Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

• URL: http://arxiv.org/abs/2207.10939v1
• Date: Fri, 22 Jul 2022 08:30:10 GMT
• Title: Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis
• Authors: Paolo Braca, Leonardo M. Millefiori, Augusto Aubry, Stefano Marano, Antonio De Maio and Peter Willett
• Abstract summary: We study the performance -- and specifically the rate at which the error probability converges to zero -- of Machine Learning (ML) classification techniques. We provide the mathematical conditions for a ML to exhibit error probabilities that vanish exponentially, say $sim expleft(-n,I + o(n) right) In other words, the classification error probability convergence to zero and its rate can be computed on a portion of the dataset available for training.
• Score: 15.605887551756933
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the performance -- and specifically the rate at which the error probability converges to zero -- of Machine Learning (ML) classification techniques. Leveraging the theory of large deviations, we provide the mathematical conditions for a ML classifier to exhibit error probabilities that vanish exponentially, say $\sim \exp\left(-n\,I + o(n) \right)$, where $n$ is the number of informative observations available for testing (or another relevant parameter, such as the size of the target in an image) and $I$ is the error rate. Such conditions depend on the Fenchel-Legendre transform of the cumulant-generating function of the Data-Driven Decision Function (D3F, i.e., what is thresholded before the final binary decision is made) learned in the training phase. As such, the D3F and, consequently, the related error rate $I$, depend on the given training set, which is assumed of finite size. Interestingly, these conditions can be verified and tested numerically exploiting the available dataset, or a synthetic dataset, generated according to the available information on the underlying statistical model. In other words, the classification error probability convergence to zero and its rate can be computed on a portion of the dataset available for training. Coherently with the large deviations theory, we can also establish the convergence, for $n$ large enough, of the normalized D3F statistic to a Gaussian distribution. This property is exploited to set a desired asymptotic false alarm probability, which empirically turns out to be accurate even for quite realistic values of $n$. Furthermore, approximate error probability curves $\sim \zeta_n \exp\left(-n\,I \right)$ are provided, thanks to the refined asymptotic derivation (often referred to as exact asymptotics), where $\zeta_n$ represents the most representative sub-exponential terms of the error probabilities.

Related papers

• A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models [45.60426164657739]
  We develop non-asymptotic convergence theory for a diffusion-based sampler. We prove that $d/varepsilon$ are sufficient to approximate the target distribution to within $varepsilon$ total-variation distance. Our results also characterize how $ell$ score estimation errors affect the quality of the data generation processes.
  arXiv  Detail & Related papers  (2024-08-05T09:02:24Z)
• Doubly Robust Conditional Independence Testing with Generative Neural Networks [8.323172773256449]
  This article addresses the problem of testing the conditional independence of two generic random vectors $X$ and $Y$ given a third random vector $Z$. We propose a new non-parametric testing procedure that avoids explicitly estimating any conditional distributions.
  arXiv  Detail & Related papers  (2024-07-25T01:28:59Z)
• Scaling Laws in Linear Regression: Compute, Parameters, and Data [86.48154162485712]
  We study the theory of scaling laws in an infinite dimensional linear regression setup. We show that the reducible part of the test error is $Theta(-(a-1) + N-(a-1)/a)$. Our theory is consistent with the empirical neural scaling laws and verified by numerical simulation.
  arXiv  Detail & Related papers  (2024-06-12T17:53:29Z)
• Convergence Analysis of Probability Flow ODE for Score-based Generative Models [5.939858158928473]
  We study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. We prove the total variation between the target and the generated data distributions can be bounded above by $mathcalO(d3/4delta1/2)$ in the continuous time level.
  arXiv  Detail & Related papers  (2024-04-15T12:29:28Z)
• Byzantine-resilient Federated Learning With Adaptivity to Data Heterogeneity [54.145730036889496]
  This paper deals with Gradient learning (FL) in the presence of malicious attacks Byzantine data. A novel Average Algorithm (RAGA) is proposed, which leverages robustness aggregation and can select a dataset.
  arXiv  Detail & Related papers  (2024-03-20T08:15:08Z)
• Large Deviations for Classification Performance Analysis of Machine Learning Systems [16.74271332025289]
  We show that under appropriate conditions the error probabilities vanish exponentially, as $sim expleft(-n,I + o(n) right)$, where $I$ is the error rate and $n$ is the number of observations available for testing. The theoretical findings are finally tested using the popular MNIST dataset.
  arXiv  Detail & Related papers  (2023-01-16T10:48:12Z)
• How Does Pseudo-Labeling Affect the Generalization Error of the Semi-Supervised Gibbs Algorithm? [73.80001705134147]
  We provide an exact characterization of the expected generalization error (gen-error) for semi-supervised learning (SSL) with pseudo-labeling via the Gibbs algorithm. The gen-error is expressed in terms of the symmetrized KL information between the output hypothesis, the pseudo-labeled dataset, and the labeled dataset.
  arXiv  Detail & Related papers  (2022-10-15T04:11:56Z)
• Understanding the Under-Coverage Bias in Uncertainty Estimation [58.03725169462616]
  quantile regression tends to emphunder-cover than the desired coverage level in reality. We prove that quantile regression suffers from an inherent under-coverage bias. Our theory reveals that this under-coverage bias stems from a certain high-dimensional parameter estimation error.
  arXiv  Detail & Related papers  (2021-06-10T06:11:55Z)
• SLOE: A Faster Method for Statistical Inference in High-Dimensional Logistic Regression [68.66245730450915]
  We develop an improved method for debiasing predictions and estimating frequentist uncertainty for practical datasets. Our main contribution is SLOE, an estimator of the signal strength with convergence guarantees that reduces the computation time of estimation and inference by orders of magnitude.
  arXiv  Detail & Related papers  (2021-03-23T17:48:56Z)
• A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent [85.77233010209368]
  This article characterizes the exacts of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples $n$ is all large and comparable. This analysis also provides accurate estimates of training and test regression errors for large $n,p,N$.
  arXiv  Detail & Related papers  (2020-06-09T02:05:40Z)
• Error bounds in estimating the out-of-sample prediction error using leave-one-out cross validation in high-dimensions [19.439945058410203]
  We study the problem of out-of-sample risk estimation in the high dimensional regime. Extensive empirical evidence confirms the accuracy of leave-one-out cross validation. One technical advantage of the theory is that it can be used to clarify and connect some results from the recent literature on scalable approximate LO.
  arXiv  Detail & Related papers  (2020-03-03T20:07:07Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.