  In high stake applications, active experimentation may be considered too
risky and thus data are often collected passively. While in simple cases, such
as in bandits, passive and active data collection are similarly effective, the
price of passive sampling can be much higher when collecting data from a system
with controlled states. The main focus of the current paper is the
characterization of this price. For example, when learning in episodic finite
state-action Markov decision processes (MDPs) with $\mathrm{S}$ states and
$\mathrm{A}$ actions, we show that even with the best (but passively chosen)
logging policy, $\Omega(\mathrm{A}^{\min(\mathrm{S}-1, H)}/\varepsilon^2)$
episodes are necessary (and sufficient) to obtain an $\epsilon$-optimal policy,
where $H$ is the length of episodes. Note that this shows that the sample
complexity blows up exponentially compared to the case of active data
collection, a result which is not unexpected, but, as far as we know, have not
been published beforehand and perhaps the form of the exact expression is a
little surprising. We also extend these results in various directions, such as
other criteria or learning in the presence of function approximation, with
similar conclusions. A remarkable feature of our result is the sharp
characterization of the exponent that appears, which is critical for
understanding what makes passive learning hard.
