Randomly initialized neural networks induce a prior over functions, but the predictor used in practice is produced only after training. We ask how much of this initial bias survives the training pipeline. To make the question measurable, we introduce initialization memory: the dependence of the validation-selected predictor on the scale of the random initialization. We perform controlled CIFAR-10 experiments on ResNets where initialization memory already sharply separates training regimes. Low-learning-rate SGD can interpolate while still remembering its initialization: on ResNet-9 with batch size $b=128$, test accuracy varies by $26.5$ percentage points across initialization scales despite $\ge99.5\%$ training accuracy. This is not undertraining: extending the same low-learning-rate regime to $5{,}000$ epochs leaves the spread essentially unchanged. In contrast, Adam-family methods largely erase the dependence. SGD can also be made to forget when larger learning rates are paired with explicit $L_2$ norm control. We interpret these findings in terms of the time scale of forgetting: gradient-flow-like dynamics can preserve initialization memory, whereas stochastic finite-step effects, explicit norm decay, and adaptive preconditioning erase it on scales governed by the size of explicit or implicit regularization. The practical inductive bias of a trained network is therefore not the architectural prior alone, but the architectural prior after being filtered by the forgetting dynamics of the training pipeline; and the same regularizers that improve generalization are precisely those that erase memory of initialization.

We develop a natural Bayesian multiplicity-correcting prior distribution within the probabilistic forward stepwise representation of model space priors for regression problems. The proposed prior, obtained from making an analogy to the Holm procedure, exhibits behavior closely aligned with that of the Matryoshka doll prior. We compare both priors to several other priors, including some recently put forward as objective choices for model space prior probabilities. Our comparisons indicate that adequate multiplicity correction requires a degree of sparsity that many recommended priors do not provide, and we argue that multiplicity correction itself offers a principled and transparent criterion for specifying model space priors in regression.

Fraud detection models in payment networks train on chargeback labels that are systematically biased. Every label must survive three sequential gates: authorization (declined transactions generate no labels), issuer reporting (unreported fraud is invisible), and delay (pending chargebacks are missing at training time). Labels that do arrive may be corrupted by first-party misuse or issuer misclassification. A companion paper [arXiv:2605.27557] proved that these four impairments impose a minimax lower bound on detection performance. This paper asks: can that bound be achieved? We formalize the observation pipeline as a sequential missing-data problem with three propensity stages and a corruption layer, and construct the Sequential Triply Robust (STR) estimator. The STR corrects for all four impairments simultaneously and achieves the semiparametric efficiency bound -- no estimator can have lower asymptotic variance. It is sequentially triply robust: at each gate, consistency requires only that either the propensity model or the outcome regression is correctly specified, not both. We provide corruption correction via noise-rate-adjusted pseudo-labels, empirical Bayes shrinkage to stabilize inverse-propensity weights for small issuers, a plug-in variance estimator yielding valid confidence intervals, and a Bernstein concentration inequality for finite-sample guarantees. On the operational side, we derive the optimal training delay -- the maturity window that minimizes the sum of label-quality loss and model staleness -- and prove that the STR permits training on data that is days old rather than months old, decoupling model freshness from the chargeback maturity cycle. The STR provably dominates naive chargeback-based training in mean squared error for any sample size.

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For this subclass we prove a sample-level almost-sure convergence theorem and an explicit almost-sure rate of convergence, under coupled rate conditions on the penalty parameter, the random-feature count, the sample size, and the optimization tolerance. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

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.

Pairwise human-preference platforms such as Chatbot Arena have become central to large language model (LLM) evaluation, yet reliable task-specific ranking remains challenging. Global leaderboards mask task heterogeneity, while ranking each fine-grained task independently is unstable under sparse, imbalanced comparisons. We propose a low-rank framework for task-specific LLM ranking from sparse pairwise comparisons, modeling the task-by-model ability matrix $Θ^\star \in \mathbb{R}^{d_t \times d_m}$ as low rank so that information is shared across related tasks while task-specific differences are preserved. We first develop a max-norm ($\ell_\infty$) accurate estimator for the latent scores, combining a convex initializer with alternating-minimization refinement, and prove task-wise top-$K$ recovery guarantees under sparse sampling. Our main contribution is an uncertainty quantification framework for task-specific ranking. We construct cross-fitted one-step debiased estimators for fixed score contrasts -- such as the task-specific ability gap between two models -- yielding asymptotically valid confidence intervals that attain the semiparametric efficiency bound. We then extend the inference to the high-dimensional ranking regime, where per-task ranks and top-$K$ membership are determined by many dependent score-gap hypotheses. Using Gaussian and multiplier-bootstrap calibration, we obtain simultaneous confidence sets for per-task ranks and valid top-$K$ membership tests across many tasks and models. Experiments on synthetic data and Chatbot Arena show that low-rank sharing improves sample efficiency over independent task-wise Bradley-Terry estimation and produces tighter, better-calibrated ranking certificates, with the largest gains in the sparse regime typical of real LLM benchmarks.

