Immune repertoire classification, a typical multiple instance learning (MIL) problem, is a frontier research topic in computational biology that makes transformative contributions to new vaccines and immune therapies. However, the traditional instance-space MIL, directly assigning bag-level labels to instances, suffers from the massive amount of noisy labels and extremely low witness rate. In this work, we propose a noisy-label-learning formulation to solve the immune repertoire classification task. To remedy the inaccurate supervision of repertoire-level labels for a sequence-level classifier, we design a robust training strategy: The initial labels are smoothed to be asymmetric and are progressively corrected using the model's predictions throughout the training process. Furthermore, two models with the same architecture but different parameter initialization are co-trained simultaneously to remedy the known ``confirmation bias'' problem in the self-training-like schema. As a result, we obtain accurate sequence-level classification and, subsequently, repertoire-level classification. Experiments on the Cytomegalovirus (CMV) and Cancer datasets demonstrate our method's effectiveness and superior performance on sequence-level and repertoire-level tasks. Code available at https://github.com/TencentAILabHealthcare/NLL-IRC.