We study nonparametric estimation and forecasting for GARCH-type processes and propose an iterated neural estimator that alternates between estimating the recursion map and updating the unobserved conditional variances. Under mild conditions, we establish finite-sample bounds for the time-averaged in-sample variance-path error and show that the estimator is minimax rate-optimal up to logarithmic factors. Under stationarity and mixing conditions, we establish the same rate for recursion-map estimation and derive finite-sample bounds for out-of-sample variance forecasts and expected excess Gaussian quasi-likelihood (QLIKE) loss. The convergence rates depend on the true lag orders and remain unchanged when the fitted orders are overspecified. Simulation studies illustrate the effectiveness of our estimators under nonlinear and time-varying volatility dynamics. Real-data applications suggest that our neural methods perform competitively and can incorporate option-price and high-frequency information.
Learn-then-Test (LTT) provides a general framework for certifying deployment policies against a prescribed risk target, but its standard formulation targets a fixed population risk estimated from calibration losses. In sequential deployment, losses may instead be dependent and nonstationary, so the risk of a fixed policy can evolve with the observed history. We introduce StreamLTT, which replaces this fixed population risk with average predictable risk, defined by conditioning each decision's loss on the information available before its outcome is revealed. Conditional centering turns empirical deviations from this target into a martingale sum, enabling both a test that adapts to predictable variance and a finite-sample Hoeffding alternative. Combined with family-wise error rate control, StreamLTT yields a simultaneously certified policy region that supports subsequent deployment selection. Across seven forecasting datasets and seven deployment risks, StreamLTT places the certified frontier substantially closer to the prescribed risk target than IID, robust-variance, block-bootstrap, and concentration-based alternatives, while empirical family-wise error closely follows the prescribed level. A finite-sample ablation further shows that the predictable-risk formulation drives the tighter risk frontier, while adaptation to predictable variance makes more effective use of the available family-wise error budget. StreamLTT therefore extends risk-certified policy selection to dependent deployment streams by addressing both what risk should be certified and how it should be tested.
Compositional priors describe the generic properties of layered functions in deep Bayesian models, where deep neural networks with random weights are a canonical example. In the wide-network limit, the prior is a Gaussian process (GP) with a depth-dependent kernel, and its behaviour as depth grows has been extensively studied through this kernel. Here, we study another case, where each layer itself is a vector valued Gaussian process, and our aim is similarly to understand the limiting behaviour of the prior as depth grows. Previous GP work has established that for the radial basis function (RBF) kernel and a certain range of bandwidths $r$, the prior degenerates in the limit, converging to the set of constant functions, which is not useful as a probabilistic model. In this paper we establish several new results. First, we identify a sharp bandwidth threshold $r_c(d) = \Theta(\sqrt{d})$ above which the limit is degenerate, strengthening the earlier bounds. Second, and more importantly, we show that for $r$ below the threshold $r_c(d)$ the prior converges to a limit distribution $\pi_{\bar{Z}}$. We also prove that these distributions are non-degenerate and non-Gaussian, with non-vanishing dependence between coordinates. In contrast to the previously known degenerate regime, deep Gaussian process priors can therefore admit non-trivial limits. Empirically, we verify the threshold across a range of dimensions $d$, and demonstrate a complex multimodal behaviour of the limit distributions $\pi_{\bar{Z}}$, a regime that becomes increasingly narrow with $d$ and would be hard to identify without knowing the threshold.
