Non-omniscient backdoor injection with a single poison sample: Proving the one-poison hypothesis for linear regression and linear classification

Backdoor injection attacks are a threat to machine learning models that are trained on large data collected from untrusted sources; these attacks enable attackers to inject malicious behavior into the model that can be triggered by specially crafted inputs. Prior work has established bounds on the success of backdoor attacks and their impact on the benign learning task, however, an open question is what amount of poison data is needed for a successful backdoor attack. Typical attacks either use few samples, but need much information about the data points or need to poison many data points.
In this paper, we formulate the one-poison hypothesis: An adversary with one poison sample and limited background knowledge can inject a backdoor with zero backdooring-error and without significantly impacting the benign learning task performance. Moreover, we prove the one-poison hypothesis for linear regression and linear classification. For adversaries that utilize a direction that is unused by the benign data distribution for the poison sample, we show that the resulting model is functionally equivalent to a model where the poison was excluded from training. We build on prior work on statistical backdoor learning to show that in all other cases, the impact on the benign learning task is still limited. We also validate our theoretical results experimentally with realistic benchmark data sets.