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1. Non-asymptotic Performance Guarantees for Neural Estimation of f-divergences

   Sreekumar, Sreejith; Zhang, Zhengxin; Goldfeld, Ziv (January 2022, Proceedings of Machine Learning Research)

   Statistical distances (SDs), which quantify the dissimilarity between probability distributions, are central to machine learning and statistics. A modern method for estimating such distances from data relies on parametrizing a variational form by a neural network (NN) and optimizing it. These estimators are abundantly used in practice, but corresponding performance guarantees are partial and call for further exploration. In particular, there seems to be a fundamental tradeoff between the two sources of error involved: approximation and estimation. While the former needs the NN class to be rich and expressive, the latter relies on controlling complexity. This paper explores this tradeoff by means of non-asymptotic error bounds, focusing on three popular choices of SDs—Kullback-Leibler divergence, chi-squared divergence, and squared Hellinger distance. Our analysis relies on non-asymptotic function approximation theorems and tools from empirical process theory. Numerical results validating the theory are also provided. 

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2. Non-asymptotic Performance Guarantees for Neural Estimation of f-Divergences

   Sreejith Sreekumar, Zhengxin Zhang (January 2021, Proeedings of the International Workshop on Artificial Intelligence and Statistics)

   null (Ed.)

   Statistical distances (SDs), which quantify the dissimilarity between probability distri- butions, are central to machine learning and statistics. A modern method for esti- mating such distances from data relies on parametrizing a variational form by a neu- ral network (NN) and optimizing it. These estimators are abundantly used in prac- tice, but corresponding performance guar- antees are partial and call for further ex- ploration. In particular, there seems to be a fundamental tradeoff between the two sources of error involved: approximation and estimation. While the former needs the NN class to be rich and expressive, the latter relies on controlling complexity. This paper explores this tradeoff by means of non-asymptotic error bounds, focusing on three popular choices of SDs—Kullback- Leibler divergence, chi-squared divergence, and squared Hellinger distance. Our analysis relies on non-asymptotic function approxima- tion theorems and tools from empirical pro- cess theory. Numerical results validating the theory are also provided. 

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3. Indirect inference for time series using the empirical characteristic function and control variates

   https://doi.org/10.1111/jtsa.12582  

   Davis, Richard A.; do Rêgo Sousa, Thiago; Klüppelberg, Claudia (February 2021, Journal of Time Series Analysis)

   We estimate the parameter of a stationary time series process by minimizing the integrated weighted mean squared error between the empirical and simulated characteristic function, when the true characteristic functions cannot be explicitly computed. Motivated by Indirect Inference, we use a Monte Carlo approximation of the characteristic function based on i.i.d. simulated blocks. As a classical variance reduction technique, we propose the use of control variates for reducing the variance of this Monte Carlo approximation. These two approximations yield two new estimators that are applicable to a large class of time series processes. We show consistency and asymptotic normality of the parameter estimators under strong mixing, moment conditions, and smoothness of the simulated blocks with respect to its parameter. In a simulation study we show the good performance of these new simulation based estimators, and the superiority of the control variates based estimator for Poisson driven time series of counts. 

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4. Consistency of empirical distributions of sequences of graph statistics in networks with dependent edges

   https://doi.org/10.1016/j.jmva.2025.105420  

   Stewart, Jonathan R (February 2025, Journal of Multivariate Analysis)

   One of the first steps in applications of statistical network analysis is frequently to produce summary charts of important features of the network. Many of these features take the form of sequences of graph statistics counting the number of realized events in the network, examples of which are degree distributions, edgewise shared partner distributions, and more. We provide conditions under which the empirical distributions of sequences of graph statistics are consistent in the L-infinity-norm in settings where edges in the network are dependent. We accomplish this task by deriving concentration inequalities that bound probabilities of deviations of graph statistics from the expected value under weak dependence conditions. We apply our concentration inequalities to empirical distributions of sequences of graph statistics and derive non-asymptotic bounds on the L-infinity-error which hold with high probability. Our non-asymptotic results are then extended to demonstrate uniform convergence almost surely in selected examples. We illustrate theoretical results through examples, simulation studies, and an application. 

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5. Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks

   https://doi.org/10.1109/TIT.2022.3208653  

   Parhi, Rahul; Nowak, Robert D (February 2023, IEEE Transactions on Information Theory)

   We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared Euclidean norm of the network weights. This minimization corresponds to the common approach of training a neural network with weight decay. We quantify the performance (mean-squared error) of these neural network estimators when the data-generating function belongs to the second-order Radon-domain bounded variation space. This space of functions was recently proposed as the natural function space associated with shallow ReLU neural networks. We derive a minimax lower bound for the estimation problem for this function space and show that the neural network estimators are minimax optimal up to logarithmic factors. This minimax rate is immune to the curse of dimensionality. We quantify an explicit gap between neural networks and linear methods (which include kernel methods) by deriving a linear minimax lower bound for the estimation problem, showing that linear methods necessarily suffer the curse of dimensionality in this function space. As a result, this paper sheds light on the phenomenon that neural networks seem to break the curse of dimensionality. 

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