Replicability in Learning: Geometric Partitions and KKM-Sperner Lemma

This paper studies replicability in machine learning tasks from a geometric viewpoint. Recent works have revealed the role of geometric partitions and Sperner's lemma (and its variations) in designing replicable learning algorithms and in establishing impossibility results. A partition $\mathcal{P}$ of $\mathbb{R}^d$ is called a $(k,\epsilon)$-secluded partition if for every $\vec{p}\in\mathbb{R}^d$, an $\varepsilon$-radius ball (with respect to the $\ell_{\infty}$ norm) centered at $\vec{p}$ intersects at most $k$ members of $\mathcal{P}$. In relation to replicable learning, the parameter $k$ is closely related to the $\textit{list complexity}$, and the parameter $\varepsilon$ is related to the sample complexity of the replicable learner. Construction of secluded partitions with better parameters (small $k$ and large $\varepsilon$) will lead to replicable learning algorithms with small list and sample complexities. Motivated by this connection, we undertake a comprehensive study of secluded partitions and establish near-optimal relationships between $k$ and $\varepsilon$. 1. We show that for any $(k,\epsilon)$-secluded partition where each member has at most unit measure, it must be that $k \geq(1+2\varepsilon)^d$, and consequently, for the interesting regime $k\in[2^d]$ it must be that $\epsilon\leq\frac{\log_4(k)}{d}$. 2. To complement this upper bound on $\epsilon$, we show that for each $d\in\mathbb{N}$ and each viable $k\in[2^d]$, a construction of a $(k,\epsilon)$-secluded (unit cube) partition with $\epsilon\geq\frac{\log_4(k)}{d}\cdot\frac{1}{8\log_4(d+1)}$. This establishes the optimality of $\epsilon$ within a logarithmic factor.3. Finally, we adapt our proof techniques to obtain a new ``neighborhood'' variant of the cubical KKM lemma (or cubical Sperner's lemma): For any coloring of $[0,1]^d$ in which no color is used on opposing faces, it holds for each $\epsilon\in(0,\frac12]$ that there is a point where the open $\epsilon$-radius $\ell_\infty$-ball intersects at least $(1+\frac23\epsilon)^d$ colors. While the classical Sperner/KKM lemma guarantees the existence of a point that is "adjacent" to points with $(d+1)$ distinct colors, the neighborhood version guarantees the existence of a small neighborhood with exponentially many points with distinct colors.