Hardness of Maximum Likelihood Learning of DPPs.

Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively correlated sets. DPPs are used in Machine Learning applications to select a diverse, yet representative subset of data. In these applications, the parameters of the DPP need to be fit to match the data; typically, we seek a set of parameters that maximize the likelihood of the data. The algorithms used for this task either optimize over a limited family of DPPs, or else use local improvement heuristics that do not provide theoretical guarantees of optimality. It is natural to ask if there exist efficient algorithms for finding a maximum likelihood DPP model for a given data set. In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD Thesis (2012) that the problem is NP-complete. In this work we prove Kulesza’s conjecture: we prove moreover, that even computing a $1-\frac{1}{\mathrm{poly} \log N}$-approximation to the maximum log-likelihood of a DPP on a set of $N$ items is NP-complete. At the same time, we also obtain the first polynomial-time algorithm obtaining a nontrivial worst-case approximation to the optimal likelihood: we present a polynomial-time $1/\log m$-approximation algorithm (for data sets of size $m$), which moreover obtains a $1-\frac{1}{\log N}$-approximation if all $N$ elements appear in a $O(1/N)$-fraction of the subsets. In terms of techniques, the hardness result reduces to solving a gap instance of a “vector coloring" problem on a hypergraph obtained from an adaptation of the constructions of Bogdanov, Obata and Trevisan (FOCS 2002), using the strong expanders of Alon and Capalbo (FOCS 2007).