On the Monotonicity of Information Aging

In this paper, we analyze the monotonicity of information aging in a remote estimation system, where historical observations of a Gaussian autoregressive AR(p) process are used to predict its future values. We consider two widely used loss functions in estimation: (i) logarithmic loss function for maximum likelihood estimation and (ii) quadratic loss function for MMSE estimation. The estimation error of the AR(p) process is written as a generalized conditional entropy which has closed-form expressions. By using a new information-theoretic tool called $\epsilon$-Markov chain, we can evaluate the divergence of the AR(p) process from being a Markov chain. When the divergence $\epsilon$ is large, the estimation error of the AR(p) process can be far from a non-decreasing function of the Age of Information (AoI). Conversely, for small divergence $\epsilon$, the inference error is close to a non-decreasing AoI function. Each observation is a short sequence taken from the AR(p) process. As the observation sequence length increases, the parameter $\epsilon$ progressively reduces to zero, and hence the estimation error becomes a non-decreasing AoI function. These results underscore a connection between the monotonicity of information aging and the divergence of from being a Markov chain.

Learning-augmented Online Minimization of Age of Information and Transmission Costs

We consider a discrete-time system where a resource-constrained source (e.g., a small sensor) transmits its time-sensitive data to a destination over a time-varying wireless channel. Each transmission incurs a fixed transmission cost (e.g., energy cost), and no transmission results in a staleness cost represented by the Age-of-Information. The source must balance the tradeoff between transmission and staleness costs. To address this challenge, we develop a robust online algorithm to minimize the sum of transmission and staleness costs, ensuring a worst-case performance guarantee. While online algorithms are robust, they are usually overly conservative and may have a poor average performance in typical scenarios. In contrast, by leveraging historical data and prediction models, machine learning (ML) algorithms perform well in average cases. However, they typically lack worst-case performance guarantees. To achieve the best of both worlds, we design a learning-augmented online algorithm that exhibits two desired properties: (i) consistency: closely approximating the optimal offline algorithm when the ML prediction is accurate and trusted; (ii) robustness: ensuring worst-case performance guarantee even ML predictions are inaccurate. Finally, we perform extensive simulations to show that our online algorithm performs well empirically and that our learning-augmented algorithm achieves both consistency and robustness.

Hoeffding's Inequality for Markov Chains under Generalized Concentrability Condition

This paper studies Hoeffding's inequality for Markov chains under the generalized concentrability condition defined via integral probability metric (IPM). The generalized concentrability condition establishes a framework that interpolates and extends the existing hypotheses of Markov chain Hoeffding-type inequalities. The flexibility of our framework allows Hoeffding's inequality to be applied beyond the ergodic Markov chains in the traditional sense. We demonstrate the utility by applying our framework to several non-asymptotic analyses arising from the field of machine learning, including (i) a generalization bound for empirical risk minimization with Markovian samples, (ii) a finite sample guarantee for Ployak-Ruppert averaging of SGD, and (iii) a new regret bound for rested Markovian bandits with general state space.

How Does Data Freshness Affect Real-time Supervised Learning?

In this paper, we analyze the impact of data freshness on real-time supervised learning, where a neural network is trained to infer a time-varying target (e.g., the position of the vehicle in front) based on features (e.g., video frames) observed at a sensing node (e.g., camera or lidar). One might expect that the performance of real-time supervised learning degrades monotonically as the feature becomes stale. Using an information-theoretic analysis, we show that this is true if the feature and target data sequence can be closely approximated as a Markov chain; it is not true if the data sequence is far from Markovian. Hence, the prediction error of real-time supervised learning is a function of the Age of Information (AoI), where the function could be non-monotonic. Several experiments are conducted to illustrate the monotonic and non-monotonic behaviors of the prediction error. To minimize the inference error in real-time, we propose a new "selection-from-buffer" model for sending the features, which is more general than the "generate-at-will" model used in earlier studies. By using Gittins and Whittle indices, low-complexity scheduling strategies are developed to minimize the inference error, where a new connection between the Gittins index theory and Age of Information (AoI) minimization is discovered. These scheduling results hold (i) for minimizing general AoI functions (monotonic or non-monotonic) and (ii) for general feature transmission time distributions. Data-driven evaluations are presented to illustrate the benefits of the proposed scheduling algorithms.

A Local Geometric Interpretation of Feature Extraction in Deep Feedforward Neural Networks

In this paper, we present a local geometric analysis to interpret how deep feedforward neural networks extract low-dimensional features from high-dimensional data. Our study shows that, in a local geometric region, the optimal weight in one layer of the neural network and the optimal feature generated by the previous layer comprise a low-rank approximation of a matrix that is determined by the Bayes action of this layer. This result holds (i) for analyzing both the output layer and the hidden layers of the neural network, and (ii) for neuron activation functions with non-vanishing gradients. We use two supervised learning problems to illustrate our results: neural network based maximum likelihood classification (i.e., softmax regression) and neural network based minimum mean square estimation. Experimental validation of these theoretical results will be conducted in our future work.

The Age of Correlated Features in Supervised Learning based Forecasting

In this paper, we analyze the impact of information freshness on supervised learning based forecasting. In these applications, a neural network is trained to predict a time-varying target (e.g., solar power), based on multiple correlated features (e.g., temperature, humidity, and cloud coverage). The features are collected from different data sources and are subject to heterogeneous and time-varying ages. By using an information-theoretic approach, we prove that the minimum training loss is a function of the ages of the features, where the function is not always monotonic. However, if the empirical distribution of the training data is close to the distribution of a Markov chain, then the training loss is approximately a non-decreasing age function. Both the training loss and testing loss depict similar growth patterns as the age increases. An experiment on solar power prediction is conducted to validate our theory. Our theoretical and experimental results suggest that it is beneficial to (i) combine the training data with different age values into a large training dataset and jointly train the forecasting decisions for these age values, and (ii) feed the age value as a part of the input feature to the neural network.

Multi-Armed Bandits with Dependent Arms

We study a variant of the classical multi-armed bandit problem (MABP) which we call as Multi-Armed Bandits with dependent arms. More specifically, multiple arms are grouped together to form a cluster, and the reward distributions of arms belonging to the same cluster are known functions of an unknown parameter that is a characteristic of the cluster. Thus, pulling an arm $i$ not only reveals information about its own reward distribution, but also about all those arms that share the same cluster with arm $i$. This "correlation" amongst the arms complicates the exploration-exploitation trade-off that is encountered in the MABP because the observation dependencies allow us to test simultaneously multiple hypotheses regarding the optimality of an arm. We develop learning algorithms based on the UCB principle which utilize these additional side observations appropriately while performing exploration-exploitation trade-off. We show that the regret of our algorithms grows as $O(K\log T)$, where $K$ is the number of clusters. In contrast, for an algorithm such as the vanilla UCB that is optimal for the classical MABP and does not utilize these dependencies, the regret scales as $O(M\log T)$ where $M$ is the number of arms.