Papers

CMinHash promises a new direction for designing MinHash algorithms with reduced variance and storage space via circulant permutations. However, the original variance analysis is difficult to extend to other MinHash estimators. We present a new approach that gives explicit and easily comparable approximate variance expressions, and use it to analyze the Minner estimator. Numerical experiments show that our variance expression matches the empirical mean square error (MSE) of estimates. Our work thereby demonstrates how to apply circulant permutation to other MinHash estimators and compute their approximate variance.

Post-training pruning enables efficient deployment, but conventional methods do not explicitly preserve internal-gradient alignment between sparse and dense models. We study novel connections between downstream performance of compressed models and the ability to retain gradient information via influence-function preservation. Central to this connection is a curvature-aware objective that accounts for both layer activations and backpropagated gradients. Despite its promise, this objective remains underexplored in pruning due to a lack of efficient solvers. We introduce $\texttt{CACTUS}$, a novel ADMM-based optimizer supporting both pruning and sparse-plus-low-rank decompositions of pre-trained weights. Compared to LLM-Surgeon which targets the same objective, $\texttt{CACTUS}$ can achieve over an order-of-magnitude speedup. At $50\%$ pruning of Qwen3-4B, $\texttt{CACTUS}$ reduces the C4 perplexity gap to the dense model by $45\%$, improves compressed-to-dense per-sample gradient cosine similarity on WikiText-2 from $0.670$ to $0.755$, and runs over $28\times$ faster than LLM-Surgeon on substantially weaker hardware. By preserving influence functions, $\texttt{CACTUS}$ supports effective data selection. On Llama-3.2-3B, fine-tuning a fresh dense model on the $5\%$ of data selected by $\texttt{CACTUS}$ achieves $49.44\%$ accuracy on BIG-Bench Hard, compared with $47.22\%$ using ALPS and $49.54\%$ using dense-model selection. Finally, integrating $\texttt{CACTUS}$ with LoGRA, a recent gradient-projection strategy, on Qwen3-4B with MLP layers pruned to 2:4 sparsity and a rank-64 correction retains a $0.711$ cosine alignment with dense LoGRA-projected C4 gradients.

Recently, Morris et al. introduced TinyLoRA, a strategy for LLM fine-tuning which introduces arbitrarily few parameters, and demonstrated accuracy improvements using less than $10$ parameters rather than the ${\gtrsim 10,000}$ typically demanded by LoRA-based methods. However, that study continued to use gradient-based local search methods of the kind deployed for training multi-billion parameter LLMs in the first place. This article is predicated on the observation that, in the low dimensional setting, global optimization methods become tractable, unlocking Large language Model Bayesian Optimization (LlMBO). LlMBO is sample-efficient, gradient-free, memory-light, parallelizes across devices without synchronization, and, unlike policy-gradient methods, can optimize greedy-decoding accuracy directly. We describe numerical experiments comparing LlMBO against GRPO on fine-tuning tasks for a variety of models on a math benchmark suite which show that LlMBO can be a suitable alternative in this regime, matching or exceeding GRPO's fine tuning gains, while using a fraction of the compute and memory.

Many related prediction problems, which we call questions, come with few labels each. What information can be shared across them, and what is actually necessary for good decisions? We answer both for recalibration, motivated by decision models such as Jev (TypeSafe AI, 2026) that return a probability for any yes/no question a developer writes. A predictor reports a probability $P$ with output law $\mu$ on $m$ questions, each with $n$ labels and a nondecreasing map $g_j(p)=\mathbb{P}(Y=1\mid P=p)$; the compound risk is the excess Brier score over the true maps, averaged over questions. Over maps within sup-distance $\tau\le 1$ and total variation $V\le 2$ of one nondecreasing shape (fixed effects), the minimax compound risk is $(mn)^{-2/3}+\min\{\tau^2,1/n\}+\min\{\tau^2,V^2,(V/n)^{2/3}\}$ up to universal constants, for every non-atomic $\mu$. A pooled isotonic fit with a per-question correction sized once, on held-out labels of all questions, attains it without knowing $\tau$, $V$ or $\mu$ ($n\ge 2$, $m\ge 3$); tuning each question on its own labels loses a factor of order $\min\{n,\log(m^2/n)\}$ when the maps differ by constants and $m^2/n$ is large. When the maps are drawn from an unknown prior (random effects), other questions need not identify the Bayes rule. The loss, the identification regret, is at most $2^{-n}$ once the law of a question's data is known, extending a bound of Skibinsky (1968) for coins, and in the worst case at least $\beta_\star^{-n}/(72\sqrt{n})$ at every finite $m$, where $\beta_\star\approx 3.583$. Beyond a few labels per question, what cannot be learned is nearly unnecessary. For constant maps (an uninformative predictor) and constant-map priors, the exponential base is exactly $\beta_\star$, not the 4 suggested by the moment range; for monotone maps it lies in $[2,\beta_\star]$.

