We consider the problem of distribution-free learning for Boolean function classes in the PAC and agnostic models. Generalizing a recent beautiful work of Malach and Shalev-Shwartz \mbox{%DIFAUXCMD malach2020hardness who gave the first tight correlational SQ (CSQ) lower bounds for learning DNF formulas, we show that lower bounds on the threshold or approximate degree of any function class directly imply CSQ lower bounds for PAC or agnostic learning respectively. These match corresponding positive results using upper bounds on the threshold or approximate degree in the SQ model for PAC or agnostic learning , and in this sense our results show that the polynomial method is a universal, best-possible approach for distribution-free CSQ learning. Our results imply the first exponential (in the dimension) CSQ lower bounds for PAC learning intersections of two halfspaces and constant depth circuits as well as the first exponential lower bounds for agnostically learning conjunctions and halfspaces .
We consider the problem of distribution-free learning for Boolean function classes in the PAC and agnostic models. Generalizing a recent beautiful work of Malach and Shalev-Shwartz (2020) who gave the first tight correlational SQ (CSQ) lower bounds for learning DNF formulas, we show that lower bounds on the threshold or approximate degree of any function class directly imply CSQ lower bounds for PAC or agnostic learning respectively. These match corresponding positive results using upper bounds on the threshold or approximate degree in the SQ model for PAC or agnostic learning . Many of these results were implicit in earlier works of Feldman and Sherstov .
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\mbox{%DIFAUXCMD malach2020hardness
(2020)
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, and in this sense our results show that the polynomial method is a universal, best-possible approach for distribution-free CSQ learning. Our results imply the first exponential (in the dimension) CSQ lower bounds for PAC learning intersections of two halfspaces and constant depth circuits as well as the first exponential lower bounds for agnostically learning conjunctions and halfspaces
. Many of these results were implicit in earlier works of Feldman and Sherstov