  Recently, Mahloujifar and Mahmoody (TCC'17) studied attacks against learning
algorithms using a special case of Valiant's malicious noise, called
$p$-tampering, in which the adversary gets to change any training example with
independent probability $p$ but is limited to only choose malicious examples
with correct labels. They obtained $p$-tampering attacks that increase the
error probability in the so called targeted poisoning model in which the
adversary's goal is to increase the loss of the trained hypothesis over a
particular test example. At the heart of their attack was an efficient
algorithm to bias the expected value of any bounded real-output function
through $p$-tampering.
  In this work, we present new biasing attacks for increasing the expected
value of bounded real-valued functions. Our improved biasing attacks, directly
imply improved $p$-tampering attacks against learners in the targeted poisoning
model. As a bonus, our attacks come with considerably simpler analysis. We also
study the possibility of PAC learning under $p$-tampering attacks in the
non-targeted (aka indiscriminate) setting where the adversary's goal is to
increase the risk of the generated hypothesis (for a random test example). We
show that PAC learning is possible under $p$-tampering poisoning attacks
essentially whenever it is possible in the realizable setting without the
attacks. We further show that PAC learning under "correct-label" adversarial
noise is not possible in general, if the adversary could choose the (still
limited to only $p$ fraction of) tampered examples that she substitutes with
adversarially chosen ones. Our formal model for such "bounded-budget" tampering
attackers is inspired by the notions of (strong) adaptive corruption in secure
multi-party computation.
