Attention Entropy

The projection of queries and keys are central to the attention mechanism in Transformer architectures. While they are mathematically symmetric, they play different roles in attention mechanisms. The question of whether there is an effect from their functional distinction on their geometric development in training remains unanswered. We investigate the problem through the training of small GPT-like Transformers on character-level WikiText-103 for three different depths (4, 6, and 8 layers), three types of initialization for queries and keys, and four random seeds, resulting in 36 runs and 54 trajectories of average layers across seeds. We track the effective dimensionality of those layers using participation ratios and discover that effective dimension of queries expand while keys shrink, and that $PR_Q - PR_K$ is positive in all trajectories studied. In connection to attention, the shrinking of keys leads to a narrower spectrum of $QK^\top$ and more peaked attention weights. In order to determine if this connection is causal or coincidental, we directly control the spectrum of keys during training across five seeds: restricting it to make it shrink sharpens the attention with high directional confidence, while keeping it constant to the level of initial dispersion makes attention softer. Additional token-level checkpoint analyses show that the monotonic paired-contrast trend is not universal across pretrained families, but survives as an early-training regime that later decays over a full pretraining run, and the link between interaction-rank geometry and attention entropy remains visible in several models.

Each row of a transformer's attention matrix is a probability distribution over tokens, and in trained models most of that probability lands on a single sink token, usually the first. Standard tools for comparing attention rows (cosine similarity, Jensen--Shannon divergence, Shannon entropy) therefore hinge on a choice papers rarely report: keep the sink, or drop it and renormalize. This choice can reverse conclusions. On ten pretrained models from five families, 17--47% of verdicts about which of two heads is more similar flip with the convention, and the most prominent structure in a standard BERT head-clustering pipeline is an artifact of it. The reason is that one-number summaries mix two questions: how much attention the sink takes, and how the rest is divided among the content tokens. Treating rows as compositional data separates them exactly: the Aitchison distance splits orthogonally into a sink term and a content term, entropy splits by an exact identity, and the content distance is characterized by invariances the transformer itself possesses. The separation matters in practice: most measured entropy collapse during training is the sink growing, not attention sharpening (30% of the drop at 70M parameters, 95% at 1B, 79% at 1.4B), and pruning heads with the wrong channel can inflate perplexity more than a hundredfold. We map where each convention is safe, test a frozen out-of-sample predictor (one confirmation, one abstention, one failure), and release code regenerating every number.

Large Vision-Language Models (LVLMs) have demonstrated remarkable capabilities across various tasks, yet they remain vulnerable to backdoor attacks. Existing defense methods predominantly focus on sample-level defense, which relies on the knowledge of training data or triggers. However, identifying whether a given model is backdoored remains a critical but unexplored task. To fill this gap, we propose EntropyScan, a lightweight and trigger-agnostic method for model-level backdoor detection in LVLMs. We first observe that backdoor injection disrupts the cross-modal alignment, resulting in pronounced structural anomalies in visual attention allocation on benign samples. Based on this insight, EntropyScan detects the backdoor models by quantifying such attention deviations. Specifically, it extracts visual attention distributions from the initial layers of the Large Language Model (LLM) and applies Tsallis entropy to capture these structural distortions. By employing a reference-anchored Z-score normalization on a small set of benign samples, it effectively identifies the backdoored model. Extensive experiments across two LVLMs architectures and three advanced attack scenarios show that EntropyScan achieves an F1 score of 98.5% in average and an AUC of 96.6%. Our code will be publicly available soon.

Reinforcement-learning-based post-training has become a key approach for improving the reasoning ability of large language models, but its token-level learning signals remain poorly understood. This work studies their heterogeneity through attention entropy, which measures how concentrated or diffuse the contextual support is for each response token. We first show that token-level RL objectives are sparsely estimable: uniformly random 20 percent token subsets preserve much of the full-token held-out performance, suggesting substantial redundancy in token-level updates. However, entropy-structured subsets behave very differently. Low-attention-entropy tokens, which we call anchors, rely on concentrated support, produce stable gradients aligned with full-token updates, and provide a reliable optimization backbone, but tend to plateau on harder benchmarks. High-attention-entropy tokens, which we call explorers, aggregate more diffuse context and induce larger but more volatile gradients. Explorer-only training is unstable on average, though rare successful runs suggest that these tokens may contain useful hard-reasoning signals when optimization remains stable. We support this anchor-explorer spectrum with evidence-gathering analyses, entropy dynamics, gradient-geometry diagnostics, and controls showing that position, predictive entropy, and loss normalization do not explain the observed asymmetry. Finally, a dynamic entropy-aware soft-reweighting intervention improves Qwen3-8B-Base from 34.39 to 37.40 held-out average in the strongest setting. These findings suggest that attention entropy reveals optimization-relevant structure in token-level RL signals, and that uniform token averaging can obscure meaningful heterogeneity in reasoning post-training.

