Automatic Differentiation is no Panacea for Phylogenetic Gradient Computation

Many-core algorithms for high-dimensional gradients on phylogenetic trees

MOTIVATION

Advancements in high-throughput genomic sequencing are delivering genomic pathogen data at an unprecedented rate, positioning statistical phylogenetics as a critical tool to monitor infectious diseases globally. This rapid growth spurs the need for efficient inference techniques, such as Hamiltonian Monte Carlo (HMC) in a Bayesian framework, to estimate parameters of these phylogenetic models where the dimensions of the parameters increase with the number of sequences N. HMC requires repeated calculation of the gradient of the data log-likelihood with respect to (wrt) all branch-length-specific (BLS) parameters that traditionally takes O(N2) operations using the standard pruning algorithm. A recent study proposes an approach to calculate this gradient in O(N), enabling researchers to take advantage of gradient-based samplers such as HMC. The CPU implementation of this approach makes the calculation of the gradient computationally tractable for nucleotide-based models but falls short in performance for larger state-space size models, such as Markov-modulated and codon models. Here, we describe novel massively parallel algorithms to calculate the gradient of the log-likelihood wrt all BLS parameters that take advantage of graphics processing units (GPUs) and result in many fold higher speedups over previous CPU implementations.

RESULTS

We benchmark these GPU algorithms on three computing systems using three evolutionary inference examples exploring complete genomes from 997 dengue viruses, 62 carnivore mitochondria and 49 yeasts, and observe a >128-fold speedup over the CPU implementation for codon-based models and >8-fold speedup for nucleotide-based models. As a practical demonstration, we also estimate the timing of the first introduction of West Nile virus into the continental Unites States under a codon model with a relaxed molecular clock from 104 full viral genomes, an inference task previously intractable.

AVAILABILITY AND IMPLEMENTATION

We provide an implementation of our GPU algorithms in BEAGLE v4.0.0 (https://github.com/beagle-dev/beagle-lib), an open-source library for statistical phylogenetics that enables parallel calculations on multi-core CPUs and GPUs. We employ a BEAGLE-implementation using the Bayesian phylogenetics framework BEAST (https://github.com/beast-dev/beast-mcmc).

A topology-marginal composite likelihood via a generalized phylogenetic pruning algorithm

Bayesian phylogenetics is a computationally challenging inferential problem. Classical methods are based on random-walk Markov chain Monte Carlo (MCMC), where random proposals are made on the tree parameter and the continuous parameters simultaneously. Variational phylogenetics is a promising alternative to MCMC, in which one fits an approximating distribution to the unnormalized phylogenetic posterior. Previous work fit this variational approximation using stochastic gradient descent, which is the canonical way of fitting general variational approximations. However, phylogenetic trees are special structures, giving opportunities for efficient computation. In this paper we describe a new algorithm that directly generalizes the Felsenstein pruning algorithm (a.k.a. sum-product algorithm) to compute a composite-like likelihood by marginalizing out ancestral states and subtrees simultaneously. We show the utility of this algorithm by rapidly making point estimates for branch lengths of a multi-tree phylogenetic model. These estimates accord with a long MCMC run and with estimates obtained using a variational method, but are much faster to obtain. Thus, although generalized pruning does not lead to a variational algorithm as such, we believe that it will form a useful starting point for variational inference.