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.
