Improved learning of k-parities

We consider the problem of learning k-parities in the online mistake-bound model: given a hidden vector x∈{0,1}ⁿ where the hamming weight of x is k and a sequence of “questions” a₁,a₂,…∈{0,1}ⁿ, where the algorithm must reply to each question with 〈a_i,x〉(mod2), what is the best trade-off between the number of mistakes made by the algorithm and its time complexity? We improve the previous best result of Buhrman et al. [3] by an exp⁡(k) factor in the time complexity. Next, we consider the problem of learning k-parities in the PAC model in the presence of random classification noise of rate [Formula Presented]. Here, we observe that even in the presence of classification noise of non-trivial rate, it is possible to learn k-parities in time better than (nk/2), whereas the current best algorithm for learning noisy k-parities, due to Grigorescu et al. [9], inherently requires time (nk/2) even when the noise rate is polynomially small.