Title: Learnability and Expressiveness in Self-Supervised Learning. Abstract: In this work, we argue that representations induced by self-supervised learning (SSL) methods should both be expressive and learnable. To measure expressiveness, we propose to use the Intrinsic Dimension (ID) of the dataset in representation space. Inspired by the human study of Laina et al. (2020), we introduce Cluster Learnability (CL), defined in terms of the learning speed of a KNN classifier trained to predict K-means cluster labels for held-out representations. By collecting 30 state-of-art checkpoints, both supervised and self-supervised, using different architectures, we show that ID and CL can be combined to predict downstream classification performance better than the existing techniques based on contrastive losses or pretext tasks, while having no requirements on data augmentation, model architecture or human labels. To further demonstrate the utility of our framework, we propose modifying DeepCluster (Caron et al., 2018) to improve the learnability of the representations. Using our modification, we are able to outperform DeepCluster on both STL10 and ImageNet benchmarks. The performance of the intermediate checkpoints can also be well predicted under our framework, suggesting the possibility of developing new SSL algorithms without labels.