We study the long-context limit of softmax self-attention with a fixed query and a random context of $n$ i.i.d. keys on the sphere with the intrinsic dimension of the key distribution held fixed, viewing the inverse temperature $\beta_n$ as the scaling parameter that decides whether attention degenerates into uniform averaging or collapses onto the single closest key. We show that the critical scale at which selectivity emerges is determined by the local exponent of the distance-to-query distribution near zero rather than by global features of the context, and scales like $\beta_n^\ast \asymp n^{2/(d-1)}$ for uniform keys on $\mathbb{S}^{d-1}$. Furthermore, we characterize the limiting laws of the ordered attention weights and of the attention output across all regimes of $\beta_n$: a subcritical regime in which the output reduces to a local average around $q$ with explicit deterministic bias and Gaussian fluctuations; a critical regime in which a finite collection of nearest keys retains macroscopic mass without single-key collapse; and a supercritical regime in which all mass concentrates on the closest key. Of notable interest is the subcritical case with identity value matrix where the attention map approximately implements a backward heat equation.
Standard dropout applies a fixed, uniform noise rate to all units, ignoring available gradient information. We propose momentum adaptive gradient dropout (MAGDrop), a Bayesian linear layer that adapts per-unit weight noise by tracking an exponential moving average of gradient magnitudes. Units with large recent gradient magnitudes receive less noise to preserve learned features; units with below-average gradient magnitudes receive more noise to promote exploration. This adaptation transforms the classifier head into a data-dependent Bayesian predictor whose mean predictor has no additional inference cost. Our primary contribution is a certificate-aware low-rank posterior, MAGDrop-Subspace, that addresses the tension between strong accuracy and tight bounds from probably approximately correct (PAC)-Bayes theory. A full-head stochastic posterior incurs a Kullback--Leibler (KL) divergence of approximately 8,625 nats. By confining the posterior to an $r$-dimensional subspace constructed from an independent prior-construction sample $S_0$ and certifying on an independent sample $S$ with $S_0\perp S$, MAGDrop-Subspace reduces KL by approximately a factor of 800 while keeping accuracy close to the baseline. On CIFAR-10 with a 13-layer convolutional neural network and $r=50$, MAGDrop-Subspace achieves a PAC-Bayes Gibbs-risk bound of 0.1882 at 90\% confidence using binary KL inversion, compared with 0.4439 for the full-head baseline. Mean-predictor accuracy is 82.31\%, compared with 83.74\% for the baseline. Complementary accuracy experiments on ImageNet-1K show that MAGDrop reaches top-1 accuracy similar to the strong baseline: 78.57\% compared with 78.51\% on the base Vision Transformer with $16\times16$ patches, and 82.14\% compared with 82.23\% on ConvNeXt-Base.
Conformal Prediction (CP) is a popular framework for constructing prediction bands with valid coverage in finite samples, while being free of any distributional assumption. A well-known limitation of conformal prediction is the lack of adaptivity, although several works introduced practically efficient alternate procedures. In this work, we build upon recent ideas that rely on recasting the CP problem as a statistical learning problem, directly targeting coverage and adaptivity. This statistical learning problem is based on reproducible kernel Hilbert spaces (RKHS) and kernel sum-of-squares (SoS) methods. First, we extend previous results with a general representer theorem and exhibit the dual formulation of the learning problem. Crucially, such dual formulation can be solved efficiently by accelerated gradient methods with several hundreds or thousands of samples, unlike previous strategies based on off-the-shelf semidefinite programming algorithms. Second, we introduce a new hyperparameter tuning strategy tailored specifically to target adaptivity through bounds on test-conditional coverage. This strategy, based on the Hilbert-Schmidt Independence Criterion (HSIC), is introduced here to tune kernel lengthscales in our framework, but has broader applicability since it could be used in any CP algorithm where the score function is learned. Finally, extensive experiments are conducted to show how our method compares to related work. All figures can be reproduced with the accompanying code at \href{https://www.gitlab.com/drti/ksos-bands}{gitlab.com/drti/ksos-bands}.