Expected improvement based on stochastic kriging (SK) surrogate is widely used in continuous simulation optimization, but unknown heteroscedastic noise makes three basic design choices consequential: the incumbent to define improvement, replication allocation, and final recommendation. Existing methods typically either assume the noise levels are known or choose these components separately through heuristic rules or empirical calibration. We instead derive a high-probability upper bound on the simple regret and systematically choose the three components within general rule classes to tighten this bound. We first establish a finite-sample confidence bound for the SK predictor as a foundation for the three design choices. Within general posterior-quantile rule classes, tightening the regret certificate selects a lower-quantile incumbent and an upper-quantile recommendation among sampled designs. The opposing quantiles reflect the different roles of exploratory experiments and the final implemented decision. Under mild conditions satisfied by common kernels, minimizing a tractable kernel-specific bound yields a logarithmic replication order. Based on these design choices, the resulting simple regret matches the existing results under known homoscedastic noise up to logarithmic factors. To our knowledge, this is the first simple-regret guarantee of this form for a continuous-domain EI-type simulation optimization method that learns unknown heteroscedastic variances through replications.

Random features provide a standard approach to scaling kernel methods to large datasets by approximating the kernel with a finite-dimensional feature map. Existing works of random feature methods primarily focus on well-specified models and light-tailed noise conditions, leaving a gap in understanding their performance under more general conditions, such as misspecified models and heavy-tailed noise. In this paper, we provide a comprehensive and improved analysis of kernel ridge regression with random features (KRR-RF) under general conditions. In the light-tailed noise setting, we improve upon the worst-case uniform bounds of the feature map used in the previous works and lead to the minimax optimal learning rates with $2r+\gamma>\alpha_0$, where $(r,\gamma,\alpha_0)$ denote the regularity of the target function, the effective dimension of the kernel, and the embedding index of the RKHS, respectively. In the heavy-tailed noise setting, we leverage a recently developed Hilbert-valued Fuk-Nagaev inequality and extend the same optimal results under finite weak moment conditions. The additional cost to achieve robustness is a mildly stronger explicit sample-size requirement than in the light-tailed case. For both settings, our results recover the conditions random features budgets (up to log factors) required in the previous works. In the most benign case when $\alpha_0=\gamma$, KRR-RF are minimax optimal for all $r \in (0,1]$. To the best of our knowledge, this is the first work to establish optimal learning rates for KRR-RF under weak moment conditions in the misspecified setting. Numerical experiments and a real data analysis validate the derived learning rates and demonstrate their practical relevance in the presence of mis-specification and heavy-tailed noise.

This chapter provides a mathematical introduction to operator surrogates, which are finite-parametric maps approximating complex (typically nonlinear) operators between infinite-dimensional (function) spaces. A typical approach to constructing operator surrogates is the encoder-approximator-decoder paradigm, whereby infinite-dimensional inputs are mapped to a finite-dimensional latent space, processed, and then mapped back to the infinite-dimensional output space. We discuss various methods for creating efficient representation systems, including stable linear frameworks like Banach frames and data-driven techniques such as Principal Component Analysis (PCA), Active Subspaces, and nonlinear autoencoders. Subsequently, we survey key surrogate architectures, contrasting spectral methods, based on polynomial chaos expansions and related technologies, with more recent neural network approaches. We further discuss architectures that do not rely on representation systems, such as Fourier Neural Operators and transformers. We elaborate on their relationship and how they relate to other neural network operator surrogates, particularly those obtained by so-called "algorithm-unrolling". Additionally, we provide an introduction to the theoretical analysis of approximation and generalization errors. We explain how underlying low-dimensional structures, provided for example by holomorphy, are essential to overcome the so-called "curse of dimensionality": this refers to the fact that the complexity of numerical algorithms tends to increase exponentially in the problem dimensionality. We present convergence rate bounds for holomorphic operators that depend on the number of trainable parameters and the regularity of the input and output data. Throughout, we prioritize intuition over detailed technical proofs and provide precise pointers to the literature for detailed arguments.

