Deep learning models can exhibit what appears to be a sudden ability to solve a new problem as training time, training data, or model size increases, a phenomenon known as emergence. In this paper, we present a framework where each new ability (a skill) is represented as a basis function. We solve a simple multi-linear model in this skill-basis, finding analytic expressions for the emergence of new skills, as well as for scaling laws of the loss with training time, data size, model size, and optimal compute. We compare our detailed calculations to direct simulations of a two-layer neural network trained on multitask sparse parity, where the tasks in the dataset are distributed according to a power-law. Our simple model captures, using a single fit parameter, the sigmoidal emergence of multiple new skills as training time, data size or model size increases in the neural network.

Recent work by Baratin et al. (2021) sheds light on an intriguing pattern that occurs during the training of deep neural networks: some layers align much more with data compared to other layers (where the alignment is defined as the normalize euclidean product of the tangent features matrix and the data labels matrix). The curve of the alignment as a function of layer index (generally) exhibits a ascent-descent pattern where the maximum is reached for some hidden layer. In this work, we provide the first explanation for this phenomenon. We introduce the Equilibrium Hypothesis which connects this alignment pattern to signal propagation in deep neural networks. Our experiments demonstrate an excellent match with the theoretical predictions.

Deep neural networks (DNNs) generalise remarkably well in the overparameterised regime, suggesting a strong inductive bias towards functions with low generalisation error. We empirically investigate this bias by calculating, for a range of architectures and datasets, the probability $P_{SGD}(f\mid S)$ that an overparameterised DNN, trained with stochastic gradient descent (SGD) or one of its variants, converges on a function $f$ consistent with a training set $S$. We also use Gaussian processes to estimate the Bayesian posterior probability $P_{B}(f\mid S)$ that the DNN expresses $f$ upon random sampling of its parameters, conditioned on $S$. Our main findings are that $P_{SGD}(f\mid S)$ correlates remarkably well with $P_{B}(f\mid S)$ and that $P_{B}(f\mid S)$ is strongly biased towards low-error and low complexity functions. These results imply that strong inductive bias in the parameter-function map (which determines $P_{B}(f\mid S)$), rather than a special property of SGD, is the primary explanation for why DNNs generalise so well in the overparameterised regime. While our results suggest that the Bayesian posterior $P_{B}(f\mid S)$ is the first order determinant of $P_{SGD}(f\mid S)$, there remain second order differences that are sensitive to hyperparameter tuning. A function probability picture, based on $P_{SGD}(f\mid S)$ and/or $P_{B}(f\mid S)$, can shed light on the way that variations in architecture or hyperparameter settings such as batch size, learning rate, and optimiser choice, affect DNN performance.