Progress in artificial intelligence and machine learning over the past decade has been driven by the ability to train larger deep neural networks (DNNs), leading to a compute demand that far exceeds the growth in hardware performance afforded by Moore's law. Training DNNs is an extremely memory-intensive process, requiring not just the model weights but also activations and gradients for an entire minibatch to be stored. The need to provide high-density and low-leakage on-chip memory motivates the exploration of emerging non-volatile memory for training accelerators. Spin-Transfer-Torque MRAM (STT-MRAM) offers several desirable properties for training accelerators, including 3-4x higher density than SRAM, significantly reduced leakage power, high endurance and reasonable access time. On the one hand, MRAM write operations require high write energy and latency due to the need to ensure reliable switching. In this study, we perform a comprehensive device-to-system evaluation and co-optimization of STT-MRAM for efficient ML training accelerator design. We devised a cross-layer simulation framework to evaluate the effectiveness of STT-MRAM as a scratchpad replacing SRAM in a systolic-array-based DNN accelerator. To address the inefficiency of writes in STT-MRAM, we propose to reduce write voltage and duration. To evaluate the ensuing accuracy-efficiency trade-off, we conduct a thorough analysis of the error tolerance of input activations, weights, and errors during the training. We propose heterogeneous memory configurations that enable training convergence with good accuracy. We show that MRAM provide up to 15-22x improvement in system level energy across a suite of DNN benchmarks under iso-capacity and iso-area scenarios. Further optimizing STT-MRAM write operations can provide over 2x improvement in write energy for minimal degradation in application-level training accuracy.