We study the problem of learning parity functions that depend on at most k variables (k-parities) attribute-efficiently in the mistake-bound model.
We design simple, deterministic, polynomial-time algorithms for learning k-parities with mistake bound O(n^{1-\frac{c}{k}}), for any constant c > 0. These are the first polynomial-time algorithms that learn \omega(1)-parities in the mistake-bound model with mistake bound o(n).

Using the standard conversion techniques from the mistake-bound model to the PAC model, our algorithms can also be used for learning k-parities in the PAC model. In particular, this implies a slight improvement over the results of Klivans and Servedio
for learning k-parities in the PAC model.

We also show that the \widetilde{O}(n^{k/2}) time algorithm from
Klivans and Servedio's paper that PAC-learns k-parities with optimal sample complexity can be extended to the mistake-bound model.