Thomas van Vuren

Thomas van Vuren, Thomas Cronk, Jaron Sanders

Clustering algorithms frequently require the number of clusters to be chosen in advance, but it is usually not clear how to do this. To tackle this challenge when clustering within sequential data, we present a method for estimating the number of clusters when the data is a trajectory of a Block Markov Chain. Block Markov Chains are Markov Chains that exhibit a block structure in their transition matrix. The method considers a matrix that counts the number of transitions between different states within the trajectory, and transforms this into a spectral embedding whose dimension is set via singular value thresholding. The number of clusters is subsequently estimated via density-based clustering of this spectral embedding, an approach inspired by literature on the Stochastic Block Model. By leveraging and augmenting recent results on the spectral concentration of random matrices with Markovian dependence, we show that the method is asymptotically consistent - in spite of the dependencies between the count matrix's entries, and even when the count matrix is sparse. We also present a numerical evaluation of our method, and compare it to alternatives.

Thomas van Vuren, Fiona Sloothaak, Maarten G. Wolf et al.

The curse of dimensionality renders Reinforcement Learning (RL) impractical in many real-world settings with exponentially large state and action spaces. Yet, many environments exhibit exploitable structure that can accelerate learning. To formalize this idea, we study RL in Block Markov Decision Processes (BMDPs). BMDPs model problems with large observation spaces, but where transition dynamics are fully determined by latent states. Recent advances in clustering methods have enabled the efficient recovery of this latent structure. However, a regret analysis that exploits these techniques to determine their impact on learning performance remained open. We are now addressing this gap by providing a regret analysis that explicitly leverages clustering, demonstrating that accurate latent state estimation can indeed effectively speed up learning. Concretely, this paper analyzes a two-phase RL algorithm for BMDPs that first learns the latent structure through random exploration and then switches to an optimism-guided strategy adapted to the uncovered structure. This algorithm achieves a regret that is $O(\sqrt{T}+n)$ on a large class of BMDPs susceptible to clustering. Here, $T$ denotes the number of time steps, $n$ is the cardinality of the observation space, and the Landau notation $O(\cdot)$ holds up to constants and polylogarithmic factors. This improves the best prior bound, $O(\sqrt{T}+n^2)$, especially when $n$ is large. Moreover, we prove that no algorithm can achieve lower regret uniformly on this same class of BMDPs. This establishes that, on this class, the algorithm achieves asymptotic optimality.