We provide an information-theoretic framework for studying the generalization properties of machine learning algorithms. Our framework ties together existing approaches, including uniform convergence bounds and recent methods for adaptive data analysis. Specifically, we use Conditional Mutual Information (CMI) to quantify how well the input (i.e., the training data) can be recognized given the output (i.e., the trained model) of the learning algorithm. We show that bounds on CMI can be obtained from VC dimension, compression schemes, differential privacy, and other methods. We then show that bounded CMI implies various forms of generalization.
We propose a new information-theoretic bound on generalization error based on a combination of the error decomposition technique of Bu et al. and the conditional mutual information (CMI) construction of Steinke and Zakynthinou. In a previous work, Haghifam et al. proposed a different bound combining the two aforementioned techniques, which we refer to as the conditional individual mutual information (CIMI) bound. However, in a simple Gaussian setting, both the CMI and the CIMI bounds are order-wise worse than that by Bu et al.. This observation motivated us to propose the new bound, which over…
University of Toronto（多伦多大学）；Vector Institute（矢量研究所）；Element AI（Element AI公司）
The information-theoretic framework of Russo and J. Zou (2016) and Xu and Raginsky (2017) provides bounds on the generalization error of a learning algorithm in terms of the mutual information between the algorithm's output and the training sample. In this work, we study the proposal, by Steinke and Zakynthinou (2020), to reason about the generalization error of a learning algorithm by introducing a super sample that contains the training sample as a random subset and computing mutual information conditional on the super sample. We first show that these new bounds based on the conditional mutu…
CWI Amsterdam（阿姆斯特丹数学与计算机科学国家研究所）；Leiden University（莱顿大学）；Google Research（谷歌研究院）；Khoury College of Computer Sciences, Northeastern University（东北大学库里计算机科学学院）
We give a novel, unified derivation of conditional PAC-Bayesian and mutual information (MI) generalization bounds. We derive conditional MI bounds as an instance, with special choice of prior, of conditional MAC-Bayesian (Mean Approximately Correct) bounds, itself derived from conditional PAC-Bayesian bounds, where `conditional' means that one can use priors conditioned on a joint training and ghost sample. This allows us to get nontrivial PAC-Bayes and MI-style bounds for general VC classes, something recently shown to be impossible with standard PAC-Bayesian/MI bounds. Second, it allows us t…
Chalmers University of Technology（查尔姆斯理工大学）；Kings College London（伦敦国王学院）
Meta-learning optimizes an inductive bias---typically in the form of the hyperparameters of a base-learning algorithm---by observing data from a finite number of related tasks. This paper presents an information-theoretic bound on the generalization performance of any given meta-learner, which builds on the conditional mutual information (CMI) framework of Steinke and Zakynthinou (2020). In the proposed extension to meta-learning, the CMI bound involves a training \textit{meta-supersample} obtained by first sampling $2N$ independent tasks from the task environment, and then drawing $2M$ indepe…
The conditional mutual information I(X;Y|Z) measures the average information that X and Y contain about each other given Z. This is an important primitive in many learning problems including conditional independence testing, graphical model inference, causal strength estimation and time-series problems. In several applications, it is desirable to have a functional purely of the conditional distribution p_{Y|X,Z} rather than of the joint distribution p_{X,Y,Z}. We define the potential conditional mutual information as the conditional mutual information calculated with a modified joint distribut…
Virginia State University（弗吉尼亚州立大学）；Henan Normal University（河南师范大学）
Due to the advantage of achieving a better performance under weak regularization, elastic net has attracted wide attention in statistics, machine learning, bioinformatics, and other fields. In particular, a variation of the elastic net, adaptive elastic net (AEN), integrates the adaptive grouping effect. In this paper, we aim to develop a new algorithm: Adaptive Elastic Net with Conditional Mutual Information (AEN-CMI) that further improves AEN by incorporating conditional mutual information into the gene selection process. We apply this new algorithm to screen significant genes for two kinds …
IIT Delhi（印度理工学院德里分校）；University of Washington（华盛顿大学）；TIFR Mumbai（塔塔基础研究院孟买分院）
Estimation of information theoretic quantities such as mutual information and its conditional variant has drawn interest in recent times owing to their multifaceted applications. Newly proposed neural estimators for these quantities have overcome severe drawbacks of classical $k$NN-based estimators in high dimensions. In this work, we focus on conditional mutual information (CMI) estimation by utilizing its formulation as a minmax optimization problem. Such a formulation leads to a joint training procedure similar to that of generative adversarial networks. We find that our proposed estimator …
