A computational phase transition for learning-to-sample from Ising models

We study \emph{learning-to-sample} -- a basic algorithmic task underlying generative modeling -- for Ising models, a standard testbed for algorithmic ideas in both theoretical computer science and machine learning. Given i.i.d. samples of an unknown target distribution, the goal of learning-to-sample is to learn a computationally efficient generation procedure that produces new samples following approximately the same distribution. We construct a family of Ising models of constantly bounded-width which lie just beyond the spectral threshold $λ_{\max}(J)-λ_{\min}(J)=1$, and show that learning-to-sample for this family is computationally hard under standard cryptographic assumptions, even when the learner is given both polynomially many i.i.d. samples from the model and explicit access to its parameters. Combined with results of [AJKPV24,KLV25] showing tractability of learning-to-sample below the spectral threshold, this establishes a sharp computational phase transition at the spectral threshold. Moreover, combined with prior results on parameter learning for bounded-width Ising models [KM17,WSD19,VML20], this shows that learning-to-sample can be more difficult than parameter learning. Finally, we show that any efficient learner for these hard instances exhibits a natural memorization-hallucination dichotomy: the learner must either output configurations that, after a simple transformation, match the (transformed) training data or place substantial mass on configurations of negligible probability under the target distribution.

Understanding and Mitigating Premature Confidence for Better LLM Reasoning

Long chains of thought (CoT) from current language models frequently contain logical gaps and unjustified leaps, limiting the gains from additional test-time compute. Improving reasoning quality directly would require process reward models, but the step-level annotations needed to train them are expensive and scarce. We find such a signal in how the model's confidence evolves during reasoning: premature confidence, the tendency to commit to an answer early and use the remaining tokens to rationalize it, strongly predicts flawed reasoning across tasks and model scales. We exploit this in progressive confidence shaping, a reinforcement learning objective that trains models to update their confidence as they reason rather than commit early -- rewarding gradual confidence growth and penalizing early commitment, with no external labels or reward models. The method improves accuracy and reasoning quality from 1.5B to 8B parameters across arithmetic (Countdown), math (DAPO, AIME), and science (ScienceQA): on Countdown, accuracy improves 3.2x (+42.0pp) and flawed reasoning drops 48pp; on AIME, Pass@64 improves 6.6pp. Consistent with this mechanism, the method also improves faithfulness: on a safety benchmark, our models more transparently surface misleading content in their reasoning traces rather than concealing it. Controlled experiments reveal that the problem and its remedy scale together: premature confidence grows with model size and task difficulty, and so do the gains from addressing it.

Steering diffusion models with quadratic rewards: a fine-grained analysis

Inference-time algorithms are an emerging paradigm in which pre-trained models are used as subroutines to solve downstream tasks. Such algorithms have been proposed for tasks ranging from inverse problems and guided image generation to reasoning. However, the methods currently deployed in practice are heuristics with a variety of failure modes -- and we have very little understanding of when these heuristics can be efficiently improved. In this paper, we consider the task of sampling from a reward-tilted diffusion model -- that is, sampling from $p^{\star}(x) \propto p(x) \exp(r(x))$ -- given a reward function $r$ and pre-trained diffusion oracle for $p$. We provide a fine-grained analysis of the computational tractability of this task for quadratic rewards $r(x) = x^\top A x + b^\top x$. We show that linear-reward tilts are always efficiently sampleable -- a simple result that seems to have gone unnoticed in the literature. We use this as a building block, along with a conceptually new ingredient -- the Hubbard-Stratonovich transform -- to provide an efficient algorithm for sampling from low-rank positive-definite quadratic tilts, i.e. $r(x) = x^\top A x$ where $A$ is positive-definite and of rank $O(1)$. For negative-definite tilts, i.e. $r(x) = - x^\top A x$ where $A$ is positive-definite, we prove that the problem is intractable even if $A$ is of rank 1 (albeit with exponentially-large entries).

CodePDE: An Inference Framework for LLM-driven PDE Solver Generation

Partial differential equations (PDEs) are fundamental to modeling physical systems, yet solving them remains a complex challenge. Traditional numerical solvers rely on expert knowledge to implement and are computationally expensive, while neural-network-based solvers require large training datasets and often lack interpretability. In this work, we frame PDE solving as a code generation task and introduce CodePDE, the first inference framework for generating PDE solvers using large language models (LLMs). Leveraging advanced inference-time algorithms and scaling strategies, CodePDE unlocks critical capacities of LLM for PDE solving: reasoning, debugging, selfrefinement, and test-time scaling -- all without task-specific tuning. CodePDE achieves superhuman performance across a range of representative PDE problems. We also present a systematic empirical analysis of LLM generated solvers, analyzing their accuracy, efficiency, and numerical scheme choices. Our findings highlight the promise and the current limitations of LLMs in PDE solving, offering a new perspective on solver design and opportunities for future model development. Our code is available at https://github.com/LithiumDA/CodePDE.

On the Query Complexity of Verifier-Assisted Language Generation

Recently, a plethora of works have proposed inference-time algorithms (e.g. best-of-n), which incorporate verifiers to assist the generation process. Their quality-efficiency trade-offs have been empirically benchmarked on a variety of constrained generation tasks, but the algorithmic design landscape is still largely poorly understood. In this paper, we develop a mathematical framework for reasoning about constrained generation using a pre-trained language model generator oracle and a process verifier--which can decide whether a prefix can be extended to a string which satisfies the constraints of choice. We show that even in very simple settings, access to a verifier can render an intractable problem (information-theoretically or computationally) to a tractable one. In fact, we show even simple algorithms, like tokenwise rejection sampling, can enjoy significant benefits from access to a verifier. Empirically, we show that a natural modification of tokenwise rejection sampling, in which the sampler is allowed to "backtrack" (i.e., erase the final few generated tokens) has robust and substantive benefits over natural baselines (e.g. (blockwise) rejection sampling, nucleus sampling)--both in terms of computational efficiency, accuracy and diversity.

