One of the fundamental problems in machine learning is generalization. In neural network models with a large number of weights (parameters), many solutions can be found to fit the training data equally well. The key question is which solution can describe testing data not in the training set. Here, we report the discovery of an exact duality (equivalence) between changes in activities in a given layer of neurons and changes in weights that connect to the next layer of neurons in a densely connected layer in any feed forward neural network. The activity-weight (A-W) duality allows us to map variations in inputs (data) to variations of the corresponding dual weights. By using this mapping, we show that the generalization loss can be decomposed into a sum of contributions from different eigen-directions of the Hessian matrix of the loss function at the solution in weight space. The contribution from a given eigen-direction is the product of two geometric factors (determinants): the sharpness of the loss landscape and the standard deviation of the dual weights, which is found to scale with the weight norm of the solution. Our results provide an unified framework, which we used to reveal how different regularization schemes (weight decay, stochastic gradient descent with different batch sizes and learning rates, dropout), training data size, and labeling noise affect generalization performance by controlling either one or both of these two geometric determinants for generalization. These insights can be used to guide development of algorithms for finding more generalizable solutions in overparametrized neural networks.

Generative adversarial networks (GANs) are a class of machine-learning models that use adversarial training to generate new samples with the same (potentially very complex) statistics as the training samples. One major form of training failure, known as mode collapse, involves the generator failing to reproduce the full diversity of modes in the target probability distribution. Here, we present an effective model of GAN training, which captures the learning dynamics by replacing the generator neural network with a collection of particles in the output space; particles are coupled by a universal kernel valid for certain wide neural networks and high-dimensional inputs. The generality of our simplified model allows us to study the conditions under which mode collapse occurs. Indeed, experiments which vary the effective kernel of the generator reveal a mode collapse transition, the shape of which can be related to the type of discriminator through the frequency principle. Further, we find that gradient regularizers of intermediate strengths can optimally yield convergence through critical damping of the generator dynamics. Our effective GAN model thus provides an interpretable physical framework for understanding and improving adversarial training.

Generalization is one of the most important problems in deep learning (DL). In the overparameterized regime in neural networks, there exist many low-loss solutions that fit the training data equally well. The key question is which solution is more generalizable. Empirical studies showed a strong correlation between flatness of the loss landscape at a solution and its generalizability, and stochastic gradient descent (SGD) is crucial in finding the flat solutions. To understand how SGD drives the learning system to flat solutions, we construct a simple model whose loss landscape has a continuous set of degenerate (or near degenerate) minima. By solving the Fokker-Planck equation of the underlying stochastic learning dynamics, we show that due to its strong anisotropy the SGD noise introduces an additional effective loss term that decreases with flatness and has an overall strength that increases with the learning rate and batch-to-batch variation. We find that the additional landscape-dependent SGD-loss breaks the degeneracy and serves as an effective regularization for finding flat solutions. Furthermore, a stronger SGD noise shortens the convergence time to the flat solutions. However, we identify an upper bound for the SGD noise beyond which the system fails to converge. Our results not only elucidate the role of SGD for generalization they may also have important implications for hyperparameter selection for learning efficiently without divergence.

Lack of performance when it comes to continual learning over non-stationary distributions of data remains a major challenge in scaling neural network learning to more human realistic settings. In this work we propose a new conceptualization of the continual learning problem in terms of a temporally symmetric trade-off between transfer and interference that can be optimized by enforcing gradient alignment across examples. We then propose a new algorithm, Meta-Experience Replay (MER), that directly exploits this view by combining experience replay with optimization based meta-learning. This method learns parameters that make interference based on future gradients less likely and transfer based on future gradients more likely. We conduct experiments across continual lifelong supervised learning benchmarks and non-stationary reinforcement learning environments demonstrating that our approach consistently outperforms recently proposed baselines for continual learning. Our experiments show that the gap between the performance of MER and baseline algorithms grows both as the environment gets more non-stationary and as the fraction of the total experiences stored gets smaller.