ECCC-Report TR19-059https://eccc.weizmann.ac.il/report/2019/059Comments and Revisions published for TR19-059en-usWed, 18 Sep 2019 00:19:43 +0300
Revision 1
| Samplers and Extractors for Unbounded Functions |
Rohit Agrawal
https://eccc.weizmann.ac.il/report/2019/059#revision1Blasiok (SODA'18) recently introduced the notion of a subgaussian sampler, defined as an averaging sampler for approximating the mean of functions $f:\{0,1\}^m \to \mathbb{R}$ such that $f(U_m)$ has subgaussian tails, and asked for explicit constructions. In this work, we give the first explicit constructions of subgaussian samplers (and in fact averaging samplers for the broader class of subexponential functions) that match the best-known constructions of averaging samplers for $[0,1]$-bounded functions in the regime of parameters where the approximation error $\varepsilon$ and failure probability $\delta$ are subconstant. Our constructions are established via an extension of the standard notion of randomness extractor (Nisan and Zuckerman, JCSS'96) where the error is measured by an arbitrary divergence rather than total variation distance, and a generalization of Zuckerman's equivalence (Random Struct. Alg.'97) between extractors and samplers. We believe that the framework we develop, and specifically the notion of an extractor for the Kullback-Leibler (KL) divergence, are of independent interest. In particular, KL-extractors are stronger than both standard extractors and subgaussian samplers, but we show that they exist with essentially the same parameters (constructively and non-constructively) as standard extractors.Wed, 18 Sep 2019 00:19:43 +0300https://eccc.weizmann.ac.il/report/2019/059#revision1
Paper TR19-059
| Samplers and extractors for unbounded functions |
Rohit Agrawal
https://eccc.weizmann.ac.il/report/2019/059Blasiok (SODA'18) recently introduced the notion of a subgaussian sampler, defined as an averaging sampler for approximating the mean of functions $f:\{0,1\}^m \to \mathbb{R}$ such that $f(U_m)$ has subgaussian tails, and asked for explicit constructions. In this work, we give the first explicit constructions of subgaussian samplers (and in fact averaging samplers for the broader class of subexponential functions) that match the best-known constructions of averaging samplers for $[0,1]$-bounded functions in the regime of parameters where the approximation error $\varepsilon$ and failure probability $\delta$ are subconstant. Our constructions are established via an extension of the standard notion of randomness extractor (Nisan and Zuckerman, JCSS'96) where the error is measured by an arbitrary divergence rather than total variation distance, and a generalization of Zuckerman's equivalence (Random Struct. Alg.'97) between extractors and samplers. We believe that the framework we develop, and specifically the notion of an extractor for the Kullback-Leibler (KL) divergence, are of independent interest. In particular, KL-extractors are stronger than both standard extractors and subgaussian samplers, but we show that they exist with essentially the same parameters (constructively and non-constructively) as standard extractors.Thu, 18 Apr 2019 03:27:56 +0300https://eccc.weizmann.ac.il/report/2019/059