Mean-field Analysis on Two-layer Neural Networks from a Kernel Perspective

In this paper, we study the feature learning ability of two-layer neural networks in the mean-field regime through the lens of kernel methods. To focus on the dynamics of the kernel induced by the first layer, we utilize a two-timescale limit, where the second layer moves much faster than the first layer. In this limit, the learning problem is reduced to the minimization problem over the intrinsic kernel. Then, we show the global convergence of the mean-field Langevin dynamics and derive time and particle discretization error. We also demonstrate that two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than any kernel methods, and neural networks acquire data-dependent kernel which aligns with the target function. In addition, we develop a label noise procedure, which converges to the global optimum and show that the degrees of freedom appears as an implicit regularization.

Mechanistic Design and Scaling of Hybrid Architectures

The development of deep learning architectures is a resource-demanding process, due to a vast design space, long prototyping times, and high compute costs associated with at-scale model training and evaluation. We set out to simplify this process by grounding it in an end-to-end mechanistic architecture design (MAD) pipeline, encompassing small-scale capability unit tests predictive of scaling laws. Through a suite of synthetic token manipulation tasks such as compression and recall, designed to probe capabilities, we identify and test new hybrid architectures constructed from a variety of computational primitives. We experimentally validate the resulting architectures via an extensive compute-optimal and a new state-optimal scaling law analysis, training over 500 language models between 70M to 7B parameters. Surprisingly, we find MAD synthetics to correlate with compute-optimal perplexity, enabling accurate evaluation of new architectures via isolated proxy tasks. The new architectures found via MAD, based on simple ideas such as hybridization and sparsity, outperform state-of-the-art Transformer, convolutional, and recurrent architectures (Transformer++, Hyena, Mamba) in scaling, both at compute-optimal budgets and in overtrained regimes. Overall, these results provide evidence that performance on curated synthetic tasks can be predictive of scaling laws, and that an optimal architecture should leverage specialized layers via a hybrid topology.

How do Transformers perform In-Context Autoregressive Learning?

Transformers have achieved state-of-the-art performance in language modeling tasks. However, the reasons behind their tremendous success are still unclear. In this paper, towards a better understanding, we train a Transformer model on a simple next token prediction task, where sequences are generated as a first-order autoregressive process $s_{t+1} = W s_t$. We show how a trained Transformer predicts the next token by first learning $W$ in-context, then applying a prediction mapping. We call the resulting procedure in-context autoregressive learning. More precisely, focusing on commuting orthogonal matrices $W$, we first show that a trained one-layer linear Transformer implements one step of gradient descent for the minimization of an inner objective function, when considering augmented tokens. When the tokens are not augmented, we characterize the global minima of a one-layer diagonal linear multi-head Transformer. Importantly, we exhibit orthogonality between heads and show that positional encoding captures trigonometric relations in the data. On the experimental side, we consider the general case of non-commuting orthogonal matrices and generalize our theoretical findings.

Transformers Learn Nonlinear Features In Context: Nonconvex Mean-field Dynamics on the Attention Landscape

Large language models based on the Transformer architecture have demonstrated impressive capabilities to learn in context. However, existing theoretical studies on how this phenomenon arises are limited to the dynamics of a single layer of attention trained on linear regression tasks. In this paper, we study the optimization of a Transformer consisting of a fully connected layer followed by a linear attention layer. The MLP acts as a common nonlinear representation or feature map, greatly enhancing the power of in-context learning. We prove in the mean-field and two-timescale limit that the infinite-dimensional loss landscape for the distribution of parameters, while highly nonconvex, becomes quite benign. We also analyze the second-order stability of mean-field dynamics and show that Wasserstein gradient flow almost always avoids saddle points. Furthermore, we establish novel methods for obtaining concrete improvement rates both away from and near critical points. This represents the first saddle point analysis of mean-field dynamics in general and the techniques are of independent interest.

Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems

In this paper, we extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We propose mean-field Langevin averaged gradient (MFL-AG), a single-loop algorithm that implements gradient descent ascent in the distribution spaces with a novel weighted averaging, and establish average-iterate convergence to the mixed Nash equilibrium. We also study both time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result which accounts for the dependency of the particle interactions on all previous distributions. Furthermore, we propose mean-field Langevin anchored best response (MFL-ABR), a symmetric double-loop algorithm based on best response dynamics with linear last-iterate convergence. Finally, we study applications to zero-sum Markov games and conduct simulations demonstrating long-term optimality.

Scalable Federated Learning for Clients with Different Input Image Sizes and Numbers of Output Categories

Federated learning is a privacy-preserving training method which consists of training from a plurality of clients but without sharing their confidential data. However, previous work on federated learning do not explore suitable neural network architectures for clients with different input images sizes and different numbers of output categories. In this paper, we propose an effective federated learning method named ScalableFL, where the depths and widths of the local models for each client are adjusted according to the clients' input image size and the numbers of output categories. In addition, we provide a new bound for the generalization gap of federated learning. In particular, this bound helps to explain the effectiveness of our scalable neural network approach. We demonstrate the effectiveness of ScalableFL in several heterogeneous client settings for both image classification and object detection tasks.

Gradient-Based Feature Learning under Structured Data

Recent works have demonstrated that the sample complexity of gradient-based learning of single index models, i.e. functions that depend on a 1-dimensional projection of the input data, is governed by their information exponent. However, these results are only concerned with isotropic data, while in practice the input often contains additional structure which can implicitly guide the algorithm. In this work, we investigate the effect of a spiked covariance structure and reveal several interesting phenomena. First, we show that in the anisotropic setting, the commonly used spherical gradient dynamics may fail to recover the true direction, even when the spike is perfectly aligned with the target direction. Next, we show that appropriate weight normalization that is reminiscent of batch normalization can alleviate this issue. Further, by exploiting the alignment between the (spiked) input covariance and the target, we obtain improved sample complexity compared to the isotropic case. In particular, under the spiked model with a suitably large spike, the sample complexity of gradient-based training can be made independent of the information exponent while also outperforming lower bounds for rotationally invariant kernel methods.

Learning Green's Function Efficiently Using Low-Rank Approximations

Learning the Green's function using deep learning models enables to solve different classes of partial differential equations. A practical limitation of using deep learning for the Green's function is the repeated computationally expensive Monte-Carlo integral approximations. We propose to learn the Green's function by low-rank decomposition, which results in a novel architecture to remove redundant computations by separate learning with domain data for evaluation and Monte-Carlo samples for integral approximation. Using experiments we show that the proposed method improves computational time compared to MOD-Net while achieving comparable accuracy compared to both PINNs and MOD-Net.

Graph Neural Networks Provably Benefit from Structural Information: A Feature Learning Perspective

Graph neural networks (GNNs) have pioneered advancements in graph representation learning, exhibiting superior feature learning and performance over multilayer perceptrons (MLPs) when handling graph inputs. However, understanding the feature learning aspect of GNNs is still in its initial stage. This study aims to bridge this gap by investigating the role of graph convolution within the context of feature learning theory in neural networks using gradient descent training. We provide a distinct characterization of signal learning and noise memorization in two-layer graph convolutional networks (GCNs), contrasting them with two-layer convolutional neural networks (CNNs). Our findings reveal that graph convolution significantly augments the benign overfitting regime over the counterpart CNNs, where signal learning surpasses noise memorization, by approximately factor $\sqrt{D}^{q-2}$, with $D$ denoting a node's expected degree and $q$ being the power of the ReLU activation function where $q > 2$. These findings highlight a substantial discrepancy between GNNs and MLPs in terms of feature learning and generalization capacity after gradient descent training, a conclusion further substantiated by our empirical simulations.