arXiv subjects

Aolin Xu

Publications and source records attributed to Aolin Xu.

13 recordsLinked to original sources

arXiv2026

Minimum Distortion Quantization with Specified Output Distribution

We derive the optimal quantizer of a real-valued random variable $W$ with distribution $P_W$ such that 1) the distribution of the quantization output $X$ that can take $k$ values follows any specified distribution $P_X$ over $\{1,\ldots,k\}$, and 2) the minimum mean squared error (MMSE) of estimating $W$ from $X$ is minimized. It is shown that the optimal quantizer takes the form $X=\sigma\big(F_{\sigma^{-1}(X)}^{-1}(F_W(W))\big)$, where $\sigma$ is the optimal permutation of $\{1,\ldots,k\}$ among all permutations to minimize the MMSE, and $F$ is the cumulative distribution function. When $P_W$ is uniform over an interval or $P_X$ is uniform over $\{1,\ldots,k\}$, the quantizer takes a simple form $X=F_{X}^{-1}(F_W(W))$. The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information between input and output, tailored output distribution to match channel input requirements for communication, and data anonymization.

cs.IT↗

arXiv2026

Optimal Feedback Communication with Information Maximization and Distortion Minimization

We study the problem of optimally sending a real-valued source through multiple uses of a channel with feedback. First, we state a set of conditions that are sufficient for an encoder to achieve maximal mutual information between the source and all the channel outputs. This set of conditions are also necessary when the channel is input-identifiable, a condition widely satisfied by common channel models. More notably, we further study the information maximization-distortion minimization problem, where the mutual information between the source and all channel outputs still needs to be maximized, while at each step, the MMSE of estimating the source from the channel outputs so far also needs to be minimized. We derive a solution to this problem for discrete channels with certain symmetries, e.g. $k$-ary symmetric or $k$-ary erasure channels. We show that for such channels the famous posterior matching scheme, while not necessary for information maximization alone, is sufficient and essentially necessary for achieving both information maximization and distortion minimization. This work also provides a new perspective of regularizing distortion-minimizing feedback communication through information maximization, which enables us to find the optimal solution that otherwise would be intractable.

cs.IT↗

arXiv2025

One if by Land, Two if by Sea, Three if by Four Seas, and More to Come -- Values of Perception, Prediction, Communication, and Common Sense in Decision Making

This work aims to rigorously define the values of perception, prediction, communication, and common sense in decision making. The defined quantities are decision-theoretic, but have information-theoretic analogues, e.g., they share some simple but key mathematical properties with Shannon entropy and mutual information, and can reduce to these quantities in particular settings. One interesting observation is that, the value of perception without prediction can be negative, while the value of perception together with prediction and the value of prediction alone are always nonnegative. The defined quantities suggest answers to practical questions arising in the design of autonomous decision-making systems. Example questions include: Do we need to observe and predict the behavior of a particular agent? How important is it? What is the best order to observe and predict the agents? The defined quantities may also provide insights to cognitive science and neural science, toward the understanding of how natural decision makers make use of information gained from different sources and operations.

cs.IT↗

arXiv2024

Optimal Driver Warning Generation in Dynamic Driving Environment

The driver warning system that alerts the human driver about potential risks during driving is a key feature of an advanced driver assistance system. Existing driver warning technologies, mainly the forward collision warning and unsafe lane change warning, can reduce the risk of collision caused by human errors. However, the current design methods have several major limitations. Firstly, the warnings are mainly generated in a one-shot manner without modeling the ego driver's reactions and surrounding objects, which reduces the flexibility and generality of the system over different scenarios. Additionally, the triggering conditions of warning are mostly rule-based threshold-checking given the current state, which lacks the prediction of the potential risk in a sufficiently long future horizon. In this work, we study the problem of optimally generating driver warnings by considering the interactions among the generated warning, the driver behavior, and the states of ego and surrounding vehicles on a long horizon. The warning generation problem is formulated as a partially observed Markov decision process (POMDP). An optimal warning generation framework is proposed as a solution to the proposed POMDP. The simulation experiments demonstrate the superiority of the proposed solution to the existing warning generation methods.

