Diffusion of Context and Credit Information in Markovian Models

This paper studies the problem of ergodicity of transition probability matrices in Marko-vian models, such as hidden Markov models (HMMs), and how it makes very diicult the task of learning to represent long-term context for sequential data. This phenomenon hurts the forward propagation of long-term context information, as well as learning a hidden state representation to represent long-term context, which depends on propagating credit information backwards in time. Using results from Markov chain theory, we show that this problem of diiusion of context and credit is reduced when the transition probabilities approach 0 or 1, i.e., the transition probability matrices are sparse and the model essentially deterministic. The results found in this paper apply to learning approaches based on continuous optimization, such as gradient descent and the Baum-Welch algorithm.