Papers
Topics
Authors
Recent
2000 character limit reached

Mean estimation and regression under heavy-tailed distributions--a survey

Published 10 Jun 2019 in math.ST, cs.LG, stat.ML, and stat.TH | (1906.04280v1)

Abstract: We survey some of the recent advances in mean estimation and regression function estimation. In particular, we describe sub-Gaussian mean estimators for possibly heavy-tailed data both in the univariate and multivariate settings. We focus on estimators based on median-of-means techniques but other methods such as the trimmed mean and Catoni's estimator are also reviewed. We give detailed proofs for the cornerstone results. We dedicate a section on statistical learning problems--in particular, regression function estimation--in the presence of possibly heavy-tailed data.

Citations (220)
View on Semantic Scholar

Summary

  • The paper surveys robust statistical techniques, particularly Median-Of-Means and its variants, for estimating means and regression functions effectively under heavy-tailed data.
  • Median-Of-Means and similar estimators offer sub-Gaussian error rates and robust performance even with minimal distributional assumptions or high-dimensional data.
  • The surveyed robust techniques are critical for reliable inference in fields like finance and data science, paving the way for resilient machine learning under data irregularities.

Mean Estimation and Regression Under Heavy-Tailed Distributions: An Expert Overview

The paper "Mean Estimation and Regression under Heavy-Tailed Distributions" by Lugosi and Mendelson presents a comprehensive survey of methodologies for estimating means and regression functions in settings where heavy-tailed data is prevalent. The focus lies on sub-Gaussian estimators for mean estimation and the adaptation of these techniques to multivariate settings and regression problems.

Statistical inference in heavy-tailed environments poses considerable challenges due to significant data variability and the presence of outliers. Traditional mean estimators, like the empirical mean, fall short when applied to data with heavy-tailed distributions as they are not robust to outliers and can yield unreliable estimates. The methodologies surveyed in this paper address these limitations by proposing alternative estimators that demonstrate robust performance.

Key Contributions

The paper centers around the Median-Of-Means (MoM) technique and its variants as robust mean estimators in presence with little moment or distributional assumptions. The MoM estimator involves creating several subsamples, calculating the mean of each, and then taking the median of these means. This approach enhances robustness and can achieve a sub-Gaussian error rate under a finite second-moment condition, even in the presence of heavy-tailed data.

Univariate Analysis

The univariate case is explored with attention given to the fundamental limitations of conventional methods and several robust alternatives like the trimmed means and Catoni’s estimator. These alternatives are detailed with proofs, demonstrating their capacity to achieve non-asymptotic performance bounds that approach sub-Gaussian behavior for non-Gaussian data.

Multivariate and General Norms

The extension to multivariate cases outlines challenges posed by vector data and introduces notions like multivariate medians and thresholds. The geometric median-of-means and other generalized notions of medians are shown to provide strong performance in high dimensions. Of particular interest is the proposal of using median-of-means tournaments, which offer dimension-free bounds on estimation errors, demonstrating their efficacy even for non-Euclidean norms.

Regression Problems

In regression, the authors extend MoM techniques to estimate regression functions, challenging empirical risk minimization approaches in heavy-tailed scenarios. The paper presents "median-of-means tournaments" as powerful tools in tackling regression problems by leveraging robust estimations to construct reliable predictors.

Implications and Future Directions

The surveyed methodologies imply significant practical impacts on fields where robust statistical estimates are critical, such as finance, physics, and large-scale data science applications. Theoretical developments in estimators reveal critical insights into designing algorithms that remain effective under minimal distributional assumptions.

Looking ahead, these insights have the potential to inform robust learning algorithms, particularly in the development of models resilient to adversarial attacks and data corruption. Future research may explore computational efficiency improvements and adaptations for even broader classes of normed spaces and application areas.

In conclusion, Lugosi and Mendelson’s work provides a thorough examination of mean and regression function estimation in heavy-tailed contexts, offering robust techniques that are particularly suitable for modern applications faced with data irregularities. The paper outlines a clear path forward for researchers interested in advancing robust statistics and machine learning.

