  Deep learning has grown tremendously over recent years, yielding
state-of-the-art results in various fields. However, training such models
requires huge amounts of data, increasing the computational time and cost. To
address this, dataset distillation was proposed to compress a large training
dataset into a smaller synthetic one that retains its performance -- this is
usually done by (1) uniformly initializing a synthetic set and (2) iteratively
updating/learning this set according to a predefined loss by uniformly sampling
instances from the full data. In this paper, we improve both phases of dataset
distillation: (1) we present a provable, sampling-based approach for
initializing the distilled set by identifying important and removing redundant
points in the data, and (2) we further merge the idea of data subset selection
with dataset distillation, by training the distilled set on ``important''
sampled points during the training procedure instead of randomly sampling the
next batch. To do so, we define the notion of importance based on the relative
contribution of instances with respect to two different loss functions, i.e.,
one for the initialization phase (a kernel fitting function for kernel ridge
regression and $K$-means based loss function for any other distillation
method), and the relative cross-entropy loss (or any other predefined loss)
function for the training phase. Finally, we provide experimental results
showing how our method can latch on to existing dataset distillation techniques
and improve their performance.
