Exact solutions to the nonlinear dynamics of learning in deep linear neural networks

Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics on weights that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that as the depth of a network approaches infinity, learning speed can nevertheless remain finite: for a special class of initial conditions on the weights, very deep networks incur only a finite, depth independent, delay in learning speed relative to shallow networks. We show that, under certain conditions on the training data, unsupervised pretraining can find this special class of initial conditions, while scaled random Gaussian initializations cannot. We further exhibit a new class of random orthogonal initial conditions on weights that, like unsupervised pre-training, enjoys depth independent learning times. We further show that these initial conditions also lead to faithful propagation of gradients even in deep nonlinear networks, as long as they operate in a special regime known as the edge of chaos.Comment: Submission to ICLR2014. Revised based on reviewer feedbac

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

Inferring Actions, Intentions, and Causal Relations in a Deep Neural Network

From a young age, we can select actions to achieve desired goals, infer the goals of other agents, and learn causal relations in our environment through social interactions. Crucially, these abilities are productive and generative: we can impute desires to others that we have never held ourselves. These abilities are often captured by only partially overlapping models, each requiring substantial changes to fit combinations of abilities. Here, in an attempt to unify previous models, we present a neural network underpinned by the linearly solvable Markov Decision Process (LMDP) framework which permits a distributed representation of tasks. The network contains two pathways: one captures the desirability of states, and another encodes the passive dynamics of state transitions in the absence of control. Interactions between pathways are bound by a principle of rational action, enabling generative inference of actions, goals, and causal relations supported by gradient updates to parts of the network

If deep learning is the answer, then what is the question?

Neuroscience research is undergoing a minor revolution. Recent advances in machine learning and artificial intelligence (AI) research have opened up new ways of thinking about neural computation. Many researchers are excited by the possibility that deep neural networks may offer theories of perception, cognition and action for biological brains. This perspective has the potential to radically reshape our approach to understanding neural systems, because the computations performed by deep networks are learned from experience, not endowed by the researcher. If so, how can neuroscientists use deep networks to model and understand biological brains? What is the outlook for neuroscientists who seek to characterise computations or neural codes, or who wish to understand perception, attention, memory, and executive functions? In this Perspective, our goal is to offer a roadmap for systems neuroscience research in the age of deep learning. We discuss the conceptual and methodological challenges of comparing behaviour, learning dynamics, and neural representation in artificial and biological systems. We highlight new research questions that have emerged for neuroscience as a direct consequence of recent advances in machine learning.Comment: 4 Figures, 17 Page

On The Specialization of Neural Modules

A number of machine learning models have been proposed with the goal of achieving systematic generalization: the ability to reason about new situations by combining aspects of previous experiences. These models leverage compositional architectures which aim to learn specialized modules dedicated to structures in a task that can be composed to solve novel problems with similar structures. While the compositionality of these architectures is guaranteed by design, the modules specializing is not. Here we theoretically study the ability of network modules to specialize to useful structures in a dataset and achieve systematic generalization. To this end we introduce a minimal space of datasets motivated by practical systematic generalization benchmarks. From this space of datasets we present a mathematical definition of systematicity and study the learning dynamics of linear neural modules when solving components of the task. Our results shed light on the difficulty of module specialization, what is required for modules to successfully specialize, and the necessity of modular architectures to achieve systematicity. Finally, we confirm that the theoretical results in our tractable setting generalize to more complex datasets and non-linear architectures

Meta-Learning Strategies through Value Maximization in Neural Networks

Biological and artificial learning agents face numerous choices about how to learn, ranging from hyperparameter selection to aspects of task distributions like curricula. Understanding how to make these meta-learning choices could offer normative accounts of cognitive control functions in biological learners and improve engineered systems. Yet optimal strategies remain challenging to compute in modern deep networks due to the complexity of optimizing through the entire learning process. Here we theoretically investigate optimal strategies in a tractable setting. We present a learning effort framework capable of efficiently optimizing control signals on a fully normative objective: discounted cumulative performance throughout learning. We obtain computational tractability by using average dynamical equations for gradient descent, available for simple neural network architectures. Our framework accommodates a range of meta-learning and automatic curriculum learning methods in a unified normative setting. We apply this framework to investigate the effect of approximations in common meta-learning algorithms; infer aspects of optimal curricula; and compute optimal neuronal resource allocation in a continual learning setting. Across settings, we find that control effort is most beneficial when applied to easier aspects of a task early in learning; followed by sustained effort on harder aspects. Overall, the learning effort framework provides a tractable theoretical test bed to study normative benefits of interventions in a variety of learning systems, as well as a formal account of optimal cognitive control strategies over learning trajectories posited by established theories in cognitive neuroscience.Comment: Under Revie