Information theoretic limits of learning a sparse rule
This provides theoretical limits for sparse learning in high-dimensional settings, extending beyond linear regression to perceptrons, but is incremental as it builds on prior work on phase transitions.
The paper tackles the problem of learning sparse signals in generalized linear models with sublinear sparsity and data, deriving an asymptotic mutual information formula and showing that the minimum mean-square error (MMSE) is piecewise constant and can exhibit an all-or-nothing phase transition, jumping from maximum to zero at a critical sampling rate.
We consider generalized linear models in regimes where the number of nonzero components of the signal and accessible data points are sublinear with respect to the size of the signal. We prove a variational formula for the asymptotic mutual information per sample when the system size grows to infinity. This result allows us to derive an expression for the minimum mean-square error (MMSE) of the Bayesian estimator when the signal entries have a discrete distribution with finite support. We find that, for such signals and suitable vanishing scalings of the sparsity and sampling rate, the MMSE is nonincreasing piecewise constant. In specific instances the MMSE even displays an all-or-nothing phase transition, that is, the MMSE sharply jumps from its maximum value to zero at a critical sampling rate. The all-or-nothing phenomenon has previously been shown to occur in high-dimensional linear regression. Our analysis goes beyond the linear case and applies to learning the weights of a perceptron with general activation function in a teacher-student scenario. In particular, we discuss an all-or-nothing phenomenon for the generalization error with a sublinear set of training examples.