Catoni-style confidence sequences for heavy-tailed mean estimation
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arXiv
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| Format: | Preprint |
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2022
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| author | Wang, Hongjian Ramdas, Aaditya |
| author_facet | Wang, Hongjian Ramdas, Aaditya |
| contents | A confidence sequence (CS) is a sequence of confidence intervals that is valid at arbitrary data-dependent stopping times. These are useful in applications like A/B testing, multi-armed bandits, off-policy evaluation, election auditing, etc. We present three approaches to constructing a confidence sequence for the population mean, under the minimal assumption that only an upper bound $σ^2$ on the variance is known. While previous works rely on light-tail assumptions like boundedness or subGaussianity (under which all moments of a distribution exist), the confidence sequences in our work are able to handle data from a wide range of heavy-tailed distributions. The best among our three methods -- the Catoni-style confidence sequence -- performs remarkably well in practice, essentially matching the state-of-the-art methods for $σ^2$-subGaussian data, and provably attains the $\sqrt{\log \log t/t}$ lower bound due to the law of the iterated logarithm. Our findings have important implications for sequential experimentation with unbounded observations, since the $σ^2$-bounded-variance assumption is more realistic and easier to verify than $σ^2$-subGaussianity (which implies the former). We also extend our methods to data with infinite variance, but having $p$-th central moment ($1<p<2$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_01250 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Catoni-style confidence sequences for heavy-tailed mean estimation Wang, Hongjian Ramdas, Aaditya Statistics Theory Methodology 62L12, 60G42 A confidence sequence (CS) is a sequence of confidence intervals that is valid at arbitrary data-dependent stopping times. These are useful in applications like A/B testing, multi-armed bandits, off-policy evaluation, election auditing, etc. We present three approaches to constructing a confidence sequence for the population mean, under the minimal assumption that only an upper bound $σ^2$ on the variance is known. While previous works rely on light-tail assumptions like boundedness or subGaussianity (under which all moments of a distribution exist), the confidence sequences in our work are able to handle data from a wide range of heavy-tailed distributions. The best among our three methods -- the Catoni-style confidence sequence -- performs remarkably well in practice, essentially matching the state-of-the-art methods for $σ^2$-subGaussian data, and provably attains the $\sqrt{\log \log t/t}$ lower bound due to the law of the iterated logarithm. Our findings have important implications for sequential experimentation with unbounded observations, since the $σ^2$-bounded-variance assumption is more realistic and easier to verify than $σ^2$-subGaussianity (which implies the former). We also extend our methods to data with infinite variance, but having $p$-th central moment ($1<p<2$). |
| title | Catoni-style confidence sequences for heavy-tailed mean estimation |
| topic | Statistics Theory Methodology 62L12, 60G42 |
| url | https://arxiv.org/abs/2202.01250 |