We provide efficient replicable algorithms for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. We design the first dimension-independent replicable algorithms for this task which runs in polynomial time, is proper, and has strictly improved sample complexity compared to the one achieved by Impagliazzo et al. [2022] with respect to all the relevant parameters. Moreover, our first algorithm has sample complexity that is optimal with respect to the accuracy parameter $\epsilon$. We also design an SGD-based replicable algorithm that, in some parameters' regimes, achieves better sample and time complexity than our first algorithm. Departing from the requirement of polynomial time algorithms, using the DP-to-Replicability reduction of Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sorrell, and Sivakumar [STOC, 2023], we show how to obtain a replicable algorithm for large-margin halfspaces with improved sample complexity with respect to the margin parameter $\tau$, but running time doubly exponential in $1/\tau^2$ and worse sample complexity dependence on $\epsilon$ than one of our previous algorithms. We then design an improved algorithm with better sample complexity than all three of our previous algorithms and running time exponential in $1/\tau^{2}$.

翻译：我们针对大间隔半空间学习问题提出了高效的可复制算法。我们的结果改进了Impagliazzo、Lei、Pitassi和Sorrell [STOC, 2022] 提供的算法。我们设计了首个与维度无关的可复制算法，该算法运行时间为多项式时间、具有真属性，并且在所有相关参数上相对于Impagliazzo等人 [2022] 的算法实现了严格更优的样本复杂度。此外，我们的第一个算法在精度参数 $\epsilon$ 上达到了最优样本复杂度。我们还设计了一个基于SGD的可复制算法，在某些参数范围内，该算法的样本复杂度和时间复杂度均优于第一个算法。放宽多项式时间算法的要求后，利用Bun、Gaboardi、Hopkins、Impagliazzo、Lei、Pitassi、Sorrell和Sivakumar [STOC, 2023] 提出的DP到可复制性归约方法，我们展示了如何获得针对大间隔半空间的可复制算法，该算法在间隔参数 $\tau$ 上具有更优的样本复杂度，但运行时间为 $1/\tau^2$ 的双指数级，且 $\epsilon$ 上的样本复杂度依赖关系弱于我们之前的某个算法。随后，我们设计了一个改进算法，其样本复杂度优于前述三种算法，运行时间为 $1/\tau^{2}$ 的指数级。