Infinity Learning: Learning Markov Chains from Aggregate Steady-State Observations

Abstract

We consider the task of learning a parametric Continuous Time Markov Chain (CTMC) sequence model without examples of sequences, where the training data consists entirely of aggregate steady-state statistics. Making the problem harder, we assume that the states we wish to predict are unobserved in the training data. Specifically, given a parametric model over the transition rates of a CTMC and some known transition rates, we wish to extrapolate its steady state distribution to states that are unobserved. A technical roadblock to learn a CTMC from its steady state has been that the chain rule to compute gradients will not work over the arbitrarily long sequences necessary to reach steady state —from where the aggregate statistics are sampled. To overcome this optimization challenge, we propose ∞-SGD, a principled stochastic gradient descent method that uses randomly-stopped estimators to avoid infinite sums required by the steady state computation, while learning even when only a subset of the CTMC states can be observed. We apply ∞-SGD to a real-world testbed and synthetic experiments showcasing its accuracy, ability to extrapolate the steady state distribution to unobserved states under unobserved conditions (heavy loads, when training under light loads), and succeeding in difficult scenarios where even a tailor-made extension of existing methods fails.

Algorithmic Minimization of Uncertain Continuous-Time Markov Chains

“…This is essentially a birth/death process, a fundamental type of CTMCs (e.g., [45]). When used to model real systems such as computer applications, the assumptions made in a classical timehomogeneous setting may be restrictive [46]. Here we consider an M/M/1/2 queue: it has exponentially distributed arrival and service times, 1 server, and processes jobs of two classes according to a first-come first-served scheduling (FCFS).…”

“…Real-world phenomena may lead to time-or class-dependent behaviors. For example, time-dependent arrival rates account for peak/offpeak variations (e.g., [47]); service rates may degrade as the queue length increases, as would be identified by state-of-theart learning methods for CTMCs [46]. If one wishes to account more precisely for these effects, arrival and service rates can be assumed to be contained in some intervals [λ; λ] and [µ; µ], respectively.…”