Pseudo-derandomizing learning and approximation

Conference item

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.

Authors: Oliveira, I; Santhanam, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Schloss Dagstuhl
• Host title: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
• Journal: 22nd International Conference on Randomization and Approximation
• Volume: 116
• Series: Leibniz International Proceedings in Informatics
• Publication date: 2018-08-02
• Acceptance date: 2018-06-01
• DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.55
• ISSN: 1868-8969
• ISBN: 9783959770859
• Keywords: boolean circuits; derandomization; approximation; learning