Hierarchical time series forecasting requires generating accurate predictions that are coherent across multiple levels of aggregation. Existing methods either rely on two-stage pipelines with post-hoc reconciliation or on end-to-end models that often lack statistical guarantees. We introduce a unified framework that formulates hierarchical forecasting as a multivariate regression problem in a vector-valued reproducing kernel Hilbert space. By designing structured operator-valued kernels, our approach enforces coherence directly within the hypothesis space, ensuring that all predictions satisfy hierarchical constraints by construction. We further propose tailored kernel constructions for hierarchical data and derive excess risk bounds that characterize generalization performance. Empirical results on synthetic and real-world datasets demonstrate the relevance of the proposed method.

Randomly initialised neural networks of width $m$ converge to Gaussian processes as $m \to \infty$, with two types of quantitative bounds characterising this convergence. Finite-sample bounds compare the network with its Gaussian limit at a fixed finite sample set, on an arbitrary domain, in the quadratic Wasserstein distance $\mathcal W_2$, achieving a convergence rate of $m^{-1/2}$ with explicit constants. However, these constants grow with the number of samples, so the bound becomes uninformative as the sample set is refined. Functional bounds remove that dependence by comparing the two as random functions, at the price of spherical input, metrics weaker than $\mathcal W_2$, or slower rates. In this paper, we therefore lift a finite-sample bound to a functional one. The lifting refines the sample set until it is dense in the domain, so the bound must first be made independent of the number of samples. We achieve this by measuring the error per sample, which makes the constant uniform. For a polynomial activation and any compact input set, this gives a rate of $m^{-1/2}$ in the supremum norm, with an explicit constant.

We study learning an unknown element $g$ of a group acting with finite orbits on a countable set, from examples $(x,gx)$. After a sample on $A$, the consistent hypotheses are the restrictions of a coset of the pointwise stabilizer of $A$, and the sample forces exactly the labels of points this stabilizer fixes; for a Galois group, they form the field generated by the data. Through this correspondence, multiclass dimensions become group invariants: the DS and graph dimensions equal the height, the Littlestone dimension the longest irredundant sequence length, and the teaching dimension the minimal base size. Among deterministic learners, symmetry makes empirical risk minimization optimal: on nondegenerate classes every consistent learner, whatever its tie-breaking, attains the optimal realizable sample complexity up to constant factors, and every consistent online learner the optimal mistake bound. Agnostically, every ERM's sample complexity exceeds the known lower bound by at most the DS-to-Natarajan ratio, up to constants. For finite Galois extensions of countable fields, the Natarajan dimension equals a group invariant, the coset-cube number, which is at least the number of complemented factors of any chief series of any subgroup. Hence the dimensions coincide for every solvable Galois group, while for the groups $A_5$ and $S_5$ of the quintic they are $2<3$ and $3<4$, and $A_5$ is the smallest group with a gap. For the Galois group $S_n$ of the general equation of degree $n$, they coincide exactly when $n\le 4$, i.e., in characteristic zero, exactly when the equation is solvable by radicals; the implication is one-way, as the simple group $\mathrm{PSL}(2,8)$ has no gap. For linear, orthogonal and unitary maps of $\mathbb R^d$ or $\mathbb C^d$ both dimensions equal $d$, and a two-fibered pseudo-cube is a disjoint union of Natarajan cubes exactly when its monodromy group is abelian.

We study stochastic bandits without knowledge of reward scales or tail parameters. Almost-sure regret guarantees describe eventual learning but do not quantify expected performance or the risk of large regret in a finite run. We introduce $\texttt{EL-F}$, an anytime empirical-likelihood (EL) extension of the $g$-Forcing policy of Cowan and Katehakis (2020), and characterize its performance at three levels. First, for integrable rewards and non-decreasing, diverging exploration schedules $g$ with vanishing increments, $\texttt{EL-F}$ preserves the $g$-Forcing asymptotics for pseudo-regret $R(T)$: $R(T)/g(T)\to\sum_{i:\Delta_i>0}\Delta_i$ almost surely as $T\to\infty$, where $\Delta_i$ is the suboptimality gap of arm $i$. Second, on every sub-exponential instance, for non-decreasing $g(t)=\omega(\log t)$ satisfying a mild feasibility condition, expected pseudo-regret is at most $g(T)\sum_{i:\Delta_i>0}\Delta_i+C_{\underline{P}}$, where $C_{\underline{P}}$ is an instance-dependent constant independent of $T$. Moreover, we also establish finite-time tail bounds with super-polynomial decay in regret beyond the forced-exploration budget. For slowly varying $g$, we determine the exact exponent of linear regret deviations on two-armed Gaussian instances with common variance and establish a matching lower bound among policies satisfying the same expected-regret budget on every such Gaussian instance. These quantitative guarantees also cover the greedy variant obtained at zero EL radius. A fixed positive radius introduces local optimism while preserving the leading asymptotic regret. We establish coupled pathwise dominance over the greedy variant in two-armed bandits with a deterministic suboptimal arm. Experiments demonstrate reductions in prolonged suboptimal allocation compared with $g$-Forcing on challenging instances.

