TMLR Journal 2024 Journal Article
In this paper, we consider the meta learning problem for estimating the graphs associated with high-dimensional Ising models, using the method of $\ell_1$-regularized logistic regression for neighborhood selection of each node. Our goal is to use the information learned from the auxiliary tasks in the learning of the novel task to reduce its sufficient sample complexity. To this end, we propose a novel generative model as well as an improper estimation method. In our setting, all the tasks are similar in their random model parameters and supports. By pooling all the samples from the auxiliary tasks to improperly estimate a single parameter vector, we can recover the true support union, assumed small in size, with a high probability with a sufficient sample complexity of $n = O(d^3 \log p/K)$ per task, for $K$ tasks of Ising models with $p$ nodes and a maximum neighborhood size $d$. This is very relevant for meta learning where there are many tasks $K = O(d^3 \log p)$, each with very few samples, i.e., $n = O(1)$, in an scenario where multi-task learning fails. We prove a matching information-theoretic lower bound for the necessary number of samples per task, which is $n = \Omega(d^3 \log p/K)$, and thus, our algorithm is minimax optimal. Finally, with the support for the novel task restricted to the estimated support union, we prove that consistent neighborhood selection for the novel task can be obtained with a sufficient sample complexity of $O(d^3 \log d)$. This reduces the original sample complexity of $n = O(d^3 \log p)$ for learning a single task. We also prove a matching information-theoretic lower bound of $\Omega(d^3 \log d)$ for the necessary number of samples.