Invariance principles underlie many key theories in modern science. They provide the explanatory and predictive framework necessary for the rigorous study of natural phenomena ranging from the structure of crystals, to magnetism, to relativistic mechanics. Vigo (2008, 2009) introduced a new general notion and principle of invariance from which two parameter-free (ratio and exponential) models were derived to account for human conceptual behavior. Here we introduce a new parameterized exponential law based on the same invariance principle. The law accurately predicts the subjective degree of difficulty that humans experience when learning different types of concepts. In addition, it precisely fits the data from a large-scale experiment which examined a total of 84 category structures across 10 category families (R^2=.97, p<.0001; r=.98, p<.0001). Moreover, it overcomes seven key challenges that had, hitherto, been grave obstacles for theories of concept learning.