Nested dichotomies (NDs) reduce multiclass classification to a tree of binary classifiers. Tree structure can change accuracy, bias, and variance, but the full space of structures has not been mapped in aggregate. We introduce ndscape, a two-dimensional mapping of ND structures based on depth-weighted structural tokens. We use this embedding to conduct what is, to our knowledge, the first bias-variance analysis across the ND structure space. In simulations with 6 to 10 classes, NDs with similar tree structures exhibit similar squared bias, variance, and squared probability loss, regardless of whether the binary classifier is logistic regression or a decision tree. Variance, however, follows mean root-to-leaf depth when logistic regression is used at each node, so flatter, more balanced NDs are more stable, whereas with decision-tree node models, variance is not associated with depth and differs little across ND structures. On six datasets with 6 to 19 classes, held-out accuracy, log loss, and prediction variance remain spatially organised on the same topology-only map. None of the six standard construction heuristics tested selects a tree from the best-performing cluster on any dataset, although some select high-performing individual trees. The ndscape embedding turns ND selection from one opaque construction step into a readable comparison of class groupings, exposing the upper-level splits that distinguish high-performing and stable regions.

Tabular learning tasks often suffer from class imbalance, which is often addressed via weighting-based strategies. In this work, we compare the naive approach (unweighted) with the weighting-based rebalancing strategies. Somewhat surprisingly, we demonstrate that, asymptotically, as the predictor complexity increases with sample size, weighting strategies yield predictive performance identical to unweighted approaches across all evaluation metrics, up to a shift in the final threshold. To go beyond, we theoretically study the estimation of a linear regression function through weighted and unweighted CART trees with limited depths. In this setting, we prove that the points in the PR space for both strategies are distributed along the same curve, thus illustrating the similar nonasymptotic performance of the two strategies. Then, we numerically identify more complex synthetic settings where performance gaps, evaluated via fixed classifier complexity, favor weighting strategies. Thus, to guide practitioners, we introduce an indicator that aims at signaling whether a data set may benefit from weighting.

Certified machine unlearning ensures the proximity between the unlearned model and a model retrained without the data to be forgotten. Extending this framework to sequential deletion requests, typically arriving as groups of variable size (e.g., all records of one user or of coordinated users), is challenging. For unlearning methods based on fine-tuning on the retain sets, this leads to either degrading performance or increasing computation as the number of unlearning requests grows. For independent successive requests, we instead certify each model release against a counterfactual unlearning trajectory in which the group being deleted never took part. With a fixed fine-tuning budget per deletion, we show that, for smooth strongly convex objectives, this approach keeps the performance loss bounded, uniformly in the number of requests. Experiments with multiclass logistic regression on FEMNIST show that the resulting algorithm needs fewer corrective steps than existing sequential methods and outperforms training-time differential privacy, particularly when group sizes are heterogeneous.

We study parameter estimation when each sample from a continuous parametric distribution is accompanied by a binary quality label governed by the same unknown parameter. By imposing hard feasibility constraints, deterministic labels can accelerate estimation from the classical $n^{-1/2}$ rate of smooth quality-aware M-estimators to $n^{-1}$. Combining PAC guarantees with local error-to-parameter conversions, we establish this rate for both identifiable and finitely ambiguous quality classifiers, with likelihood information resolving the latter ambiguity. When the quality model is known only up to an additive constant, we construct threshold-free pairwise preferences from oppositely labeled samples and retain the accelerated rate despite the unknown threshold. We further show that pointwise qualities and pairwise preferences are incomparable: either can be strictly more informative than the other. Our assumptions hold for Gaussian, Laplace, and exponential families under thresholded log-likelihood rewards. Finally, a minimax lower bound establishes the optimal $n^{-1}$ dependence on sample size, up to logarithmic and dimension-dependent factors, while experiments confirm the benefits of constraint-based estimation.

