We consider the problem of learning k-parities in the online mistake-bound model: given a hidden vector $x\in {\{0,1\}}^n$ where the hamming weight of x is k and a sequence of “questions” $a_1,a_2,\dots \in {\{0,1\}}^n$, where the algorithm must reply to each question with $〈a_i,x〉(\mathrm{mod}2)$, 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 $\eta \in (0,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$. 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 $\left(\begin{array}{c}n\\ k/2\end{array}\right)$, whereas the current best algorithm for learning noisy k-parities, due to Grigorescu et al. [9], inherently requires time $\left(\begin{array}{c}n\\ k/2\end{array}\right)$ even when the noise rate is polynomially small.

我们考虑在线错误约束模型中学习k-奇偶性的问题：给定一个隐藏向量$X\in {\{0，\mathrm{1个}\}}^{ñ}$x的汉明权重为k以及一系列“问题”${\mathrm{一种}}_{\mathrm{1个}}，{\mathrm{一种}}_2，\dots \in {\{0，\mathrm{1个}\}}^{ñ}$，算法必须在其中回答每个问题 $〈{\mathrm{一种}}_{\mathrm{一世}}，X〉（\mathrm{模}2）$，算法所犯的错误数量与其时间复杂度之间的最佳权衡是什么？我们提高了Buhrman等人先前的最佳结果。[3]由$\mathrm{经验值}（ķ）$ 时间复杂度的因素。

接下来，我们考虑在速率随机分类噪声存在下在PAC模型中学习k-奇偶性的问题$\eta \in （0，\raisebox{1ex}{$\mathrm{1个}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.）$。在这里，我们看到，即使是在不平凡率的分类噪声的存在，有可能学习ķ -parities在时间上比好$（\begin{array}{c}ñ\\ ķ/2\end{array}）$，而由于Grigorescu等人，目前用于学习有噪k奇偶校验的最佳算法。[9]，本质上需要时间$（\begin{array}{c}ñ\\ ķ/2\end{array}）$ 即使噪声率在多项式上很小。