File Document Download Save Streamline Icon: https://streamlinehq.com Markdown

Paper to Video (Beta)

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Explain it Like I'm 14

What is this paper about?

This paper looks at a very basic task in statistics: estimating the average (the “mean”) of data, and building regression models (predicting one thing from another). It focuses on situations where the data can have “heavy tails”—that is, you sometimes see very large, rare values (outliers). In such cases, the usual average can be unreliable. The authors explain several smarter ways to estimate averages and do regression so that the answers are accurate and trustworthy even when outliers occur.

Questions the paper asks

  • How can we estimate an average from data that might include extreme, rare values without being fooled by those outliers?
  • Can we make estimators with high confidence guarantees (meaning they are accurate “with high probability”), similar to what we would get with perfectly nice, bell-shaped (Gaussian) data?
  • How do we extend these ideas from one number (univariate data) to many numbers at once (vectors, or multivariate data)?
  • How can these ideas help in machine learning, especially in building regression models when errors or inputs can be heavy-tailed?
  • Are there limits—what is possible and what is not—if we only assume the data has a finite variance?

How did the authors approach the problem?

The main challenge is that outliers can pull the ordinary average far away from the true mean. The paper surveys several “robust” estimators—methods designed to resist the influence of outliers—then analyzes how accurate they can be, with clear probability guarantees.

Key idea: “Sub-Gaussian” performance

Imagine your data was perfectly nice and followed a bell curve. Then the error of the sample mean shrinks like:

  • roughly σ1n\sigma \sqrt{\frac{1}{n}} on average, and
  • with high probability it is at most about σlog⁡(1/δ)n\sigma \sqrt{\frac{\log(1/\delta)}{n}},

where nn is the number of samples, σ\sigma is the standard deviation, and δ\delta is the allowed failure probability (say δ=0.05\delta = 0.05 means 95% confidence). The paper asks: can we get this same kind of guarantee (“sub-Gaussian”) even if the data are heavy-tailed? Surprisingly, yes—if we use the right estimators.

Median-of-means (MoM)

  • Split your data into kk groups (blocks) of roughly equal size.
  • Compute the average within each group.
  • Take the median of those group averages.

Why this works: An outlier may ruin one group’s mean, but it’s unlikely to ruin most groups. The median ignores a few bad groups and keeps the typical ones. This simple trick gives strong accuracy guarantees even when the usual average fails.

Key points:

  • It achieves sub-Gaussian-like error: about σlog⁡(1/δ)n\sigma \sqrt{\frac{\log(1/\delta)}{n}}.
  • You must choose the number of groups kk based on your desired confidence δ\delta.
  • It also works when only a slightly stronger condition holds (like having a finite 1+α1+\alpha moment), and even better when a third moment exists.

Catoni’s estimator

  • Instead of averaging raw values, it solves a modified equation that “down-weights” large deviations (outliers) using a carefully designed influence function.
  • Think of it as a smart average that grows more slowly when confronted with big values.

Key points:

  • It can match optimal sub-Gaussian guarantees.
  • It needs a tuning parameter related to δ\delta and, ideally, a bound on variance σ2\sigma^2.
  • If you don’t know σ2\sigma^2, there are adaptive tricks (like Lepski’s method) to pick a good parameter from the data.

Trimmed mean

  • Sort the data.
  • Chop off the largest and smallest few percent (potential outliers).
  • Average the rest.

Key points:

  • If you trim a fraction proportional to log⁡(1/δ)/n\log(1/\delta)/n, you get sub-Gaussian-like performance.
  • It is conceptually simple and robust, even if some data points are adversarially corrupted.

