Fine-tuning models on edge devices like mobile phones would enable privacy-preserving personalization over sensitive data. However, edge training has historically been limited to relatively small models with simple architectures because training is both memory and energy intensive. We present POET, an algorithm to enable training large neural networks on memory-scarce battery-operated edge devices. POET jointly optimizes the integrated search search spaces of rematerialization and paging, two algorithms to reduce the memory consumption of backpropagation. Given a memory budget and a run-time constraint, we formulate a mixed-integer linear program (MILP) for energy-optimal training. Our approach enables training significantly larger models on embedded devices while reducing energy consumption while not modifying mathematical correctness of backpropagation. We demonstrate that it is possible to fine-tune both ResNet-18 and BERT within the memory constraints of a Cortex-M class embedded device while outperforming current edge training methods in energy efficiency. POET is an open-source project available at https://github.com/ShishirPatil/poet