Title: MACH: Embarrassingly parallel $K$-class classification in $O(d\log{K})$ memory and $O(K\log{K} + d\log{K})$ time, instead of $O(Kd)$. Abstract: We present Merged-Averaged Classifiers via Hashing (MACH) for $K$-classification with large $K$. Compared to traditional one-vs-all classifiers that require $O(Kd)$ memory and inference cost, MACH only need $O(d\log{K})$ memory while only requiring $O(K\log{K} + d\log{K})$ operation for inference. MACH is the first generic $K$-classification algorithm, with provably theoretical guarantees, which requires $O(\log{K})$ memory without any assumption on the relationship between classes. MACH uses universal hashing to reduce classification with a large number of classes to few independent classification task with very small (constant) number of classes. We provide theoretical quantification of accuracy-memory tradeoff by showing the first connection between extreme classification and heavy hitters. With MACH we can train ODP dataset with 100,000 classes and 400,000 features on a single Titan X GPU (12GB), with the classification accuracy of 19.28\%, which is the best-reported accuracy on this dataset. Before this work, the best performing baseline is a one-vs-all classifier that requires 40 billion parameters (320 GB model size) and achieves 9\% accuracy.  In contrast, MACH can achieve 9\% accuracy with 480x reduction in the model size (of mere 0.6GB). With MACH, we also demonstrate complete training of fine-grained imagenet dataset (compressed size 104GB), with 21,000 classes, on a single GPU.