Large neural networks can improve the accuracy and generalization on tasks across many domains. However, this trend cannot continue indefinitely due to limited hardware memory. As a result, researchers have devised a number of memory optimization methods (MOMs) to alleviate the memory bottleneck, such as gradient checkpointing, quantization, and swapping. In this work, we study memory optimization methods and show that, although these strategies indeed lower peak memory usage, they can actually decrease training throughput by up to 9.3x. To provide practical guidelines for practitioners, we propose a simple but effective performance model PAPAYA to quantitatively explain the memory and training time trade-off. PAPAYA can be used to determine when to apply the various memory optimization methods in training different models. We outline the circumstances in which memory optimization techniques are more advantageous based on derived implications from PAPAYA. We assess the accuracy of PAPAYA and the derived implications on a variety of machine models, showing that it achieves over 0.97 R score on predicting the peak memory/throughput, and accurately predicts the effectiveness of MOMs across five evaluated models on vision and NLP tasks.