Unbiased estimators for random design regression Article Swipe
Michał Dereziński
,
Manfred K. Warmuth
,
Daniel Hsu
·
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1907.03411
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1907.03411
In linear regression we wish to estimate the optimum linear least squares predictor for a distribution over $d$-dimensional input points and real-valued responses, based on a small sample. Under standard random design analysis, where the sample is drawn i.i.d. from the input distribution, the least squares solution for that sample can be viewed as the natural estimator of the optimum. Unfortunately, this estimator almost always incurs an undesirable bias coming from the randomness of the input points, which is a significant bottleneck in model averaging. In this paper we show that it is possible to draw a non-i.i.d. sample of input points such that, regardless of the response model, the least squares solution is an unbiased estimator of the optimum. Moreover, this sample can be produced efficiently by augmenting a previously drawn i.i.d. sample with an additional set of $d$ points, drawn jointly according to a certain determinantal point process constructed from the input distribution rescaled by the squared volume spanned by the points. Motivated by this, we develop a theoretical framework for studying volume-rescaled sampling, and in the process prove a number of new matrix expectation identities. We use them to show that for any input distribution and $ε>0$ there is a random design consisting of $O(d\log d+ d/ε)$ points from which an unbiased estimator can be constructed whose expected square loss over the entire distribution is bounded by $1+ε$ times the loss of the optimum. We provide efficient algorithms for generating such unbiased estimators in a number of practical settings and support our claims experimentally.
Related Topics
Concepts
Estimator
Mathematics
Statistics
Bias of an estimator
Minimum-variance unbiased estimator
Mean squared error
Randomness
Stein's unbiased risk estimate
Applied mathematics
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/1907.03411
- https://arxiv.org/pdf/1907.03411
- OA Status
- green
- Cited By
- 7
- References
- 33
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W2954491832
All OpenAlex metadata
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https://openalex.org/W2954491832Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.1907.03411Digital Object Identifier
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Unbiased estimators for random design regressionWork title
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preprintOpenAlex work type
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enPrimary language
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2019Year of publication
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2019-07-08Full publication date if available
- Authors
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Michał Dereziński, Manfred K. Warmuth, Daniel HsuList of authors in order
- Landing page
-
https://arxiv.org/abs/1907.03411Publisher landing page
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https://arxiv.org/pdf/1907.03411Direct link to full text PDF
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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Estimator, Mathematics, Statistics, Bias of an estimator, Minimum-variance unbiased estimator, Mean squared error, Randomness, Stein's unbiased risk estimate, Applied mathematicsTop concepts (fields/topics) attached by OpenAlex
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7Total citation count in OpenAlex
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2025: 1, 2022: 2, 2021: 1, 2020: 1, 2019: 2Per-year citation counts (last 5 years)
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33Number of works referenced by this work
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20Other works algorithmically related by OpenAlex
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| abstract_inverted_index.them | 190 |
| abstract_inverted_index.this | 61, 86, 121 |
| abstract_inverted_index.wish | 4 |
| abstract_inverted_index.with | 134 |
| abstract_inverted_index.Under | 28 |
| abstract_inverted_index.based | 23 |
| abstract_inverted_index.drawn | 37, 131, 141 |
| abstract_inverted_index.input | 18, 41, 75, 100, 153, 196 |
| abstract_inverted_index.least | 10, 44, 110 |
| abstract_inverted_index.model | 83 |
