Tight Bounds on $\ell_1$ Approximation and Learning of SelfBounding Functions
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Proceedings of the 28th International Conference on Algorithmic Learning Theory, PMLR 76:540559, 2017.
Abstract
We study the complexity of learning and approximation of selfbounding functions over the uniform distribution on the Boolean hypercube $\{0,1\}^n$. Informally, a function $f:\{0,1\}^n \rightarrow \mathbb{R}$ is selfbounding if for every $x \in \{0,1\}^n$, $f(x)$ upper bounds the sum of all the $n$ marginal decreases in the value of the function at $x$. Selfbounding functions include such wellknown classes of functions as submodular and fractionallysubadditive (XOS) functions. They were introduced by Boucheron et al. in the context of concentration of measure inequalities. Our main result is a nearly tight $\ell_1$approximation of selfbounding functions by lowdegree juntas. Specifically, all selfbounding functions can be $\epsilon$approximated in $\ell_1$ by a polynomial of degree $\tilde{O}(1/\epsilon)$ over $2^{\tilde{O}(1/\epsilon)}$ variables. We show that both the degree and juntasize are optimal up to logarithmic terms. Previous techniques considered stronger $\ell_2$ approximation and proved nearly tight bounds of $\Theta(1/\epsilon^{2})$ on the degree and $2^{\Theta(1/\epsilon^2)}$ on the number of variables. Our bounds rely on the analysis of noise stability of selfbounding functions together with a stronger connection between noise stability and $\ell_1$ approximation by lowdegree polynomials. This technique can also be used to get tighter bounds on $\ell_1$ approximation by lowdegree polynomials and faster learning algorithm for halfspaces. \newline These results lead to improved and in several cases almost tight bounds for PAC and agnostic learning of selfbounding functions relative to the uniform distribution. In particular, assuming hardness of learning juntas, we show that PAC and agnostic learning of selfbounding functions have complexity of $n^{\tilde{\Theta}(1/\epsilon)}$.
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