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.

Conformal prediction methods enjoy strong theoretical and empirical predictive inference performance, provided the data is exchangeable, and predictors are trained in a memoryless fashion. However, these assumptions and constraints are impractical in many real-data settings, such as time series (where temporal dependence violates exchangeability, and where memoryless predictors will inevitably have poor predictive accuracy). Recent work shows that the split conformal prediction method is robust to these issues of memory-based predictors and deviations from exchangeability that are common features of time-series data. However, since using sample splitting can lead to lower accuracy, this motivates asking whether other predictive inference methods (that do not rely on data splitting) could also be reliably used in the time series setting. In this work, we show that the vanilla leave-one-out jackknife can suffer an arbitrary loss of coverage even in canonical time series models with mild temporal dependence. As a remedy, we propose a careful modification tailored to such settings, which we term the \emph{leave-a-window-out} (LWO) method, and show that it can achieve valid coverage provided that the model-fitting procedure satisfies mild stability properties. Our proofs are based on quantifying the degree to which the data departs from \emph{cyclic exchangeability}, and we introduce new coefficients to measure the extent of this departure. Experiments on time series data demonstrate that our LWO method often enjoys valid coverage when the vanilla jackknife fails to cover, while producing much narrower intervals than split conformal prediction.

Causal-discovery algorithms return a directed graph, yet provide no principled means of distinguishing edge directions identified by the data from those assigned without an identifying assumption. Under the standard Markov and faithfulness conditions, the observational distribution identifies only a Markov equivalence class; orientations within that class are not determined by the joint distribution and cannot be recovered from additional samples alone, but require either a functional restriction or an intervention. We introduce a protocol for observational causal discovery on continuous data that attaches to each candidate edge a discrete impossibility certificate: a RESOLVED code records the identifiability theorem under which the direction was committed, while an IMPOSSIBLE code records the failure mode together with the specific question a domain expert must answer to resolve it. The bivariate cascade is extended with five gated identifiability tiers LSNM, IGCI, Stein, MDL, and PEIT that abstain when their precondition test rejects. Two oracle primitives, the meta-hub query and the node-children query, jointly establish an upper bound of $1+K$ expert interactions sufficient to recover any DAG, where $K$ denotes the number of non-leaf vertices. Under an ideal-oracle assumption, the bound is met exactly on the asia, sachs, child, and alarm benchmarks.

Fairness-accuracy trade-offs are a central concern in the deployment of fairness-aware machine learning methods. When sensitive attributes are unavailable at inference time-the so called unawareness setting, principled methods for obtaining accurate predictions under relaxed fairness constraints are largely missing. In this work, we address this gap by formulating regression under a demographic parity penalty as an optimal transport problem. Our framework unifies both the \emph{aware} and \emph{unaware} settings and characterizes optimal prediction functions via optimal transport maps, under both squared Wasserstein-2 and Total Variation penalties. These results reveal that the choice of penalty reflects fundamentally different fairness philosophies: the Wasserstein penalty induces a smooth, population-wide compromise, while Total Variation enforces exact parity for a subset of individuals. Building on these theoretical characterizations, we propose an algorithm that is simple to implement, computationally efficient, and consistently matches or outperforms state-of-the-art baselines on real-world benchmarks.

Prior-data fitted networks (PFNs) have recently emerged as a powerful approach for Bayesian prediction tasks, approximating the posterior predictive distribution (PPD) through in-context learning. Despite their strong empirical performance and ability to go beyond point predictions, theoretical understandings of the algorithmic capability of transformers to learn distributions in context are still lacking. Focusing on Gaussian process regression problems, we show by construction that transformers can implement a gradient descent algorithm targeting the posterior predictive mean and variance, followed by nonlinear mappings that yield binned probabilities of PPD. We study the error bounds of the approximated PPD in terms of attention depth and bin resolution. Based on these results, we further demonstrate the key role of normalization and the choice of attention depth in enabling the extrapolation abilities of transformers beyond the pretraining sample size range. We conduct simulations that corroborate our findings, providing insight into the expressivity of PFNs targeting PPDs and how architectural choices may influence generalization capabilities.

