We develop a rigorous statistical theory of multi-head attention (MHA) as an ensemble of Nadaraya-Watson (NW) kernel regression estimators. Building on the algebraic identity between single-head softmax attention and the NW estimator, we prove that MHA is a structured ensemble of H NW estimators, each operating in a distinct learned projection subspace of the key space. We derive an explicit Bias-Variance-Covariance decomposition of the MHA mean squared error, showing that variance reduction depends not merely on the number of heads H but fundamentally on the decorrelation of head outputs. Decorrelation is governed by the principal angles between learned projection subspaces: orthogonal projections yield maximum variance reduction; aligned projections yield none. We introduce the Head Diversity Index (HDI), a computable spectral measure of inter-head decorrelation, and prove that MHA mean squared error is monotonically decreasing in HDI. This provides the first rigorous theoretical explanation for the empirically observed specialization of attention heads. Under a fixed total-dimension budget D = H * d_k, we solve the optimal head-dimension allocation problem, deriving the MSE-minimizing pair (H*, d_k*) from data distribution and regression smoothness. The solution yields a new architectural scaling law: the optimal per-head dimension grows logarithmically with training set size, while the optimal number of heads grows nearly linearly with the total budget D. Our framework unifies three strands of prior work: the NW theory of single-head attention, the general weighting theory for ensemble learning, and the decorrelation-variance-reduction isomorphism between biological and computational ensembles. Multi-head attention is the Transformer's instantiation of a universal principle: identical agents plus diversity-enforcing mechanisms yields emergent optimality.

Neural Information Processing Systems http://nips.cc/

Wefurther propose attention sharing strategies to facilitate parameter sharing and specialization in multilingual and multi-domain sequencemodeling.

In particular, we consider the sparse linear regression task, i.e., the data is generated from a

Yet, the theoretical advantages of multi-head versus single-head attention, beyond mere parallel processing, remain underex-plored. In this paper, we reframe multi-head attention as a system of potentially synergistic computational graphs, where each head functions as a feedforward directed acyclic graph (DAG) with a common sink state. We provide intuition and preliminary theoretical analysis of mixing time and minimax fidelity in this framework. Our results show that multi-head attention can synergistically enhance information propagation, yielding faster mixing times and minimax fidelity amplification under specific head-diversity conditions. Finally, we train single-head and multi-head Transformers, each with the same total number of parameters, on sequence manipulation tasks and empirically verify the predicted effects.

ABSTRACT Cell segmentation in single-shot quantitative phase microscopy (ssQPM) faces challenges from traditional thresh-olding methods that are sensitive to noise and cell density, while deep learning approaches using simple channel concatenation fail to exploit the complementary nature of polarized intensity images and phase maps. We introduce DM-QPMNet, a dual-encoder network that treats these as distinct modalities with separate encoding streams. Our architecture fuses modality-specific features at intermediate depth via multi-head attention, enabling polarized edge and texture representations to selectively integrate complementary phase information. This content-aware fusion preserves training stability while adding principled multi-modal integration through dual-source skip connections and per-modality normalization at minimal overhead. Our approach demonstrates substantial improvements over monolithic concatenation and single-modality baselines, showing that modality-specific encoding with learnable fusion effectively exploits ssQPM's simultaneous capture of complementary illumination and phase cues for robust cell segmentation.

Multi-head attention (MHA) has become the cornerstone of modern large language models, enhancing representational capacity through parallel attention heads. However, increasing the number of heads inherently weakens individual head capacity, and existing attention mechanisms - whether standard MHA or its variants like grouped-query attention (GQA) and grouped-tied attention (GTA) - simply concatenate outputs from isolated heads without strong interaction. To address this limitation, we propose knocking-heads attention (KHA), which enables attention heads to "knock" on each other - facilitating cross-head feature-level interactions before the scaled dot-product attention. This is achieved by applying a shared, diagonally-initialized projection matrix across all heads. The diagonal initialization preserves head-specific specialization at the start of training while allowing the model to progressively learn integrated cross-head representations. KHA adds only minimal parameters and FLOPs and can be seamlessly integrated into MHA, GQA, GTA, and other attention variants. We validate KHA by training a 6.1B parameter MoE model (1.01B activated) on 1T high-quality tokens. Compared to baseline attention mechanisms, KHA brings superior and more stable training dynamics, achieving better performance across downstream tasks.

The adaptation of pre-trained large language models (LLMs) to diverse downstream tasks via fine-tuning is critical for numerous applications. However, the inefficiency of parameter-efficient fine-tuning (PEFT) techniques presents significant challenges in terms of time investments and operational costs. In this paper, we first introduce a nuanced form of sparsity, termed Shadowy Sparsity, which is distinctive in fine-tuning and has not been adequately addressed for acceleration. Under Shadowy Sparsity, we propose Long Exposure, an efficient system to accelerate PEFT for LLMs. Long Exposure comprises three key components: Shadowy-sparsity Exposer employs a prolonged sensing range to capture more sparsity details under shadowy sparsity; Sequence-oriented Predictor provides efficient yet accurate predictions to handle large sequence inputs and constantly-evolving parameters; and Dynamic-aware Operator facilitates more structured computational patterns and coalesced memory accesses, addressing dynamic sparse operations. Extensive evaluations show that Long Exposure outperforms state-of-the-arts with up to a $2.49\times$ speedup in end-to-end fine-tuning, offering promising advancements in accelerating PEFT for LLMs.