (KL) divergence between two distributions. The method uses the Donsker-Varadhan representation to arrive at the estimate of the KL divergence and is better than the existing estimators in terms of scalability and flexibility. The output of MINE algorithm is not guaranteed to be a consistent estimator. We propose a new estimator that instead of searching among functions characterized by neural networks searche…
Romanian Institute of Science and Technology（罗马尼亚科学与技术研究院）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
In this paper we propose two novel bounds for the log-likelihood based on Kullback-Leibler and the Rényi divergences, which can be used for variational inference and in particular for the training of Variational AutoEncoders. Our proposal is motivated by the difficulties encountered in training VAEs on continuous datasets with high contrast images, such as those with handwritten digits and characters, where numerical issues often appear unless noise is added, either to the dataset during training or to the generative model given by the decoder. The new bounds we propose, which are obtained fro…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
In this work, we present formulations for regularized Kullback-Leibler and Rényi divergences via the Alpha Log-Determinant (Log-Det) divergences between positive Hilbert-Schmidt operators on Hilbert spaces in two different settings, namely (i) covariance operators and Gaussian measures defined on reproducing kernel Hilbert spaces (RKHS); and (ii) Gaussian processes with squared integrable sample paths. For characteristic kernels, the first setting leads to divergences between arbitrary Borel probability measures on a complete, separable metric space. We show that the Alpha Log-Det divergences …
University of Bologna（博洛尼亚大学）；Department of Informatics: Science and Engineering (DISI)（信息学系：科学与工程（DISI））
In the loss function of Variational Autoencoders there is a well known tension between two components: the reconstruction loss, improving the quality of the resulting images, and the Kullback-Leibler divergence, acting as a regularizer of the latent space. Correctly balancing these two components is a delicate issue, easily resulting in poor generative behaviours. In a recent work, Dai and Wipf obtained a sensible improvement by allowing the network to learn the balancing factor during training, according to a suitable loss function. In this article, we show that learning can be replaced by a …
University of Cambridge（剑桥大学）；Microsoft Research Cambridge（微软剑桥研究院）；Tehran Institute for Advanced Studies（德黑兰高等研究院）
Variational Autoencoders (VAEs) are known to suffer from learning uninformative latent representation of the input due to issues such as approximated posterior collapse, or entanglement of the latent space. We impose an explicit constraint on the Kullback-Leibler (KL) divergence term inside the VAE objective function. While the explicit constraint naturally avoids posterior collapse, we use it to further understand the significance of the KL term in controlling the information transmitted through the VAE channel. Within this framework, we explore different properties of the estimated posterior…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Cambridge（剑桥大学）；National University of Singapore（新加坡国立大学）
Modern machine learning approaches excel in static settings where a large amount of i.i.d. training data are available for a given task. In a dynamic environment, though, an intelligent agent needs to be able to transfer knowledge and re-use learned components across domains. It has been argued that this may be possible through causal models, aiming to mirror the modularity of the real world in terms of independent causal mechanisms. However, the true causal structure underlying a given set of data is generally not identifiable, so it is desirable to have means to quantify differences between …
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University（阿尔托大学）
We introduce a method, KL-LIME, for explaining predictions of Bayesian predictive models by projecting the information in the predictive distribution locally to a simpler, interpretable explanation model. The proposed approach combines the recent Local Interpretable Model-agnostic Explanations (LIME) method with ideas from Bayesian projection predictive variable selection methods. The information theoretic basis helps in navigating the trade-off between explanation fidelity and complexity. We demonstrate the method in explaining MNIST digit classifications made by a Bayesian deep convolutional…
We propose a method to fuse posterior distributions learned from heterogeneous datasets. Our algorithm relies on a mean field assumption for both the fused model and the individual dataset posteriors and proceeds using a simple assign-and-average approach. The components of the dataset posteriors are assigned to the proposed global model components by solving a regularized variant of the assignment problem. The global components are then updated based on these assignments by their mean under a KL divergence. For exponential family variational distributions, our formulation leads to an efficien…
