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Online A-optimal design and active linear regression

Abstract : We consider in this paper the problem of optimal experiment design where a decision maker can choose which points to sample to obtain an estimateβ of the hidden parameter β of an underlying linear model. The key challenge of this work lies in the heteroscedasticity assumption that we make, meaning that each covariate has a different and unknown variance. The goal of the decision maker is then to figure out on the fly the optimal way to allocate the total budget of T samples between covariates, as sampling several times a specific one will reduce the variance of the estimated model around it (but at the cost of a possible higher variance elsewhere). By trying to minimize the 2-loss E[ β − β 2 ] the decision maker is actually minimizing the trace of the covariance matrix of the problem, which corresponds then to online A-optimal design. Combining techniques from bandit and convex optimization we propose a new active sampling algorithm and we compare it with existing ones. We provide theoretical guarantees of this algorithm in different settings, including a O(T −2) regret bound in the case where the covariates form a basis of the feature space, generalizing and improving existing results. Numerical experiments validate our theoretical findings.
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https://hal.inria.fr/hal-03289318
Contributor : Michal Valko Connect in order to contact the contributor
Submitted on : Friday, July 16, 2021 - 5:58:51 PM
Last modification on : Thursday, April 7, 2022 - 1:58:35 PM
Long-term archiving on: : Sunday, October 17, 2021 - 7:47:47 PM

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  • HAL Id : hal-03289318, version 1

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Xavier Fontaine, Pierre Perrault, Michal Valko, Vianney Perchet. Online A-optimal design and active linear regression. International Conference on Machine Learning, Jul 2021, Vienna / Virtual, Austria. ⟨hal-03289318⟩

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