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.