The Transformer architecture has revolutionized deep learning on sequential data, becoming ubiquitous in state-of-the-art solutions for a wide variety of applications. Yet vanilla Transformers are notoriously resource-expensive, requiring O(L^2) in serial time and memory as functions of input length L. Recent works proposed various linear self-attention mechanisms, scaling only as O(L) for serial computation. We perform a thorough analysis of recent Transformer mechanisms with linear self-attention, Performers, in terms of overall computational complexity. We observe a remarkable computational flexibility: forward and backward propagation can be performed with no approximations using sub-linear memory as a function of L (in addition to negligible storage for the input sequence), at a cost of greater time complexity in the parallel setting. In the extreme case, a Performer consumes only O(1) memory during training, and still requires O(L) time. This discovered time-memory tradeoff can be used for training or, due to complete backward-compatibility, for fine-tuning on a low-memory device, e.g. a smartphone or an earlier-generation GPU, thus contributing towards decentralized and democratized deep learning.