<p>Serverless computing has become a transformative cloud paradigm due to its operational simplicity, cost efficiency, and elastic scalability. Unfortunately, it suffers from cold-start problems caused by container initialization during the first function invocation or after container reclamation. Container caching (i.e., keep-alive policies) is an effective and widely adopted solution to mitigate cold-starts. However, existing keep-alive policies rely on fixed time thresholds or static heuristics, making them inadequate to handle the dynamic and heterogeneous workloads. In this paper, we propose LACE, a learned container caching framework for resource management in serverless computing. Leveraging high-quality features and labels of irregular functions, LACE employs a Mixture-of-Experts (MoE) architecture to model dynamic function invocation patterns. It further incorporates a multi-task learning structure that combines regression for fine-grained interval prediction with auxiliary classification for coarse-grained pattern estimation. Evaluation on Azure workload traces shows that LACE outperforms the state-of-the-art keep-alive policy, reducing cold-start overhead and rate by 32.72% and 22.46%, respectively, while achieving container caching accuracy close to the optimal Belady policy, all without incurring excessive model decision delay or memory footprints.</p>