Quantization is essential for reducing the computational cost and memory usage of deep neural networks, enabling efficient inference on low-precision hardware. Despite the growing adoption of uniform and floating-point quantization schemes, selecting optimal quantization parameters remains a key challenge, particularly for diverse data distributions encountered during training and inference. This work presents a novel statistical error analysis framework for uniform and floating-point quantization, providing theoretical insight into error behavior across quantization configurations. Building on this analysis, we propose iterative quantizers designed for arbitrary data distributions and analytic quantizers tailored for Gaussian-like weight distributions. These methods enable efficient, low-error quantization suitable for both activations and weights. We incorporate our quantizers into quantization-aware training and evaluate them across integer and floating-point formats. Experiments demonstrate improved accuracy and stability, highlighting the effectiveness of our approach for training low-precision neural networks.

Training data attribution (TDA) should enable generative model interpretability and foster a variety of related downstream tasks. Nonetheless, current TDA approaches lack reliability and robustness, preventing their adoption in real-world setups. In this paper, we take a decisive step towards more reliable and robust TDA for diffusion models. We propose to perform TDA with mirrored unlearning and noise-consistent skew (MUCS). The idea is to fine-tune a second model with bounded mirrored gradient ascent, and to measure the normalized skew of this model with respect to the original one using consistent noise samples. We show that, while being conceptually simple and generic, MUCS systematically outperforms existing methods on three different datasets by a large margin. We additionally study the effect that core design choices have on final performance, and analyze novel aspects regarding the overlap of influential instances across generated items and the potential of ensembling TDA approaches. We believe that our findings may have broader implications for more general unlearning setups, as well as for tasks requiring the comparison of diffusion losses.

Uncertainty quantification (UQ) is critical for safety-critical domains like healthcare, yet it is rarely evaluated under realistic out-of-distribution (OOD) conditions. Here, we assessed predictive performance and uncertainty reliability for deep learning-based blood pressure (BP) estimation from photoplethysmography (PPG) signals under both in-distribution (ID) and OOD settings. Using an XResNet1D-50 trained on PulseDB and tested on four external datasets, we compared deep ensembles (DE) and Monte Carlo dropout (MCD) with Gaussian negative log-likelihood (GNLL) and mean squared error (MSE) losses, optionally followed by post-hoc recalibration via conformal prediction (CP), temperature scaling (TS), and isotonic regression (IR). The key findings of our study are as follows: (1) DE provides stronger predictive robustness under domain shift than MCD, an advantage that becomes clear primarily under external shift. (2) Recalibrated GNLL-based methods yield the best uncertainty calibration (e.g., GNLL+DE+CP for systolic blood pressure (SBP), GNLL+DE+TS for diastolic blood pressure (DBP)), while MSE-based uncertainty requires recalibration to become practically useful. (3) Across settings, CP and TS offer the most consistent gains, with IR remaining competitive in several cases. Overall, our results identify DE-based methods as most robust for predictive performance under domain shift, GNLL as strongest for native UQ, and recalibration as essential for making MSE-based uncertainty practical. These findings highlight the need to jointly assess predictive accuracy and calibration on external data for trustworthy cuffless BP estimation

Muon has recently emerged as a strong alternative to AdamW for training neural networks, with encouraging large-scale pretraining results and growing evidence that matrix-structured updates can be faster in practice. Yet Muon, and more generally Linear Minimization Oracle (LMO) based methods, are typically used synchronously. This is problematic in heterogeneous distributed systems, where workers complete gradient computations at different speeds and synchronous training must repeatedly wait for slower workers. In this work, we introduce Ringmaster LMO, an asynchronous LMO-based momentum method for unconstrained stochastic nonconvex optimization. Our method builds on the delay-thresholding idea of Ringmaster ASGD. For SGD-type methods, Ringmaster ASGD achieves optimal time complexity by discarding overly stale gradients. Ringmaster LMO extends this mechanism to general LMO-based updates. We establish convergence guarantees under generalized $(L_0, L_1)$-smoothness and further develop a parameter-agnostic variant with decreasing stepsizes and adaptive delay thresholds. Finally, we translate our iteration guarantees into time complexity bounds under heterogeneous worker computation times. In the classical Euclidean smooth setting, these bounds recover the optimal time complexity of Ringmaster ASGD. Experiments on stochastic quadratic problems and NanoChat language-model pretraining show that the advantages of Ringmaster LMO grow with system heterogeneity and that the method outperforms strong synchronous and asynchronous baselines.

Wide neural networks in the feature-learning regime drive modern deep learning, and yet they remain far less studied than their kernel-regime counterparts. We consider a critical yet under-explored difference between these two regimes: the regulariser and prior implied by gradient flow training. This canonical regularisation property is well-studied in kernel regime networks -- of all the infinite global minima, gradient flow selects exactly the vanishing ridge solution -- and underpins the celebrated NN-GP correspondence, precisely allowing the modelling of noise during training. However, we prove ridge regularisation biases gradient flow in feature-learning regime networks, even in the infinitesimal limit of vanishing regularisation. Over training, ridge distorts the inductive bias of the network, with a particular damage done to pretrained networks where the implicit prior is informative. We resolve this by axiomatising the canonical regulariser as a regime-agnostic function-space energy and lift, which uniquely identifies ridge in the kernel regime, and crucially generalises to the feature-learning regime. By studying the Riemannian geometry of feature-learning networks, we derive geodesic ridge from our framework, generalising ridge to the feature-learning regime. Correspondingly, we prove the canonical function-space prior is a Riemannian Gibbs Process, generalising the more familiar Gaussian Process. As a practical contribution, we propose arc ridge as a minimax-robust, scalable surrogate to geodesic ridge, revealing a deep relationship between early stopping and canonical regularisation across learning regimes. Finally, we demonstrate the consequences of our theory empirically on both image processing and NLP transfer-learning problems.

