On the Probabilistic Degrees of Symmetric Boolean functions. The probabilistic degree of a Boolean function f:0,1nrightarrow 0,1 is defined to be the smallest d such that there is a random polynomial mathbfP of degree at most d that agrees with f at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions  specifically symmetric Boolean functions  have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems.  :76,this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

Degreed Chow Parameters Robustly Determine Degreed PTFs (and Algorithmic Applications). The degreed Chow parameters of a Boolean function f: 1,1n to mathbbR are its degree at most d Fourier coefficients. It is wellknown that degreed Chow parameters uniquely characterize degreed polynomial threshold functions (PTFs) within the space of all bounded functions. In this paper, we prove a robust version of this theorem: For f any Boolean degreed PTF and g any bounded function, if the degreed Chow parameters of f are close to the degreed Chow parameters of g in ell_2norm, then f is close to g in ell_1distance. Notably, our bound relating the two distances is completely independent of the dimension n. That is, we show that Boolean degreed PTFs are em robustly identifiable from their degreed Chow parameters. Results of this form had been shown for the d1 caseciteOS11:chow, DeDFS14, but no nontrivial bound was previously known for d u003e1.  robust :142,identifiability result gives the following algorithmic applications: First, we show that Boolean degreed PTFs can be efficiently approximately reconstructed from approximations to their degreed Chow parameters. This immediately implies that degreed PTFs are efficiently learnable in the uniform distribution dRFA modelciteBenDavidDichterman:98. As a byproduct of our approach, we also obtain the first low integerweight approximations of degreed PTFs, for du003e1. As our second application, our robust :142,identifiability result gives the first efficient algorithm, with dimensionindependent error guarantees, for malicious learning of Boolean degreed PTFs under the uniform distribution.