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 /n ω(1) , then for every k ≥ 1 and ε > 0 the class C[n k ] can be learned to high accuracy in time O(2 n

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... ε). There is ε > 0 such that C[2 n ε ] can be learned in time 2 n /n ω(1) if and only if C[poly(n)] can be learned in time 2 (log n) 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 pseu-dorandom functions computable in C[poly(n)]. Lower Bounds from Nontrivial Learning. If for each k ≥ 1, (depth-d)-C[n k ] admits a ran-domized weak learning algorithm with membership queries under the uniform distribution that runs in time 2 n /n ω(1) , then for each k ≥ 1, BPE (depth-d)-C[n k ]. If for some ε > 0 there are P-natural proofs useful against C[2 n 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 k ], then BPEXP ⊆ i.o.EXP/O(log n). If ZPEXP ⊆ i.o.Circuit[2 n/3 ], then ZPEXP ⊆ i.o.ESUBEXP. Hardness Results for MCSP. All functions in non-uniform NC 1 reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC 0 circuits. In particular, if MCSP ∈ TC 0 then NC 1 = TC 0 .