Towards characterizing the value of edge embeddings in Graph Neural Networks

Graph neural networks (GNNs) are the dominant approach to solving machine learning problems defined over graphs. Despite much theoretical and empirical work in recent years, our understanding of finer-grained aspects of architectural design for GNNs remains impoverished. In this paper, we consider the benefits of architectures that maintain and update edge embeddings. On the theoretical front, under a suitable computational abstraction for a layer in the model, as well as memory constraints on the embeddings, we show that there are natural tasks on graphical models for which architectures leveraging edge embeddings can be much shallower. Our techniques are inspired by results on time-space tradeoffs in theoretical computer science. Empirically, we show architectures that maintain edge embeddings almost always improve on their node-based counterparts -- frequently significantly so in topologies that have ``hub'' nodes.

Progressive distillation induces an implicit curriculum

Knowledge distillation leverages a teacher model to improve the training of a student model. A persistent challenge is that a better teacher does not always yield a better student, to which a common mitigation is to use additional supervision from several ``intermediate'' teachers. One empirically validated variant of this principle is progressive distillation, where the student learns from successive intermediate checkpoints of the teacher. Using sparse parity as a sandbox, we identify an implicit curriculum as one mechanism through which progressive distillation accelerates the student's learning. This curriculum is available only through the intermediate checkpoints but not the final converged one, and imparts both empirical acceleration and a provable sample complexity benefit to the student. We then extend our investigation to Transformers trained on probabilistic context-free grammars (PCFGs) and real-world pre-training datasets (Wikipedia and Books). Through probing the teacher model, we identify an analogous implicit curriculum where the model progressively learns features that capture longer context. Our theoretical and empirical findings on sparse parity, complemented by empirical observations on more complex tasks, highlight the benefit of progressive distillation via implicit curriculum across setups.

On the Benefits of Memory for Modeling Time-Dependent PDEs

Data-driven techniques have emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs). These techniques frequently offer a better trade-off between computational cost and accuracy for many PDE families of interest. For time-dependent PDEs, existing methodologies typically treat PDEs as Markovian systems, i.e., the evolution of the system only depends on the ``current state'', and not the past states. However, distortion of the input signals -- e.g., due to discretization or low-pass filtering -- can render the evolution of the distorted signals non-Markovian. In this work, motivated by the Mori-Zwanzig theory of model reduction, we investigate the impact of architectures with memory for modeling PDEs: that is, when past states are explicitly used to predict the future. We introduce Memory Neural Operator (MemNO), a network based on the recent SSM architectures and Fourier Neural Operator (FNO). We empirically demonstrate on a variety of PDE families of interest that when the input is given on a low-resolution grid, MemNO significantly outperforms the baselines without memory, achieving more than 6 times less error on unseen PDEs. Via a combination of theory and experiments, we show that the effect of memory is particularly significant when the solution of the PDE has high frequency Fourier components (e.g., low-viscosity fluid dynamics), and it also increases robustness to observation noise.

Promises and Pitfalls of Generative Masked Language Modeling: Theoretical Framework and Practical Guidelines

Autoregressive language models are the currently dominant paradigm for text generation, but they have some fundamental limitations that cannot be remedied by scale-for example inherently sequential and unidirectional generation. While alternate classes of models have been explored, we have limited mathematical understanding of their fundamental power and limitations. In this paper we focus on Generative Masked Language Models (GMLMs), a non-autoregressive paradigm in which we train a model to fit conditional probabilities of the data distribution via masking, which are subsequently used as inputs to a Markov Chain to draw samples from the model, These models empirically strike a promising speed-quality trade-off as each step can be typically parallelized by decoding the entire sequence in parallel. We develop a mathematical framework for analyzing and improving such models which sheds light on questions of sample complexity and inference speed and quality. Empirically, we adapt the T5 model for iteratively-refined parallel decoding, achieving 2-3x speedup in machine translation with minimal sacrifice in quality compared with autoregressive models. We run careful ablation experiments to give recommendations on key design choices, and make fine-grained observations on the common error modes in connection with our theory. Our mathematical analyses and empirical observations characterize both potentials and limitations of this approach, and can be applied to future works on improving understanding and performance of GMLMs. Our codes are released at https://github.com/google-research/google-research/tree/master/padir

Transformers are uninterpretable with myopic methods: a case study with bounded Dyck grammars

Interpretability methods aim to understand the algorithm implemented by a trained model (e.g., a Transofmer) by examining various aspects of the model, such as the weight matrices or the attention patterns. In this work, through a combination of theoretical results and carefully controlled experiments on synthetic data, we take a critical view of methods that exclusively focus on individual parts of the model, rather than consider the network as a whole. We consider a simple synthetic setup of learning a (bounded) Dyck language. Theoretically, we show that the set of models that (exactly or approximately) solve this task satisfy a structural characterization derived from ideas in formal languages (the pumping lemma). We use this characterization to show that the set of optima is qualitatively rich; in particular, the attention pattern of a single layer can be ``nearly randomized'', while preserving the functionality of the network. We also show via extensive experiments that these constructions are not merely a theoretical artifact: even after severely constraining the architecture of the model, vastly different solutions can be reached via standard training. Thus, interpretability claims based on inspecting individual heads or weight matrices in the Transformer can be misleading.