  We design mechanisms for online procurement of data held by strategic agents
for machine learning tasks. The challenge is to use past data to actively price
future data and give learning guarantees even when an agent's cost for
revealing her data may depend arbitrarily on the data itself. We achieve this
goal by showing how to convert a large class of no-regret algorithms into
online posted-price and learning mechanisms. Our results in a sense parallel
classic sample complexity guarantees, but with the key resource being money
rather than quantity of data: With a budget constraint $B$, we give robust risk
(predictive error) bounds on the order of $1/\sqrt{B}$. Because we use an
active approach, we can often guarantee to do significantly better by
leveraging correlations between costs and data.
  Our algorithms and analysis go through a model of no-regret learning with $T$
arriving pairs (cost, data) and a budget constraint of $B$. Our regret bounds
for this model are on the order of $T/\sqrt{B}$ and we give lower bounds on the
same order.
