On the Strong Convexity of Variational Inference

Variational methods provide a tractable approximation to the intractable task of computing inference in probabilistic graphical models. Much of the literature in this field concerns the convexity of the variational objective, since convexity guarantees convergence to a global optimum. However, less attention has focused on the strength of the convexity. By this we mean the following. Definition 1. A differentiable function, φ : S → R, of a convex set S, is κ-strongly convex with respect to a norm ‖ · ‖ iff, for all s, s′ ∈ S, κ2 ‖s− s ′‖ + 〈∇φ(s), s′ − s〉 ≤ φ(s′)− φ(s).