We study the gradient query complexity of sampling from a density $\pi(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ has a $\beta$-Lipschitz gradient but need not be convex. Our approach uses the proximal sampler, which alternates Gaussian perturbation with sampling a position conditional on the noisy observation. We implement its observation chain using the high-order noisy conditional sampler of [OpenAI 2026], obtaining improved dimension dependence under functional inequalities. If $\pi$ satisfies a log-Sobolev inequality (LSI) with constant $\alpha$, then for every fixed $\nu>0$, there is a finite constant $C_\nu$ depending only on $\nu$ such that the algorithm returns a sample within total-variation distance $\varepsilon$ using at most $C_\nu\kappa(d+\kappa+\varepsilon^{-1}+\|\nabla V(x_0)\|^2/\beta)^\nu$ queries, where $\kappa=\beta/\alpha$ and $x_0$ is an arbitrary reference point for initialization. The analysis controls center-dependent local errors along exact proximal histories and removes the final observation noise by conditional decoding. Under a Poincar\'e inequality (PI), we obtain a corresponding guarantee from a warm start with a known bound on its R\'enyi divergence. An oracle lower bound shows that a stationary point alone, which suffices under LSI, is insufficient under PI.
Calibrated multiaccuracy asks that a predictor be globally calibrated and multiaccurate with respect to a bounded test family $G$; that is, its residuals should be uncorrelated with every test $g(x)\in G$. This weakening of multicalibration suffices for many applications. Its sample complexity was not known to improve on that of full multicalibration, which is $\widetilde\Theta\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{3}}\right)$ when $|G|$ is at least a sufficiently large power of $\log(1/\varepsilon)$. We resolve the sample complexity of $\varepsilon$-calibrated multiaccuracy, showing that it scales as $$ \widetilde\Theta\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{5/2}}\right), $$ separating it from the complexity of multicalibration. The upper bound is obtained by an algorithm that outputs a randomized predictor. For deterministic predictors, we establish the tight rate $$ \widetilde\Theta\left( \frac{\log|G|}{\varepsilon^2} + \frac{(\log|G|)^{1/3}}{\varepsilon^{8/3}} \right) $$ whenever $\log|G|=\Omega(\log(1/\varepsilon))$. When $\log|G|\le\varepsilon^{-1+\kappa}$ for any fixed $\kappa\in(0,1)$, this rate is polynomially larger than the rate for randomized predictors. Thus, unlike for multicalibration, calibrated multiaccuracy exhibits a genuine sample-complexity separation between deterministic and randomized predictors. The matching upper and lower bounds show that the unusual cube-root dependence on $\log|G|$ is intrinsic. At $\log|G|\asymp\varepsilon^{-1}$, the minimax rates for randomized and deterministic calibrated multiaccuracy both coincide with the minimax rate for multicalibration.
This paper presents a near-quadratic-time algorithm that correctly matches almost all nodes of two correlated Erd\H{o}s-R\'enyi graphs without any incorrect matches. Each graph has average degree $\lambda$ and its edges are subsampled from a common parent Erd\H{o}s-R\'enyi graph with $n$ nodes, with subsampling probability $s$. The nodes of one graph are then relabeled by an unknown permutation. The algorithm is built on a test for correlated versus independent Galton-Watson trees with approximated likelihood ratios. For every constant $s>\sqrt{C_{\mathrm{Otter}}}$, where $C_{\mathrm{Otter}}\approx0.338$ is Otter's tree-counting constant, and every $\lambda=\omega(1)$ with $\lambda=(\log n)^{O(1)}$, the algorithm outputs a partial matching of size $n-o(n)$ that contains no incorrect matches with high probability. If, in addition, $\lambda s\ge(1+\epsilon)\log n$ for a constant $\epsilon>0$, a seeded completion step recovers the permutation exactly. In both cases, the time complexity of the algorithm is $n^{2+o(1)}$ with high probability, whereas previous polynomial-time algorithms with the same guarantees either have a time-complexity bound whose exponent goes to $\infty$ as $s$ approaches $\sqrt{C_{\mathrm{Otter}}}$ from above, or apply only when $s$ is close to $1$.
