Arnab Bhattacharyya: Hardness of Learning Noisy Halfspaces Using Polynomial Thresholds

Thursday, May 11, 2017 - 2:00pm to 3:00pm
Arnab Bhattacharyya

We prove the hardness of weakly learning halfspaces in the presence of adversarial noise using polynomial threshold functions (PTFs), assuming Khot's Unique Games Conjecture (UGC). In particular, we prove that for any constants d in? the positive integers? and eps > 0, assuming the UGC, it is NP-hard to decide: given a set of {-1,1}-labeled points in R^n whether (YES Case) there exists a halfspace that classifies (1-eps)-fraction of the points correctly, or (NO Case) any degree-d PTF classifies at most (1/2 + eps)-fraction of the points correctly.
This strengthens to all constant degrees the previous NP-hardness of learning using degree-2 PTFs shown by Diakonikolas et al. (2011). The latter result had remained the only progress over the works of Feldman et al. (2006) and Guruswami et al. (2006) ruling out weakly proper learning adversarially noisy halfspaces.

Joint work with Suprovat Ghoshal (IISc) and Rishi Saket (IBM).