Under standard graphical assumptions, the Markov boundary of a target variable is the smallest set of features that renders every other feature redundant. Once the boundary is observed, the target is conditionally independent of the rest of the table. This is a tempting object for tabular prediction, since it names exactly the columns a model should need. Yet modern regressors are still trained on the full feature set. We ask whether the Markov boundary is genuinely useful for prediction on SCM3K, a 3,450-task synthetic SCM benchmark with feature counts from 40 to 1000 and six SCM families, evaluated with six regressors. The answer is more nuanced than the theory suggests. Restricting a regressor to the oracle boundary often improves prediction substantially, and the improvement grows as the feature space becomes larger and sparser. But the natural pipeline of recovering the boundary with causal discovery and training on the recovered mask does not deliver. Existing estimators exhaust the compute budget before reaching the regime where the boundary helps most, and even where they run they rarely beat the full feature set. We trace this to three causes. Discovery optimizes structural recovery rather than prediction. False negatives and false positives carry sharply asymmetric predictive cost. The exact boundary is only one of many feature sets that beat all features. We then develop what these facts imply for prediction-aligned feature selection and for tabular models that learn to use causal structure.

Purpose: Task-based assessment of image quality (IQ) is critically important for the design and optimization of medical imaging systems. Ideal observers, including the Bayesian Ideal Observer (IO) and the ideal linear observer, i.e., the Hotelling observer (HO), provide objective figures of merit (FOMs) that quantify system performance on signal detection tasks. However, the application of ideal observers to high-dimensional image data is often computationally intractable. Channel mechanisms provide an effective framework for dimensionality reduction that can facilitate the computation of ideal observers. This work presents a conjugate gradient (CG)-based method to construct efficient channels for approximating the IO and HO performance.

In randomized trials involving multiple treatments, bivariate survival outcomes present significant analytical challenges for making decisions. This paper addresses the problem of deriving optimal individualized treatment rules to maximize the joint survival probability beyond fixed time points $(t_1, t_2)$ through deep neural networks, while accounting for right censoring. We propose a novel approach that models treatment rules via stochastic policies, coupling marginal accelerated failure time models via link function to capture bivariate dependence. To enhance robustness and effectiveness of decision making, we introduce an adaptive prediction-powered method that leverages auxiliary predictions from machine learning models.

While parameter-efficient fine-tuning methods like low-rank adaptation (LoRA) are standard for large language models, principled estimation of epistemic uncertainty remains challenging. Recent results in the LoRA regime suggest that discrete multi-mode approaches such as deep ensembles offer little benefit over single-mode methods. This contradicts broader observations in deep learning, where ensembling independent optima typically improves generalization, and linking these modes through continuous low-loss valleys further enhances Bayesian model averaging (BMA). Whether such structure exists in the LoRA space and whether it yields functional diversity missed by local or discrete methods has not been studied. We introduce LoRA-Curve, a segmented Bézier curve parameterization in the LoRA space, with two variants: a free configuration that jointly optimizes all control points, and an anchored configuration that connects independently fine-tuned LoRA optima. We prove pathwise continuity and Lipschitz regularity of the loss along the curve and empirically show, across reasoning and classification benchmarks with Qwen2.5 7B, that linear interpolation encounters loss barriers, while our anchored multi-segment curves connect independent optima through continuous low-loss valleys. Combined with flat-minima perturbations and a Jensen-Shannon divergence regularizer, LoRA-Curve yields measurably higher mutual information of the predictive distribution without sacrificing performance, and links continuous parameter-space traversal to functional diversity.