Supplementary Material for “DivMCuts: Faster Training of Structural SVMs with Diverse M-Best Cutting-Planes”

This section reviews the notation used in the main paper and revisits cutting-plane methods for training structured-output predictors. Notation. For any positive integer n we use [n] as shorthand for the set {1, 2, . . . , n}. We use y for a structured-output, and Y = (y1, . . . ,y|Y|) for a tuple of structured-outputs. Given a training dataset of input-output pairs {(xi,yi) |xi ∈X ,yi ∈Y}, we are interested in learning a mapping f : X → Y from an input space X to a structured output space Y that is finite but typically exponentially large (e.g., the set of all segmentations of an image, or all English translations of a Chinese sentence). Structural Support Vector Machines (SSVMs). In an SSVM setting, the mapping is defined as f(x) = argmaxy∈Y w TΨ(x,y), where Ψ(x,y) is a joint feature map: Ψ : X × Y → Rd. The quality of the prediction ŷi = f(xi) is measured by a task-specific loss function ` : Y × Y → R+, where `(yi, ŷi) denotes the cost of predicting ŷi when the correct label is yi. Since the task-loss ` is typically non-convex and non-continuous, [2] proposed to optimize the hinge upper bound on `. The regularized hinge-loss SSVM learning problem can be formulated as a QP with exponentially many constraints. In this paper, we work with the 1-slack formulation of Joachims et al. [1] in the MarginRescaling variant (1). Optimization Problem 1 (OP1). 1-slack Structural SVM (Margin-Rescaling) Training (Primal) formulation,