A Survey of Weight Space Learning: Understanding, Representation, and Generation

Neural network weights are typically viewed as the end product of training, while most deep learning research focuses on data, features, and architectures. However, recent advances show that the set of all possible weight values (weight space) itself contains rich structure: pretrained models form organized distributions, exhibit symmetries, and can be embedded, compared, or even generated. Understanding such structures has tremendous impact on how neural networks are analyzed and compared, and on how knowledge is transferred across models, beyond individual training instances. This emerging research direction, which we refer to as Weight Space Learning (WSL), treats neural weights as a meaningful domain for analysis and modeling. This survey provides the first unified taxonomy of WSL. We categorize existing methods into three core dimensions: Weight Space Understanding (WSU), which studies the geometry and symmetries of weights; Weight Space Representation (WSR), which learns embeddings over model weights; and Weight Space Generation (WSG), which synthesizes new weights through hypernetworks or generative models. We further show how these developments enable practical applications, including model retrieval, continual and federated learning, neural architecture search, and data-free reconstruction. By consolidating fragmented progress under a coherent framework, this survey highlights weight space as a learnable, structured domain with growing impact across model analysis, transferring, and weight generation. We release an accompanying resource at https://github.com/Zehong-Wang/Awesome-Weight-Space-Learning.

Progressive Residual Warmup for Language Model Pretraining

Transformer architectures serve as the backbone for most modern Large Language Models, therefore their pretraining stability and convergence speed are of central concern. Motivated by the logical dependency of sequentially stacked layers, we propose Progressive Residual Warmup (ProRes) for language model pretraining. ProRes implements an "early layer learns first" philosophy by multiplying each layer's residual with a scalar that gradually warms up from 0 to 1, with deeper layers taking longer warmup steps. In this way, deeper layers wait for early layers to settle into a more stable regime before contributing to learning. We demonstrate the effectiveness of ProRes through pretraining experiments across various model scales, as well as normalization and initialization schemes. Comprehensive analysis shows that ProRes not only stabilizes pretraining but also introduces a unique optimization trajectory, leading to faster convergence, stronger generalization and better downstream performance. Our code is available at https://github.com/dandingsky/ProRes.

Why Diffusion Language Models Struggle with Truly Parallel (Non-Autoregressive) Decoding?

Diffusion Language Models (DLMs) are often advertised as enabling parallel token generation, yet practical fast DLMs frequently converge to left-to-right, autoregressive (AR)-like decoding dynamics. In contrast, genuinely non-AR generation is promising because it removes AR's sequential bottleneck, better exploiting parallel hardware to reduce synchronization/communication overhead and improve latency scaling with output length. We argue that a primary driver of AR-like decoding is a mismatch between DLM objectives and the highly sequential structure of widely used training data, including standard pretraining corpora and long chain-of-thought (CoT) supervision. Motivated by this diagnosis, we propose NAP (Non-Autoregressive Parallel DLMs), a proof-of-concept, data-centric approach that better aligns supervision with non-AR parallel decoding. NAP curates examples as multiple independent reasoning trajectories and couples them with a parallel-forced decoding strategy that encourages multi-token parallel updates. Across math reasoning benchmarks, NAP yields stronger performance under parallel decoding than DLMs trained on standard long CoT data, with gains growing as parallelism increases. Our results suggest that revisiting data and supervision is a principled direction for mitigating AR-like behavior and moving toward genuinely non-autoregressive parallel generation in DLMs. Our code is available at https://github.com/pixeli99/NAP.

Search or Accelerate: Confidence-Switched Position Beam Search for Diffusion Language Models

Diffusion Language Models (DLMs) generate text by iteratively denoising a masked sequence, repeatedly deciding which positions to commit at each step. Standard decoding follows a greedy rule: unmask the most confident positions, yet this local choice can lock the model into a suboptimal unmasking order, especially on reasoning-heavy prompts. We present SOAR, a training-free decoding algorithm that adapts its behavior to the model's uncertainty. When confidence is low, SOAR briefly widens the search over alternative unmasking decisions to avoid premature commitments; when confidence is high, it collapses the search and decodes many positions in parallel to reduce the number of denoising iterations. Across mathematical reasoning and code generation benchmarks (GSM8K, MBPP, HumanEval) on Dream-7B and LLaDA-8B, SOAR improves generation quality while maintaining competitive inference speed, offering a practical way to balance quality and efficiency in DLM decoding.

Long Chain-of-Thought Compression via Fine-Grained Group Policy Optimization

Large Language Models (LLMs) often generate unnecessarily verbose Chain-of-Thought (CoT) reasoning that increases computational costs and latency without proportional performance gains. In this paper, we propose \textbf{F}ine-grained \textbf{G}roup policy \textbf{O}ptimization (\textbf{FGO}), a Reinforcement Learning (RL) algorithm that refines group responses by subdividing them and assigning appropriate weights based on length and entropy, thereby enabling effective CoT compression. Meanwhile, as an enhanced variant of Group Relative Policy Optimization (GRPO), FGO successfully addresses two major limitations of the GRPO: inefficient data utilization and entropy collapse. We evaluate FGO on multiple reasoning LLMs and benchmarks, including MATH500, AIME24, AMC23, and Minerva. Experimental results show that FGO achieves efficient CoT compression without degrading performance, and simultaneously resolves the key limitations of GRPO.

