Near-optimal sample complexity bounds for learning Latent $k-$polytopes and applications to Ad-Mixtures

Recently near-optimal bounds on sample complexity of Mixture of Gaussians was shown in the seminal paper \cite{HSNCAY18}. No such results are known for Ad-mixtures. In this paper we show that $O^*(dk/m)$ samples are sufficient to learn each of $k-$ topic vectors of LDA, a popular Ad-mixture model, with vocabulary size $d$ and $m\in \Omega(1)$ words per document, to any constant error in $L_1$ norm. This is a corollary of the major contribution of the current paper: the first sample complexity upper bound for the problem (introduced in \cite{BK20}) of learning the vertices of a Latent $k-$ Polytope in ${\bf R}^d$, given perturbed points from it. The bound, $O^*(dk/\beta)$, is optimal and applies to many stochastic models including LDA, Mixed Membership block Models(MMBM),Dirichlet Simplex Nest, and large class of Ad-mixtures. The parameter, $\beta$ depends on the probability laws governing individual models and in many cases can be expressed very succintly, e.g. it is equal to the average degree of each node for MMBM, and equal to $m$ in LDA. The tightness is proved by a nearly matching lower of $\Omega^*(dk/\beta)$ by a combinatorial construction based on a code-design. Our upper bound proof combines two novel methods. The first is {\it vertex set certification} which, for any $k-$polytope $K$ gives convex geometry based sufficient conditions for a set of $k$ points from a larger candidate set to be close in Hausdorff distance to the set of $k$ vertices of the polytope. The second is {\it subset averaging} which uses $\beta$ to prove that the set of averages of all large subsets of data is a good candidate set.