Sliding down the stairs: how correlated latent variables accelerate learning with neural networks

Neural networks extract features from data using stochastic gradient descent (SGD). In particular, higher-order input cumulants (HOCs) are crucial for their performance. However, extracting information from the $p$th cumulant of $d$-dimensional inputs is computationally hard: the number of samples required to recover a single direction from an order-$p$ tensor (tensor PCA) using online SGD grows as $d^{p-1}$, which is prohibitive for high-dimensional inputs. This result raises the question of how neural networks extract relevant directions from the HOCs of their inputs efficiently. Here, we show that correlations between latent variables along the directions encoded in different input cumulants speed up learning from higher-order correlations. We show this effect analytically by deriving nearly sharp thresholds for the number of samples required by a single neuron to weakly-recover these directions using online SGD from a random start in high dimensions. Our analytical results are confirmed in simulations of two-layer neural networks and unveil a new mechanism for hierarchical learning in neural networks.

Learning from higher-order statistics, efficiently: hypothesis tests, random features, and neural networks

Neural networks excel at discovering statistical patterns in high-dimensional data sets. In practice, higher-order cumulants, which quantify the non-Gaussian correlations between three or more variables, are particularly important for the performance of neural networks. But how efficient are neural networks at extracting features from higher-order cumulants? We study this question in the spiked cumulant model, where the statistician needs to recover a privileged direction or "spike" from the order-$p\ge 4$ cumulants of~$d$-dimensional inputs. We first characterise the fundamental statistical and computational limits of recovering the spike by analysing the number of samples~$n$ required to strongly distinguish between inputs from the spiked cumulant model and isotropic Gaussian inputs. We find that statistical distinguishability requires $n\gtrsim d$ samples, while distinguishing the two distributions in polynomial time requires $n \gtrsim d^2$ samples for a wide class of algorithms, i.e. those covered by the low-degree conjecture. These results suggest the existence of a wide statistical-to-computational gap in this problem. Numerical experiments show that neural networks learn to distinguish the two distributions with quadratic sample complexity, while "lazy" methods like random features are not better than random guessing in this regime. Our results show that neural networks extract information from higher-order correlations in the spiked cumulant model efficiently, and reveal a large gap in the amount of data required by neural networks and random features to learn from higher-order cumulants.