Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness

Abstract

We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let ℭ be any typical class of Boolean circuits, and ℭ[s(n)] denote n-variable ℭ-circuits of size ≤ s(n). We show:Learning Speedups. If ℭ[poly(n)] admits a randomized weak learning algorithm under the uniform distribution with membership queries that runs in time 2n/nω(1), then for every k ≥ 1 and e > 0 the class ℭ[nk] can be learned to high accuracy in time O(2ne). There is e > 0 such that ℭ[2ne] can be learned in time 2n/nω(1) if and only if ℭ[poly(n)] can be learned in time 2(logn)o(1).Equivalences between Learning Models. We use learning speedups to obtain equivalences between various randomized learning and compression models, including sub-exponential time learning with membership queries, sub-exponential time learning with membership and equivalence queries, probabilistic function compression and probabilistic average-case function compression.A Dichotomy between Learnability and Pseudorandomness. In the non-uniform setting, there is non-trivial learning for ℭ[poly(n)] if and only if there are no exponentially secure pseudorandom functions computable in ℭ[poly(n)].Lower Bounds from Nontrivial Learning. If for each k ≥ 1, (depth-d)-ℭ[nk] admits a randomized weak learning algorithm with membership queries under the uniform distribution that runs in time 2n/nω(1), then for each k ≥ 1, BPE n (depth-d)-ℭ[nk]. If for some e > 0 there are P-natural proofs useful against ℭ[2ne], then ZPEXP n ℭ[poly(n)].Karp-Lipton Theorems for Probabilistic Classes. If there is a k > 0 such that BPE ⊆ i.o.Circuit[nk], then BPEXP ⊆ i.o.EXP/O(log n). If ZPEXP ⊆ i.o.Circuit[2n/3], then ZPEXP ⊆ i.o.ESUBEXP.Hardness Results for MCSP. All functions in non-uniform NC1 reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC0 circuits. In particular, if MCSP e TC0 then NC1 = TC0.