Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness

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 2/n, then for every k ≥ 1 and ε > 0 the class C[n] can be learned to high accuracy in timeO(2 ε ). There is ε > 0 such that C[2 ε ] can be learned in time 2/n if and only if C[poly(n)] can be learned in time 2 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[n] admits a randomized weak learning algorithm with membership queries under the uniform distribution that runs in time 2/n, then for each k ≥ 1, BPE * (depth-d)-C[n]. If for some ε > 0 there are P-natural proofs useful against C[2 ε ], then ZPEXP * C[poly(n)]. Karp-Lipton Theorems for Probabilistic Classes. If there is a k > 0 such that BPE ⊆ i.o.Circuit[n], then BPEXP ⊆ i.o.EXP/O(log n). If ZPEXP ⊆ i.o.Circuit[2], then ZPEXP ⊆ i.o.ESUBEXP. Hardness Results for MCSP. All functions in non-uniform NC reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC circuits. In particular, if MCSP ∈ TC then NC = TC. 1 ar X iv :1 61 1. 01 19 0v 1 [ cs .C C ] 3 N ov 2 01 6