  Auditors need robust methods to assess the compliance of web platforms with
the law. However, since they hardly ever have access to the algorithm,
implementation, or training data used by a platform, the problem is harder than
a simple metric estimation. Within the recent framework of manipulation-proof
auditing, we study in this paper the feasibility of robust audits in realistic
settings, in which models exhibit large capacities. We first prove a
constraining result: if a web platform uses models that may fit any data, no
audit strategy -- whether active or not -- can outperform random sampling when
estimating properties such as demographic parity. To better understand the
conditions under which state-of-the-art auditing techniques may remain
competitive, we then relate the manipulability of audits to the capacity of the
targeted models, using the Rademacher complexity. We empirically validate these
results on popular models of increasing capacities, thus confirming
experimentally that large-capacity models, which are commonly used in practice,
are particularly hard to audit robustly. These results refine the limits of the
auditing problem, and open up enticing questions on the connection between
model capacity and the ability of platforms to manipulate audit attempts.
