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.

Attention is at the heart of the popular Transformer architecture, yet suffers from quadratic time and memory complexity. In a recent significant development, FlashAttention shows that the I/O complexity of attention is the true bottleneck in scaling Transformers. Given two levels of memory hierarchy, a fast cache (e.g. GPU on-chip SRAM) where computation happens and a slow memory (e.g. GPU high-bandwidth memory) where the data resides, the I/O complexity measures the number of accesses to the slow memory. FlashAttention is an I/O-aware algorithm for self-attention that requires $\frac{N^2d^2}{M}$ I/O operations where $N$ is the dimension of the attention matrix, $d$ is the head-dimension and $M$ is the size of cache. Naturally, to further reduce the computational costs of Attention, the authors ask the question: is FlashAttention's I/O complexity optimal for every value of $M$? We resolve the above question in its full generality by showing an I/O complexity lower bound that matches the upper bound provided by FlashAttention for any values of $M \geq d^2$ within any constant factors. Moreover, our lower bounds do not rely on using combinatorial matrix multiplication for computing the attention matrix: even if one uses fast matrix multiplication, the above I/O complexity bounds cannot be improved. Further, we give a better algorithm with lower I/O complexity for $M < d^2$, and show that it is optimal for combinatorial algorithms. We do so by introducing a new communication complexity protocol for matrix compression, and connecting communication complexity to I/O complexity. We believe this connection could be of independent interest and will find more applications in proving I/O complexity lower bounds in future.

Multi-Head Attention (MHA) is a key component of Transformer. In MHA, attention heads work independently, causing problems such as low-rank bottleneck of attention score matrices and head redundancy. We propose Dynamically Composable Multi-Head Attention (DCMHA), a parameter and computation efficient attention architecture that tackles the shortcomings of MHA and increases the expressive power of the model by dynamically composing attention heads. At the core of DCMHA is a Compose function that transforms the attention score and weight matrices in an input-dependent way. DCMHA can be used as a drop-in replacement of MHA in any transformer architecture to obtain the corresponding DCFormer. DCFormer significantly outperforms Transformer on different architectures and model scales in language modeling, matching the performance of models with 1.7x-2.0x compute. For example, DCPythia-6.9B outperforms open source Pythia-12B on both pretraining perplexity and downstream task evaluation.

Graph Transformer, due to its global attention mechanism, has emerged as a new tool in dealing with graph-structured data. It is well recognized that the global attention mechanism considers a wider receptive field in a fully connected graph, leading many to believe that useful information can be extracted from all the nodes. In this paper, we challenge this belief: does the globalizing property always benefit Graph Transformers? We reveal the over-globalizing problem in Graph Transformer by presenting both empirical evidence and theoretical analysis, i.e., the current attention mechanism overly focuses on those distant nodes, while the near nodes, which actually contain most of the useful information, are relatively weakened. Then we propose a novel Bi-Level Global Graph Transformer with Collaborative Training (CoBFormer), including the inter-cluster and intra-cluster Transformers, to prevent the over-globalizing problem while keeping the ability to extract valuable information from distant nodes. Moreover, the collaborative training is proposed to improve the model's generalization ability with a theoretical guarantee. Extensive experiments on various graphs well validate the effectiveness of our proposed CoBFormer.