Transformer self-attention can be interpreted as a gradient flow on the unit sphere, in which tokens evolve under softmax interaction potentials and tend to form clusters. While prior work has established clustering behavior for single-head attention, the multi-head setting remains less understood due to geometric interference between heads, which invalidates standard monotonicity arguments. In this work, we develop a theoretical framework for multi-head self-attention dynamics and resolve several open questions. We show that, under suitable conditions on the score matrices, a natural multi-head energy functional is non-decreasing along both flat and spherical dynamics. We identify the key obstruction to per-head monotonicity as radial shadow terms, which are projections of each head's output onto token directions, persisting even under orthogonality assumptions. We introduce a sufficient condition ensuring monotonicity and establish robustness to approximate orthogonality. In a simplified scalar-head regime with equiangular token configurations, we derive a closed-form expression for the critical inverse temperature governing clustering behavior, and show that heterogeneous heads exhibit super-additive clustering rates. In this regime, we also prove a separation in clustering time between ReLU and softmax attention in the linearized dynamics. Finally, we establish an entropy production identity and show that attention entropy increases monotonically toward equilibrium as clustering progresses. Our results provide a unified perspective on the dynamics of multi-head attention and clarify the mechanisms underlying clustering and stability in transformer models.

Transformer-based architectures have established a dominant paradigm in global semantic perception; however, they remain fundamentally constrained by the profound spatial heterogeneity inherent in natural images. Specifically, the imposition of a uniform global receptive field across regions of varying information density inevitably leads to local feature degradation, particularly in dense conflict zones populated by microscopic targets. To address this mechanistic limitation, we propose ViCrop-Det, a training-free inference framework that introduces adaptive spatial trust region shrinkage. Inspired by the use of attention entropy in anomaly segmentation, ViCrop-Det leverages the detection decoder's cross-attention distribution as an endogenous probe. By utilizing Spatial Attention Entropy (SAE) to heuristically evaluate local spatial ambiguity, the framework executes dynamic spatial routing, allocating a fixed computational budget exclusively to regions exhibiting both high target saliency and high cognitive uncertainty. By shrinking the spatial trust region and injecting high-frequency localized observations, ViCrop-Det actively resolves spatial ambiguity and recovers fine-grained features without requiring architectural modifications. Extensive evaluations on VisDrone and DOTA-v1.5 demonstrate that ViCrop-Det yields competitive performance enhancements, consistently adding +1-3 mAP@50 to RT-DETR-R50 and Deformable DETR with a marginal 20-23% latency overhead. On MS COCO, $AP_{S}$ improves while $AP_{M}/AP_{L}$ remains stable, indicating precise fine-scale refinement without compromising the global spatial prior. Under compute-matched settings, our adaptive routing strategy comprehensively surpasses uniform slicing baselines, achieving a highly optimized accuracy-speed trade-off.

Transformers are strong baselines in both vision and language because self-attention captures long-range dependencies across tokens. However, the cost of self-attention grows quadratically with the number of tokens. Patch pruning mitigates this cost by estimating per-patch importance and removing redundant patches. To identify informative patches for pruning, we introduce a criterion based on the Shannon entropy of the attention distribution. Low-entropy patches, which receive selective and concentrated attention, are kept as important, while high-entropy patches with attention spread across many locations are treated as redundant. We also extend the criterion from Shannon to Rényi entropy, which emphasizes sharp attention peaks and supports pruning strategies that adapt to task needs and computational limits. In experiments on fine-grained image recognition, where patch selection is critical, our method reduced computation while preserving accuracy. Moreover, adjusting the pruning policy through the Rényi entropy measure yields further gains and improves the trade-off between accuracy and computation.

Understanding the intricate non-convex training dynamics of softmax-based models is crucial for explaining the empirical success of transformers. In this article, we analyze the gradient flow dynamics of the value-softmax model, defined as ${L}(\mathbf{V} σ(\mathbf{a}))$, where $\mathbf{V}$ and $\mathbf{a}$ are a learnable value matrix and attention vector, respectively. As the matrix times softmax vector parameterization constitutes the core building block of self-attention, our analysis provides direct insight into transformer's training dynamics. We reveal that gradient flow on this structure inherently drives the optimization toward solutions characterized by low-entropy outputs. We demonstrate the universality of this polarizing effect across various objectives, including logistic and square loss. Furthermore, we discuss the practical implications of these theoretical results, offering a formal mechanism for empirical phenomena such as attention sinks and massive activations.

Visual Autoregressive(VAR) models enhance generation quality but face a critical efficiency bottleneck in later stages. In this paper, we present a novel optimization framework for VAR models that fundamentally differs from prior approaches such as FastVAR and SkipVAR. Instead of relying on heuristic skipping strategies, our method leverages attention entropy to characterize the semantic projections across different dimensions of the model architecture. This enables precise identification of parameter dynamics under varying token granularity levels, semantic scopes, and generation scales. Building on this analysis, we further uncover sparsity patterns along three critical dimensions-token, layer, and scale-and propose a set of fine-grained optimization strategies tailored to these patterns. Extensive evaluation demonstrates that our approach achieves aggressive acceleration of the generation process while significantly preserving semantic fidelity and fine details, outperforming traditional methods in both efficiency and quality. Experiments on Infinity-2B and Infinity-8B models demonstrate that ToProVAR achieves up to 3.4x acceleration with minimal quality loss, effectively mitigating the issues found in prior work. Our code will be made publicly available.