We study in-context learning problems where a Transformer is pretrained on tasks drawn from a mixture distribution $\pi=\sum_{\alpha\in\mathcal{A}} \lambda_{\alpha} \pi_{\alpha}$, called the pretraining prior, in which each mixture component $\pi_{\alpha}$ is a distribution on tasks of a specific difficulty level indexed by $\alpha$. Our goal is to understand the performance of the pretrained Transformer when evaluated on a different test distribution $\mu$, consisting of tasks of fixed difficulty $\beta\in\mathcal{A}$, and with potential distribution shift relative to $\pi_\beta$, subject to the chi-squared divergence $\chi^2(\mu,\pi_{\beta})$ being at most $\kappa$. In particular, we consider nonparametric regression problems with random smoothness, and multi-index models with both random smoothness and random effective dimension. We prove that a large Transformer pretrained on sufficient data achieves the optimal rate of convergence corresponding to the difficulty level $\beta$, uniformly over test distributions $\mu$ in the chi-squared divergence ball. Thus, the pretrained Transformer is able to achieve faster rates of convergence on easier tasks and is robust to distribution shift at test time. Finally, we prove that even if an estimator had access to the test distribution $\mu$, the convergence rate of its expected risk over $\mu$ could not be faster than that of our pretrained Transformers, thereby providing a more appropriate optimality guarantee than minimax lower bounds.
Stochastic differential equations (SDEs) are a fundamental tool for modelling dynamic processes, including gene regulatory networks (GRNs), contaminant transport, financial markets, and image generation. However, learning the underlying SDE from data is a challenging task, especially if individual trajectories are not observable. Motivated by burgeoning research in single-cell datasets, we present the first comprehensive approach for jointly identifying the drift and diffusion of an SDE from its temporal marginals. Assuming linear drift and additive diffusion, we show that non-identifiability can only arise if the initial distribution possesses generalized rotational symmetries. We further prove that even if this condition holds, the drift and diffusion can almost always be recovered from the marginals. Additionally, we show that the causal graph of any SDE with additive diffusion can be recovered from the identified SDE parameters. To complement this theory, we adapt entropy-regularized optimal transport to handle anisotropic diffusion and introduce APPEX (Alternating Projection Parameter Estimation from $X_0$), an iterative algorithm designed to estimate the drift, diffusion, and causal graph of an additive noise SDE, solely from temporal marginals, and demonstrate its effectiveness on simulated data from linear additive noise SDEs.
A source can make a training mixture easier to learn while making its accuracy less valuable for the target. We study how the training problem a source creates and the finite-sample error of learning it jointly determine target-risk guarantees under reweighting. For binary classification, for each pair of laws and each positive target weight, we derive the sharpest target excess hinge-risk bound implied by mixture excess alone, uniformly over clipped predictors. This exact conversion curve is determined by density-weighted local votes. A two-sample oracle inequality for weighted support vector machines supplies the mixture excess, and the composition becomes explicit for Gaussian kernels. On a tilted-boundary class, we establish the all-procedure minimax classification rate $[n+s_+^2/(m^{-1}+\tau^2)]^{-1}$ in the transferable strength $s_+$ and the tilt $\tau$, separating source sampling uncertainty ($m^{-1}$) from relation uncertainty ($\tau^2$). A target-checked Gaussian-SVM selector adapts to unknown $s_+$ and $\tau$ up to a logarithmic factor for fixed dimension, and retains the target-only rate up to the same factor against arbitrary independent sources under the stated target model. On a separate smooth opposition class, the minimax risk within the family of exact Gaussian SVMs fitted to the original labels with nonnegative weights is exponentially larger than the unrestricted minimax risk whenever the target sample is at most half the source sample, despite arbitrary data-dependent weighting and tuning. The value of a source thus depends on the training problem it creates, on what accuracy there means for the target, and on whether the permitted use of the source can implement the needed correction.
