Abstract: The transfer of tensors from/to memory during neural network training dominates time and energy. To improve energy efficiency and performance, research has been exploring ways to use narrower data representations. So far, these attempts relied on user-directed trial-and-error to achieve convergence. We present methods that relieve users from this responsibility. Our methods dynamically adjust the size and format of the floating-point containers used for activations and weights during training, achieving adaptivity across three dimensions: i) which datatype to use, ii) on which tensor, and iii) how it changes over time. The different meanings and distributions of exponent and mantissas lead us to tailored approaches for each. We present two lossy pairs of methods to eliminate as many mantissa and exponent bits as possible without affecting accuracy. Quantum Mantissa and Quantum Exponent are machine learning compression methods that tap into the gradient descent algorithm to learn the minimal mantissa and exponent bitlengths on a per-layer granularity. They automatically learn that many tensors can use just 1 or 2 mantissa bits and 3 or 4 exponent bits. Overall, the two machine learning methods reduce the footprint by $4.73\times$. Alternatively, BitWave observes changes in the loss function during training to adjust mantissa and exponent bitlengths network-wide, yielding a $3.17\times$ reduction in footprint. Finally, we present an optional method, Gecok, to exploit the naturally emerging, lop-sided exponent distribution to losslessly compress resulting exponents from Quantum Exponent or BitWave and, on average, improve compression rates to $5.61\times$ and $4.53\times$.