We consider classification in high-dimensional feature spaces where only a vanishing fraction of features are useful and the number of classes may be large. While this rare-and-weak useful-feature regime is well understood in the binary setting, its multiclass version raises new questions: a feature can be useful for class discrimination without separating any specific pair of classes. We focus on a rare feature diversity model, in which most features are useless, but a small fraction promote discrimination among classes in a way that can be detected by a per-feature diversity test, such as one-way ANOVA or the Kruskal-Wallis test. For this setting, we propose Diversity Pursuit Higher Criticism Thresholding (DP-HCT), a feature-screening procedure that applies a higher-criticism threshold to the collection of p-values obtained from per-feature diversity tests. We justify DP-HCT under a Gaussian multiclass Rare-and-Weak Useful Feature model with class-mean contrasts of fixed magnitude in random directions, extending the binary model of Donoho and Jin (2008). When the number of classes is fixed, our model recovers the binary phase transition. When the number of classes grows logarithmically with the number of features, however, we derive a new and strictly harder phase transition, driven by the growing degrees of freedom and their effect on the useful-feature tail rate. We show that whenever classification beyond chance is possible, DP-HCT with ANOVA, followed by a simple classifier based on similarity to class means, achieves asymptotically perfect accuracy. Experiments on real multiclass classification benchmarks demonstrate substantial improvements, and numerical simulations confirm that the derived phase-transition curve accurately predicts the finite-sample behavior of DP-HCT in the growing-class regime.
When testing a composite null $\mathcal{P}$ against a composite alternative $\mathcal{Q}$, it has been known since Huber and Strassen (1973) that if a least favorable distribution (LFD) pair exists, then a version of its likelihood ratio yields a minimax optimal test, and Huber (1965) derived a similar result in the sequential setting for a specific robust testing problem that is the focus of the current paper; the same likelihood ratio is log-optimal in the worst case; its expected logarithm under the alternative equals $\inf_{P \in \mathcal{P},S \in \mathcal{Q}} KL(S\|P)$, whose inverse also determines the stopping time of the test under the alternative. These results focus on optimizing worst-case performance, but the resulting methods have lower power if the data come from simpler alternatives. This paper studies an arguably more practical goal of achieving asymptotic \emph{instance optimality} for ``power-one'' sequential tests, which remarkably may even be possible when $\mathcal{Q} = \mathcal{P}^c$. This means that when the data are drawn from any $S \in \mathcal{P}^c$, the test has power one, and its stopping time is determined by the larger information number $\inf_{P \in \mathcal{P}} KL(S\|P)$, which governs the performance of an oracle that knows $S$. To demonstrate this point, our paper focuses on the case of testing a TV ball around a discrete distribution with $d$ components, and designs a computationally and memory efficient algorithm that is also asymptotically instance-optimal. Specifically, we consider two different asymptotic settings: one is the "Wald setting", where the type-1 error level $\alpha$ approaches zero for a fixed alternative $S$ and the second is the "Farrell setting", where $\alpha$ is fixed and $\inf_{P \in \mathcal{P}} KL(S\|P)$ approaches $0$. Our sequential test is simultaneously optimal in both regimes.
Spectral shrinkage encompasses a broad family of estimators for high-dimensional linear regression that regularize estimation and prediction by applying spectral filters to the sample covariance matrix. Despite its widespread use, achieving oracle-optimal prediction risk under general covariance structures remains challenging, as the oracle filter requires unknown population quantities and has a closed form only for specific covariance models such as isotropic and spiked models. We propose \textit{risk-targeted Krylov fitting} (RTK), a data-driven spectral estimator that adapts to the unknown covariance structure and attains the oracle risk asymptotically. Like partial least squares, RTK operates in an $m$-dimensional Krylov space, but it targets the oracle risk directly by minimizing the leave-one-out risk, which we compute exactly without refitting the model $n$ times. Under general distributions and covariance structures, RTK attains the optimal prediction risk in the Krylov space up to $O(1/n)$. Furthermore, its risk approaches that of the optimal analytic filter with $m$ of order $\log n$. In simulations over a wide range of covariance structures, RTK and a truncated variant were robust to the choice of $m$ and consistently among the spectral estimators closest to the oracle. In applications to single-cell sequencing data and large-scale neural recordings, the proposed methods achieved higher prediction accuracy than competing estimators on most tasks.
