The scaling exponent $α$ in neural scaling laws $L(N) \propto N^{-α}$ is commonly treated as a fixed constant set by architecture and data. We present evidence that $α$ depends systematically on the optimizer. In controlled random-feature regression experiments -- the canonical theoretical framework for neural scaling -- we measure $α$ across five optimizer variants and six spectral conditions. Preconditioned optimizers consistently yield steeper scaling (larger $α$), with the $α$-shift increasing across most of the tested spectral range, peaking near $s = 1.5$, and remaining large at $s = 2.0$. At $s \approx 1.0$ (characteristic of natural language), the full natural gradient achieves $α\approx 0.31$ versus $α\approx 0.12$ for gradient descent -- a $2.6\times$ larger fitted exponent that, within the random-feature model, compounds with each model-size doubling. Whether and how this exponent shift transfers to large-scale LLM training -- where recent evidence suggests the advantage may attenuate with scale -- remains an important open question. Our results imply that scaling-law forecasts should account for optimizer choice, and we provide a spectral diagnostic predicting when advanced optimizers will pay off.

Language model reasoning traces are rarely all-or-nothing; they frequently contain valid intermediate steps before a critical error occurs. Existing uncertainty quantification methods typically certify final answers or entire responses, failing to provide statistical guarantees for the proportion of a sequential trace that can be safely retained. To address this, we introduce CROP (Conformal Reasoning Output Prefixes), a verifier-agnostic calibration procedure for clean-prefix certification. Given any step-level risk proxy, CROP selects a calibrated threshold and returns the longest contiguous prefix whose step risk proxies remain below it, routing the uncertified suffix for downstream review or repair. Assuming exchangeability, CROP rigorously controls the marginal probability that the returned prefix contains an annotated error. Across six process-labeled reasoning datasets, we demonstrate that standard step-level metrics such as AUROC do not fully capture prefix utility, suggesting verifiers should instead be evaluated by certified prefix length. Furthermore, CROP balances over- and under-withholding, improving downstream repair accuracy by preserving valid intermediate reasoning while discarding misleading suffixes. Ultimately, this work positions prefix certification as a rigorous, practical bridge between process supervision, abstention, and repair.

Uniform sampling on implicitly defined manifolds is a core primitive in motion planning, constrained simulation, and probabilistic machine learning. MASEM addresses this problem by entropy-maximizing resampling, but its resampling weights depend on a local k-nearest-neighbour density estimate whose errors can be amplified by aggressive resampling temperatures. We ask whether a polynomial-maximization moment estimator can replace the plug-in density rule without changing the surrounding MASEM architecture. The proposed PMM-MASEM module computes shell spacings from nested k-nearest-neighbour radii, estimates their standardized cumulants, and uses a gated PMM2/PMM3 estimator only when the spacing distribution departs from the flat Exp(1) regime; otherwise it falls back to the plug-in/MLE rule. This fallback is essential: on a flat homogeneous manifold the plug-in estimator is already the MLE, so PMM should not outperform it. A local Known-DGP Monte Carlo experiment confirms this gate: the selector returns MLE on flat Exp(1) spacings and reduces density MSE by 22--36% on asymmetric gamma and boundary-spacing regimes. The evidence is not uniformly positive: PMM3 worsens a platykurtic uniform spacing law, and a lightweight resampling-proxy experiment improves seven-lobes coverage but degrades the sine and swiss-roll proxies. The current evidence therefore supports an applicability-boundary result rather than a general MASEM improvement claim.

Unobserved confounding is a fundamental challenge for estimating causal effects. To address unobserved confounding, recent literature has turned to two different approaches -- proxy variables and the use of multiple treatments. The first approach, commonly referred to as proximal causal inference, requires proxies to be assigned to specific asymmetric roles: treatment-inducing proxies (negative control exposures), variables that act as common causes of the treatment and outcome, and outcome-inducing proxies (negative control outcomes). In practice, however, identifying variables that satisfy these asymmetric roles can be difficult depending on the application domain. The second approach, commonly referred to as the ``Deconfounder," deals with multiple conditionally independent treatments. There has been limited progress towards developing a consistent estimation method for this setting. As the primary contribution of this work, we establish that causal effects are identifiable in both settings when the unobserved confounder is categorical under suitable conditions. Our approach builds on a mixture learning perspective: we show that the underlying confounding structure can be recovered by identifying the corresponding mixture distribution. We propose an estimation procedure based on tensor decomposition, which allows consistent recovery of the latent structure and comes with non-asymptotic guarantees. Simulation studies and real data experiments demonstrate that the proposed method performs well even with limited data.