We give the first adversarial linear bandit algorithm based on Follow-the-Perturbed-Leader (FTPL) that achieves $\sqrt{T}$ regret over general convex action sets. Our algorithm attains $\widetilde{\mathcal{O}}(d^{5/4}\sqrt{T})$ regret using only linear optimization oracles, and the near-optimal $\widetilde{\mathcal{O}}(d\sqrt{T})$ regret when an optimal-design exploration distribution is further available. This improves on the $T^{2/3}$ regret of the FTPL algorithm of Dani and Hayes (2006) for general convex sets, and generalizes the $\sqrt{T}$ regret of Lévy et al. (2026) over two special action sets (hypercube and Euclidean ball). Compared to previous oracle-efficient linear bandit algorithms based on continuous exponential weights, the FTPL approach generally has better computational efficiency, and is much simpler to implement. The key algorithmic element is a perturbation distribution that adapts to the past reward estimators. Its simplest form, an adaptive Gaussian perturbation, is straightforward to implement and already attains the regret bound above. We further extend it to two general constructions, random-weighting and elliptical perturbations, each covering distributions from light tails (Gaussian) to heavy tails (Student-$t_2$). On the technical side, we develop a systematic way to analyze the stability term in the FTPL regret through the Fisher information of the perturbation distribution. This builds a general framework for FTPL beyond existing ones such as the hazard rate and the FTRL-FTPL duality.

Least squares summarizes the joint distribution of a response and regressor through second-order information. When an observed regressor is composed of latent components carrying different relationships with the response, this summary can average over distinct linear effects. We study the additional linear information contained in higher-order mixed cumulants and formalize it through the latent slope spectrum. Ordinary least squares is a variance-weighted barycenter of this spectrum, whereas higher-order cumulants can separate its components when the spectrum is sufficiently parsimonious. In the scalar one-confounder model, fourth-order cumulants identify the unordered pair $\{\beta,\beta+\gamma\}$ through a $2\times2$ linear system and a quadratic equation. We develop estimation and root-$n$ inference under regular separation and show that detection and estimation diverge near degeneracy: slope multiplicity remains detectable on an $n^{-1/4}$ local scale even when the individual slopes are not uniformly regularly estimable. In multiple regression, a quadratic tensor recurrence yields a test of one versus multiple effective coefficient vectors that does not require the vectors themselves to be identified; multiplicity can therefore be consistently detectable even when the recurrence operator is noninjective. The same quadratic diagnostic can retain signal in directions where a first-order higher-cumulant restriction cancels exactly. The results distinguish representation, identification, detection, and estimation of latent linear heterogeneity, while making explicit the limits of structural and causal interpretation from observational data alone.

Gaussian process bandits exploit similarities among configurations to optimize prompts, retrieval systems, and agent workflows. In these applications, however, inexpensive model-based evaluations can disagree with trusted task outcomes. We study selective verification when an action is chosen before a diagnostic is observed and trusted feedback is informative on an unknown set of diagnostic states. A finite reliability class and a finite RKHS signal dictionary describe these two sources of uncertainty. We construct a fixed audit design that combines reliability coverage with reward separation and establish a finite-time regret bound that accounts for a hard audit budget and the rounds needed to obtain informative observations. A safe adaptive extension mixes this robust design with predictable Chernoff allocations while preserving a uniform identification floor and exact expected acquisition-cost accounting. For signed-coordinate kernels, structured and diffuse reliability classes require different audit scales despite sharing the same reward geometry. Synthetic experiments distinguish these certified designs from an adaptive posterior heuristic and GP controls. Safe adaptive allocation reduces aggregate regret relative to the fixed design under the exact dictionary, but its finite-budget error certificate is numerically loose. Conditional-feedback errors can remove the heuristic's advantage and target perturbations can reverse it. Separate AI-task replays assess feedback correction and fitted-dictionary policies without establishing end-to-end validation of a learned selective-verification method.

We propose a structured continuous-time stochastic volatility model for one-day-ahead integrated-variance forecasting. The model introduces a jointly evolving realized-measure state that feeds directly into latent variance dynamics, while correlated Brownian innovations capture return-linked dependence. Parameters are estimated through simulation-based predictive learning, and Monte Carlo simulation forecasts integrated variance. Across 17 equity indices, the proposed model shows strong out-of-sample performance relative to GARCH, GARCH-Itô, Realized GARCH, HAR, and HEAVY benchmarks under QLIKE, MSE, and MAE. Ablation results show that realized-measure feedback provides substantial incremental forecasting value within the proposed framework, while return-linked Brownian dependence contributes more modest complementary gains. The results support incorporating measurement-driven information directly into continuous-time latent volatility dynamics to improve integrated-variance forecasting.

