DPRM: A Plug-in Doob h transform-induced Token-Ordering Module for Diffusion Language Models

Diffusion language models generate without a fixed left-to-right order, making token ordering a central algorithmic choice: which tokens should be revealed, retained, revised or verified at each step? Existing systems mainly use random masking or confidence-driven ordering. Random masking creates train--test mismatch, while confidence-only rules are efficient but can be myopic and suppress useful exploration. We introduce DPRM (Doob h-transform Process Reward Model), a plug-in token-ordering module for diffusion language models. DPRM keeps the host architecture, denoising objective and supervision unchanged, and changes only the ordering policy. It starts from confidence-driven progressive ordering and gradually shifts to Doob h transform Process Reward guided ordering through online estimates. We characterize the exact DPRM policy as a reward-tilted Gibbs reveal law, prove O(1/N) convergence of the stagewise Soft-BoN approximation, and show that the online bucketized controller tracks the exact DPRM score at empirical-Bernstein rates. Under tractable optimization assumptions, DPRM also yields a sample-complexity advantage over random and confidence-only ordering. DPRM improves over confidence-based baselines in pretraining, post-training, test-time scaling, and single-cell masked diffusion, with particularly strong gains on harder reasoning subsets. In protein, molecular generation and DNA design, the effect is more multi-objective: ordering-aware variants significantly improve selected structural or fragment-constrained metrics while not uniformly dominating the host baseline on every quality metric. These results identify token ordering as a fundamental control axis in diffusion language models and establish DPRM as a general-purpose module for improving it. Code is available at https://github.com/DakeBU/DPRM-DLLM.

On the Learning Dynamics of Two-layer Linear Networks with Label Noise SGD

One crucial factor behind the success of deep learning lies in the implicit bias induced by noise inherent in gradient-based training algorithms. Motivated by empirical observations that training with noisy labels improves model generalization, we delve into the underlying mechanisms behind stochastic gradient descent (SGD) with label noise. Focusing on a two-layer over-parameterized linear network, we analyze the learning dynamics of label noise SGD, unveiling a two-phase learning behavior. In \emph{Phase I}, the magnitudes of model weights progressively diminish, and the model escapes the lazy regime; enters the rich regime. In \emph{Phase II}, the alignment between model weights and the ground-truth interpolator increases, and the model eventually converges. Our analysis highlights the critical role of label noise in driving the transition from the lazy to the rich regime and minimally explains its empirical success. Furthermore, we extend these insights to Sharpness-Aware Minimization (SAM), showing that the principles governing label noise SGD also apply to broader optimization algorithms. Extensive experiments, conducted under both synthetic and real-world setups, strongly support our theory. Our code is released at https://github.com/a-usually/Label-Noise-SGD.

Takeuchi's Information Criteria as Generalization Measures for DNNs Close to NTK Regime

Generalization measures have been studied extensively in the machine learning community to better characterize generalization gaps. However, establishing a reliable generalization measure for statistically singular models such as deep neural networks (DNNs) is difficult due to their complex nature. This study focuses on Takeuchi's information criterion (TIC) to investigate the conditions under which this classical measure can effectively explain the generalization gaps of DNNs. Importantly, the developed theory indicates the applicability of TIC near the neural tangent kernel (NTK) regime. In a series of experiments, we trained more than 5,000 DNN models with 12 architectures, including large models (e.g., VGG-16), on four datasets, and estimated the corresponding TIC values to examine the relationship between the generalization gap and the TIC estimates. We applied several TIC approximation methods with feasible computational costs and assessed the accuracy trade-off. Our experimental results indicate that the estimated TIC values correlate well with the generalization gap under conditions close to the NTK regime. However, we show both theoretically and empirically that outside the NTK regime such correlation disappears. Finally, we demonstrate that TIC provides better trial pruning ability than existing methods for hyperparameter optimization.

Inference-Aware Meta-Alignment of LLMs via Non-Linear GRPO

Aligning large language models (LLMs) to diverse human preferences is fundamentally challenging since criteria can often conflict with each other. Inference-time alignment methods have recently gained popularity as they allow LLMs to be aligned to multiple criteria via different alignment algorithms at inference time. However, inference-time alignment is computationally expensive since it often requires multiple forward passes of the base model. In this work, we propose inference-aware meta-alignment (IAMA), a novel approach that enables LLMs to be aligned to multiple criteria with limited computational budget at inference time. IAMA trains a base model such that it can be effectively aligned to multiple tasks via different inference-time alignment algorithms. To solve the non-linear optimization problems involved in IAMA, we propose non-linear GRPO, which provably converges to the optimal solution in the space of probability measures.

Transformers as Measure-Theoretic Associative Memory: A Statistical Perspective and Minimax Optimality

Transformers excel through content-addressable retrieval and the ability to exploit contexts of, in principle, unbounded length. We recast associative memory at the level of probability measures, treating a context as a distribution over tokens and viewing attention as an integral operator on measures. Concretely, for mixture contexts $ν= I^{-1} \sum_{i=1}^I μ^{(i^*)}$ and a query $x_{\mathrm{q}}(i^*)$, the task decomposes into (i) recall of the relevant component $μ^{(i^*)}$ and (ii) prediction from $(μ_{i^*},x_\mathrm{q})$. We study learned softmax attention (not a frozen kernel) trained by empirical risk minimization and show that a shallow measure-theoretic Transformer composed with an MLP learns the recall-and-predict map under a spectral assumption on the input densities. We further establish a matching minimax lower bound with the same rate exponent (up to multiplicative constants), proving sharpness of the convergence order. The framework offers a principled recipe for designing and analyzing Transformers that recall from arbitrarily long, distributional contexts with provable generalization guarantees.

