Robust Learning in Bayesian Parallel Branching Graph Neural Networks: The Narrow Width Limit

The infinite width limit of random neural networks is known to result in Neural Networks as Gaussian Process (NNGP) (Lee et al. [2018]), characterized by task-independent kernels. It is widely accepted that larger network widths contribute to improved generalization (Park et al. [2019]). However, this work challenges this notion by investigating the narrow width limit of the Bayesian Parallel Branching Graph Neural Network (BPB-GNN), an architecture that resembles residual networks. We demonstrate that when the width of a BPB-GNN is significantly smaller compared to the number of training examples, each branch exhibits more robust learning due to a symmetry breaking of branches in kernel renormalization. Surprisingly, the performance of a BPB-GNN in the narrow width limit is generally superior or comparable to that achieved in the wide width limit in bias-limited scenarios. Furthermore, the readout norms of each branch in the narrow width limit are mostly independent of the architectural hyperparameters but generally reflective of the nature of the data. Our results characterize a newly defined narrow-width regime for parallel branching networks in general.

Order parameters and phase transitions of continual learning in deep neural networks

Continual learning (CL) enables animals to learn new tasks without erasing prior knowledge. CL in artificial neural networks (NNs) is challenging due to catastrophic forgetting, where new learning degrades performance on older tasks. While various techniques exist to mitigate forgetting, theoretical insights into when and why CL fails in NNs are lacking. Here, we present a statistical-mechanics theory of CL in deep, wide NNs, which characterizes the network's input-output mapping as it learns a sequence of tasks. It gives rise to order parameters (OPs) that capture how task relations and network architecture influence forgetting and knowledge transfer, as verified by numerical evaluations. We found that the input and rule similarity between tasks have different effects on CL performance. In addition, the theory predicts that increasing the network depth can effectively reduce overlap between tasks, thereby lowering forgetting. For networks with task-specific readouts, the theory identifies a phase transition where CL performance shifts dramatically as tasks become less similar, as measured by the OPs. Sufficiently low similarity leads to catastrophic anterograde interference, where the network retains old tasks perfectly but completely fails to generalize new learning. Our results delineate important factors affecting CL performance and suggest strategies for mitigating forgetting.

Coding schemes in neural networks learning classification tasks

Neural networks posses the crucial ability to generate meaningful representations of task-dependent features. Indeed, with appropriate scaling, supervised learning in neural networks can result in strong, task-dependent feature learning. However, the nature of the emergent representations, which we call the `coding scheme', is still unclear. To understand the emergent coding scheme, we investigate fully-connected, wide neural networks learning classification tasks using the Bayesian framework where learning shapes the posterior distribution of the network weights. Consistent with previous findings, our analysis of the feature learning regime (also known as `non-lazy', `rich', or `mean-field' regime) shows that the networks acquire strong, data-dependent features. Surprisingly, the nature of the internal representations depends crucially on the neuronal nonlinearity. In linear networks, an analog coding scheme of the task emerges. Despite the strong representations, the mean predictor is identical to the lazy case. In nonlinear networks, spontaneous symmetry breaking leads to either redundant or sparse coding schemes. Our findings highlight how network properties such as scaling of weights and neuronal nonlinearity can profoundly influence the emergent representations.

Dissecting the Interplay of Attention Paths in a Statistical Mechanics Theory of Transformers

Despite the remarkable empirical performance of Transformers, their theoretical understanding remains elusive. Here, we consider a deep multi-head self-attention network, that is closely related to Transformers yet analytically tractable. We develop a statistical mechanics theory of Bayesian learning in this model, deriving exact equations for the network's predictor statistics under the finite-width thermodynamic limit, i.e., $N,P\rightarrow\infty$, $P/N=\mathcal{O}(1)$, where $N$ is the network width and $P$ is the number of training examples. Our theory shows that the predictor statistics are expressed as a sum of independent kernels, each one pairing different 'attention paths', defined as information pathways through different attention heads across layers. The kernels are weighted according to a 'task-relevant kernel combination' mechanism that aligns the total kernel with the task labels. As a consequence, this interplay between attention paths enhances generalization performance. Experiments confirm our findings on both synthetic and real-world sequence classification tasks. Finally, our theory explicitly relates the kernel combination mechanism to properties of the learned weights, allowing for a qualitative transfer of its insights to models trained via gradient descent. As an illustration, we demonstrate an efficient size reduction of the network, by pruning those attention heads that are deemed less relevant by our theory.

Probing Biological and Artificial Neural Networks with Task-dependent Neural Manifolds

Recently, growth in our understanding of the computations performed in both biological and artificial neural networks has largely been driven by either low-level mechanistic studies or global normative approaches. However, concrete methodologies for bridging the gap between these levels of abstraction remain elusive. In this work, we investigate the internal mechanisms of neural networks through the lens of neural population geometry, aiming to provide understanding at an intermediate level of abstraction, as a way to bridge that gap. Utilizing manifold capacity theory (MCT) from statistical physics and manifold alignment analysis (MAA) from high-dimensional statistics, we probe the underlying organization of task-dependent manifolds in deep neural networks and macaque neural recordings. Specifically, we quantitatively characterize how different learning objectives lead to differences in the organizational strategies of these models and demonstrate how these geometric analyses are connected to the decodability of task-relevant information. These analyses present a strong direction for bridging mechanistic and normative theories in neural networks through neural population geometry, potentially opening up many future research avenues in both machine learning and neuroscience.