One-step generative modeling has emerged as a leading approach to amortize the inference cost of diffusion and flow-matching models. Among distillation-free methods, MeanFlow training is notoriously unstable, with non-decreasing loss and unbounded gradient variance. In this work, we establish a theory that attributes this pathology to a misuse of the conditional velocity field: it plays two distinct statistical roles in the loss, both as an unbiased regression target and as a Monte Carlo control variate inside a Jacobi-vector product, with the original loss assigning the wrong coefficient to the latter. We derive the optimal coefficient in closed form, and show that a family of fixes in concurrent works corresponds to different practical realizations of the same optimum. A controlled sweep of this coefficient on two-dimensional benchmarks and on a latent Diffusion Transformer recovers the predicted bias-variance ordering. The optimal coefficient yields up to a %54 improvement in sample quality on two-dimensional benchmarks and a monotone FID trend at every matched-step DiT checkpoint. Crucially, the same DiT measurement also reveals a quantitative FID-MSE landscape mismatch: although gradient variance is minimized at an interior coefficient value, the coefficient that minimizes FID prefers the direct use of conditional velocity.

Understanding why trained Transformers generalize well is a fundamental problem in modern machine learning theory, and complexity-based generalization bounds provide a principled way to study this question. While existing norm-based bounds for Transformers remove the explicit polynomial dependence on the hidden dimension, they typically impose fixed norm constraints specified a priori and can exhibit unfavorable exponential dependence on depth. In this paper, we derive spectrum-adaptive post hoc generalization bounds for multi-layer Transformers. Under layerwise spectral norm control, the bounds are expressed in terms of layerwise Schatten quantities of the query-key, value, and feedforward weight matrices. Since the Schatten indices need not be fixed a priori and can instead be selected after training, separately for each matrix type and layer, the bounds adaptively trade off spectral complexity against the dimension- and depth-dependent factors according to the learned singular-value profiles. Empirical comparisons of BERT-adapted proxies for the leading complexity factors suggest that the proxies induced by our bounds grow more slowly with depth and hidden dimension than the corresponding norm-based proxies. Overall, our results provide a complexity-based perspective on how the spectral structure of trained Transformers is reflected in generalization analyses.

In many scientific domains, including experimentation, researchers rely on measurements of proxy outcomes to achieve faster and more frequent reads, especially when the primary outcome of interest is challenging to measure directly. While proxies offer a more readily accessible observation for inference, the ultimate goal is to draw statistical inferences about the primary outcome parameter and proxy data are typically imperfect in some ways. To correct for these imperfections, current statistical inference methods often depend on strict identifying assumptions (such as surrogacy, covariate/label shift, or missingness assumptions). These assumptions can be difficult to validate and may be violated by various additional sources of distribution shift, potentially leading to biased parameter estimates and miscalibrated uncertainty quantification. We introduce an estimate-level framework, inspired by domain adaptation techniques, to empirically calibrate proxy-based inference. This framework models the proxy-primary metric discrepancy as a random effect at the parameter level, estimating its distribution from aggregated historical observations across past domains (e.g., experiments, time periods, or distinct segments). This method avoids the requirement for retaining individual-level response data. Additionally, this adjustment can be layered on top of existing proxy-correction methods (such as prediction-powered inference or importance weighting) to account for additional biases not addressed by those corrections. To manage uncertainty when the number of historical domains is limited, we provide both a method-of-moments estimator and a domain bootstrap procedure. We further validate this approach using publicly available datasets and real-world experiments.

Few-shot segmentation (FSS) aims at performing semantic segmentation on novel classes given a few annotated support samples. With a rethink of recent advances, we find that the current FSS framework has deviated far from the supervised segmentation framework: Given the deep features, FSS methods typically use an intricate decoder to perform sophisticated pixel-wise matching, while the supervised segmentation methods use a simple linear classification head. Due to the intricacy of the decoder and its matching pipeline, it is not easy to follow such an FSS framework. This paper revives the straightforward framework of "feature extractor + linear classification head" and proposes a novel Feature-Proxy Transformer (FPTrans) method, in which the "proxy" is the vector representing a semantic class in the linear classification head. FPTrans has two keypoints for learning discriminative features and representative proxies: 1) To better utilize the limited support samples, the feature extractor makes the query interact with the support features from bottom to top layers using a novel prompting strategy.