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Since training a large-scale backdoored model from scratch requires a largetraining dataset, several recent attacks have considered to inject backdoorsinto a trained clean model without altering model behaviors on the clean data.Previous work finds that backdoors can be injected into a trained clean modelwith Adversarial Weight Perturbation (AWP). Here AWPs refers to the variationsof parameters that are small in backdoor learning. In this work, we observe aninteresting phenomenon that the variations of parameters are always AWPs whentuning the trained clean model to inject backdoors. We further providetheoretical analysis to explain this phenomenon. We formulate the behavior ofmaintaining accuracy on clean data as the consistency of backdoored models,which includes both global consistency and instance-wise consistency. Weextensively analyze the effects of AWPs on the consistency of backdooredmodels. In order to achieve better consistency, we propose a novel anchoringloss to anchor or freeze the model behaviors on the clean data, with atheoretical guarantee. Both the analytical and the empirical results validatethe effectiveness of the anchoring loss in improving the consistency,especially the instance-wise consistency.

How to Inject Backdoors with Better Consistency: Logit Anchoring on Clean Data