School of Mathematics and Physics（数学与物理学院）；The University of Queensland（昆士兰大学）；Institute of Mathematics for Industry（产业数学研究所）；Kyushu University（九州大学）；School of Computing, Engineering, and Mathematical Sciences（计算、工程与数学科学学院）；La Trobe University（拉筹伯大学）
We consider the problem of estimating probability density functions based on sample data, using a finite mixture of densities from some component class. To this end, we introduce the $h$-lifted Kullback--Leibler (KL) divergence as a generalization of the standard KL divergence and a criterion for conducting risk minimization. Under a compact support assumption, we prove an $\mathcal{O}(1/{\sqrt{n}})$ bound on the expected estimation error when using the $h$-lifted KL divergence, which extends the results of Rakhlin et al. (2005, ESAIM: Probability and Statistics, Vol. 9) and Li and Barron (199…
Learning word representations has garnered greater attention in the recent past due to its diverse text applications. Word embeddings encapsulate the syntactic and semantic regularities of sentences. Modelling word embedding as multi-sense gaussian mixture distributions, will additionally capture uncertainty and polysemy of words. We propose to learn the Gaussian mixture representation of words using a Kullback-Leibler (KL) divergence based objective function. The KL divergence based energy function provides a better distance metric which can effectively capture entailment and distribution sim…
Kullback-Leibler Penalized Sparse Discriminant Analysis for Event-Related Potential Classification
Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional, UNL, CONICET, FICH（信号、系统与智能计算研究所，国立利托拉尔大学，国家科学与技术研究理事会，水文与工程系）；Facultad de Ingeniería, Universidad Nacional de Entre Ríos（国立恩特雷里奥斯大学工程学院）；Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQ（滨海应用数学研究所，国立利托拉尔大学，国家科学与技术研究理事会，化学工程系）
This short note is on a property of the Kullback-Leibler (KL) divergence which indicates that independent Gaussian distributions minimize the KL divergence from given independent Gaussian distributions. The primary purpose of this note is for the referencing of papers that need to make use of this property entirely or partially.
CAKD: A Correlation-Aware Knowledge Distillation Framework Based on Decoupling Kullback-Leibler Divergence
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Faculty of Engineering, The University of Sydney（悉尼大学工程学院）；School of Mechanical and Electronic Engineering, Wuhan University of Technology（武汉理工大学机电工程学院）
University of California Berkeley（加州大学伯克利分校）；Duke University（杜克大学）；Google Research（谷歌研究院）；University of California San Diego（加州大学圣迭戈分校）
Variational Inference (VI) is a popular alternative to asymptotically exact sampling in Bayesian inference. Its main workhorse is optimization over a reverse Kullback-Leibler divergence (RKL), which typically underestimates the tail of the posterior leading to miscalibration and potential degeneracy. Importance sampling (IS), on the other hand, is often used to fine-tune and de-bias the estimates of approximate Bayesian inference procedures. The quality of IS crucially depends on the choice of the proposal distribution. Ideally, the proposal distribution has heavier tails than the target, whic…
Fraunhofer-Institute for Intelligent Analysis and Information Systems (IAIS)（弗劳恩霍夫智能分析与信息系统研究所（IAIS））；University of Bonn（波恩大学）
The concept of Label Distribution Learning (LDL) is a technique to stabilize classification and regression problems with ambiguous and/or imbalanced labels. A prototypical use-case of LDL is human age estimation based on profile images. Regarding this regression problem, a so called Deep Label Distribution Learning (DLDL) method has been developed. The main idea is the joint regression of the label distribution and its expectation value. However, the original DLDL method uses loss components with different mathematical motivation and, thus, different scales, which is why the use of a hyperpara…
Nonnegative matrix factorization (NMF) is a standard linear dimensionality reduction technique for nonnegative data sets. In order to measure the discrepancy between the input data and the low-rank approximation, the Kullback-Leibler (KL) divergence is one of the most widely used objective function for NMF. It corresponds to the maximum likehood estimator when the underlying statistics of the observed data sample follows a Poisson distribution, and KL NMF is particularly meaningful for count data sets, such as documents or images. In this paper, we first collect important properties of the KL …
We study an extension of the classic stochastic multi-armed bandit problem which involves multiple plays and Markovian rewards in the rested bandits setting. In order to tackle this problem we consider an adaptive allocation rule which at each stage combines the information from the sample means of all the arms, with the Kullback-Leibler upper confidence bound of a single arm which is selected in round-robin way. For rewards generated from a one-parameter exponential family of Markov chains, we provide a finite-time upper bound for the regret incurred from this adaptive allocation rule, which …
