We show that all non-negative submodular functions have high noise-stability. As a consequence, we obtain a polynomial-time learning algorithm for this class with respect to any product distribution on $\{-1,1\}^n$ (for any constant accuracy parameter $\epsilon$ ). Our algorithm also succeeds in the agnostic setting. Previous work on learning submodular ... more >>>

We formulate a notion of evolvability for functions with domain and range that are real-valued vectors, a compelling way of expressing many natural biological processes. We show that linear and fixed degree polynomial functions are evolvable in the following dually robust sense: There is a single evolution algorithm that for ... more >>>

Let $f$ be a non-commutative polynomial such that $f=0$ if we assume that the variables in $f$ commute. Let $Q(f)$ be the smallest $k$ such that there exist polynomials $g_1,g_1', g_2, g_2',\dots, g_k, g_k' $ with \[f\in I([g_1,g_1'], [g_2, g_2'],\dots, [g_k, g_k'] )\,,\]
where $[g,h]=gh-hg$. Then $Q(f)\leq {n\choose 2}$, where ... more >>>