A Relative-Budget Theory for Reinforcement Learning with Verifiable Rewards in Large Language Model Reasoning

Reinforcement learning (RL) is a dominant paradigm for improving the reasoning abilities of large language models, yet its effectiveness varies across tasks and compute budgets. We propose a \emph{relative-budget} theory explaining this variation through a single quantity called relative budget $ξ:= H/\mathbb{E}[T]$, where $H$ is the generation horizon (token budget) and $T$ denotes the number of tokens until the first correct solution under a base policy. We show that $ξ$ determines sample efficiency by controlling reward variance and the likelihood of informative trajectories. Our analysis reveals three regimes: in the \emph{deficient} regime ($ξ\to 0$), informative trajectories are rare and the sample complexity explodes; in the \emph{balanced} regime ($ξ=Θ(1)$), informative trajectories occur with non-negligible probability and RL is maximally sample-efficient; and in the \emph{ample} regime ($ξ\to \infty$), learning remains stable but marginal gains per iteration diminish. We further provide finite-sample guarantees for online RL that characterize learning progress across these regimes. Specifically, in a case study under idealized distributional assumptions, we show that the relative budget grows linearly over iterations. Our empirical results confirm these predictions in realistic settings, identifying a budget $ξ\in [1.5, 2.0]$ that maximizes learning efficiency and coincides with peak reasoning performance.

Zero-Flow Encoders

Flow-based methods have achieved significant success in various generative modeling tasks, capturing nuanced details within complex data distributions. However, few existing works have exploited this unique capability to resolve fine-grained structural details beyond generation tasks. This paper presents a flow-inspired framework for representation learning. First, we demonstrate that a rectified flow trained using independent coupling is zero everywhere at $t=0.5$ if and only if the source and target distributions are identical. We term this property the \emph{zero-flow criterion}. Second, we show that this criterion can certify conditional independence, thereby extracting \emph{sufficient information} from the data. Third, we translate this criterion into a tractable, simulation-free loss function that enables learning amortized Markov blankets in graphical models and latent representations in self-supervised learning tasks. Experiments on both simulated and real-world datasets demonstrate the effectiveness of our approach. The code reproducing our experiments can be found at: https://github.com/probabilityFLOW/zfe.

From Shortcut to Induction Head: How Data Diversity Shapes Algorithm Selection in Transformers

Transformers can implement both generalizable algorithms (e.g., induction heads) and simple positional shortcuts (e.g., memorizing fixed output positions). In this work, we study how the choice of pretraining data distribution steers a shallow transformer toward one behavior or the other. Focusing on a minimal trigger-output prediction task -- copying the token immediately following a special trigger upon its second occurrence -- we present a rigorous analysis of gradient-based training of a single-layer transformer. In both the infinite and finite sample regimes, we prove a transition in the learned mechanism: if input sequences exhibit sufficient diversity, measured by a low ``max-sum'' ratio of trigger-to-trigger distances, the trained model implements an induction head and generalizes to unseen contexts; by contrast, when this ratio is large, the model resorts to a positional shortcut and fails to generalize out-of-distribution (OOD). We also reveal a trade-off between the pretraining context length and OOD generalization, and derive the optimal pretraining distribution that minimizes computational cost per sample. Finally, we validate our theoretical predictions with controlled synthetic experiments, demonstrating that broadening context distributions robustly induces induction heads and enables OOD generalization. Our results shed light on the algorithmic biases of pretrained transformers and offer conceptual guidelines for data-driven control of their learned behaviors.

Sliding Window Recurrences for Sequence Models

Multi-hybrid architectures are poised to take over language modeling due to better quality and performance. We introduce a hierarchical decomposition framework for linear recurrences that allows us to develop algorithms aligned with GPU memory hierarchies, yielding Sliding Window Recurrences. We focus specifically on truncating recurrences to hardware-aligned windows which are naturally jagged, limiting costly inter-warp communication. Using SWR, we develop Phalanx layers that serve as drop-in replacements for windowed attention or linear recurrences. In 1B parameter multi-hybrid models, Phalanx achieves over 10-40% speedup across 4K to 32K context length over optimized Transformers while matching perplexity.

Towards a Unified Analysis of Neural Networks in Nonparametric Instrumental Variable Regression: Optimization and Generalization

We establish the first global convergence result of neural networks for two stage least squares (2SLS) approach in nonparametric instrumental variable regression (NPIV). This is achieved by adopting a lifted perspective through mean-field Langevin dynamics (MFLD), unlike standard MFLD, however, our setting of 2SLS entails a \emph{bilevel} optimization problem in the space of probability measures. To address this challenge, we leverage the penalty gradient approach recently developed for bilevel optimization which formulates bilevel optimization as a Lagrangian problem. This leads to a novel fully first-order algorithm, termed \texttt{F$^2$BMLD}. Apart from the convergence bound, we further provide a generalization bound, revealing an inherent trade-off in the choice of the Lagrange multiplier between optimization and statistical guarantees. Finally, we empirically validate the effectiveness of the proposed method on an offline reinforcement learning benchmark.