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 $1/\tau^{2}$.

我们提供了学习大间隔半空间问题的高效可复制算法，改进了 Impagliazzo 等人提供的算法。我们首次设计了这个任务的无维度依赖的可复制算法，其在多项式时间内运行，是合适的，并且在所有相关参数方面与 Impagliazzo 等人取得的结果相比，样本复杂度有明显提高。此外，我们的第一个算法在精度参数 ε 上具有最优的样本复杂度。我们还设计了一种基于 SGD 的可复制算法，在某些参数区间内，比我们第一个算法具有更好的样本和时间复杂度。最后，我们设计了一种改进的算法，其在样本复杂度上优于我们以前的三种算法，并且运行时间呈指数关系于 1/τ^2。