cs.RO↗

arXiv2022

Bayesian Learning for Dynamic Inference

The traditional statistical inference is static, in the sense that the estimate of the quantity of interest does not affect the future evolution of the quantity. In some sequential estimation problems however, the future values of the quantity to be estimated depend on the estimate of its current value. This type of estimation problems has been formulated as the dynamic inference problem. In this work, we formulate the Bayesian learning problem for dynamic inference, where the unknown quantity-generation model is assumed to be randomly drawn according to a random model parameter. We derive the optimal Bayesian learning rules, both offline and online, to minimize the inference loss. Moreover, learning for dynamic inference can serve as a meta problem, such that all familiar machine learning problems, including supervised learning, imitation learning and reinforcement learning, can be cast as its special cases or variants. Gaining a good understanding of this unifying meta problem thus sheds light on a broad spectrum of machine learning problems as well.

cs.LG↗

arXiv2021

Minimum Excess Risk in Bayesian Learning

We analyze the best achievable performance of Bayesian learning under generative models by defining and upper-bounding the minimum excess risk (MER): the gap between the minimum expected loss attainable by learning from data and the minimum expected loss that could be achieved if the model realization were known. The definition of MER provides a principled way to define different notions of uncertainties in Bayesian learning, including the aleatoric uncertainty and the minimum epistemic uncertainty. Two methods for deriving upper bounds for the MER are presented. The first method, generally suitable for Bayesian learning with a parametric generative model, upper-bounds the MER by the conditional mutual information between the model parameters and the quantity being predicted given the observed data. It allows us to quantify the rate at which the MER decays to zero as more data becomes available. Under realizable models, this method also relates the MER to the richness of the generative function class, notably the VC dimension in binary classification. The second method, particularly suitable for Bayesian learning with a parametric predictive model, relates the MER to the minimum estimation error of the model parameters from data. It explicitly shows how the uncertainty in model parameter estimation translates to the MER and to the final prediction uncertainty. We also extend the definition and analysis of MER to the setting with multiple model families and the setting with nonparametric models. Along the discussions we draw some comparisons between the MER in Bayesian learning and the excess risk in frequentist learning.

cs.LG↗

arXiv2021

Continuity of Generalized Entropy and Statistical Learning

We study the continuity property of the generalized entropy as a function of the underlying probability distribution, defined with an action space and a loss function, and use this property to answer the basic questions in statistical learning theory: the excess risk analyses for various learning methods. We first derive upper and lower bounds for the entropy difference of two distributions in terms of several commonly used f-divergences, the Wasserstein distance, a distance that depends on the action space and the loss function, and the Bregman divergence generated by the entropy, which also induces bounds in terms of the Euclidean distance between the two distributions. Examples are given along with the discussion of each general result, comparisons are made with the existing entropy difference bounds, and new mutual information upper bounds are derived based on the new results. We then apply the entropy difference bounds to the theory of statistical learning. It is shown that the excess risks in the two popular learning paradigms, the frequentist learning and the Bayesian learning, both can be studied with the continuity property of different forms of the generalized entropy. The analysis is then extended to the continuity of generalized conditional entropy. The extension provides performance bounds for Bayes decision making with mismatched distributions. It also leads to excess risk bounds for a third paradigm of learning, where the decision rule is optimally designed under the projection of the empirical distribution to a predefined family of distributions. We thus establish a unified method of excess risk analysis for the three major paradigms of statistical learning, through the continuity of generalized entropy.

cs.LG↗

arXiv2021

Dynamic Inference

Traditional statistical estimation, or statistical inference in general, is static, in the sense that the estimate of the quantity of interest does not change the future evolution of the quantity. In some sequential estimation problems however, we encounter the situation where the future values of the quantity to be estimated depend on the estimate of its current value. Examples include stock price prediction by big investors, interactive product recommendation, and behavior prediction in multi-agent systems. We may call such problems as dynamic inference. In this work, a formulation of this problem under a Bayesian probabilistic framework is given, and the optimal estimation strategy is derived as the solution to minimize the overall inference loss. How the optimal estimation strategy works is illustrated through two examples, stock trend prediction and vehicle behavior prediction. When the underlying models for dynamic inference are unknown, we can consider the problem of learning for dynamic inference. This learning problem can potentially unify several familiar machine learning problems, including supervised learning, imitation learning, and reinforcement learning.