Watermarking embeds an imperceptible signal in text from large language models that is statistically detectable with a secret key. The recent textGrain watermark (Li et. al. (2026b)) calibrates this signal against the model's sampling entropy through an explicit entropy budget, combining the stochasticity of Red-Green and the unbiasedness of Gumbel-max watermarks, and its accompanying report proposes a detector that sums pivotal statistics. We study when, and how efficiently, textGrain can be detected after human edits, modeled as a sparse mixture in which a fraction $\varepsilon_n\asymp n^{-p}$ of positions retain the watermark and token uncertainty decays as $n^{-q}$. First, entropy calibration comes at a cost: the minimax detection boundary of textGrain is $p+q=1/2$, so its detectable region is strictly smaller than that of Gumbel-max and maximal-coupling watermarks, whose boundary is $2p+q=1$. This cost stems from textGrain's construction rather than from entropy calibration itself: an unbiased scheme with the same entropy loss remains detectable where textGrain is not. Second, perhaps surprisingly, this loss makes the sum detector optimal at the boundary when the number of transport columns is fixed, unlike for Gumbel-max. When the number of columns grows and the entropy budget does not vanish, the detectable region expands while that of the sum detector does not. Third, for a fixed edit fraction, we characterize the optimal exponential rate of type II error decay and show that the sum detector falls strictly short of it at every fixed alternative. Finally, we propose a weighted multiscale binomial scan, valid in finite samples and needing no knowledge of the next-token distributions, entropy budget, or edit fraction; it attains every detection boundary above and the optimal exponent.

Inference-time alignment steers diffusion models toward downstream objectives without costly fine-tuning, but existing methods typically optimize expected cost (or reward) without explicitly controlling tail risk. In this work, we propose a risk-averse framework for inference-time alignment of diffusion models that balances expected cost and Conditional Value-at-Risk (CVaR) under trajectory-level Kullback-Leibler regularization. We characterize the optimal path distribution and reverse transitions as functions of the CVaR threshold, reducing the alignment problem to a one-dimensional optimization problem over the threshold. We show that this one-dimensional optimization problem can be nonconvex in general, establish sufficient conditions for convexity, and introduce quadratic regularization to enable efficient derivative-free ternary search. Under suitable assumptions, we provide finite-sample guarantees for threshold optimization and value estimation, and combine them into a terminal total-variation bound relative to the regularized target under exact resampling. Experiments on synthetic data and image generation using Stable Diffusion demonstrate improved tail performance over risk-neutral baselines while maintaining competitive average cost.

We propose a continuous learning formulation for $T$ non-forgetting tasks with the spherical parameter space $\mathbb{S}^{d-1}(\sqrt{d})$ in asymptotic regime $d\rightarrow \infty$. Each task $i$ has an optimum $\bm{\Theta}_i$, minimizing its corresponding loss function $R_i(\bm{w})$ on sphere. We reduce each task $i$ to an acceptable loss region modeled by a geodesic ball $B_{\bm{\Theta}_i}(r_i(d))$ on $\mathbb{S}^{d-1}(\sqrt{d})$ centered at $\bm{\Theta}_i$ with radius $r_i(d)$. To model task dependencies (similarities), task optima $\bm{\Theta}_i$ are distributed by a Neyman-Scott clustered point process $\Phi$. Cluster centers $\{\bm{Y}_j\}_{j=1}^{N_p}$ form a homogeneous Poisson point process on sphere, while intra-cluster task optima follow a von Mises-Fisher distribution with mean $\bm{Y}_j$ and concentration $\kappa$. Each task $t\in \{1, ..., T\}$ learning algorithm is considered to be a channel $Q_{t}(\mathrm{d}\bm{w}_t | \bm{w}_{t-1}, \mathcal{K}_{t-1}(\bm{\theta}_{1:t-1}), \bm{\theta}_t)$, where $\bm{W}_t$ is the learned parameter of task $t$ and $\mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})=\bigcap_{i=1}^{t-1}B_{\bm{\Theta}_i}(r_i(d))$. Since in continual learning, the learner does not need to reach $\bm{\Theta}_t$ exactly to successfully learn task $t$, to avoid forgetting as well as reliable learning, it only needs for the learned parameter vector $\bm{W}_{t}$ to enter $B_{\bm{\Theta}_t}(r_t(d)) \cap \mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})$. Using this model, we derive an upper bound on $T$ ensuring $\mathbb{P}[\mathcal{K}_{T}(\bm{\Theta}_{1:T}) \neq \emptyset] \rightarrow 1$ as $d\rightarrow \infty$ and $\kappa/d \rightarrow \alpha$, where $\alpha>0$ is a constant. Finally, for $T$ tasks sampled uniformly from $\Phi$, we derive an upper bound for total information theoretic capacity under the non-forgetting constraint for each task.

