Costly Zero Order Oracles

Renato Paes Leme, Jon Schneider
Proceedings of Thirty Third Conference on Learning Theory, PMLR 125:3120-3132, 2020.

Abstract

We study optimization with an approximate zero order oracle where there is a cost $c(\epsilon)$ associated with querying the oracle with $\epsilon$ accuracy. We consider the task of reconstructing a linear function: given a linear function $f : X \subseteq \R^d \rightarrow [-1,1]$ defined on a not-necessarily-convex set $X$, the goal is to reconstruct $\hat f$ such that $\vert{f(x) - \hat f(x)}\vert \leq \epsilon$ for all $x \in X$. We show that this can be done with cost $O(d \cdot c(\epsilon/d))$. The algorithm is based on a (poly-time computable) John-like theorem for simplices instead of ellipsoids, which may be of independent interest. This question is motivated by optimization with threshold queries, which are common in economic applications such as pricing. With threshold queries, approximating a number up to precision $\epsilon$ requires $c(\epsilon) = \log(1/\epsilon)$. For this, our algorithm has cost $O(d \log(d/\epsilon))$ which matches the $\Omega(d \log(1/\epsilon))$ lower bound up to log factors.

Cite this Paper

BibTeX
@InProceedings{pmlr-v125-paes-leme20a, title = {Costly Zero Order Oracles}, author = {Paes Leme, Renato and Schneider, Jon}, booktitle = {Proceedings of Thirty Third Conference on Learning Theory}, pages = {3120--3132}, year = {2020}, editor = {Abernethy, Jacob and Agarwal, Shivani}, volume = {125}, series = {Proceedings of Machine Learning Research}, month = {09--12 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v125/paes-leme20a/paes-leme20a.pdf}, url = {https://proceedings.mlr.press/v125/paes-leme20a.html}, abstract = { We study optimization with an approximate zero order oracle where there is a cost $c(\epsilon)$ associated with querying the oracle with $\epsilon$ accuracy. We consider the task of reconstructing a linear function: given a linear function $f : X \subseteq \R^d \rightarrow [-1,1]$ defined on a not-necessarily-convex set $X$, the goal is to reconstruct $\hat f$ such that $\vert{f(x) - \hat f(x)}\vert \leq \epsilon$ for all $x \in X$. We show that this can be done with cost $O(d \cdot c(\epsilon/d))$. The algorithm is based on a (poly-time computable) John-like theorem for simplices instead of ellipsoids, which may be of independent interest. This question is motivated by optimization with threshold queries, which are common in economic applications such as pricing. With threshold queries, approximating a number up to precision $\epsilon$ requires $c(\epsilon) = \log(1/\epsilon)$. For this, our algorithm has cost $O(d \log(d/\epsilon))$ which matches the $\Omega(d \log(1/\epsilon))$ lower bound up to log factors. } }
Endnote
%0 Conference Paper %T Costly Zero Order Oracles %A Renato Paes Leme %A Jon Schneider %B Proceedings of Thirty Third Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2020 %E Jacob Abernethy %E Shivani Agarwal %F pmlr-v125-paes-leme20a %I PMLR %P 3120--3132 %U https://proceedings.mlr.press/v125/paes-leme20a.html %V 125 %X We study optimization with an approximate zero order oracle where there is a cost $c(\epsilon)$ associated with querying the oracle with $\epsilon$ accuracy. We consider the task of reconstructing a linear function: given a linear function $f : X \subseteq \R^d \rightarrow [-1,1]$ defined on a not-necessarily-convex set $X$, the goal is to reconstruct $\hat f$ such that $\vert{f(x) - \hat f(x)}\vert \leq \epsilon$ for all $x \in X$. We show that this can be done with cost $O(d \cdot c(\epsilon/d))$. The algorithm is based on a (poly-time computable) John-like theorem for simplices instead of ellipsoids, which may be of independent interest. This question is motivated by optimization with threshold queries, which are common in economic applications such as pricing. With threshold queries, approximating a number up to precision $\epsilon$ requires $c(\epsilon) = \log(1/\epsilon)$. For this, our algorithm has cost $O(d \log(d/\epsilon))$ which matches the $\Omega(d \log(1/\epsilon))$ lower bound up to log factors.
APA
Paes Leme, R. & Schneider, J.. (2020). Costly Zero Order Oracles. Proceedings of Thirty Third Conference on Learning Theory, in Proceedings of Machine Learning Research 125:3120-3132 Available from https://proceedings.mlr.press/v125/paes-leme20a.html.

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