  A function $f$ is $d$-resilient if all its Fourier coefficients of degree at
most $d$ are zero, i.e., $f$ is uncorrelated with all low-degree parities. We
study the notion of $\mathit{approximate}$ $\mathit{resilience}$ of Boolean
functions, where we say that $f$ is $\alpha$-approximately $d$-resilient if $f$
is $\alpha$-close to a $[-1,1]$-valued $d$-resilient function in $\ell_1$
distance. We show that approximate resilience essentially characterizes the
complexity of agnostic learning of a concept class $C$ over the uniform
distribution. Roughly speaking, if all functions in a class $C$ are far from
being $d$-resilient then $C$ can be learned agnostically in time $n^{O(d)}$ and
conversely, if $C$ contains a function close to being $d$-resilient then
agnostic learning of $C$ in the statistical query (SQ) framework of Kearns has
complexity of at least $n^{\Omega(d)}$. This characterization is based on the
duality between $\ell_1$ approximation by degree-$d$ polynomials and
approximate $d$-resilience that we establish. In particular, it implies that
$\ell_1$ approximation by low-degree polynomials, known to be sufficient for
agnostic learning over product distributions, is in fact necessary.
  Focusing on monotone Boolean functions, we exhibit the existence of
near-optimal $\alpha$-approximately
$\widetilde{\Omega}(\alpha\sqrt{n})$-resilient monotone functions for all
$\alpha>0$. Prior to our work, it was conceivable even that every monotone
function is $\Omega(1)$-far from any $1$-resilient function. Furthermore, we
construct simple, explicit monotone functions based on ${\sf Tribes}$ and ${\sf
CycleRun}$ that are close to highly resilient functions. Our constructions are
based on a fairly general resilience analysis and amplification. These
structural results, together with the characterization, imply nearly optimal
lower bounds for agnostic learning of monotone juntas.
