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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.55
URN: urn:nbn:de:0030-drops-94598
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Carboni Oliveira, Igor ; Santhanam, Rahul

Pseudo-Derandomizing Learning and Approximation

LIPIcs-APPROX-RANDOM-2018-55.pdf (0.5 MB)


We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser [Eran Gat and Shafi Goldwasser, 2011]. A pseudo-deterministic algorithm is a probabilistic algorithm which produces a fixed output with high probability. We explore pseudo-determinism in the settings of learning and approximation. Our goal is to simulate known randomized algorithms in these settings by pseudo-deterministic algorithms in a generic fashion - a goal we succinctly term pseudo-derandomization. Learning. In the setting of learning with membership queries, we first show that randomized learning algorithms can be derandomized (resp. pseudo-derandomized) under the standard hardness assumption that E (resp. BPE) requires large Boolean circuits. Thus, despite the fact that learning is an algorithmic task that requires interaction with an oracle, standard hardness assumptions suffice to (pseudo-)derandomize it. We also unconditionally pseudo-derandomize any {quasi-polynomial} time learning algorithm for polynomial size circuits on infinitely many input lengths in sub-exponential time.
Next, we establish a generic connection between learning and derandomization in the reverse direction, by showing that deterministic (resp. pseudo-deterministic) learning algorithms for a concept class C imply hitting sets against C that are computable deterministically (resp. pseudo-deterministically). In particular, this suggests a new approach to constructing hitting set generators against AC^0[p] circuits by giving a deterministic learning algorithm for AC^0[p]. Approximation. Turning to approximation, we unconditionally pseudo-derandomize any poly-time randomized approximation scheme for integer-valued functions infinitely often in subexponential time over any samplable distribution on inputs. As a corollary, we get that the (0,1)-Permanent has a fully pseudo-deterministic approximation scheme running in sub-exponential time infinitely often over any samplable distribution on inputs.
Finally, we {investigate} the notion of approximate canonization of Boolean circuits. We use a connection between pseudodeterministic learning and approximate canonization to show that if BPE does not have sub-exponential size circuits infinitely often, then there is a pseudo-deterministic approximate canonizer for AC^0[p] computable in quasi-polynomial time.

BibTeX - Entry

  author =	{Igor Carboni Oliveira and Rahul Santhanam},
  title =	{{Pseudo-Derandomizing Learning and Approximation}},
  booktitle =	{Approximation, Randomization, and Combinatorial  Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{55:1--55:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-94598},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.55},
  annote =	{Keywords: derandomization, learning, approximation, boolean circuits}

Keywords: derandomization, learning, approximation, boolean circuits
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
Issue Date: 2018
Date of publication: 13.08.2018

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