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Title: A Data-Efficient Mutual Information Neural Estimator for Statistical Dependency Testing

Paper content:
Measuring Mutual Information (MI) between high-dimensional, continuous, random variables from observed samples has wide theoretical and practical applications.
Recent works have developed accurate MI estimators through provably low-bias approximations and tight variational lower bounds assuming abundant supply of samples, but require an unrealistic number of samples to guarantee statistical significance of the estimation.
In this work, we focus on improving data efficiency and propose a Data-Efficient MINE Estimator (DEMINE) that can provide a tight lower confident interval of MI under limited data, through adding cross-validation to the MINE lower bound (Belghazi et al., 2018).
Hyperparameter search is employed and a novel meta-learning approach with task augmentation is developed to increase robustness to hyperparamters, reduce overfitting and improve accuracy.
With improved data-efficiency, our DEMINE estimator enables statistical testing of dependency at practical dataset sizes.
We demonstrate the effectiveness of DEMINE on synthetic benchmarks and a real world fMRI dataset, with application of inter-subject correlation analysis.
Mutual Information (MI) is an important, theoretically grounded measure of similarity between random variables.
MI captures general, non-linear, statistical dependencies between random variables.
MI estimators that estimate MI from samples are important tools widely used in not only subjects such as physics and neuroscience, but also machine learning ranging from feature selection and representation learning to explaining decisions and analyzing generalization of neural networks.
Existing studies on MI estimation between general random variables focus on deriving asymptotic lower bounds and approximations to MI under infinite data, and techniques for reducing estimator bias such as bias correction, improved signal modeling with neural networks and tighter lower bounds.
Widely used approaches include the k-NN-based KSG estimator (Kraskov et al., 2004) and the variational lower-bound-based Mutual Information Neural Estimator (MINE) family (Belghazi et al., 2018; Poole et al., 2018) .
Despite the empirical and asymptotic bias improvements, MI estimation has not seen wide adoption.
The challenges are two-fold.
First, the analysis of dependencies among variables -let alone any MI analyses for scientific studies -requires not only an MI estimate, but also confidence intervals (Holmes & Nemenman, 2019) around the estimate to quantify uncertainty and statistical significance.
Existing MI estimators, however, do not provide confidence intervals.
As low probability events may still carry a significant amount of information, the MI estimates could vary greatly given additional observations (Poole et al., 2018) .
Towards providing upper and lower bounds of true MI under limited number of observations, existing MI lower bound techniques assume infinite data and would need further relaxations when a limited number of observations are provided.
Closest to our work, Belghazi et al. (2018) studied the lower bound of the MINE estimator under limited data, but it involves bounds on generalization error of the signal model and would not yield useful confidence intervals for realistic datasets.
Second, practical MI estimators should be insensitive to the choice of hyperparameters.
An estimator should return a single MI estimate with its confidence interval irrespective of the type of the data and the number of observations.
For learning-based approaches, this means that the model design and optimization hyperparameters need to not only be determined automatically but also taken into account when computing the confidence interval.
Towards addressing these challenges, our estimator, DEMINE, introduces a predictive MI lower bound for limited samples that enables statistical dependency testing under practical dataset sizes.
Our estimator builds on top of the MINE estimator family, but performs cross-validation to remove the need to bound generalization error.
This yields a much tighter lower bound agnostic to hyperparameter search.
We automatically selected hyperparameters through hyperparameter search, and a new cross-validation meta-learning approach is developed, based upon few-shot meta-learning, to automatically decide initialization of model parameters.
Meta-overfitting is strongly controlled through task augmentation, a new task generation approach for meta-learning.
With these improvements, we show that DEMINE enables practical statistical testing of dependency for not only synthetic datasets but also for real world functional Magnetic Resonance Imaging (fMRI) data analysis capturing nonlinear and higher-order brain-to-brain coupling.
Our contributions are summarized as follows:
1) A data-efficient Mutual Information Neural Estimator (DEMINE) for statistical dependency testing;
2) A new formulation of meta-learning using Task Augmentation (Meta-DEMINE);
3) Application to real life, data-scarce applications (fMRI).
We illustrated that a predictive view of the MI lower bounds coupled with meta-learning results in data-efficient variational MI estimators, DEMINE and Meta-DEMINE, that are capable of performing statistical test of dependency.
We also showed that our proposed task augmentation reduces overfitting and improves generalization in meta-learning.
We successfully applied MI estimation to real world, data-scarce, fMRI datasets.
Our results suggest a greater avenue of using neural networks and meta-learning to improve MI analysis and applying neural network-based information theory tools to enhance the analysis of information processing in the brain.
Model-agnostic, high-confidence, MI lower bound estimation approaches -including MINE, DEMINE and Meta-DEMINE-are limited to estimating small MI lower bounds up to O(log N ) as pointed out in (McAllester & Statos, 2018) , where N is the number of samples.
In real fMRI datasets, however, strong dependency is rare and existing MI estimation tools are limited more by their ability to accurately characterize the dependency.
Nevertheless, when quantitatively measuring strong dependency, cross-entropy (McAllester & Statos, 2018) Sample a batch of (
Update θ (i) using Adam (Kingma & Ba, 2014) with η 7: end for

assistant: text: A new & practical statistical test of dependency using neural networks, benchmarked on synthetic and a real fMRI datasets.