Several recent works in communication systems have proposed to leverage the power of neural networks in the design of encoders and decoders. In this approach, these blocks can be tailored to maximize the transmission rate based on aggregated samples from the channel. Motivated by the fact that, in many communication schemes, the achievable transmission rate is determined by a conditional mutual information term, this paper focuses on neural-based estimators for this information-theoretic quantity. Our results are based on variational bounds for the KL-divergence and, in contrast to some previo…
University of Coimbra（科英布拉大学）；Institute of Systems and Robotics（系统与机器人研究所）
This paper presents a novel feature selection method based on the conditional mutual information (CMI). The proposed High Order Conditional Mutual Information Maximization (HOCMIM) incorporates high order dependencies into the feature selection procedure and has a straightforward interpretation due to its bottom-up derivation. The HOCMIM is derived from the CMI's chain expansion and expressed as a maximization optimization problem. The maximization problem is solved using a greedy search procedure, which speeds up the entire feature selection process. The experiments are run on a set of benchm…
Conditional Mutual Information (CMI) is a measure of conditional dependence between random variables X and Y, given another random variable Z. It can be used to quantify conditional dependence among variables in many data-driven inference problems such as graphical models, causal learning, feature selection and time-series analysis. While k-nearest neighbor (kNN) based estimators as well as kernel-based methods have been widely used for CMI estimation, they suffer severely from the curse of dimensionality. In this paper, we leverage advances in classifiers and generative models to design metho…
The estimation of mutual information (MI) or conditional mutual information (CMI) from a set of samples is a long-standing problem. A recent line of work in this area has leveraged the approximation power of artificial neural networks and has shown improvements over conventional methods. One important challenge in this new approach is the need to obtain, given the original dataset, a different set where the samples are distributed according to a specific product density function. This is particularly challenging when estimating CMI. In this paper, we introduce a new technique, based on k neare…
National Chung Hsing University（国立中兴大学）；IBM Research（IBM研究院）
Recent studies have found that deep learning systems are vulnerable to adversarial examples; e.g., visually unrecognizable adversarial images can easily be crafted to result in misclassification. The robustness of neural networks has been studied extensively in the context of adversary detection, which compares a metric that exhibits strong discriminate power between natural and adversarial examples. In this paper, we propose to characterize the adversarial subspaces through the lens of mutual information (MI) approximated by conditional generation methods. We use MI as an information-theoreti…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Markov blanket feature selection, while theoretically optimal, is generally challenging to implement. This is due to the shortcomings of existing approaches to conditional independence (CI) testing, which tend to struggle either with the curse of dimensionality or computational complexity. We propose a novel two-step approach which facilitates Markov blanket feature selection in high dimensions. First, neural networks are used to map features to low-dimensional representations. In the second step, CI testing is performed by applying the $k$-NN conditional mutual information estimator to the le…
We derive information theoretic generalization bounds for supervised learning algorithms based on a new measure of leave-one-out conditional mutual information (loo-CMI). Contrary to other CMI bounds, which are black-box bounds that do not exploit the structure of the problem and may be hard to evaluate in practice, our loo-CMI bounds can be computed easily and can be interpreted in connection to other notions such as classical leave-one-out cross-validation, stability of the optimization algorithm, and the geometry of the loss-landscape. It applies both to the output of training algorithms as…
Towards Dynamic Feature Acquisition on Medical Time Series by Maximizing Conditional Mutual Information
ETH Zurich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）；Helmholtz AI（亥姆霍兹人工智能研究所）；Technical University of Munich（慕尼黑工业大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）
KTH Royal Institute of Technology（瑞典皇家理工学院）；RISE Research Institutes of Sweden（瑞典RISE研究院）
We present a representation learning framework for financial time series forecasting. One challenge of using deep learning models for finance forecasting is the shortage of available training data when using small datasets. Direct trend classification using deep neural networks trained on small datasets is susceptible to the overfitting problem. In this paper, we propose to first learn compact representations from time series data, then use the learned representations to train a simpler model for predicting time series movements. We consider a class-conditioned latent variable model. We train …
University of Texas at Dallas（德克萨斯大学达拉斯分校）；University of Washington（华盛顿大学）；Tata Institute of Fundamental Research（塔塔基础研究院）
