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Scalable backpropagation for Gaussian Processes using celerite
Daniel Foreman-Mackey
TL;DR
GP likelihood gradients can be computationally expensive for large datasets, even when celerite makes covariance operations scalable. This note derives reverse-accumulation algorithms for celerite and reports efficient gradient computation suitable for integration with automatic-differentiation libraries.
Problem
General GP likelihood calculations can scale as O(N^3), while gradient computation can remain prohibitive at O(N^2) for large datasets.
Method
The note derives reverse-accumulation, or backpropagation, algorithms that differentiate celerite factorization and solve procedures.
Results
The resulting algorithms efficiently compute gradients for GP models on large datasets using celerite.
Takeaways & Limitations
The reverse-accumulation algorithms enable integration of celerite gradients into Stan, TensorFlow, and other automatic-differentiation libraries.
Abstract
from arXiv · showhide
This research note presents a derivation and implementation of efficient and scalable gradient computations using the celerite algorithm for Gaussian Process (GP) modeling. The algorithms are derived in a "reverse accumulation" or "backpropagation" framework and they can be easily integrated into existing automatic differentiation frameworks to provide a scalable method for evaluating the gradients of the GP likelihood with respect to all input parameters. The algorithm derived in this note uses less memory and is more efficient than versions using automatic differentiation and the computational cost scales linearly with the number of data points.
INTRODUCTION
Gaussian Processes are broadly used for astrophysical stochastic modeling, but general GP likelihood evaluations scale as O(N^3). Celerite reduces this cost for a class of one-dimensional models through factorization and inverse application algorithms whose cost scales as O(N J^2) and O(N J), respectively.
- These GP methods support applications including astronomical variability, light-curve and stellar-spectrum modeling, and cosmic microwave background analysis.
- General GP likelihood evaluation requires a log-determinant and covariance-matrix inverse multiplication, with O(N^3) computational cost.
- Celerite computes GP likelihoods with O(N) scaling for a class of one-dimensional models.
- The celerite factorization computes the log-determinant using an algorithm scaling as O(N J^2).
- Celerite applies the covariance inverse to compute Z = K^-1 Y using an O(N J) algorithm.
GRADIENTS OF GP MODELS USING CELERITE
The note derives reverse-mode gradients for celerite because standard GP gradient computation can become prohibitive even when covariance application is scalable. The resulting backpropagation algorithms differentiate the celerite factorization and solve procedures directly.
- Gradient computation can scale as O(N^2), making celerite-based inference prohibitive for large datasets.
- The algorithms are motivated by direct differentiation of Cholesky factorizations, which can provide substantial gains over general-case gradient computation.
- The note presents reverse-accumulation gradients for celerite using matrix-gradient notation and backpropagation.
- The celerite factor reverse pass propagates adjoints through the factorization to obtain gradients with respect to U, P, a, and V.
- A corresponding reverse-accumulation function is provided for celerite solve, using stored forward quantities and adjoints.
DISCUSSION
The note supplies efficient gradient algorithms for GP models using celerite and supports their integration into automatic-differentiation libraries. Its empirical scaling is reported in Figure 1, and reference implementations, benchmarks, and tests are available online.
- The note’s algorithms efficiently compute gradients for GP models on large datasets using celerite.
- The reverse-accumulation derivations support integration with Stan, TensorFlow, and other automatic-differentiation libraries.
- Reference implementations, benchmarks, and tests are available through the cited celerite-grad repository and Foreman-Mackey (2018).
- Figure 1 measures empirical costs for likelihood and gradient computations across data-point count N and model complexity J.