Continuous monitoring of generative AI systems requires detecting distribution shifts in the unstructured data they ingest and generate. Existing monitoring methods typically control false alarms by requiring the discrepancy measure used to detect shifts to be *pre-specified*. However, for unstructured data, pre-specification creates a sensitivity–breadth trade-off: a narrowly defined measure is sensitive to the anticipated shift but may miss unexpected changes, whereas a broadly defined measure may detect a wider range of shifts but with reduced sensitivity. Although adaptively hypothesizing the shift direction can overcome this limitation, such data-dependent adaptation generally invalidates conventional false-alarm guarantees. We reconcile adaptivity with statistical validity through testing-by-betting, reframing sequential changepoint detection as a betting game in which the bettor continually updates its hypothesized shift direction, or witness function, as new observations arrive. We introduce *asymptotic e-detectors*, which approximate the pre-change distribution using a calibration set of size $m$, and prove that they asymptotically control the average run length to a false alarm as $m$ increases. Empirically, calibration sets as small as $m=30$ provide effective control. We then develop adaptive monitoring strategies ranging from black-box bettors that learn from embeddings to interpretable bettors based on LLM-proposed concepts. Across three real-world datasets, including data from a clinical AI scribe deployment, adaptive monitors detect shifts faster than fixed monitors at the same false-alarm level. Further, when the true shifts are sparse and interpretable, LLM-concept monitors provide the fastest and most interpretable detection, while also guiding root-cause diagnosis for complex shifts.

We introduce and define measure neural mappings (MNMs), which combine measure-dependent neural transport with ensemble consistency for learning operators on probability measures. Ensemble consistency requires outputs from growing ensembles sampled from an input law to converge to the specified population output in probability. We establish quantitative continuum limits for attention on noncompact laws under strong sub-Gaussian and tail-integrability assumptions. These laws accommodate unbounded state spaces in scientific machine learning, including Bayesian inference and uncertainty quantification. For fixed self-attention transformers, the expected output Wasserstein error is bounded by the empirical input Wasserstein error plus an additional $N^{-1/2}$ attention term. These self-attention transformers are MNMs under the stated assumptions: pre-norm architectures follow from compact stability after normalization, while post-norm architectures are covered under the stated tail conditions. We prove corresponding cross-attention estimates and extend the results to multihead attention and grouped transformer evaluation, with convergence as the group size grows. For Set Transformers viewed as measure-to-vector mappings, we obtain an $N^{-1/2}$ bound on the expected Euclidean output error under the stated assumptions. Experiments on seven operators in dimensions $d=1,2,3,5,10$ support the predicted particle-count dependence, using one-dimensional Wasserstein computations and Sinkhorn approximations in higher dimensions. The observed Wasserstein and sliced Wasserstein decay rates are close to their respective theoretical sampling benchmarks.

We revisit the statistical mechanics of Transformer attention via modern Hopfield models. We present a free-energy formalism to characterize parameter regimes and critical thresholds for robust attention inference. The resulting free-energy landscape incorporates randomness in query states and attention weights. Attention then emerges as the equation of motion of the corresponding statistical-mechanical system (i.e., modern Hopfield model) through the standard free-energy minimization principle. This allows us to identify first-order phase transitions in attention between glassy and robust-inference phases. We further extend to attentions with stochastic queries and keys. Together, our results provide a rigorous statistical-mechanics framework for characterizing phase structure and inference reliability across a family of Transformer attention mechanisms.

Conditional conformal test martingales provide sequential tests for detecting distribution shifts, using a fixed reference dataset to avoid contamination from test-time samples. Prior work used a penalized betting function via online optimization with smoothing, clipping, and other hyperparameters, which can adapt slowly when a shift occurs after a long stationary period. In this work, we introduce an approach that requires no hyperparameters and provably adapts quickly to shifts occurring at unknown times. Our key idea is to frame the penalized betting problem as portfolio selection in a three-asset market. Then, we use a restarting universal portfolio, with a heavy-tailed wealth allocation over time. We establish anytime false-alarm control and derive optimal finite-sample bounds on detection delay under detectable alternatives. We also develop an FFT-based implementation that computes the portfolio sequence over $T$ rounds in $\mathcal{O}(T^2\ln T)$ time, instead of $\mathcal{O}(T^3)$ of the naive implementation. Synthetic and real-world experiments suggest that the proposed approach adapts rapidly after long periods of null samples, with detection delays depending substantially less on shift time than prior baselines.

