Minimax experimental design: Bridging the gap between statistical and worst-case approaches to least squares regression Article Swipe

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Michał Dereziński , Kenneth L. Clarkson , Michael W. Mahoney , Manfred K. Warmuth ·

YOU? · · 2019 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1902.00995

In experimental design, we are given a large collection of vectors, each with a hidden response value that we assume derives from an underlying linear model, and we wish to pick a small subset of the vectors such that querying the corresponding responses will lead to a good estimator of the model. A classical approach in statistics is to assume the responses are linear, plus zero-mean i.i.d. Gaussian noise, in which case the goal is to provide an unbiased estimator with smallest mean squared error (A-optimal design). A related approach, more common in computer science, is to assume the responses are arbitrary but fixed, in which case the goal is to estimate the least squares solution using few responses, as quickly as possible, for worst-case inputs. Despite many attempts, characterizing the relationship between these two approaches has proven elusive. We address this by proposing a framework for experimental design where the responses are produced by an arbitrary unknown distribution. We show that there is an efficient randomized experimental design procedure that achieves strong variance bounds for an unbiased estimator using few responses in this general model. Nearly tight bounds for the classical A-optimality criterion, as well as improved bounds for worst-case responses, emerge as special cases of this result. In the process, we develop a new algorithm for a joint sampling distribution called volume sampling, and we propose a new i.i.d. importance sampling method: inverse score sampling. A key novelty of our analysis is in developing new expected error bounds for worst-case regression by controlling the tail behavior of i.i.d. sampling via the jointness of volume sampling. Our result motivates a new minimax-optimality criterion for experimental design which can be viewed as an extension of both A-optimal design and sampling for worst-case regression.

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Minimax Estimator Mathematics Mathematical optimization Minimum-variance unbiased estimator Bias of an estimator Computer science Bridging (networking) Variance (accounting) Algorithm Applied mathematics Statistics Accounting Computer network Business

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/1902.00995
PDF: https://arxiv.org/pdf/1902.00995
OA Status: green
Cited By: 8
References: 24
Related Works: 10
OpenAlex ID: https://openalex.org/W2914330359

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DOI: https://doi.org/10.48550/arxiv.1902.00995

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Title: Minimax experimental design: Bridging the gap between statistical and worst-case approaches to least squares regression

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Type: preprint

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Language: en

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Publication year: 2019

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Publication date: 2019-02-04

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Authors: Michał Dereziński, Kenneth L. Clarkson, Michael W. Mahoney, Manfred K. Warmuth

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Landing page: https://arxiv.org/abs/1902.00995

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PDF URL: https://arxiv.org/pdf/1902.00995

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Concepts: Minimax, Estimator, Mathematics, Mathematical optimization, Minimum-variance unbiased estimator, Bias of an estimator, Computer science, Bridging (networking), Variance (accounting), Algorithm, Applied mathematics, Statistics, Accounting, Computer network, Business

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Cited by: 8

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abstract_inverted_index.hidden	14
abstract_inverted_index.i.i.d.	66, 230, 259
abstract_inverted_index.linear	24
abstract_inverted_index.model,	25
abstract_inverted_index.model.	51, 185
abstract_inverted_index.noise,	68
abstract_inverted_index.proven	137
abstract_inverted_index.result	268
abstract_inverted_index.strong	172
abstract_inverted_index.subset	33
abstract_inverted_index.viewed	280
abstract_inverted_index.volume	223, 265
abstract_inverted_index.Despite	126
abstract_inverted_index.address	140
abstract_inverted_index.between	132
abstract_inverted_index.derives	20
abstract_inverted_index.design,	2
abstract_inverted_index.develop	213
abstract_inverted_index.general	184
abstract_inverted_index.inputs.	125
abstract_inverted_index.inverse	234
abstract_inverted_index.linear,	63
abstract_inverted_index.method:	233
abstract_inverted_index.novelty	239
abstract_inverted_index.propose	227
abstract_inverted_index.provide	76
abstract_inverted_index.quickly	120
abstract_inverted_index.related	88
abstract_inverted_index.result.	208
abstract_inverted_index.special	204
abstract_inverted_index.squared	83
abstract_inverted_index.squares	114
abstract_inverted_index.unknown	157
abstract_inverted_index.vectors	36
abstract_inverted_index.Gaussian	67
abstract_inverted_index.achieves	171
abstract_inverted_index.analysis	242
abstract_inverted_index.approach	54
abstract_inverted_index.behavior	257
abstract_inverted_index.computer	93
abstract_inverted_index.design).	86
abstract_inverted_index.elusive.	138
abstract_inverted_index.estimate	111
abstract_inverted_index.expected	247
abstract_inverted_index.improved	197
abstract_inverted_index.process,	211
abstract_inverted_index.produced	153
abstract_inverted_index.querying	39
abstract_inverted_index.response	15
abstract_inverted_index.sampling	220, 232, 260, 289
abstract_inverted_index.science,	94
abstract_inverted_index.smallest	81
abstract_inverted_index.solution	115
abstract_inverted_index.unbiased	78, 177
abstract_inverted_index.variance	173
abstract_inverted_index.vectors,	10
abstract_inverted_index.A-optimal	286
abstract_inverted_index.algorithm	216
abstract_inverted_index.approach,	89
abstract_inverted_index.arbitrary	101, 156
abstract_inverted_index.attempts,	128
abstract_inverted_index.classical	53, 191
abstract_inverted_index.criterion	273
abstract_inverted_index.efficient	165
abstract_inverted_index.estimator	48, 79, 178
abstract_inverted_index.extension	283
abstract_inverted_index.framework	145
abstract_inverted_index.jointness	263
abstract_inverted_index.motivates	269
abstract_inverted_index.possible,	122
abstract_inverted_index.procedure	169
abstract_inverted_index.proposing	143
abstract_inverted_index.responses	42, 61, 99, 151, 181
abstract_inverted_index.sampling,	224
abstract_inverted_index.sampling.	236, 266
abstract_inverted_index.zero-mean	65
abstract_inverted_index.(A-optimal	85
abstract_inverted_index.approaches	135
abstract_inverted_index.collection	8
abstract_inverted_index.criterion,	193
abstract_inverted_index.developing	245
abstract_inverted_index.importance	231
abstract_inverted_index.randomized	166
abstract_inverted_index.regression	252
abstract_inverted_index.responses,	118, 201
abstract_inverted_index.statistics	56
abstract_inverted_index.underlying	23
abstract_inverted_index.worst-case	124, 200, 251, 291
abstract_inverted_index.controlling	254
abstract_inverted_index.regression.	292
abstract_inverted_index.A-optimality	192
abstract_inverted_index.distribution	221
abstract_inverted_index.experimental	1, 147, 167, 275
abstract_inverted_index.relationship	131
abstract_inverted_index.corresponding	41
abstract_inverted_index.distribution.	158
abstract_inverted_index.characterizing	129
abstract_inverted_index.minimax-optimality	272
cited_by_percentile_year
countries_distinct_count	1
institutions_distinct_count	4
citation_normalized_percentile