Attention Mechanisms

I show that ordinary least squares (OLS) predictions can be rewritten as the output of a restricted attention module, akin to those forming the backbone of large language models. The connection comes from viewing OLS as a similarity-based prediction rule in a learned embedding space. In this representation, least squares does not estimate coefficients per se. Instead, it selects an embedding that minimizes squared prediction error by matching training and test vectors through inner products. This maps directly onto the query-key-value structure of attention mechanisms. I then discuss extensions to dimensionality reduction, nonlinearity, and time series econometrics. Monte Carlo simulations and real-data experiments on UCI/OpenML benchmarks show that nonlinear Attention Regression performs competitively against standard machine learning baselines. In the reverse direction, I replace the attention sublayer of a transformer for tabular data with an explicit regression on polynomial features. The resulting model performs comparably to the standard transformer at a fraction of its parameter count.

In contrast to RNNs, which compress their history into a single hidden state, Transformers can attend to all past tokens directly. However, standard Transformers rely solely on the hidden state from the previous layer to represent the entire context. We show that this design creates pressure toward representation collapse and can degrade performance. To address this issue, we introduce Layer-Integrated Memory (LIMe), a lightweight extension that leverages existing key-value buffers and learns per-head, per-layer routing weights to integrate representations from previous layers. Across language modeling, synthetic reasoning, and deep architectures, LIMe improves perplexity per FLOP in the studied regimes and yields strong gains on synthetic tasks while preserving higher value-vector entropy and token separability. Finally, learned routing weights reveal systematic reuse of local and long-distance features, showing how LIMe enriches attention-time memory without increasing hidden-state size. Code is available at https://github.com/corl-team/lime.

Pretrained Transformers keep what they learned in their weights and lose what they observe once a session ends. The first version of this paper proposed a Retention Layer, a persistent memory that a Transformer block reads with attention and writes during use. Because most of what a deployed model could retain is produced by other agents, this revision treats deciding what to keep as a social learning problem: when to rely on observed behaviour, whom to learn from and how much independent agreement to require. We give a corrected specification of the layer, which reduces exactly to the base Transformer when its memory is empty. We derive the memory's lifecycle from social learning strategies: encoding gated by surprise, observed outcomes and earned credibility; consolidation by a credibility weighted quorum of distinct, recent sources that must also outweigh every rival behaviour; and reconsolidation by the outcomes of reproduction. We prove that raising the quorum lowers the risk of consolidating a coordinated false template exponentially while delaying true templates only linearly, and that relative consolidation protects only while credible honest evidence arrives faster than adversarial evidence. In a simulation with world drift and three memory-poisoning attacks, the lifecycle reached accuracies of 0.989 to 0.996, against 0.62 to 0.63 for the ungated first version design, and kept attack success at or below 0.07 when 30% of the observations about a target were adversarial. As predicted, it amplified attacks once adversarial evidence outpaced honest evidence. Experience with a long running assistant adds two rules: a model's own outputs must not count as support, and a user's testimony should be kept after one mention. We close with an evaluation protocol for language models.

Large-scale pretraining of transformers has been central to the success of foundation models. However, the scale of those models limits our understanding of the mechanisms at play during optimization. In this work, we study the training dynamics of transformers in a controlled and interpretable setting. On the sparse modular addition task, we demonstrate that specialized attention circuits, called clustering heads, can be implemented during gradient descent to solve the problem. Our experiments show that such pathways naturally emerge during training. By monitoring the evolution of tokens via a visual sandbox, we uncover a two-stage learning and the occurrences of loss spikes due to the high curvature of normalization layers. Our findings provide several insights into patterns observed in more practical settings, such as the pretraining of large language models.

Web development involves turning UI designs into functional webpages, which can be difficult for both beginners and experienced developers due to the complexity of HTML's hierarchical structures and styles. While Large Language Models (LLMs) have shown promise in generating source code, two major challenges persist in UI-to-HTML code generation: (1) effectively representing HTML's hierarchical structure for LLMs, and (2) bridging the gap between the visual nature of UI designs and the text-based format of HTML code. To tackle these challenges, we introduce Waffle, a new fine-tuning strategy that uses a structure-aware attention mechanism to improve LLMs' understanding of HTML's structure and a contrastive fine-tuning approach to align LLMs' understanding of UI images and HTML code. Models fine-tuned with Waffle show up to 9.00 pp (percentage point) higher HTML match, 0.0982 higher CW-SSIM, 32.99 higher CLIP, and 27.12 pp higher LLEM on our new benchmark WebSight-Test and an existing benchmark Design2Code, outperforming current fine-tuning methods.

Large language models (LLMs) routinely fail to output the correct option in multiple-choice question answering (MCQA) while encoding the answer internally. We expose this latent knowledge via the Query--Key (QK) score, defined for an attention head as the inner product between the last-token query and the key at the end-of-line token following option $i$, evaluated before rotary positional embedding is applied. Its argmax identifies a universal class of select-and-copy heads in middle layers that perform option selection through semantic query--key alignment, mechanistically distinct from induction and copy-suppression heads (Olsson et al., 2022): they are invariant to label symbols, and solve a synthetic task with zero surface overlap---properties no positional-copy account explains and that critically require stripping RoPE. Across 24 models from 1.5B to 72B parameters (LLaMA-2/3/3.1/3.3, Qwen-2.5, Gemma, Phi-3.5, DeepSeek-R1-Distill), a single head's QK-score exceeds the model's own zero-shot accuracy by up to $+27.4$ pp on HellaSwag and $+49.8$ pp on HaluDialogue; causal zero-ablation collapses MCQA accuracy to near-random. To remove any dependence on labeled validation data, we introduce an unsupervised HeadScore that ranks heads from unlabeled inputs and recovers the supervised top-$k$ heads on every tested model. Against four positional-debiasing baselines (e.g., PriDe, Wiegrefe, Wang), QK-score is complementary by construction: debiasing re-weights output logits, whereas QK-score reads the model's selection from a middle-layer head before decoding. We release a one-line drop-in HeadScore script and per-model head indices, making every result one-command reproducible across all 24 models and four benchmarks.

2D irregular packing is a classic combinatorial optimization problem with various applications, such as material utilization and texture atlas generation. Due to its NP-hard nature, conventional numerical approaches typically encounter slow convergence and high computational costs. Previous research (GFPack) introduced a generative method for gradient-based packing, providing early evidence of its feasibility but faced limitations such as insufficient rotation support, poor boundary adaptability, and high overlap ratios. In this paper, we propose GFPack++, a deeply investigated framework that adopts attention-based geometry and relation encoding, enabling more comprehensive modeling of complex packing relationships. We further design a constrained gradient and a weighting function to enhance both the feasibility of the produced solutions and the learning effectiveness. Experimental results on multiple datasets demonstrate that GFPack++ achieves higher space utilization, supports continuous rotation, generalizes well to arbitrary boundaries, and infers orders of magnitude faster than previous approaches. Codes for this paper are at https://github.com/TimHsue/GFPack-pp.