  The top-k classification accuracy is one of the core metrics in machine
learning. Here, k is conventionally a positive integer, such as 1 or 5, leading
to top-1 or top-5 training objectives. In this work, we relax this assumption
and optimize the model for multiple k simultaneously instead of using a single
k. Leveraging recent advances in differentiable sorting and ranking, we propose
a differentiable top-k cross-entropy classification loss. This allows training
the network while not only considering the top-1 prediction, but also, e.g.,
the top-2 and top-5 predictions. We evaluate the proposed loss function for
fine-tuning on state-of-the-art architectures, as well as for training from
scratch. We find that relaxing k does not only produce better top-5 accuracies,
but also leads to top-1 accuracy improvements. When fine-tuning publicly
available ImageNet models, we achieve a new state-of-the-art for these models.
