We study the basic problem of learning overcomplete HMMs---those that have many hidden states but a small output alphabet. Despite having significant practical importance, such HMMs are poorly understood with no known positive or negative results for efficient learning. In this paper, we present several new results---both positive and negative---which help define the boundaries between the tractable-learning setting and the intractable setting. We show positive results for a large subclass of HMMs whose transition matrices are sparse, well-conditioned and have small probability mass on short cycles. We also show that learning is impossible given only a polynomial number of samples for HMMs with a small output alphabet and whose transition matrices are random regular graphs with large degree. We also discuss these results in the context of learning HMMs which can capture long-term dependencies.

We study one-sided and $α$-correct sequential hypothesis testing for data generated by an ergodic Markov chain. The null hypothesis is that the unknown transition matrix belongs to a prescribed set $P$ of stochastic matrices, and the alternative corresponds to a disjoint set $Q$. We establish a tight non-asymptotic instance-dependent lower bound on the expected stopping time of any valid sequential test under the alternative. Our novel analysis improves the existing lower bounds, which are either asymptotic or provably sub-optimal in this setting. Our lower bound incorporates both the stationary distribution and the transition structure induced by the unknown Markov chain. We further propose an optimal test whose expected stopping time matches this lower bound asymptotically as $α\to 0$. We illustrate the usefulness of our framework through applications to sequential detection of model misspecification in Markov Chain Monte Carlo and to testing structural properties, such as the linearity of transition dynamics, in Markov decision processes. Our findings yield a sharp and general characterization of optimal sequential testing procedures under Markovian dependence.

In label-noise learning, the noise transition matrix reveals how an instance transitions from its clean label to its noisy label. Accurately estimating an instance's noise transition matrix is crucial for estimating its clean label.

Both algorithms yield state-of-the-art performance guarantees.