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The performance of neural networks depends on precise relationships between four distinct ingredients: the architecture, the loss function, the statistical structure of inputs, and the ground truth target function. Much theoretical work has focused on understanding the role of the first two ingredients under highly simplified models of random uncorrelated data and target functions. In contrast, performance likely relies on a conspiracy between the statistical structure of the input distribution and the structure of the function to be learned. To understand this better we revisit ridge regression in high dimensions, which corresponds to an exceedingly simple architecture and loss function, but we analyze its performance under arbitrary correlations between input features and the target function. We find a rich mathematical structure that includes: (1) a dramatic reduction in sample complexity when the target function aligns with data anisotropy; (2) the existence of multiple descent curves; (3) a sequence of phase transitions in the performance, loss landscape, and optimal regularization as a function of the amount of data that explains the first two effects.

While contrastive approaches of selfsupervised learning (SSL) learn representations by minimizing the distance between two augmented views of the same data point (positive pairs) and maximizing views from different data points (negative pairs), recent \emph{noncontrastive} SSL (e.g., BYOL and SimSiam) show remarkable performance {\it without} negative pairs, with an extra learnable predictor and a stopgradient operation. A fundamental question rises: why they do not collapse into trivial representation? In this paper, we answer this question via a simple theoretical study and propose a novel approach, \ourmethod{}, that \emph{directly} sets the linear predictor based on the statistics of its inputs, rather than trained with gradient update. On ImageNet, it performs comparably with more complex twolayer nonlinear predictors that employ BatchNorm and outperforms linear predictor by 2.5 in 300epoch training (and 5 in 60epoch). \ourmethod{} is motivated by our theoretical study of the nonlinear learning dynamics of noncontrastive SSL in simple linear networks. Our study yields conceptual insights into how noncontrastive SSL methods learn, how they avoid representational collapse, and how multiple factors, like predictor networks, stopgradients, exponential moving averages, and weight decay all come into play. Our simple theory recapitulates the results of realworld ablation studies in both STL10 and ImageNet.more »

Biological neural networks face a formidable task: performing reliable computations in the face of intrinsic stochasticity in individual neurons, imprecisely specified synaptic connectivity, and nonnegligible delays in synaptic transmission. A common approach to combatting such biological heterogeneity involves averaging over large redundant networks of N neurons resulting in coding errors that decrease classically as the square root of N. Recent work demonstrated a novel mechanism whereby recurrent spiking networks could efficiently encode dynamic stimuli achieving a superclassical scaling in which coding errors decrease as 1/N. This specific mechanism involved two key ideas: predictive coding, and a tight balance, or cancellation between strong feedforward inputs and strong recurrent feedback. However, the theoretical principles governing the efficacy of balanced predictive coding and its robustness to noise, synaptic weight heterogeneity and communication delays remain poorly understood. To discover such principles, we introduce an analytically tractable model of balanced predictive coding, in which the degree of balance and the degree of weight disorder can be dissociated unlike in previous balanced network models, and we develop a meanfield theory of coding accuracy. Overall, our work provides and solves a general theoretical framework for dissecting the differential contributions neural noise, synaptic disorder, chaos, synaptic delays, andmore »

The brain modifies its synaptic strengths during learning in order to better adapt to its environment. However, the underlying plasticity rules that govern learning are unknown. Many proposals have been suggested, including Hebbian mechanisms, explicit error backpropagation, and a variety of alternatives. It is an open question as to what specific experimental measurements would need to be made to determine whether any given learning rule is operative in a real biological system. In this work, we take a "virtual experimental" approach to this problem. Simulating idealized neuroscience experiments with artificial neural networks, we generate a largescale dataset of learning trajectories of aggregate statistics measured in a variety of neural network architectures, loss functions, learning rule hyperparameters, and parameter initializations. We then take a discriminative approach, training linear and simple nonlinear classifiers to identify learning rules from features based on these observables. We show that different classes of learning rules can be separated solely on the basis of aggregate statistics of the weights, activations, or instantaneous layerwise activity changes, and that these results generalize to limited access to the trajectory and heldout architectures and learning curricula. We identify the statistics of each observable that are most relevant for rule identification, findingmore »

In suitably initialized wide networks, small learning rates transform deep neural networks (DNNs) into neural tangent kernel (NTK) machines, whose training dynamics is wellapproximated by a linear weight expansion of the network at initialization. Standard training, however, diverges from its linearization in ways that are poorly understood. We study the relationship between the training dynamics of nonlinear deep networks, the geometry of the loss landscape, and the time evolution of a datadependent NTK. We do so through a largescale phenomenological analysis of training, synthesizing diverse measures characterizing loss landscape geometry and NTK dynamics. In multiple neural architectures and datasets, we find these diverse measures evolve in a highly correlated manner, revealing a universal picture of the deep learning process. In this picture, deep network training exhibits a highly chaotic rapid initial transient that within 2 to 3 epochs determines the final linearly connected basin of low loss containing the end point of training. During this chaotic transient, the NTK changes rapidly, learning useful features from the training data that enables it to outperform the standard initial NTK by a factor of 3 in less than 3 to 4 epochs. After this rapid chaotic transient, the NTK changes at constant velocity,more »

Pruning the parameters of deep neural networks has generated intense interest due to potential savings in time, memory and energy both during training and at test time. Recent works have identified, through an expensive sequence of training and pruning cycles, the existence of winning lottery tickets or sparse trainable subnetworks at initialization. This raises a foundational question: can we identify highly sparse trainable subnetworks at initialization, without ever training, or indeed without ever looking at the data? We provide an affirmative answer to this question through theory driven algorithm design. We first mathematically formulate and experimentally verify a conservation law that explains why existing gradientbased pruning algorithms at initialization suffer from layercollapse, the premature pruning of an entire layer rendering a network untrainable. This theory also elucidates how layercollapse can be entirely avoided, motivating a novel pruning algorithm Iterative Synaptic Flow Pruning (SynFlow). This algorithm can be interpreted as preserving the total flow of synaptic strengths through the network at initialization subject to a sparsity constraint. Notably, this algorithm makes no reference to the training data and consistently competes with or outperforms existing stateoftheart pruning algorithms at initialization over a range of models (VGG and ResNet), datasets (CIFAR10/100 and Tinymore »