Optimal Sub-Gaussian Mean Estimation in $\mathbb{R}$

Abstract: We revisit and settle a fundamental problem in statistics: given access to independent samples from a 1D random variable (with finite but unknown mean and variance), what is the best way to estimate the mean in the high probability regime, in terms of error convergence with respect to sample size? The conventional wisdom is to use the empirical mean as our estimate. However, it is known that the empirical mean can in fact have exponentially sub-optimal convergence for certain heavy-tailed distributions. On the other hand, the median-of-means estimator (invented and reinvented in various literature) does have sub-Gaussian convergence for all finite-variance distributions, albeit only in the big-O sense with a sub-optimal multiplicative constant. The natural remaining question then, is whether it is possible to bridge the gap, and have an estimator that has optimal convergence with the right constant for all finite-variance distributions.

In this talk, we answer the question affirmatively by giving an estimator that converges with the optimal constant inside the big-O, up to a 1+o(1) multiplicative factor. The estimator is also easy to compute. The convergence analysis involves deriving tail bounds using linear and convex-concave programming dualities, which may be of independent interest.

Based on joint work with Paul Valiant.

computational complexitycomputational geometrycryptography and securitydiscrete mathematicsdata structures and algorithmsgame theorymachine learningquantum computing and informationcombinatoricsinformation theoryoptimization and controlprobability

Audience: researchers in the topic

( paper )

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TCS+

Series comments: Description: Theoretical Computer Science

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