This paper is concerned with the computational complexity of learning the Hidden Markov Model (HMM). Although HMMs are some of the most widely used tools in sequential and time series modeling, they are cryptographically hard to learn in the standard setting where one has access to i.i.d. samples of observation sequences. In this paper, we depart from this setup and consider an interactive access model, in which the algorithm can query for samples from the conditional distributions of the HMMs. We show that interactive access to the HMM enables computationally efficient learning algorithms, thereby bypassing cryptographic hardness. Specifically, we obtain efficient algorithms for learning HMMs in two settings: (a) An easier setting where we have query access to the exact conditional probabilities. Here our algorithm runs in polynomial time and makes polynomially many queries to approximate any HMM in total variation distance. (b) A harder setting where we can only obtain samples from the conditional distributions. Here the performance of the algorithm depends on a new parameter, called the fidelity of the HMM. We show that this captures cryptographically hard instances and previously known positive results. We also show that these results extend to a broader class of distributions with latent low rank structure. Our algorithms can be viewed as generalizations and robustifications of Angluin's $L^*$ algorithm for learning deterministic finite automata from membership queries.

翻译：本文研究学习隐马尔可夫模型（HMM）的计算复杂度。尽管HMM是序列和时间序列建模中使用最广泛的工具之一，但在标准设置（即算法可获取观测序列的独立同分布样本）下，其学习问题在密码学意义上是困难的。本文突破这一设定，考虑一种交互式访问模型，其中算法可查询HMM的条件分布样本。我们证明，对HMM的交互式访问能够实现计算高效的学习算法，从而绕过密码学上的困难性。具体而言，我们在两种设置下获得了学习HMM的高效算法：(a) 较简单设置：可查询精确的条件概率。在此设置下，我们的算法在多项式时间内运行，并通过多项式次查询以总变差距离逼近任意HMM。(b) 较困难设置：仅能从条件分布中获取样本。在此设置下，算法的性能取决于一个新参数——HMM的保真度。我们证明该参数能刻画密码学困难实例以及先前已知的正面结果。我们还表明这些结果可推广至具有潜在低秩结构的一类更广泛分布。我们的算法可视为Angluin用于从成员查询中学习确定性有限自动机的$L^*$算法的泛化与鲁棒化。