In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator an…
Hunan University（湖南大学）；National University of Defense Technology（国防科技大学）；Southwest University（西南大学）；National Research Center of Parallel Computer Engineering and Technology（国家并行计算机工程技术研究中心）
Recent research has revealed that deep generative models including flow-based models and Variational Autoencoders may assign higher likelihoods to out-of-distribution (OOD) data than in-distribution (ID) data. However, we cannot sample OOD data from the model. This counterintuitive phenomenon has not been satisfactorily explained and brings obstacles to OOD detection with flow-based models. In this paper, we prove theorems to investigate the Kullback-Leibler divergence in flow-based model and give two explanations for the above phenomenon. Based on our theoretical analysis, we propose a new me…
Probabilistic Forecast Reconciliation with Kullback-Leibler Divergence Regularization
AI Lab Lenovo Research（联想研究院AI实验室）；School of Statistics and Mathematics Central University of Finance and Economics（中央财经大学统计与数学学院）；School of Economics and Management Beihang University（北京航空航天大学经济管理学院）
Nara Institute of Science and Technology（奈良先端科学技术大学院大学）
This paper addresses a new interpretation of the traditional optimization method in reinforcement learning (RL) as optimization problems using reverse Kullback-Leibler (KL) divergence, and derives a new optimization method using forward KL divergence, instead of reverse KL divergence in the optimization problems. Although RL originally aims to maximize return indirectly through optimization of policy, the recent work by Levine has proposed a different derivation process with explicit consideration of optimality as stochastic variable. This paper follows this concept and formulates the traditio…
Kullback-Leibler (KL) control enables efficient numerical methods for nonlinear optimal control problems. The crucial assumption of KL control is the full controllability of the transition distribution. However, this assumption is often violated when the dynamics evolves in a continuous space. Consequently, applying KL control to problems with continuous spaces requires some approximation, which leads to the lost of the optimality. To avoid such approximation, in this paper, we reformulate the KL control problem for continuous spaces so that it does not require unrealistic assumptions. The key…
VŠB-Technical University of Ostrava（俄斯特拉发技术大学）；Wroclaw University of Science and Technology（弗罗茨瓦夫科技大学）
Recently, continual learning has received a lot of attention. One of the significant problems is the occurrence of \emph{concept drift}, which consists of changing probabilistic characteristics of the incoming data. In the case of the classification task, this phenomenon destabilizes the model's performance and negatively affects the achieved prediction quality. Most current methods apply statistical learning and similarity analysis over the raw data. However, similarity analysis in streaming data remains a complex problem due to time limitation, non-precise values, fast decision speed, scalab…
This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution $U$ which maximizes the KL divergence from a given distribution $P$ under the assumption that $P$ and $U$ have been generated by distributing a given discrete quantity, a quantum. Quantum distributions naturally represent a wide range of probability distributions that are used in practical applications. Moreover, such a class of distributions can be obtained as an approximation of any probabi…
Hefei University of Technology（合肥工业大学）；University of Science and Technology of China（中国科学技术大学）；Nanyang Technological University（南洋理工大学）；The Chinese University of Hong Kong（香港中文大学）；The University of Hong Kong（香港大学）；Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））
Knowledge distillation (KD), transferring knowledge from a cumbersome teacher model to a lightweight student model, has been investigated to design efficient neural architectures. Generally, the objective function of KD is the Kullback-Leibler (KL) divergence loss between the softened probability distributions of the teacher model and the student model with the temperature scaling hyperparameter tau. Despite its widespread use, few studies have discussed the influence of such softening on generalization. Here, we theoretically show that the KL divergence loss focuses on the logit matching when…
University of Charleston（查尔斯顿大学）；Apple Inc.（苹果公司）；Cerence B. V.（赛轮思公司）
Efficient textual data distributions (TDD) alignment and generation are open research problems in textual analytics and NLP. It is presently difficult to parsimoniously and methodologically confirm that two or more natural language datasets belong to similar distributions, and to identify the extent to which textual data possess alignment. This study focuses on addressing a segment of the broader problem described above by applying multiple supervised and unsupervised machine learning (ML) methods to explore the behavior of TDD by (i) topical alignment, and (ii) by sentiment alignment. Further…