SemPA: Improving Sentence Embeddings of Large Language Models through Semantic Preference Alignment

Traditional sentence embedding methods employ token-level contrastive learning on non-generative pre-trained models. Recently, there have emerged embedding methods based on generative large language models (LLMs). These methods either rely on fixed prompt templates or involve modifications to the model architecture. The former lacks further optimization of the model and results in limited performance, while the latter alters the internal computational mechanisms of the model, thereby compromising its generative capabilities. We propose SemPA, a novel approach that boosts the sentence representations while preserving the generative ability of LLMs via semantic preference alignment. We leverage sentence-level Direct Preference Optimization (DPO) to efficiently optimize LLMs on a paraphrase generation task, where the model learns to discriminate semantically equivalent sentences while preserving inherent generative capacity. Theoretically, we establish a formal connection between DPO and contrastive learning under the Plackett-Luce model framework. Empirically, experimental results on both semantic textual similarity tasks and various benchmarks for LLMs show that SemPA achieves better semantic representations without sacrificing the inherent generation capability of LLMs.

Demystifying the Roles of LLM Layers in Retrieval, Knowledge, and Reasoning

Recent studies suggest that the deeper layers of Large Language Models (LLMs) contribute little to representation learning and can often be removed without significant performance loss. However, such claims are typically drawn from narrow evaluations and may overlook important aspects of model behavior. In this work, we present a systematic study of depth utilization across diverse dimensions, including evaluation protocols, task categories, and model architectures. Our analysis confirms that very deep layers are generally less effective than earlier ones, but their contributions vary substantially with the evaluation setting. Under likelihood-based metrics without generation, pruning most layers preserves performance, with only the initial few being critical. By contrast, generation-based evaluation uncovers indispensable roles for middle and deeper layers in enabling reasoning and maintaining long-range coherence. We further find that knowledge and retrieval are concentrated in shallow components, whereas reasoning accuracy relies heavily on deeper layers -- yet can be reshaped through distillation. These results highlight that depth usage in LLMs is highly heterogeneous and context-dependent, underscoring the need for task-, metric-, and model-aware perspectives in both interpreting and compressing large models.

Diffusion Language Models Know the Answer Before Decoding

Diffusion language models (DLMs) have recently emerged as an alternative to autoregressive approaches, offering parallel sequence generation and flexible token orders. However, their inference remains slower than that of autoregressive models, primarily due to the cost of bidirectional attention and the large number of refinement steps required for high quality outputs. In this work, we highlight and leverage an overlooked property of DLMs early answer convergence: in many cases, the correct answer can be internally identified by half steps before the final decoding step, both under semi-autoregressive and random remasking schedules. For example, on GSM8K and MMLU, up to 97% and 99% of instances, respectively, can be decoded correctly using only half of the refinement steps. Building on this observation, we introduce Prophet, a training-free fast decoding paradigm that enables early commit decoding. Specifically, Prophet dynamically decides whether to continue refinement or to go "all-in" (i.e., decode all remaining tokens in one step), using the confidence gap between the top-2 prediction candidates as the criterion. It integrates seamlessly into existing DLM implementations, incurs negligible overhead, and requires no additional training. Empirical evaluations of LLaDA-8B and Dream-7B across multiple tasks show that Prophet reduces the number of decoding steps by up to 3.4x while preserving high generation quality. These results recast DLM decoding as a problem of when to stop sampling, and demonstrate that early decode convergence provides a simple yet powerful mechanism for accelerating DLM inference, complementary to existing speedup techniques. Our code is publicly available at https://github.com/pixeli99/Prophet.

AlphaDecay:Module-wise Weight Decay for Heavy-Tailed Balancing in LLMs

Weight decay is a standard regularization technique for training large language models (LLMs). While it is common to assign a uniform decay rate to every layer, this approach overlooks the structural diversity of LLMs and the varying spectral properties across modules. In this paper, we introduce AlphaDecay, a simple yet effective method that adaptively assigns different weight decay strengths to each module of an LLM. Our approach is guided by Heavy-Tailed Self-Regularization (HT-SR) theory, which analyzes the empirical spectral density (ESD) of weight correlation matrices to quantify "heavy-tailedness." Modules exhibiting more pronounced heavy-tailed ESDs, reflecting stronger feature learning, are assigned weaker decay, while modules with lighter-tailed spectra receive stronger decay. Our method leverages tailored weight decay assignments to balance the module-wise differences in spectral properties, leading to improved performance. Extensive pre-training tasks with various model sizes from 60M to 1B demonstrate that AlphaDecay achieves better perplexity and generalization than conventional uniform decay and other adaptive decay baselines.

AgentAlign: Navigating Safety Alignment in the Shift from Informative to Agentic Large Language Models

The acquisition of agentic capabilities has transformed LLMs from "knowledge providers" to "action executors", a trend that while expanding LLMs' capability boundaries, significantly increases their susceptibility to malicious use. Previous work has shown that current LLM-based agents execute numerous malicious tasks even without being attacked, indicating a deficiency in agentic use safety alignment during the post-training phase. To address this gap, we propose AgentAlign, a novel framework that leverages abstract behavior chains as a medium for safety alignment data synthesis. By instantiating these behavior chains in simulated environments with diverse tool instances, our framework enables the generation of highly authentic and executable instructions while capturing complex multi-step dynamics. The framework further ensures model utility by proportionally synthesizing benign instructions through non-malicious interpretations of behavior chains, precisely calibrating the boundary between helpfulness and harmlessness. Evaluation results on AgentHarm demonstrate that fine-tuning three families of open-source models using our method substantially improves their safety (35.8% to 79.5% improvement) while minimally impacting or even positively enhancing their helpfulness, outperforming various prompting methods. The dataset and code have both been open-sourced.