ABSTRACT Discrete diffusion models are a powerful class of generative models with strong performance across many domains. For efficiency, however, discrete diffusion typically parameterizes the generative (reverse) process with factorized distributions, which makes it difficult for the model to learn the target process in a small number of steps and necessitates a long, computationally expensive sampling procedure. To reduce the gap between the target and model distributions and enable few-step generation, we propose Forward-Learned Discrete Diffusion (FLDD), which introduces discrete diffusion with a learnable forward (noising) process. Rather than fixing a Markovian forward chain, we adopt a non-Markovian formulation with learnable marginal and posterior distributions. This allows the generative process to remain factorized while matching the target defined by the noising process. We train all parameters end-to-end under the standard variational objective. Experiments on various benchmarks show that, for a given number of sampling steps, our approach produces a higher quality samples than conventional discrete diffusion models using the same reverse parameterization. 1 INTRODUCTION In the last years, diffusion models have demonstrated strong performance across many continuous (Hoogeboom et al., 2024) and discrete (Lou et al.) domains . Recent work has shown that distillation approaches and advanced training techniques allow learning a few-step (Salimans et al., 2024), or sometimes even a single-step, generative (Xu et al., 2025) procedure in the continuous domain.

Generative models can produce nonsensical text, unrealistic images, and unstable materials faster than simulation or human review can absorb; without per-sample confidence, trust erodes. Existing fixes run $k$ ensembles or stochastic trajectories at $k\times$ compute, measuring variability between models, not model confidence. We propose Flow Matching with Confidence (FMwC). FMwC injects input-dependent multiplicative noise at selected layers, propagates its variance through the network in closed form, and integrates it along the ODE trajectory, yielding a per-sample confidence score at standard sampling cost. The score supports multiple uses: filtering improves image quality and thermodynamic stability of crystals; editing rewinds trajectories to the points where the model commits and redirects them; and adaptive stepping concentrates ODE compute where the flow is ambiguous. We find that the confidence score correlates with the magnitude of the divergence of the learned velocity field, which gives us a window to understand the generative process, opening up surgical forms of guidance that target the moments that matter, new sampling algorithms and interpretability of generative models.

While diffusion has drawn considerable recent attention from the language modeling community, continuous diffusion has appeared less scalable than discrete approaches. To challenge this belief we revisit Plaid, a likelihood-based continuous diffusion language model (DLM), and construct RePlaid by aligning the architecture of Plaid with modern discrete DLMs. In this unified setting, we establish the first scaling law for continuous DLMs that rivals discrete DLMs: RePlaid exhibits a compute gap of only $20\times$ compared to autoregressive models, outperforms Duo while using fewer parameters, and outperforms MDLM in the over-trained regime. We benchmark RePlaid against recent continuous DLMs: on OpenWebText, RePlaid achieves a new state-of-the-art PPL bound of $22.1$ among continuous DLMs and superior generation quality. These results suggest that continuous diffusion, when trained via likelihood, is a highly competitive and scalable alternative to discrete DLMs. Moreover, we offer theoretical insights to understand the advantage of likelihood-based training. We show that optimizing the noise schedule to minimize the ELBO's variance naturally yields linear cross-entropy (information loss) over time. This evenly distributes denoising difficulty without any case-specific time reparameterization. In addition, we find that optimizing embeddings via likelihood creates structured geometries and drives the most significant likelihood gain.

Federated Learning is a leading framework for training ML and AI models collaboratively across numerous user devices or databases. We study the trade-offs among estimation accuracy, privacy constraints, and communication cost for differentially private (DP) federated M estimation. The two standard methods in the literature are FedAvg, which may suffer from high federation bias, and FedSGD, which can incur high communication cost. Aimed at improving accuracy at a reduced communication cost, we propose FedHybrid, which uses FedSGD starting with an improved initialization by the FedAvg estimator. We propose FedNewton, which averages local Newton iterations to reduce bias in FedAvg, achieving an estimation accuracy comparable to FedSGD with much fewer communication rounds when the number of clients grows sufficiently slowly. We establish finite sample upper bounds on the mean-squared error rates of the DP versions of these estimators as functions of the number of clients, local sample sizes, privacy budget, and number of iterations. We further derive a minimax lower bound on the MSE of any iterative private federated procedure that provides a benchmark to assess the optimality gap of these methods. We numerically evaluate our methods for training a logistic regression and a neural network on the computer vision datasets MNIST and CIFAR-10.

Elon Musk, left, and Sam Altman. Elon Musk, left, and Sam Altman. How Sam Altman's victory over Elon Musk clears way for OpenAI's trillion-dollar ambitions OpenAI's plans now seem all but guaranteed, given that the world's richest man couldn't put a stop to them On Monday morning, a jury in Oakland, California, handed a resounding victory to Sam Altman and OpenAI in their long, bitter courtroom battle with Elon Musk. The federal jury found Altman, OpenAI and its president, Greg Brockman, not liable for Elon Musk's claims that they unjustly enriched themselves and broke a founding contract made with Musk when founding the startup. The unanimous verdict, delivered after less than two hours of deliberation, is a stark rebuke of Musk and his lawyer's claims that Altman "stole a charity" through his leadership of OpenAI.