We study finite-horizon reinforcement learning with a safety constraint on the probability of reaching a failure set. In contrast to the well-studied safe reinforcement learning literature, we consider problems where the probability of reaching the failure set is vanishingly small, but the consequences of reaching the failure set are catastrophic. Direct simulation and estimation of the optimal policy is naturally ineffective when the failure probability is extremely small. Moreover, on a real system the controller can only choose actions, and this is not enough to make failures frequent. In contrast to the rare event simulation literature, our goal is not to estimate the probability of the rare event occurrence, but to upper bound it such that the failure constraint is satisfied. Thus, instead of sampling failures, we use the large deviations principle. We define a damage variable that on exceeding a certain threshold indicates failure. We study the cumulant generating function (CGF) as its positive root provides an upper bound on the failure probability via Cramer's theorem. We extend this approach to Markov decision processes with hitting times through state-action CGFs, and give finite-horizon and state-dependent versions of the bound. The state-action CGFs do not depend on the policy and can be estimated from regular transitions, so the bound can be checked for any policy a priori. This gives a policy improvement algorithm in which every policy satisfies the failure constraint with high probability.
Tensor factor models provide low-dimensional representations of tensor-valued observations while preserving their multiway structure. When these observations are held by different clients, privacy constraints may prevent them from being pooled for centralized analysis. We study estimation of common loading spaces in Tucker tensor factor models under federated differential privacy, allowing heterogeneous sample sizes, privacy budgets, and factor strengths. The proposed method combines private stability testing, Huber aggregation, and Gaussian perturbation at each client with weighted aggregation at the server. We establish non-asymptotic upper bounds governed by a scaled harmonic mean of the squared local error rates, and prove minimax optimality up to logarithmic factors under strong factors, fixed factor ranks, and fixed noise variance. We further develop a private calibration procedure that selects the local tuning parameters and aggregation weights without knowledge of the noise level or client-specific factor strengths. The resulting adaptive estimator preserves the privacy guarantee and attains the error rate obtained with oracle tuning. We illustrate the finite-sample performance of the proposed estimators through simulation experiments and an application to a Meditation Traits EEG dataset.
Conditional sampling is central to Bayesian inference, generative modeling, and causal inference: given a joint law for random variables $X$ and $Y$ and an observation $Y=y$, one seeks to sample from the conditional law of $X\mid Y=y$. Even when the joint distribution is accurately represented, this task can be computationally difficult as it may require evaluating intractable normalizing constants or solving a new inference problem for each observation. Measure transport provides an approach to this problem by identifying a map with triangular structure that directly pushes forward a reference to the conditional of a joint distribution for any observation. This article reviews the theoretical, methodological, and algorithmic foundations of triangular transports, with the Knothe-Rosenblatt rearrangement as a canonical construction. We discuss methods for estimating triangular maps from finite samples, as well as their use in simulation-based inference and causal inference. While our primary aim is to review existing formulations, we also present new ideas and open questions for conditional sampling using transport.
Bayesian inverse problems require both informative prior models and computationally feasible posterior inference. Score-based diffusion models provide expressive learned priors, but their use for inversion introduces a conditioning problem: the score of a noised posterior is generally unavailable from a prior score alone. This selective survey reviews the principal approaches to that problem, including likelihood guidance, spectral and projection methods, sequential Monte Carlo corrections, and amortized conditional training. We organize methods by how they split cost between offline preparation and online inference, and compare their assumptions, advantages and limitations, with emphasis on computational speedups. Some ongoing directions and open questions are gathered at the end.