cs.LG↗

arXiv2021

Anytime Decoding by Monte-Carlo Tree Search

An anytime decoding algorithm for tree codes using Monte-Carlo tree search is proposed. The meaning of anytime decoding here is twofold: 1) the decoding algorithm is an anytime algorithm, whose decoding performance improves as more computational resource, measured by decoding time, is allowed, and 2) the proposed decoding algorithm can approximate the maximum-likelihood sequence decoding of tree codes, which has the anytime reliability when the code is properly designed. The above anytime properties are demonstrated through experiments. The proposed method may be extended to the decoding of convolutional codes and block codes by Monte-Carlo trellis search, to enable smooth complexity-performance trade-offs in these decoding tasks. Some other extensions and possible improvements are also discussed.

cs.IT↗

arXiv2017

Information-theoretic analysis of generalization capability of learning algorithms

We derive upper bounds on the generalization error of a learning algorithm in terms of the mutual information between its input and output. The bounds provide an information-theoretic understanding of generalization in learning problems, and give theoretical guidelines for striking the right balance between data fit and generalization by controlling the input-output mutual information. We propose a number of methods for this purpose, among which are algorithms that regularize the ERM algorithm with relative entropy or with random noise. Our work extends and leads to nontrivial improvements on the recent results of Russo and Zou.

cs.LG↗

arXiv2017

Information-theoretic lower bounds for distributed function computation

We derive information-theoretic converses (i.e., lower bounds) for the minimum time required by any algorithm for distributed function computation over a network of point-to-point channels with finite capacity, where each node of the network initially has a random observation and aims to compute a common function of all observations to a given accuracy with a given confidence by exchanging messages with its neighbors. We obtain the lower bounds on computation time by examining the conditional mutual information between the actual function value and its estimate at an arbitrary node, given the observations in an arbitrary subset of nodes containing that node. The main contributions include: 1) A lower bound on the conditional mutual information via so-called small ball probabilities, which captures the dependence of the computation time on the joint distribution of the observations at the nodes, the structure of the function, and the accuracy requirement. For linear functions, the small ball probability can be expressed by Lévy concentration functions of sums of independent random variables, for which tight estimates are available that lead to strict improvements over existing lower bounds on computation time. 2) An upper bound on the conditional mutual information via strong data processing inequalities, which complements and strengthens existing cutset-capacity upper bounds. 3) A multi-cutset analysis that quantifies the loss (dissipation) of the information needed for computation as it flows across a succession of cutsets in the network. This analysis is based on reducing a general network to a line network with bidirectional links and self-links, and the results highlight the dependence of the computation time on the diameter of the network, a fundamental parameter that is missing from most of the existing lower bounds on computation time.

cs.IT↗

arXiv2016

Information-Theoretic Lower Bounds on Bayes Risk in Decentralized Estimation

We derive lower bounds on the Bayes risk in decentralized estimation, where the estimator does not have direct access to the random samples generated conditionally on the random parameter of interest, but only to the data received from local processors that observe the samples. The received data are subject to communication constraints, due to quantization and the noise in the communication channels from the processors to the estimator. We first derive general lower bounds on the Bayes risk using information-theoretic quantities, such as mutual information, information density, small ball probability, and differential entropy. We then apply these lower bounds to the decentralized case, using strong data processing inequalities to quantify the contraction of information due to communication constraints. We treat the cases of a single processor and of multiple processors, where the samples observed by different processors may be conditionally dependent given the parameter, for noninteractive and interactive communication protocols. Our results recover and improve recent lower bounds on the Bayes risk and the minimax risk for certain decentralized estimation problems, where previously only conditionally independent sample sets and noiseless channels have been considered. Moreover, our results provide a general way to quantify the degradation of estimation performance caused by distributing resources to multiple processors, which is only discussed for specific examples in existing works.

cs.IT↗

arXiv2015

Converses for distributed estimation via strong data processing inequalities

We consider the problem of distributed estimation, where local processors observe independent samples conditioned on a common random parameter of interest, map the observations to a finite number of bits, and send these bits to a remote estimator over independent noisy channels. We derive converse results for this problem, such as lower bounds on Bayes risk. The main technical tools include a lower bound on the Bayes risk via mutual information and small ball probability, as well as strong data processing inequalities for the relative entropy. Our results can recover and improve some existing results on distributed estimation with noiseless channels, and also capture the effect of noisy channels on the estimation performance.

cs.IT↗