A Fine-grained Characterization of PAC Learnability

In the multiclass PAC setting, even when full learnability is unattainable, meaningful information can often be extracted to guide predictions. However, classical learning theory has mainly focused on the dichotomy “learnable vs. non-learnable”, leaving notions of partial learnability largely unexplored. Indeed, even for a non-learnable class, a learner may still achieve partial success-for example, by making reliable predictions whenever the true label belongs to a fixed subset of the label space, even if it fails otherwise. Similarly, the rigid nature of PAC learnability makes it impossible to distinguish between classes where one can achieve favorable trade-offs between, say, false-positive and false-negative rates, and classes where such trade-offs are fundamentally unattainable. In a nutshell, standard PAC learnability precludes a fine-grained exploration of learnability. To overcome this limitation, we develop a fine-grained theory of PAC learnability. For any hypothesis class H, given a loss function (which quantifies the penalty for predicting y^ instead of the true label y) and a target loss threshold z, our theory determines whether it is possible to achieve a loss of at most z. In contrast, classical PAC learning considers only the special case of the zero-one loss and z=0, corresponding to a near perfect classification guarantee. We give a complete characterization of all attainable guarantees, captured by a \emph{finite family} of combinatorial dimensions, which we term the \emph{J-cube dimensions} of H. These dimensions are defined for every subset J of at least two labels. This extends the fundamental theorem of realizable PAC learning based on the VC dimension. In fact, our results hold in a more general multi-objective setting where we fully characterize the Pareto frontier of guarantees attainable for the class H.