Optimizing Black-box Metrics with Adaptive Surrogates

We address the problem of training models with black-box and hard-to-optimize metrics by expressing the metric as a monotonic function of a small number of easy-to-optimize surrogates. We pose the training problem as an optimization over a relaxed surrogate space, which we solve by estimating local gradients for the metric and performing inexact convex projections. We analyze gradient estimates based on finite differences and local linear interpolations, and show convergence of our approach under smoothness assumptions with respect to the surrogates. Experimental results on classification and ranking problems verify the proposal performs on par with methods that know the mathematical formulation, and adds notable value when the form of the metric is unknown. 1 , . . . , K : R d → R + where K d, and express M as an unknown non-decreasing function of the K surrogates, with an unknown slack: where ψ : R K + → [0, 1] is monotonic but possibly nonconvex, and the slack : R d → [-1, 1] determines how well the metric can be approximated by the K surrogates. Note that this decomposition of M is not unique. Our results hold for any such decomposition, but to enable a tighter analysis we consider a ψ for which the associated worst-case slack over all θ, i.e., max θ∈R d | (θ)| is the minimum.