| abstract_inverted_index.paper | 87 |
| abstract_inverted_index.point | 148 |
| abstract_inverted_index.prove | 180 |
| abstract_inverted_index.small | 26 |
| abstract_inverted_index.that, | 103 |
| abstract_inverted_index.there | 200 |
| abstract_inverted_index.this, | 166 |
| abstract_inverted_index.times | 231 |
| abstract_inverted_index.where | 33 |
| abstract_inverted_index.which | 77, 212 |
| abstract_inverted_index.whose | 219 |
| abstract_inverted_index.$1+ε$ | 230 |
| abstract_inverted_index.almost | 63 |
| abstract_inverted_index.always | 64 |
| abstract_inverted_index.claims | 255 |
| abstract_inverted_index.coming | 69 |
| abstract_inverted_index.d/ε)$ | 209 |
| abstract_inverted_index.design | 31, 204 |
| abstract_inverted_index.entire | 225 |
| abstract_inverted_index.i.i.d. | 38, 132 |
| abstract_inverted_index.incurs | 65 |
| abstract_inverted_index.linear | 1, 9 |
| abstract_inverted_index.matrix | 185 |
| abstract_inverted_index.model, | 108 |
| abstract_inverted_index.number | 182, 248 |
| abstract_inverted_index.points | 19, 101, 210 |
| abstract_inverted_index.random | 30, 203 |
| abstract_inverted_index.sample | 35, 49, 98, 122, 133 |
| abstract_inverted_index.square | 221 |
| abstract_inverted_index.viewed | 52 |
| abstract_inverted_index.volume | 159 |
| abstract_inverted_index.bounded | 228 |
| abstract_inverted_index.certain | 146 |
| abstract_inverted_index.develop | 168 |
| abstract_inverted_index.jointly | 142 |
| abstract_inverted_index.natural | 55 |
| abstract_inverted_index.optimum | 8 |
| abstract_inverted_index.points, | 76, 140 |
| abstract_inverted_index.points. | 163 |
| abstract_inverted_index.process | 149, 179 |
| abstract_inverted_index.provide | 238 |
| abstract_inverted_index.sample. | 27 |
| abstract_inverted_index.spanned | 160 |
| abstract_inverted_index.squared | 158 |
| abstract_inverted_index.squares | 11, 45, 111 |
| abstract_inverted_index.support | 253 |
| abstract_inverted_index.$O(d\log | 207 |
| abstract_inverted_index.estimate | 6 |
| abstract_inverted_index.expected | 220 |
| abstract_inverted_index.optimum. | 59, 119, 236 |
| abstract_inverted_index.possible | 93 |
| abstract_inverted_index.produced | 125 |
| abstract_inverted_index.rescaled | 155 |
| abstract_inverted_index.response | 107 |
| abstract_inverted_index.settings | 251 |
| abstract_inverted_index.solution | 46, 112 |
| abstract_inverted_index.standard | 29 |
| abstract_inverted_index.studying | 173 |
| abstract_inverted_index.unbiased | 115, 214, 244 |
| abstract_inverted_index.$ε>0$ | 199 |
| abstract_inverted_index.Moreover, | 120 |
| abstract_inverted_index.Motivated | 164 |
| abstract_inverted_index.according | 143 |
| abstract_inverted_index.analysis, | 32 |
| abstract_inverted_index.efficient | 239 |
| abstract_inverted_index.estimator | 56, 62, 116, 215 |
| abstract_inverted_index.framework | 171 |
| abstract_inverted_index.practical | 250 |
| abstract_inverted_index.predictor | 12 |
| abstract_inverted_index.sampling, | 175 |
| abstract_inverted_index.additional | 136 |
| abstract_inverted_index.algorithms | 240 |
| abstract_inverted_index.augmenting | 128 |
| abstract_inverted_index.averaging. | 84 |
| abstract_inverted_index.bottleneck | 81 |
| abstract_inverted_index.consisting | 205 |
| abstract_inverted_index.estimators | 245 |
| abstract_inverted_index.generating | 242 |
| abstract_inverted_index.non-i.i.d. | 97 |
| abstract_inverted_index.previously | 130 |
| abstract_inverted_index.randomness | 72 |
| abstract_inverted_index.regardless | 104 |
| abstract_inverted_index.regression | 2 |
| abstract_inverted_index.responses, | 22 |
| abstract_inverted_index.constructed | 150, 218 |
| abstract_inverted_index.efficiently | 126 |
| abstract_inverted_index.expectation | 186 |
| abstract_inverted_index.identities. | 187 |
| abstract_inverted_index.real-valued | 21 |
| abstract_inverted_index.significant | 80 |
| abstract_inverted_index.theoretical | 170 |
| abstract_inverted_index.undesirable | 67 |
| abstract_inverted_index.distribution | 15, 154, 197, 226 |
| abstract_inverted_index.determinantal | 147 |
| abstract_inverted_index.distribution, | 42 |
| abstract_inverted_index.Unfortunately, | 60 |
| abstract_inverted_index.$d$-dimensional | 17 |
| abstract_inverted_index.experimentally. | 256 |
| abstract_inverted_index.volume-rescaled | 174 |
| cited_by_percentile_year | |
| countries_distinct_count | 1 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile |