We continue the study of doubly-efficient proof systems for verifying agnostic PAC learning, for which we obtain the following results. • We construct an interactive protocol for learning the t largest Fourier characters of a given function f : {0, 1}n → {0, 1} up to an arbitrarily small error, wherein the verifier uses poly(t) random examples. This improves upon the Interactive Goldreich-Levin protocol of [GRSY21], whose sample complexity is poly(t, n). • For agnostically learning the class AC0[2] under the uniform distribution, we build on the work of [CIKK17] and design an interactive protocol, where given a function f : {0, 1}n → {0, 1}, the verifier learns the closest hypothesis up to polylog(n) multiplicative factor, using quasi- polynomially many random examples. In contrast, this class has been notoriously resistant even for constructing realisable learners (without a prover) using random examples. • For agnostically learning k-juntas under the uniform distribution, we obtain an interactive protocol, where the verifier uses O(2k ) random examples to a given function f : {0, 1}n → {0, 1}. Crucially, the sample complexity of the verifier is independent of n. We also show that if we do not insist on doubly-efficient proof systems, then the model becomes trivial. Specifically, we show a protocol for an arbitrary class C of Boolean functions in the distribution-free setting, where the verifier uses O(1) labeled examples to learn f.