What functions can Neural Processes represent? We analyze the representational capacity of popular NP architectures: Conditional Neural Processes (CNPs), Attentive Neural Processes (ANPs), Transformer Neural Processes (TNPs), and their latent variants. We prove these architectures form a strict hierarchy. CNP-representable functions are exactly those depending on finitely many expected features of the context distribution. ANPs strictly generalize CNPs via query-dependent reweighting, enabling kernel smoothers. ConvCNPs and ANPs are incomparable; each contains functions outside the other, separated by stationarity versus translation equivariance. TNPs with $L$ self-attention layers capture $L$-hop context interactions. For latent NPs, we show finite-dimensional latents provide coherent sampling but do not circumvent encoder limitations; matching GP posterior distributions requires latent dimension scaling with context size. These results provide a theoretical foundation for architecture selection based on task structure.

Reward modeling is not only a prediction problem: in KL-regularized policy optimization, the learned reward is exponentiated to define the deployed policy, so downstream value depends on errors in reward-tilted regions. We study this feedback in a Gaussian single-index model with $r^*(x) = σ^*(\langle θ^*, x\rangle)$ and $x \sim N(0, I_d)$. We analyze a two-stage neural reward model that first learns the hidden direction $θ^*$ from reward-weighted samples and then fits the readout layer by weighted ridge regression. Exponential reward weighting changes the Hermite signal available to the first layer; for any feature-learning temperature $β_1$ above a dimension-free $O(1)$ threshold, a constant fraction of neurons recover the hidden direction, with weak-recovery complexity governed by the generative exponent. After feature recovery, we derive tilted-policy value-gap bounds for an idealized label-weighted fit with weights $e^{y/β_2}$ and a more practical surrogate-weighted fit with weights $e^{r_{a_0}(x)/β_2}$. Keeping the $β_2$-dependence explicit yields an admissible set of deployment temperatures, balancing the gain from lowering $β_2$ against the learning cost amplified by exponential weighting; in the surrogate-weighted case, proxy-dependent factors shrink this admissible set.

We study a stochastic multi-armed bandit problem where an agent is granted a free exploration budget before regret accumulates, a setting not captured by the classic regret minimization or pure exploration paradigms. The goal is to design an adaptive policy that strategically explores the bandit instance in the initial free exploration phase and minimizes the cumulative regret in the subsequent phase. We formalize this regret minimization with free exploration problem and identify an interesting regime where the free exploration budget scales logarithmically with the time horizon. To quantify the amount of regret saved with high probability as a result of the availability of the free exploration phase, we introduce a novel set of policies known as $(α,β)$-probably saving policies. We propose a two-phase, probably saving algorithm, UFE-KLUCB-H, which consists of a principled free exploration policy, UFE, and a history-aware regret minimization policy KLUCB-H. Instance-dependent upper bounds on UFE-KLUCB-H are derived, showing that UFE-KLUCB-H accumulates strictly less regret than policies that do not have access to a free exploration phase. Complementarily, we derive instance-dependent lower bounds based on novel multi-instance perturbation arguments tailored to the free-exploration setting, demonstrating the near-optimality of UFE-KLUCB-H for two-valued bandits. Our upper and lower bounds reveal sharp phase transitions in the accumulated regret depending on the amount of available free exploration. Simulations are conducted to demonstrate that forced exploration and adaptivity in the algorithm lead to greater regret savings.

Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version.

We introduce a model for neural scaling laws under sparse activations. In the model, test loss is often dominated by rare coordinates that are never observed in the training input. This mechanism induces a novel bottleneck absent from dense models. We derive the asymptotic population loss in both the underparameterized and overparameterized regimes, and show that the loss exhibits a double-descent peak near the interpolation threshold -- where the number of parameters is just sufficient to fit the training data -- resulting in a loss curve governed by two distinct scaling exponents -- one for the overparameterized regime and one for the underparameterized regime -- with a gap determined by the degree of sparsity. Additionally, we derive a compute-optimal frontier that favors increasing dataset size over model capacity under fixed compute budgets. We also analyze gradient-descent dynamics and identify a scaling law for the probability that fixed-step gradient descent becomes unstable. We further show that the sparsity-induced effect persists under nonlinear activations.