Dynamic feature transformation (the rich regime) does not always align with predictive performance (better representation), yet accuracy is often used as a proxy for richness, limiting analysis of their relationship. We propose a computationally efficient, performance-independent metric of richness grounded in the low-rank bias of rich dynamics, which recovers neural collapse as a special case. The metric is empirically more stable than existing alternatives and captures known lazy-to-rich transitions (e.g., grokking) without relying on accuracy. We further use it to examine how training factors (e.g., learning rate) relate to richness, confirming recognized assumptions and highlighting new observations (e.g., batch normalization promote rich dynamics). An eigendecomposition-based visualization is also introduced to support interpretability, together providing a diagnostic tool for studying the relationship between training factors, dynamics, and representations.

The remarkable performance of overparameterized deep neural networks (DNNs) must arise from an interplay between network architecture, training algorithms, and structure in the data. To disentangle these three components for supervised learning, we apply a Bayesian picture based on the functions expressed by a DNN. The prior over functions is determined by the network architecture, which we vary by exploiting a transition between ordered and chaotic regimes. For Boolean function classification, we approximate the likelihood using the error spectrum of functions on data. Combining this with the prior yields an accurate prediction for the posterior, measured for DNNs trained with stochastic gradient descent. This analysis shows that structured data, together with a specific Occam's razor-like inductive bias towards (Kolmogorov) simple functions that exactly counteracts the exponential growth of the number of functions with complexity, is a key to the success of DNNs.

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.

Quantum machine learning models based on parametrized quantum circuits, also called quantum neural networks (QNNs), are considered to be among the most promising candidates for applications on near-term quantum devices. Here we explore the expressivity and inductive bias of QNNs by exploiting an exact mapping from QNNs with inputs x to classical perceptrons acting on x ⊗ x (generalised to complex inputs). The simplicity of the perceptron architecture allows us to provide clear examples of the shortcomings of current QNN models, and the many barriers they face to becoming useful general-purpose learning algorithms. For example, a QNN with amplitude encoding cannot express the Boolean parity function for n≥ 3, which is but one of an exponential number of data structures that such a QNN is unable to express. Mapping a QNN to a classical perceptron simplifies training, allowing us to systematically study the inductive biases of other, more expressive embeddings on Boolean data. Several popular embeddings primarily produce an inductive bias towards functions with low class balance, reducing their generalisation performance compared to deep neural network architectures which exhibit much richer inductive biases. We explore two alternate strategies that move beyond standard QNNs. In the first, we use a QNN to help generate a classical DNN-inspired kernel. In the second we draw an analogy to the hierarchical structure of deep neural networks and construct a layered non-linear QNN that is provably fully expressive on Boolean data, while also exhibiting a richer inductive bias than simple QNNs. Finally, we discuss characteristics of the QNN literature that may obscure how hard it is to achieve quantum advantage over deep learning algorithms on classical data.

The architecture of a deep neural network is defined explicitly in terms of the number of layers, the width of each layer and the general network topology. Existing optimisation frameworks neglect this information in favour of implicit architectural information (e.g. second-order methods) or architecture-agnostic distance functions (e.g. mirror descent). Meanwhile, the most popular optimiser in practice, Adam, is based on heuristics. This paper builds a new framework for deriving optimisation algorithms that explicitly leverage neural architecture. The theory extends mirror descent to non-convex composite objective functions: the idea is to transform a Bregman divergence to account for the non-linear structure of neural architecture. Working through the details for deep fully-connected networks yields automatic gradient descent: a first-order optimiser without any hyperparameters. Automatic gradient descent trains both fully-connected and convolutional networks out-of-the-box and at ImageNet scale. A PyTorch implementation is available at https://github.com/jxbz/agd and also in Appendix B. Overall, the paper supplies a rigorous theoretical foundation for a next-generation of architecture-dependent optimisers that work automatically and without hyperparameters.

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 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 an 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.

Modern Deep Neural Networks (DNNs) exhibit impressive generalization properties on a variety of tasks without explicit regularization, suggesting the existence of hidden regularization effects. Recent work by Baratin et al. (2021) sheds light on an intriguing implicit regularization effect, showing that some layers are much more aligned with data labels than other layers. This suggests that as the network grows in depth and width, an implicit layer selection phenomenon occurs during training. In this work, we provide the first explanation for this alignment hierarchy. We introduce and empirically validate the Equilibrium Hypothesis which states that the layers that achieve some balance between forward and backward information loss are the ones with the highest alignment to data labels. 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 vertical bar 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 vertical bar S) that the DNN expresses f upon random sampling of its parameters, conditioned on S. Our main findings are that P-SGD(f vertical bar S) correlates remarkably well with P-B(f vertical bar S) and that P-B(f vertical bar 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 vertical bar 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 vertical bar S) is the first order determinant of P-SGD(f vertical bar S), there remain second order differences that are sensitive to hyperparameter tuning. A function probability picture, based on P(SGD()f vertical bar S) and/or P-B(f vertical bar 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.

Overparameterised deep neural networks (DNNs) are highly expressive and so can, in principle, generate almost any function that fits a training dataset with zero error. The vast majority of these functions will perform poorly on unseen data, and yet in practice DNNs often generalise remarkably well. This success suggests that a trained DNN must have a strong inductive bias towards functions with low generalisation error. Here we empirically investigate this inductive bias by calculating, for a range of architectures and datasets, the probability P_SGD(f| 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| S) that the DNN expresses f upon random sampling of its parameters, conditioned on S. Our main findings are that P_SGD(f| S) correlates remarkably well with P_B(f| S) and that P_B(f| 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| 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| S) is the first order determinant of P_SGD(f| S), there remain second order differences that are sensitive to hyperparameter tuning. A function probability picture, based on P_SGD(f| S) and/or P_B(f| S), can shed new light on the way that variations in architecture or hyperparameter settings such as batch size, learning rate, and optimiser choice, affect DNN performance.

Understanding the inductive bias of neural networks is critical to explaining their ability to generalise. Here, for one of the simplest neural networks -- a single-layer perceptron with n input neurons, one output neuron, and no threshold bias term -- we prove that upon random initialisation of weights, the a priori probability P(t) that it represents a Boolean function that classifies t points in {0,1}^n as 1 has a remarkably simple form: P(t) = 2^{-n} for 0\leq t < 2^n. Since a perceptron can express far fewer Boolean functions with small or large values of t (low entropy) than with intermediate values of t (high entropy) there is, on average, a strong intrinsic a-priori bias towards individual functions with low entropy. Furthermore, within a class of functions with fixed t, we often observe a further intrinsic bias towards functions of lower complexity. Finally, we prove that, regardless of the distribution of inputs, the bias towards low entropy becomes monotonically stronger upon adding ReLU layers, and empirically show that increasing the variance of the bias term has a similar effect.