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Register a new userIndividually Conditional Individual Mutual Information Bound on Generalization Error
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Added by
Ruida Zhou
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 overcomes this issue by reducing the conditioning terms in the conditional mutual information. In the process of establishing this bound, a conditional decoupling lemma is established, which also leads to a meaningful dichotomy and comparison among these information-theoretic bounds.
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The Mutual Information (MI) is an often used measure of dependency between two random variables utilized in information theory, statistics and machine learning. Recently several MI estimators have been proposed that can achieve parametric MSE convergence rate. However, most of the previously proposed estimators have the high computational complexity of at least $O(N^2)$. We propose a unified method for empirical non-parametric estimation of general MI function between random vectors in $mathbb{R}^d$ based on $N$ i.i.d. samples. The reduced complexity MI estimator, called the ensemble dependency graph estimator (EDGE), combines randomized locality sensitive hashing (LSH), dependency graphs, and ensemble bias-reduction methods. We prove that EDGE achieves optimal computational complexity $O(N)$, and can achieve the optimal parametric MSE rate of $O(1/N)$ if the density is $d$ times differentiable. To the best of our knowledge EDGE is the first non-parametric MI estimator that can achieve parametric MSE rates with linear time complexity. We illustrate the utility of EDGE for the analysis of the information plane (IP) in deep learning. Using EDGE we shed light on a controversy on whether or not the compression property of information bottleneck (IB) in fact holds for ReLu and other rectification functions in deep neural networks (DNN).
To provide an efficient approach to characterize the input-output mutual information (MI) under additive white Gaussian noise (AWGN) channel, this short report fits the curves of exact MI under multilevel quadrature amplitude modulation (M-QAM) signal inputs via multi-exponential decay curve fitting (M-EDCF). Even though the definition expression for instanious MI versus Signal to Noise Ratio (SNR) is complex and the containing integral is intractable, our new developed fitting formula holds a neat and compact form, which possesses high precision as well as low complexity. Generally speaking, this approximation formula of MI can promote the research of performance analysis in practical communication system under discrete inputs.
Estimators for mutual information are typically biased. However, in the case of the Kozachenko-Leonenko estimator for metric spaces, a type of nearest neighbour estimator, it is possible to calculate the bias explicitly.
The mutual information between two jointly distributed random variables $X$ and $Y$ is a functional of the joint distribution $P_{XY},$ which is sometimes difficult to handle or estimate. A coarser description of the statistical behavior of $(X,Y)$ is given by the marginal distributions $P_X, P_Y$ and the adjacency relation induced by the joint distribution, where $x$ and $y$ are adjacent if $P(x,y)>0$. We derive a lower bound on the mutual information in terms of these entities. The bound is obtained by viewing the channel from $X$ to $Y$ as a probability distribution on a set of possible actions, where an action determines the output for any possible input, and is independently drawn. We also provide an alternative proof based on convex optimization, that yields a generally tighter bound. Finally, we derive an upper bound on the mutual information in terms of adjacency events between the action and the pair $(X,Y)$, where in this case an action $a$ and a pair $(x,y)$ are adjacent if $y=a(x)$. As an example, we apply our bounds to the binary deletion channel and show that for the special case of an i.i.d. input distribution and a range of deletion probabilities, our lower and upper bounds both outperform the best known bounds for the mutual information.
Besides mimicking bio-chemical and multi-scale communication mechanisms, molecular communication forms a theoretical framework for virus infection processes. Towards this goal, aerosol and droplet transmission has recently been modeled as a multiuser scenario. In this letter, the infection performance is evaluated by means of a mutual information analysis, and by an even simpler probabilistic performance measure which is closely related to absorbed viruses. The so-called infection rate depends on the distribution of the channel input events as well as on the transition probabilities between channel input and output events. The infection rate is investigated analytically for five basic discrete memoryless channel models. Numerical results for the transition probabilities are obtained by Monte Carlo simulations for pathogen-laden particle transmission in four typical indoor environments: two-person office, corridor, classroom, and bus. Particle transfer contributed significantly to infectious diseases like SARS-CoV-2 and influenza.
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