  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 $f \colon \{0,1\}^n \to \{0,1\}$ up to an
arbitrarily small error, wherein the verifier uses $\mathsf{poly}(t)$ random
examples. This improves upon the Interactive Goldreich-Levin protocol of
Goldwasser, Rothblum, Shafer, and Yehudayoff (ITCS 2021) whose sample
complexity is $\mathsf{poly}(t,n)$.
  - For agnostically learning the class $\mathsf{AC}^0[2]$ under the uniform
distribution, we build on the work of Carmosino, Impagliazzo, Kabanets, and
Kolokolova (APPROX/RANDOM 2017) and design an interactive protocol, where given
a function $f \colon \{0,1\}^n \to \{0,1\}$, the verifier learns the closest
hypothesis up to $\mathsf{polylog}(n)$ 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 $O(2^k)$ random
examples to a given function $f \colon \{0,1\}^n \to \{0,1\}$. 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 $\mathcal{C}$ of Boolean functions in the distribution-free setting,
where the verifier uses $O(1)$ labeled examples to learn $f$.
