Multi-party Poisoning through Generalized $p$-Tampering

In a poisoning attack against a learning algorithm, an adversary tampers with a fraction of the training data $T$ with the goal of increasing the classification error of the constructed hypothesis/model over the final test distribution. In the distributed setting, $T$ might be gathered gradually from $m$ data providers $P_1,\dots,P_m$ who generate and submit their shares of $T$ in an online way. In this work, we initiate a formal study of $(k,p)$-poisoning attacks in which an adversary controls $k\in[n]$ of the parties, and even for each corrupted party $P_i$, the adversary submits some poisoned data $T'_i$ on behalf of $P_i$ that is still "$(1-p)$-close" to the correct data $T_i$ (e.g., $1-p$ fraction of $T'_i$ is still honestly generated). For $k=m$, this model becomes the traditional notion of poisoning, and for $p=1$ it coincides with the standard notion of corruption in multi-party computation. We prove that if there is an initial constant error for the generated hypothesis $h$, there is always a $(k,p)$-poisoning attacker who can decrease the confidence of $h$ (to have a small error), or alternatively increase the error of $h$, by $\Omega(p \cdot k/m)$. Our attacks can be implemented in polynomial time given samples from the correct data, and they use no wrong labels if the original distributions are not noisy. At a technical level, we prove a general lemma about biasing bounded functions $f(x_1,\dots,x_n)\in[0,1]$ through an attack model in which each block $x_i$ might be controlled by an adversary with marginal probability $p$ in an online way. When the probabilities are independent, this coincides with the model of $p$-tampering attacks, thus we call our model generalized $p$-tampering. We prove the power of such attacks by incorporating ideas from the context of coin-flipping attacks into the $p$-tampering model and generalize the results in both of these areas.