Widely observed neural scaling laws, in which error falls off as a power of the training set size, model size, or both, have driven substantial performance improvements in deep learning. However, these improvements through scaling alone require considerable costs in compute and energy. Here we focus on the scaling of error with dataset size and show how in theory we can break beyond power law scaling and potentially even reduce it to exponential scaling instead if we have access to a high-quality data pruning metric that ranks the order in which training examples should be discarded to achieve any pruned dataset size. We then test this improved scaling prediction with pruned dataset size empirically, and indeed observe better than power law scaling in practice on ResNets trained on CIFAR-10, SVHN, and ImageNet. Next, given the importance of finding high-quality pruning metrics, we perform the first large-scale benchmarking study of ten different data pruning metrics on ImageNet. We find most existing high performing metrics scale poorly to ImageNet, while the best are computationally intensive and require labels for every image. We therefore developed a new simple, cheap and scalable self-supervised pruning metric that demonstrates comparable performance to the best supervised metrics. Overall, our work suggests that the discovery of good data-pruning metrics may provide a viable path forward to substantially improved neural scaling laws, thereby reducing the resource costs of modern deep learning.

A central question in computational neuroscience is how structure determines function in neural networks. Recent large-scale connectomic studies have started to provide a wealth of structural information such as the distribution of excitatory/inhibitory cell and synapse types as well as the distribution of synaptic weights in the brains of different species. The emerging high-quality large structural datasets raise the question of what general functional principles can be gleaned from them. Motivated by this question, we developed a statistical mechanical theory of learning in neural networks that incorporates structural information as constraints. We derived an analytical solution for the memory capacity of the perceptron, a basic feedforward model of supervised learning, with constraint on the distribution of its weights. Interestingly, the theory predicts that the reduction in capacity due to the constrained weight-distribution is related to the Wasserstein distance between the cumulative distribution function of the constrained weights and that of the standard normal distribution. To test the theoretical predictions, we use optimal transport theory and information geometry to develop an SGD-based algorithm to find weights that simultaneously learn the input-output task and satisfy the distribution constraint. We show that training in our algorithm can be interpreted as geodesic flows in the Wasserstein space of probability distributions. Given a parameterized family of weight distributions, our theory predicts the shape of the distribution with optimal parameters. We apply our theory to map out the experimental parameter landscape for the estimated distribution of synaptic weights in mammalian cortex and show that our theory’s prediction for optimal distribution is close to the experimentally measured value. We further developed a statistical mechanical theory for teacher-student perceptron rule learning and ask for the best way for the student to incorporate prior knowledge of the rule (i.e., the teacher). Our theory shows that it is beneficial for the learner to adopt different prior weight distributions during learning, and shows that distribution-constrained learning outperforms unconstrained and sign-constrained learning. Our theory and algorithm provide novel strategies for incorporating prior knowledge about weights into learning, and reveal a powerful connection between structure and function in neural networks.

Medial entorhinal cortex (MEC) supports a wide range of navigational and memory related behaviors.Well-known experimental results have revealed specialized cell types in MEC --- e.g. grid, border, and head-direction cells --- whose highly stereotypical response profiles are suggestive of the role they might play in supporting MEC functionality. However, the majority of MEC neurons do not exhibit stereotypical firing patterns.How should the response profiles of these more "heterogeneous" cells be described, and how do they contribute to behavior?In this work, we took a computational approach to addressing these questions.We first performed a statistical analysis that shows that heterogeneous MEC cells are just as reliable in their response patterns as the more stereotypical cell types, suggesting that they have a coherent functional role.Next, we evaluated a spectrum of candidate models in terms of their ability to describe the response profiles of both stereotypical and heterogeneous MEC cells.We found that recently developed task-optimized neural network models are substantially better than traditional grid cell-centric models at matching most MEC neuronal response profiles --- including those of grid cells themselves --- despite not being explicitly trained for this purpose.Specific choices of network architecture (such as gated nonlinearities and an explicit intermediate place cell representation) have an important effect on the ability of the model to generalize to novel scenarios, with the best of these models closely approaching the noise ceiling of the data itself.We then performed in silico experiments on this model to address questions involving the relative functional relevance of various cell types, finding that heterogeneous cells are likely to be just as involved in downstream functional outcomes (such as path integration) as grid and border cells.Finally, inspired by recent data showing that, going beyond their spatial response selectivity, MEC cells are also responsive to non-spatial rewards, we introduce a new MEC model that performs reward-modulated path integration.We find that this unified model matches neural recordings across all variable-reward conditions.Taken together, our results point toward a conceptually principled goal-driven modeling approach for moving future experimental and computational efforts beyond overly-simplistic single-cell stereotypes.

Grid cells in the brain fire in strikingly regular hexagonal patterns across space. There are currently two seemingly unrelated frameworks for understanding these patterns. Mechanistic models account for hexagonal firing fields as the result of pattern-forming dynamics in a recurrent neural network with hand-tuned center-surround connectivity. Normative models specify a neural architecture, a learning rule, and a navigational task, and observe that grid-like firing fields emerge due to the constraints of solving this task. Here we provide an analytic theory that unifies the two perspectives by casting the learning dynamics of neural networks trained on navigational tasks as a pattern forming dynamical system. This theory provides insight into the optimal solutions of diverse formulations of the normative task, and shows that symmetries in the representation of space correctly predict the structure of learned firing fields in trained neural networks. Further, our theory proves that a nonnegativity constraint on firing rates induces a symmetry-breaking mechanism which favors hexagonal firing fields. We extend this theory to the case of learning multiple grid maps and demonstrate that optimal solutions consist of a hierarchy of maps with increasing length scales. These results unify previous accounts of grid cell firing and provide a novel framework for predicting the learned representations of recurrent neural networks.