Mice identify subgoal locations through an action-driven mapping process.

Mammals form mental maps of the environments by exploring their surroundings. Here, we investigate which elements of exploration are important for this process. We studied mouse escape behavior, in which mice are known to memorize subgoal locations-obstacle edges-to execute efficient escape routes to shelter. To test the role of exploratory actions, we developed closed-loop neural-stimulation protocols for interrupting various actions while mice explored. We found that blocking running movements directed at obstacle edges prevented subgoal learning; however, blocking several control movements had no effect. Reinforcement learning simulations and analysis of spatial data show that artificial agents can match these results if they have a region-level spatial representation and explore with object-directed movements. We conclude that mice employ an action-driven process for integrating subgoals into a hierarchical cognitive map. These findings broaden our understanding of the cognitive toolkit that mammals use to acquire spatial knowledge.

Strategically managing learning during perceptual decision making.

Making optimal decisions in the face of noise requires balancing short-term speed and accuracy. But a theory of optimality should account for the fact that short-term speed can influence long-term accuracy through learning. Here, we demonstrate that long-term learning is an important dynamical dimension of the speed-accuracy trade-off. We study learning trajectories in rats and formally characterize these dynamics in a theory expressed as both a recurrent neural network and an analytical extension of the drift-diffusion model that learns over time. The model reveals that choosing suboptimal response times to learn faster sacrifices immediate reward, but can lead to greater total reward. We empirically verify predictions of the theory, including a relationship between stimulus exposure and learning speed, and a modulation of reaction time by future learning prospects. We find that rats' strategies approximately maximize total reward over the full learning epoch, suggesting cognitive control over the learning process.

Exact learning dynamics of deep linear networks with prior knowledge.

Learning in deep neural networks is known to depend critically on the knowledge embedded in the initial network weights. However, few theoretical results have precisely linked prior knowledge to learning dynamics. Here we derive exact solutions to the dynamics of learning with rich prior knowledge in deep linear networks by generalising Fukumizu's matrix Riccati solution (Fukumizu 1998 Gen 1 1E-03). We obtain explicit expressions for the evolving network function, hidden representational similarity, and neural tangent kernel over training for a broad class of initialisations and tasks. The expressions reveal a class of task-independent initialisations that radically alter learning dynamics from slow non-linear dynamics to fast exponential trajectories while converging to a global optimum with identical representational similarity, dissociating learning trajectories from the structure of initial internal representations. We characterise how network weights dynamically align with task structure, rigorously justifying why previous solutions successfully described learning from small initial weights without incorporating their fine-scale structure. Finally, we discuss the implications of these findings for continual learning, reversal learning and learning of structured knowledge. Taken together, our results provide a mathematical toolkit for understanding the impact of prior knowledge on deep learning.

High-dimensional dynamics of generalization error in neural networks.

We perform an analysis of the average generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on the order of or even larger than the number of examples in the dataset. Using random matrix theory and exact solutions in linear models, we derive the generalization error and training error dynamics of learning and analyze how they depend on the dimensionality of data and signal to noise ratio of the learning problem. We find that the dynamics of gradient descent learning naturally protect against overtraining and overfitting in large networks. Overtraining is worst at intermediate network sizes, when the effective number of free parameters equals the number of samples, and thus can be reduced by making a network smaller or larger. Additionally, in the high-dimensional regime, low generalization error requires starting with small initial weights. We then turn to non-linear neural networks, and show that making networks very large does not harm their generalization performance. On the contrary, it can in fact reduce overtraining, even without early stopping or regularization of any sort. We identify two novel phenomena underlying this behavior in overcomplete models: first, there is a frozen subspace of the weights in which no learning occurs under gradient descent; and second, the statistical properties of the high-dimensional regime yield better-conditioned input correlations which protect against overtraining. We demonstrate that standard application of theories such as Rademacher complexity are inaccurate in predicting the generalization performance of deep neural networks, and derive an alternative bound which incorporates the frozen subspace and conditioning effects and qualitatively matches the behavior observed in simulation.

Dynamics of stochastic gradient descent for two-layer neural networks in the teacher-student setup.

Deep neural networks achieve stellar generalisation even when they have enough parameters to easily fit all their training data. We study this phenomenon by analysing the dynamics and the performance of over-parameterised two-layer neural networks in the teacher-student setup, where one network, the student, is trained on data generated by another network, called the teacher. We show how the dynamics of stochastic gradient descent (SGD) is captured by a set of differential equations and prove that this description is asymptotically exact in the limit of large inputs. Using this framework, we calculate the final generalisation error of student networks that have more parameters than their teachers. We find that the final generalisation error of the student increases with network size when training only the first layer, but stays constant or even decreases with size when training both layers. We show that these different behaviours have their root in the different solutions SGD finds for different activation functions. Our results indicate that achieving good generalisation in neural networks goes beyond the properties of SGD alone and depends on the interplay of at least the algorithm, the model architecture, and the data set.

A mathematical theory of semantic development in deep neural networks.

An extensive body of empirical research has revealed remarkable regularities in the acquisition, organization, deployment, and neural representation of human semantic knowledge, thereby raising a fundamental conceptual question: What are the theoretical principles governing the ability of neural networks to acquire, organize, and deploy abstract knowledge by integrating across many individual experiences? We address this question by mathematically analyzing the nonlinear dynamics of learning in deep linear networks. We find exact solutions to this learning dynamics that yield a conceptual explanation for the prevalence of many disparate phenomena in semantic cognition, including the hierarchical differentiation of concepts through rapid developmental transitions, the ubiquity of semantic illusions between such transitions, the emergence of item typicality and category coherence as factors controlling the speed of semantic processing, changing patterns of inductive projection over development, and the conservation of semantic similarity in neural representations across species. Thus, surprisingly, our simple neural model qualitatively recapitulates many diverse regularities underlying semantic development, while providing analytic insight into how the statistical structure of an environment can interact with nonlinear deep-learning dynamics to give rise to these regularities.