A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential Families
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We consider the problem of computing the maximum likelihood multivariate log-concave distribution for a set of points. Specifically, we present an algorithm which, given n points in R^d and an accuracy parameter ϵ>0, runs in time poly(n,d,1/ϵ), and returns a log-concave distribution which, with high probability, has the property that the likelihood of the n points under the returned distribution is at most an additive ϵ less than the maximum likelihood that could be achieved via any log-concave distribution. This is the first computationally efficient (polynomial time) algorithm for this fundamental and practically important task. Our algorithm rests on a novel connection with exponential families: the maximum likelihood log-concave distribution belongs to a class of structured distributions which, while not an exponential family, "locally" possesses key properties of exponential families. This connection then allows the problem of computing the log-concave maximum likelihood distribution to be formulated as a convex optimization problem, and solved via an approximate first-order method. Efficiently approximating the (sub) gradients of the objective function of this optimization problem is quite delicate, and is the main technical challenge in this work.
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