Many approaches have recently been proposed to combine data assimilation with machine learning for improved forecasts with geophysical models. Here, we focus on combining advanced sequential data assimilation schemes with a generative forecasting model which accounts for complex model errors. This generative model provides a stochastic approximation of the conditional distribution of the successor state given the current state. The proposed variational formulation is a novel one-step weak-constraint 4D-Var scheme in which a learned conditional stochastic forecast replaces the prescribed additive Gaussian model error representation. We derive and implement this formulation as an ensemble variational algorithm, the generative iterative ensemble Kalman filter (IEnKF-G), which generalises the iterative ensemble Kalman filter with additive Gaussian model error (IEnKF-Q). We extend the method to a smoother with an assimilation window spanning multiple time steps and propose a simplified variant. The IEnKF-G is evaluated on a stochastic variant of the Lorenz 1963 model to assess its robustness. It is then tested on a multi-scale variant of the one-dimensional Lorenz 1996 model, in which the resolved variables are affected by model error originating from the hidden small-scale variables. The algorithm is then compared to an ensemble Kalman filter that adaptively mitigates model and sampling errors, and to the IEnKF-Q. The IEnKF-G is generally more accurate in the tested configurations, with substantial gains under partial observations and significant model error. This demonstrates the potential of generative data-driven models when paired with classical data assimilation methods, particularly when model error is substantial.

Stochastic optimization models sequential decision-making under uncertainty, typically assuming known input distributions. When these distributions are unknown, \citet{agarwal2026semibandit} give an online-to-offline reduction for monotone stochastic optimization under semi-bandit feedback, where only probed variables are observed. Its expected regret relative to the offline guarantee is $\widetilde O(n\sqrt T)$, where $n$ is the number of variables and $T$ the horizon. For optimal stopping and Pandora's box, \citet{liu2025improved} achieve $\widetilde O(\sqrt{nT})$ using problem-specific algorithms. Can a general online-to-offline reduction achieve this improved rate? We give such a reduction for strongly monotone stochastic optimization, answering the corresponding open question. Strong monotonicity is defined relative to the chosen offline algorithm. Our oracle-efficient reduction achieves $\widetilde O(\sqrt{nT})$ expected regret while preserving approximation guarantees for oracles satisfying the same condition. It combines Bernstein-type optimistic distribution estimates with a Hellinger stability bound for adaptive policy transcripts, aggregating squared marginal errors weighted by the executed policy's probing probabilities. The rate is optimal up to logarithmic factors for $T\ge n$, even under full feedback. Under monotonicity and uniformly bounded coordinate-wise variation, the same algorithm achieves $\widetilde O(n^{3/4}\sqrt T)$ expected approximation regret, improving the previous linear dependence on $n$. The exponent $3/4$ is sharp for its ordinary regret over instances and exact oracle choices. Our framework applies to prophet inequality, Pandora's box, best-of-$K$ selection, stochastic knapsack, stochastic orienteering, and stochastic submodular maximization.

Deep unrolling methods for inverse problems augment traditional reconstruction algorithms with learned priors to improve their performance. These methods have been developed under the implicit assumption that the geometry of the measurement operator does not change between training and deployment; when this is no longer the case, the quality of obtained reconstructions can degrade significantly. In this paper, we argue that this is not an inherent limitation of the deep unrolling paradigm and can be mitigated by a careful choice of the underlying iterative algorithm. We propose a deep unrolling framework for the phase retrieval problem that augments the alternating projections method with a learned denoising step. Our method obtains high-quality reconstructions without the need for retraining in settings where conventional model-based alternatives struggle, such as when the dynamic range of the measurement operator changes significantly at test time.

The success of autoregressive language models trained with Chain-of-Thought~(CoT) supervision raises a fundamental question: How simple can a next-token predictor be while still enabling the efficient simulation of general computation through its autoregressively generated CoT? Surprisingly, we show that a single neuron suffices. For a broad family of activations, including threshold and ReLU, a single autoregressive neuron can simulate any feedforward circuit with the same activation. Specifically, a size-$s$ circuit on $n$ inputs can be simulated by a neuron of dimension $O(ns+s^2)$ using $O(s^2)$ generation steps, with both bounds independent of circuit depth and tight under natural assumptions. For threshold activations, this improves upon the simulation of Joshi et al. (COLT 2025), whose bounds scale exponentially with circuit depth. Consequently, autoregressive threshold neurons can simulate polynomial-time Turing machines, while remaining efficiently learnable from CoT supervision. Our main expressivity result relies on a novel connection to Sidon sets from additive combinatorics. As an experimental illustration, we efficiently train a single autoregressive ReLU neuron to mimic a neural network teacher with nontrivial test accuracy on CIFAR-10.