  Streaming algorithms are typically analyzed in the oblivious setting, where
we assume that the input stream is fixed in advance. Recently, there is a
growing interest in designing adversarially robust streaming algorithms that
must maintain utility even when the input stream is chosen adaptively and
adversarially as the execution progresses. While several fascinating results
are known for the adversarial setting, in general, it comes at a very high cost
in terms of the required space. Motivated by this, in this work we set out to
explore intermediate models that allow us to interpolate between the oblivious
and the adversarial models. Specifically, we put forward the following two
models:
  (1) *The advice model*, in which the streaming algorithm may occasionally ask
for one bit of advice.
  (2) *The bounded interruptions model*, in which we assume that the adversary
is only partially adaptive.
  We present both positive and negative results for each of these two models.
In particular, we present generic reductions from each of these models to the
oblivious model. This allows us to design robust algorithms with significantly
improved space complexity compared to what is known in the plain adversarial
model.