Operators choosing among decision rules, such as when a battery charges or where an idle taxi goes, rank them by estimated long-run value. A standard estimator, linear temporal-difference (TD) learning, fits values from one stream of correlated observations. Large fits often use an adaptive optimizer, such as adaptive moment estimation (Adam), with fixed averaging rates. We show that these rates can change what is estimated, not just how fast: the same observations drive the updates and the optimizer's step-size scaling, moving the fit away from the intended TD value. The shift differs across rules, so it need not cancel in a comparison, and more data only makes a wrong ranking look more certain. Our remedy refits the same data at a second averaging rate that halves the effective gain, the number to which the leading shift is proportional. Twice the new fit minus the original removes that shift. In New York taxi and battery models calibrated to real data, common optimizer settings leave the selected rule unchanged but order close rules wrongly and misstate its value. Our remedy corrects nearly every wrong ordering, with one extra fit when the shift is moderate and two when it is larger.
In this paper, we establish sharp lower bounds for Gaussian approximation of Polyak-Ruppert averaged stochastic gradient descent with polynomially decreasing step sizes. For every fixed exponent $\gamma\in(1/2,1)$, a single one-dimensional smooth, strongly convex example has approximation error with the limiting variance of order $n^{1/2-\gamma}+n^{\gamma-1}$ in Kolmogorov and convex distances. This matches the existing upper bound throughout the range of $\gamma$. The nonlinear bias contributes $n^{1/2-\gamma}$ even when the Gaussian reference uses the finite linearized variance. We also derive a lower bound of order $n^{-(1-\gamma)/2}$ for the relative mean error of estimating the limiting variance from one SGD trajectory, uniformly over a scalar quadratic family. With the limiting variance known, achieving approximation error $\varepsilon$ on this example takes order $\varepsilon^{-4}$ iterations for the best choice $\gamma=3/4$. Achieving both Gaussian approximation and relative mean variance-estimation errors at most $\varepsilon$ uniformly over the model class requires at least order $\varepsilon^{-6}$ iterations for every fixed $\gamma$.
Mitigating unfair disparities in sequential decision-making is a growing concern in machine learning. We study contextual bandits under a global covariate-based Lipschitz fairness constraint, pioneered by \cite{dwork2012fairness}, requiring individuals with similar contexts to receive similar action distributions. We show that this constraint induces a randomisation region near the decision boundary and explicitly quantify the population level cost of fairness. We further establish a canonical representation of the optimal fair policy, which reduces the fair policy design to a lower-dimensional boundary optimisation problem. We develop a lazy fair greedy algorithm that exploits fairness-induced covariate diversity under continuous contexts, with policy updates computed via finite-dimensional linear programming. For general context distributions, including discrete and mixed contexts, we propose a lazy polyhedral upper confidence bound (UCB) algorithm, whose optimistic optimisation reduces to finitely many linear programs over confidence-set vertices. Both methods satisfy fairness at every round and achieve regret of order $O(\sqrt{T} + T^{(d-2)/(d-1)}{\indc\{d\geq 4\}})$ up to poly-logarithmic factors, where $d$ is the dimension of the contexts. A matching minimax lower bound for $\ell_1$-based fairness establishes optimality, and reveals an unavoidable nonparametric cost of learning the decision boundary, which dominates the parametric learning cost when $d \geq 4$. Experiments on synthetic and real data support our theoretical findings and demonstrate the practical effectiveness of our methods.
In this paper, we develop a projected instrumental variable framework that relaxes conventional assumptions for causal identification with instrumental variables. Building on the interpretation of conditional expectation as an orthogonal projection in a Hilbert space, the framework replaces the conditional expectation operator with a general projection operator. We provide an alternative inverse problem formulation based on the Riesz representation. Drawing on semiparametric efficiency theory, we further develop debiased and adaptive machine learning methods for estimation and inference. The proposed framework connects invalid instrumental variables to instrumented difference-in-differences. We also examine invalid projected instrumental variables and discuss their relationship to existing methods. Simulation studies and real data analysis demonstrate that the proposed methods perform favorably compared with existing approaches.