Many real-world high-dimensional time series exhibit long-memory, but Gaussian graphical model testing in this regime remains understudied. We develop a direct, data-adaptive test statistic for assessing conditional independence in the graph structure of stationary Gaussian time series. We establish a finite-sample, Berry--Esseen type Gaussian approximation bound for the statistic, which applies to both short-memory and long-memory time series. The testing procedure uses a block bootstrap method, on which we provide a finite-sample validity result including in the ultra-high-dimensional scenario, and can be extended to comparing graphical structures in two-sample tests. We also develop a consistency-empowered correction to the statistic and show that such tests attain asymptotic consistency in both size and power. Our proposed method is applied to real-world fMRI data to understand functional connectivities within brain in different periods.
We comment on the optimally-transported generalized method of moments (OTGMM) of Schennach and Starck (2026) and give counterexamples to Theorems 2-6. In the small-error analysis, the assumptions do not ensure consistency or asymptotic normality (Theorems 2-3). In the large-error analysis, Theorem 4 states the equivalence between the OTGMM estimator and the GMM estimator with modified moments, and Theorems 5-6 assert consistency and asymptotic normality. Through counterexamples, we show that the equivalence in Theorem 4 fails and a component of the Lagrange multiplier has different limits under the two estimators (Theorems 5-6). We also point out that, under misspecification, the population value selected by OTGMM depends on the metric. We give sufficient conditions and establish results following the original motivation for OTGMM. Our corrections to Theorems 4-6 change their assumptions or interpretation more substantially than those to Theorems 2 and 3. Furthermore, we correct Assumption 16 and the test's null distribution.
We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliably when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of setting.
Generalized linear models (GLMs) are fundamental tools for statistical modeling, with maximum likelihood estimation (MLE) serving as the classical approach for parameter inference. While MLE performs well for canonical GLMs, it can become computationally challenging in more general settings with non-canonical, non-smooth, or saturating link functions, where the resulting optimization landscape may be ill-conditioned, non-convex, or non-differentiable. In this paper, we study an alternative estimation framework based on variational inequalities (VIs), which formulates GLM estimation through an operator-based equilibrium condition rather than likelihood minimization. We analyze the VI estimator from a statistical perspective and establish finite-sample error bounds and asymptotic normality, together with convergence guarantees for fixed-point and stochastic approximation algorithms. The framework accommodates a broad class of link functions, including non-canonical, non-smooth, and non-monotone cases satisfying a strong Minty-type condition, and extends naturally to generalized additive models (GAMs) via basis expansion. Numerical experiments demonstrate that under a fixed iteration budget, the VI approach achieves competitive estimation accuracy and faster, more stable convergence than MLE, particularly in GLMs and GAMs with non-canonical or non-smooth link functions.
We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the standard generative setting underlying Partial Least Squares (PLS), a widely used yet theoretically underdeveloped method. We show that the leading singular values of the sample cross-covariance matrix undergo a Baik-Ben Arous-Peche (BBP)-type phase transition, and we characterize the precise thresholds for the emergence of informative components. Our results yield the first sharp asymptotic description of the signal recovery capabilities of PLS in this setting, revealing a fundamental performance gap between PLS and the Bayes-optimal estimator. In particular, we identify the SNR and correlation regimes where PLS fails to recover any signal, despite detectability being possible in principle. These findings clarify the theoretical limits of PLS and provide guidance for the design of reliable multi-modal inference methods in high dimensions.
We study the fundamental limits of jointly estimating the signal amplitude and the atmospheric phase distortion in complex-valued multi-look coherent imaging, where the measurements are corrupted by multiplicative speckle and additive Gaussian noise, all looks share a known Gaussian sensing matrix, and both parameters belong to structured classes. By analyzing a constrained maximum likelihood estimator, we establish an upper bound on the joint minimax risk that holds uniformly over the admissible parameter classes, while oracle reductions and Fano-type arguments yield worst-case lower bounds over the admissible parameter classes. These results quantify the effects of structural complexity, signal and measurement dimensions, the number of looks, and the noise level, and establish matching minimax rates in several settings, with a remaining gap for phase estimation at low noise when the measurement dimension exceeds the signal dimension. We also construct a two-stage estimator for signal recovery under arbitrary diagonal sign distortions in the real-valued analogue. Under additional conditions, it recovers the signs up to a global sign with high probability and yields a uniform signal risk bound without a distortion-complexity term. Numerical experiments illustrate the dependence of the joint, signal, and phase errors on the model parameters and provide further support for the theoretical analysis.
