Bayesian Optimization using Student-t Processes

Finding the global minimum of a function is often difficult. We consider efficiently minimizing functions which are computationally expensive to evaluate. A Bayesian approach to the global function optimization problem places a prior distribution on the function and chooses where to evaluate the function based on its posterior distribution given a set of observations. While many recent applications use Gaussian processes as a prior for the objective function, here we show that a Student-t process is an ideal prior for such a problem, as it is also nonparametric, but naturally models heavy tailed behaviour and has a predictive covariance which explicitly depends on observations. 1 The Student-t Process We begin by deriving the Student-t process1, and its marginal likelihood and predictive distribution, starting from a hierarchical Gaussian process model. We define a prior over continous functions using the following generative model r−1 ∼ Γ (ν 2 , ρ 2 ) y|r ∼ GP ( 0, r(ν − 2)k/ρ ) , (1) where ν > 2, ρ > 0 and k : R × R → R is a kernel function. If we marginalize over r, y is a scaled mixture of Gaussian processes. Suppose y = (y1, ..., yN ) is a finite collection of observations at input points x1, ...,xN ∈ R and let K be the Gram matrix such that Kij = k(xi,xj). We can compute the marginal probability of these observations under the generative prior above as follows p(y) = ∫ p(y|r)p(r)dr = (π(ν − 2)) −N/2(ρ/2) ν+N 2 |K|1/2Γ(ν/2) ∫ r−(ν+N)/2−1 exp ( − ρ 2r ( 1 + y>K−1y ν − 2 )) dr = (π(ν − 2))−N/2 Γ((ν +N)/2) Γ(ν/2) |K|−1/2 ( 1 + y>K−1y ν − 2 )−(ν+N)/2 . (2) The Student-t process [O’Hagan, 1991, O’Hagan et al., 1999] has been used in a number of applications [Yu et al., 2007, Zhang and Yeung, 2010, Xu et al., 2011]. Our parameterization differs slightly from previous constructions.