As collaborative learning allows joint training of a model using multiple sources of data, the security problem has been a central concern. Malicious users can upload poisoned data to prevent the models convergence or inject hidden backdoors. The so-called backdoor attacks are especially difficult to detect since the model behaves normally on standard test data but gives wrong outputs when triggered by certain backdoor keys. Although Byzantine-tolerant training algorithms provide convergence guarantee, provable defense against backdoor attacks remains largely unsolved. Methods based on randomized smoothing can only correct a small number of corrupted pixels or labels; methods based on subset aggregation cause a severe drop in classification accuracy due to low data utilization. We propose a novel framework that generalizes existing subset aggregation methods. The framework shows that the subset selection process, a deciding factor for subset aggregation methods, can be viewed as a code design problem. We derive the theoretical bound of data utilization ratio and provide optimal code construction. Experiments on non-II
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