Connecting NTK and NNGP: A Unified Theoretical Framework for Neural Network Learning Dynamics in the Kernel Regime

Artificial neural networks have revolutionized machine learning in recent years, but a complete theoretical framework for their learning process is still lacking. Substantial progress has been made for infinitely wide networks. In this regime, two disparate theoretical frameworks have been used, in which the network's output is described using kernels: one framework is based on the Neural Tangent Kernel (NTK) which assumes linearized gradient descent dynamics, while the Neural Network Gaussian Process (NNGP) kernel assumes a Bayesian framework. However, the relation between these two frameworks has remained elusive. This work unifies these two distinct theories using a Markov proximal learning model for learning dynamics in an ensemble of randomly initialized infinitely wide deep networks. We derive an exact analytical expression for the network input-output function during and after learning, and introduce a new time-dependent Neural Dynamical Kernel (NDK) from which both NTK and NNGP kernels can be derived. We identify two learning phases characterized by different time scales: gradient-driven and diffusive learning. In the initial gradient-driven learning phase, the dynamics is dominated by deterministic gradient descent, and is described by the NTK theory. This phase is followed by the diffusive learning stage, during which the network parameters sample the solution space, ultimately approaching the equilibrium distribution corresponding to NNGP. Combined with numerical evaluations on synthetic and benchmark datasets, we provide novel insights into the different roles of initialization, regularization, and network depth, as well as phenomena such as early stopping and representational drift. This work closes the gap between the NTK and NNGP theories, providing a comprehensive framework for understanding the learning process of deep neural networks in the infinite width limit.

Globally Gated Deep Linear Networks

Recently proposed Gated Linear Networks present a tractable nonlinear network architecture, and exhibit interesting capabilities such as learning with local error signals and reduced forgetting in sequential learning. In this work, we introduce a novel gating architecture, named Globally Gated Deep Linear Networks (GGDLNs) where gating units are shared among all processing units in each layer, thereby decoupling the architectures of the nonlinear but unlearned gatings and the learned linear processing motifs. We derive exact equations for the generalization properties in these networks in the finite-width thermodynamic limit, defined by $P,N\rightarrow\infty, P/N\sim O(1)$, where P and N are the training sample size and the network width respectively. We find that the statistics of the network predictor can be expressed in terms of kernels that undergo shape renormalization through a data-dependent matrix compared to the GP kernels. Our theory accurately captures the behavior of finite width GGDLNs trained with gradient descent dynamics. We show that kernel shape renormalization gives rise to rich generalization properties w.r.t. network width, depth and L2 regularization amplitude. Interestingly, networks with sufficient gating units behave similarly to standard ReLU networks. Although gatings in the model do not participate in supervised learning, we show the utility of unsupervised learning of the gating parameters. Additionally, our theory allows the evaluation of the network's ability for learning multiple tasks by incorporating task-relevant information into the gating units. In summary, our work is the first exact theoretical solution of learning in a family of nonlinear networks with finite width. The rich and diverse behavior of the GGDLNs suggests that they are helpful analytically tractable models of learning single and multiple tasks, in finite-width nonlinear deep networks.

A theory of learning with constrained weight-distribution

A central question in computational neuroscience is how structure determines function in neural networks. The emerging high-quality large-scale connectomic datasets raise the question of what general functional principles can be gleaned from structural information such as the distribution of excitatory/inhibitory synapse types and the distribution of synaptic weights. 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. Our theory predicts that the reduction in capacity due to the constrained weight-distribution is related to the Wasserstein distance between the imposed distribution 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. 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. 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.

Temporal support vectors for spiking neuronal networks

When neural circuits learn to perform a task, it is often the case that there are many sets of synaptic connections that are consistent with the task. However, only a small number of possible solutions are robust to noise in the input and are capable of generalizing their performance of the task to new inputs. Finding such good solutions is an important goal of learning systems in general and neuronal circuits in particular. For systems operating with static inputs and outputs, a well known approach to the problem is the large margin methods such as Support Vector Machines (SVM). By maximizing the distance of the data vectors from the decision surface, these solutions enjoy increased robustness to noise and enhanced generalization abilities. Furthermore, the use of the kernel method enables SVMs to perform classification tasks that require nonlinear decision surfaces. However, for dynamical systems with event based outputs, such as spiking neural networks and other continuous time threshold crossing systems, this optimality criterion is inapplicable due to the strong temporal correlations in their input and output. We introduce a novel extension of the static SVMs - The Temporal Support Vector Machine (T-SVM). The T-SVM finds a solution that maximizes a new construct - the dynamical margin. We show that T-SVM and its kernel extensions generate robust synaptic weight vectors in spiking neurons and enable their learning of tasks that require nonlinear spatial integration of synaptic inputs. We propose T-SVM with nonlinear kernels as a new model of the computational role of the nonlinearities and extensive morphologies of neuronal dendritic trees.

Optimal quadratic binding for relational reasoning in vector symbolic neural architectures

Binding operation is fundamental to many cognitive processes, such as cognitive map formation, relational reasoning, and language comprehension. In these processes, two different modalities, such as location and objects, events and their contextual cues, and words and their roles, need to be bound together, but little is known about the underlying neural mechanisms. Previous works introduced a binding model based on quadratic functions of bound pairs, followed by vector summation of multiple pairs. Based on this framework, we address following questions: Which classes of quadratic matrices are optimal for decoding relational structures? And what is the resultant accuracy? We introduce a new class of binding matrices based on a matrix representation of octonion algebra, an eight-dimensional extension of complex numbers. We show that these matrices enable a more accurate unbinding than previously known methods when a small number of pairs are present. Moreover, numerical optimization of a binding operator converges to this octonion binding. We also show that when there are a large number of bound pairs, however, a random quadratic binding performs as well as the octonion and previously-proposed binding methods. This study thus provides new insight into potential neural mechanisms of binding operations in the brain.