Moving to multivariate data (vectors)

When each data point is a vector (many numbers at once), we want the estimate to be close in Euclidean distance. The paper adapts robust ideas to this case:

  • Coordinate-wise median-of-means: take MoM for each coordinate separately. This is simple but can be less sharp in high dimensions.
  • Geometric median-of-means: instead of per-coordinate medians, take the geometric median of block means (the point minimizing the sum of distances). This is a convex optimization problem, so it’s computable and yields dimension-free guarantees (no explicit dependence on the number of coordinates).
  • Thresholding the norm (Catoni–Giulini): shrink data points with very large norms toward zero before averaging. This is easy to compute, but needs a prior bound on how spread out the data are, and care to avoid issues like not being translation-invariant.

Median-of-means tournaments

This is a clever way to get truly sub-Gaussian behavior for vectors under very mild assumptions (just finite covariance). Imagine comparing candidate mean points by how well they fit the data across blocks, and choosing the one that consistently “wins” these comparisons:

  • Partition data into blocks and compute block means.
  • For any two candidate points aa and bb, see which one is closer to the block means on most blocks (“a defeats b” if it is closer more often).
  • Choose a point that minimizes the radius of the region where it “wins” against others.

Key points:

  • It achieves a high-probability error bound of the form

∥μ^−μ∥≲Tr⁡(Σ)n  +  λmax⁡(Σ) log⁡(1/δ)n,\| \hat{\mu} - \mu \| \lesssim \sqrt{\frac{\operatorname{Tr}(\Sigma)}{n}} \;+\; \sqrt{\frac{\lambda_{\max}(\Sigma)\,\log(1/\delta)}{n}},

where Σ\Sigma is the covariance matrix, Tr⁡(Σ)\operatorname{Tr}(\Sigma) is its trace (sum of variances), and λmax⁡(Σ)\lambda_{\max}(\Sigma) is its largest eigenvalue.

  • This is “dimension-free” and mirrors what you would expect in the Gaussian case.

Main findings

  • The usual sample mean can be unreliable with heavy-tailed data; its high-probability error often scales like σ/δn\sigma / \sqrt{\delta n} (worse dependence on δ\delta) rather than the desired σlog⁡(1/δ)/n\sigma \sqrt{\log(1/\delta)/n}.
  • There exist simple, robust estimators—median-of-means, Catoni’s estimator, trimmed mean—that recover sub-Gaussian-like guarantees even under heavy tails, assuming at least finite variance.
  • In many cases, the estimator must depend on the target confidence level δ\delta (for example, through the number of blocks kk in MoM). The paper proves you cannot have a single estimator that is optimally sub-Gaussian for a wide range of δ\delta without extra information.
  • For multivariate data, geometric median-of-means and MoM tournaments provide strong, dimension-free guarantees, very close to the ideal Gaussian case.
  • These robust ideas extend to regression (learning a function that predicts outcomes). Using median-of-means style comparisons, one can build learning algorithms that remain accurate even when errors have heavy tails.

Why this matters

In real life—finance, web traffic, sensor networks, medical data—outliers happen. A simple average can be badly thrown off by a few extreme values. Robust estimators give you protection: they keep the estimate reliable, with clear high-probability guarantees. This is especially crucial in machine learning, where training data often contain noise, rare spikes, or even malicious corruption.

Takeaways and impact

  • Robust mean estimators are practical and powerful. Median-of-means and trimmed mean are simple to implement and offer strong guarantees.
  • Catoni-type methods achieve optimal accuracy with thoughtful tuning and can be adapted when variance is unknown.
  • For high-dimensional data, geometric median-of-means and tournament-based methods provide “Gaussian-like” performance without assuming the data are Gaussian.
  • You typically need to decide in advance how confident you want to be (choose δ\delta), and tune the estimator accordingly; this dependence is often unavoidable.
  • These tools carry over to regression and learning, helping build models that remain dependable even when data are messy.

In short: the paper shows that, with the right robust methods, you can get the kind of reliability you expect from ideal bell-shaped data, even when your real data have heavy tails and outliers. This makes statistics and machine learning more trustworthy in the wild.

Ai Sparkles Streamline Icon: https://streamlinehq.com View Paper Prompt  List Streamline Icon: https://streamlinehq.com View All Prompts 

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (2)

  1. Gabor Lugosi 
  2. Shahar Mendelson 

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up