Information-theoretic quantities like entropy and mutual information have found numerous uses in machine learning. It is well known that there is a strong connection between these entropic quantities and submodularity since entropy over a set of random variables is submodular. In this paper, we study combinatorial information measures that generalize independence, (conditional) entropy, (conditional) mutual information, and total correlation defined over sets of (not necessarily random) variables. These measures strictly generalize the corresponding entropic measures since they are all paramet…
Multimodal image-to-image translation (I2IT) aims to learn a conditional distribution that explores multiple possible images in the target domain given an input image in the source domain. Conditional generative adversarial networks (cGANs) are often adopted for modeling such a conditional distribution. However, cGANs are prone to ignore the latent code and learn a unimodal distribution in conditional image synthesis, which is also known as the mode collapse issue of GANs. To solve the problem, we propose a simple yet effective method that explicitly estimates and maximizes the mutual informat…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Technion（以色列理工学院）；Google Research（谷歌研究院）；Mila（米拉研究所）；McGill（麦吉尔大学）
We study the mutual information between (certain summaries of) the output of a learning algorithm and its $n$ training data, conditional on a supersample of $n+1$ i.i.d. data from which the training data is chosen at random without replacement. These leave-one-out variants of the conditional mutual information (CMI) of an algorithm (Steinke and Zakynthinou, 2020) are also seen to control the mean generalization error of learning algorithms with bounded loss functions. For learning algorithms achieving zero empirical risk under 0-1 loss (i.e., interpolating algorithms), we provide an explicit c…
Normalized Conditional Mutual Information Surrogate Loss for Deep Neural Classifiers
Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Canada（滑铁卢大学电气与计算机工程系）；The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, Canada（多伦多大学Edward S. Rogers Sr.电气与计算机工程系）
Recent methods for learning unsupervised visual representations, dubbed contrastive learning, optimize the noise-contrastive estimation (NCE) bound on mutual information between two views of an image. NCE uses randomly sampled negative examples to normalize the objective. In this paper, we show that choosing difficult negatives, or those more similar to the current instance, can yield stronger representations. To do this, we introduce a family of mutual information estimators that sample negatives conditionally -- in a "ring" around each positive. We prove that these estimators lower-bound mut…
Pruning Deep Convolutional Neural Network Using Conditional Mutual Information
Faculty of Computer Science and Engineering, Ho Chi Minh City University of Technology（胡志明市技术大学计算机科学与工程学院）；Loyola University Maryland（马里兰州洛约拉大学）；Tufts University（塔夫茨大学）
Arizona State University（亚利桑那州立大学）；Bosch Research and Technology Center（博世研究与技术中心）；University of Michigan（密歇根大学）
Mutual Information (MI) is often used for feature selection when developing classifier models. Estimating the MI for a subset of features is often intractable. We demonstrate, that under the assumptions of conditional independence, MI between a subset of features can be expressed as the Conditional Mutual Information (CMI) between pairs of features. But selecting features with the highest CMI turns out to be a hard combinatorial problem. In this work, we have applied two unique global methods, Truncated Power Method (TPower) and Low Rank Bilinear Approximation (LowRank), to solve the feature s…
Efficient Sample-optimal Learning of Gaussian Tree Models via Sample-optimal Testing of Gaussian Mutual Information
Indian Institute of Technology, Kanpur（坎普尔印度理工学院）；Fujitsu Research（富士通研究所）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
We propose a mutual information-based sufficient representation learning (MSRL) approach, which uses the variational formulation of the mutual information and leverages the approximation power of deep neural networks. MSRL learns a sufficient representation with the maximum mutual information with the response and a user-selected distribution. It can easily handle multi-dimensional continuous or categorical response variables. MSRL is shown to be consistent in the sense that the conditional probability density function of the response variable given the learned representation converges to the …
Efficient Process Reward Modeling via Contrastive Mutual Information
Department of Electrical and Computer Engineering, Seoul National University（首尔大学电气与计算机工程系）；Department of Computer Science and Engineering, Konkuk University（建国大学计算机科学与工程系）
Towards Undistillable Models by Minimizing Conditional Mutual Information
Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto（多伦多大学电气与计算机工程系）；Department of Electrical Engineering Stanford University（斯坦福大学电气工程系）；Department of Electrical and Computer Engineering University of Waterloo（滑铁卢大学电气与计算机工程系）
Information sharing is key in building team cognition and enables coordination and cooperation. High-performing human teams also benefit from acting strategically with hierarchical levels of iterated communication and rationalizability, meaning a human agent can reason about the actions of their teammates in their decision-making. Yet, the majority of prior work in Multi-Agent Reinforcement Learning (MARL) does not support iterated rationalizability and only encourage inter-agent communication, resulting in a suboptimal equilibrium cooperation strategy. In this work, we show that reformulating…
