TR18-122 Authors: Igor Carboni Oliveira, Rahul Santhanam

Publication: 3rd July 2018 19:24

Downloads: 295

Keywords:

We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser

[GG11]. 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 ap-

proximation. 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 as-

sumptions 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 AC0[p] circuits by giving a deterministic learning algorithm for AC0[p].

Approximation: Turning to approximation, we unconditionally pseudo-derandomize any poly-

time randomized approximation scheme for integer-valued functions infinitely often in sub-

exponential 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 AC0[p] computable in quasi-polynomial time.