We study the parameter complexity of robust memorization for $\mathrm{ReLU}$ networks: the number of parameters required to interpolate any given dataset with $\epsilon$-separation between differently labeled points, while ensuring predictions remain consistent within a $\mu$-ball around each training sample. We establish upper and lower bounds on the parameter count as a function of the robustness ratio $\rho = \mu / \epsilon$. Unlike prior work, we provide a fine-grained analysis across the entire range $\rho \in (0,1)$ and obtain tighter upper and lower bounds that improve upon existing results. Our findings reveal that the parameter complexity of robust memorization matches that of non-robust memorization when $\rho$ is small, but grows with increasing $\rho$.

翻译：我们研究了ReLU网络鲁棒记忆的参数复杂度：即在确保每个训练样本周围μ球内预测保持一致的条件下，以ε间隔区分不同标签点的方式插值任意给定数据集所需的参数数量。我们建立了参数数量关于鲁棒比ρ = μ/ε的函数的上界和下界。与先前工作不同，我们对整个范围ρ ∈ (0,1)进行了细粒度分析，获得了比现有结果更紧的上界和下界。我们的研究结果表明，当ρ较小时，鲁棒记忆的参数复杂度与非鲁棒记忆相当，但随着ρ增大而增长。