我们研究各臂奖励分布的支撑集均为[0,1]区间的K臂老虎机问题。在该设定下，设计具有高效后悔性能的随机探索算法一直是一项挑战。Maillard采样（首次提出于文献[maillard13apprentissage]）是Thompson采样的一种有吸引力的替代方案，近期研究（文献[bian2022maillard]）表明其在次高斯奖励设定下能取得具有竞争力的后悔保证，同时保留闭式动作概率，这一特性对离线策略评估十分有用。在本工作中，我们提出Kullback-Leibler Maillard采样（KL-MS）算法，它是Maillard采样的自然扩展，旨在实现KL形式的依赖间隙的后悔界。我们证明，当奖励服从伯努利分布时，KL-MS具有渐近最优性，其最坏情况后悔界形式为O(√(μ*(1-μ*)KT lnK) + K lnT)，其中μ*为最优臂的期望奖励，T为时间步长总数。
英文摘要：
We study $K$-armed bandit problems where the reward distributions of the arms are all supported on the $[0,1]$ interval. It has been a challenge to design regret-efficient randomized exploration algorithms in this setting. Maillard sampling \cite{maillard13apprentissage}, an attractive alternative to Thompson sampling, has recently been shown to achieve competitive regret guarantees in the sub-Gaussian reward setting \cite{bian2022maillard} while maintaining closed-form action probabilities, which is useful for offline policy evaluation. In this work, we propose the Kullback-Leibler Maillard Sampling (KL-MS) algorithm, a natural extension of Maillard sampling for achieving KL-style gap-dependent regret bound. We show that KL-MS enjoys the asymptotic optimality when the rewards are Bernoulli and has a worst-case regret bound of the form $O(\sqrt{μ^*(1-μ^*) K T \ln K} + K \ln T)$, where $μ^*$ is the expected reward of the optimal arm, and $T$ is the time horizon length.
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…
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 …
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 …
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 …
We consider model-based reinforcement learning in finite Markov De- cision Processes (MDPs), focussing on so-called optimistic strategies. In MDPs, optimism can be implemented by carrying out extended value it- erations under a constraint of consistency with the estimated model tran- sition probabilities. The UCRL2 algorithm by Auer, Jaksch and Ortner (2009), which follows this strategy, has recently been shown to guarantee near-optimal regret bounds. In this paper, we strongly argue in favor of using the Kullback-Leibler (KL) divergence for this purpose. By studying the linear maximization pr…
We consider a Kullback-Leibler-based algorithm for the stochastic multi-armed bandit problem in the case of distributions with finite supports (not necessarily known beforehand), whose asymptotic regret matches the lower bound of \cite{Burnetas96}. Our contribution is to provide a finite-time analysis of this algorithm; we get bounds whose main terms are smaller than the ones of previously known algorithms with finite-time analyses (like UCB-type algorithms).
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…
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…
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…
Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We …
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…
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）
A brain computer interface (BCI) is a system which provides direct communication between the mind of a person and the outside world by using only brain activity (EEG). The event-related potential (ERP)-based BCI problem consists of a binary pattern recognition. Linear discriminant analysis (LDA) is widely used to solve this type of classification problems, but it fails when the number of features is large relative to the number of observations. In this work we propose a penalized version of the sparse discriminant analysis (SDA), called Kullback-Leibler penalized sparse discriminant analysis (…
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）
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室）
Due to the success of the bag-of-word modeling paradigm, clustering histograms has become an important ingredient of modern information processing. Clustering histograms can be performed using the celebrated $k$-means centroid-based algorithm. From the viewpoint of applications, it is usually required to deal with symmetric distances. In this letter, we consider the Jeffreys divergence that symmetrizes the Kullback-Leibler divergence, and investigate the computation of Jeffreys centroids. We first prove that the Jeffreys centroid can be expressed analytically using the Lambert $W$ function for…
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.
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…
Université Paris Descartes（巴黎笛卡尔大学）；CNRS（法国国家科学研究中心）
We measure the influence of individual observations on the sequence of the hidden states of the Hidden Markov Model (HMM) by means of the Kullback-Leibler distance (KLD). Namely, we consider the KLD between the conditional distribution of the hidden states' chain given the complete sequence of observations and the conditional distribution of the hidden chain given all the observations but the one under consideration. We introduce a linear complexity algorithm for computing the influence of all the observations. As an illustration, we investigate the application of our algorithm to the problem …
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 …
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 …
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…
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…
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）
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…
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 …
Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrices. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses such as the least-squares loss. In this article, we propose a first-order primal-dual algorithm for non-negative decomposition problems (where one factor is fixed) with the KL divergence, based on the Chambolle-Pock algorithm. All required computations may be …
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…
