Decay of correlation in network flow problems

Abstract

We develop a general theory for the local sensitivity of optimal points of constrained network optimization problems under perturbations of the constraints. For the network flow problem, we show that local perturbations on the constraints have an impact on the components of the optimal point that decreases exponentially with the graph-theoretical distance. The exponential rate is controlled by the spectral radius of a substochastic transition matrix of a killed random walk associated to the network. For graphs where this spectral radius is wellbehaved (bounded, for instance) as a function of the dimension of the network, our theory yields the first-known incarnation of the decay of correlation principle in constrained optimization.Index Terms-sensitivity of optimal point, decay of correlation, network flow, killed random walk

The infinite Viterbi alignment and decay-convexity

“…In the broader literature on convex optimization, a theory of sensitivity of optimal points with respect to constraints in convex network optimization problems has been introduced by [15,16]. The notions of scale-free optimization developed there are similar in spirit to the objectives of the present paper, but the results are not directly comparable to ours since they concern a constrained optimization problem.…”

“…In these studies the Viterbi process appears to be primarily of theoretical interest. Here we consider also a practical motivation in a similar spirit to distributed optimization methods, e.g., [12,15,16]: the existence of the limit in (3) suggests that (2) can be solved approximately using a collection of optimization algorithms which process data segments in parallel. To sketch the idea, with ∆, δ and ℓ = (n + 1)/∆ integers, consider the index sets:…”

“…, n − 1. Then writing Φ n t,0 (x) for the first d elements of the vector Φ n t (x), it follows from (15) that (12) ensures that as n → ∞, the influence of x n on Φ n t,0 (x) decays as n → ∞ with rate given by γ.…”

“…This reflects the fact that on an infinite time horizon, the prior and posterior probability measures over the entire state sequence (X n ) n∈N0 are typically singular, so that a density "p(x 0:∞ |y ⋆ 0:∞ )" does not exist. Hence there is no sense in characterizing the Viterbi process as: "ξ ∞ = arg max p(x 0:∞ |y ⋆ 0:∞ )", the correct characterization is: ∇U ∞ (ξ ∞ ) = 0, as Theorem 2 shows via a counter-part of ( 14)- (15) in the case n = ∞.…”

Scale-free network optimization: foundations and algorithms

“…Remark 1 (Connection with previous work) Some of the results presented in this paper will appear in a weaker form and without full proofs in Rebeschini and Tatikonda (2016). There, the sensitivity analysis is developed for matrices A's that are full row rank, so that the matrix AΣ(b)A T is invertible under the assumption that f is strongly convex.…”