The maximum mean discrepancy (MMD) test is a popular nonparametric test on general domains. Two lines of work have studied its optimality through structurally different approaches. The first, a convolution approach on $\mathbb{R}^d$, uses translation-invariant kernels with a shrinking bandwidth and is minimax optimal over Sobolev balls under $L^2$-separation; the second, a regularization approach on general measurable spaces, uses spectral regularization of the kernel covariance operator and is minimax optimal over Hellinger- and $\chi^2$-separated alternatives. In this paper, we study the regularization approach over Sobolev balls under $L^2$-separation, which reveals the connection between the two approaches. We show that, with a translation-invariant kernel on $\mathbb{R}^d$, both are ways of meeting a single condition on the kernel's spectral density: the spectral regularized discrepancy captures the $L^2$-separation $\Delta$ of a Sobolev alternative of smoothness $s$ as soon as the regularization parameter $\lambda$ is below the spectral density on frequencies up to $\Delta^{-1/(2s)}$. A kernel with a shrinking bandwidth $h$ meets this condition with $\lambda \asymp h^d$, and under this coupling the constant regularizer, which gives the MMD test itself, attains the Sobolev minimax rate. This explains, from a spectral perspective, the need for a shrinking bandwidth in the convolution approach. With a non-constant spectral regularizer, our analysis requires the bandwidth to shrink only when the spectral density decays faster than any polynomial, as for the Gaussian kernel. Thus, for Mat\'ern and Laplace kernels, a fixed bandwidth with $\lambda \to 0$ attains the Sobolev minimax rate. Finally, an adaptive test over a grid of bandwidths and regularization parameters is minimax optimal, up to a $\log\log$ factor, over both classes without prior knowledge of the smoothness.
We study the query complexity of sampling from smooth distributions on $\mathbb R^d$ using first-order oracles. For $d$-dimensional strongly log-concave targets with condition number $\kappa=2$, we prove that any randomized sampler requires $\exp({\Omega(\log^{1/3} d)})$ queries even from a constant warm start. This rules out polylogarithmic dimension dependence and strictly separates strongly log-concave sampling from Gaussian sampling. For targets with $L$-smooth potentials satisfying an $\alpha$-log-Sobolev inequality with $\kappa= L/\alpha$, we prove an $\Omega(\kappa)$ query lower bound for $d\gtrsim\log^2\kappa$. This shows that log-Sobolev sampling is strictly harder than strongly log-concave sampling. Our proofs construct instance-dependent tests that target samples typically pass but outputs of algorithms making few queries typically fail. For strongly log-concave targets, we embed a sequence of hidden Gaussian into high-degree tensor polynomial with smooth gates, forcing any algorithm to uncover the layers sequentially. The log-Sobolev construction hides a constant fraction of the target mass in angular caps while controlling the log-Sobolev constant via delicate estimates.
Classical statistical theory is insufficient to explain the successes of general-purpose AI models, because it depends on handcrafted inductive biases that it cannot justify. No Free Lunch (NFL) theorems force any learner that beats chance on some environments to underperform on others. We might hope that past experience informs which environments to expect, but NFL applies equally to meta-learning. Thus, any method that makes meaningful predictions necessarily begins with an inductive bias external to the data. Choosing to bias toward short programs yields Solomonoff induction (SI), whose performance is competitive against all computable learners - albeit up to "constants" that become large when comparing against specialized methods that exploit background information. We therefore relativize SI to an information vantage point, biasing toward short programs with access to all preexisting information. This reframes the inductive bias: instead of seeking some absolute notion of simplicity, we favour accessibility with respect to our vantage point. An algorithm can only outpredict the relativized SI to the extent that its code contains additional information about the data, and no algorithm can generate such information. While SI is incomputable and hence not a practical algorithm, it provides a formal optimum for inference in the limit of infinite compute, and there is evidence to suggest that frontier AI systems roughly approximate it. Thus, the only known answer to meta-NFL is rooted in algorithmic information theory, which we should expect to play a fundamental role in explaining the generalization behaviour of modern (and future) AI systems.
We study the challenging problem of distribution-free sequential changepoint detection for independent observations with unknown and \emph{completely unrestricted} pre- and post-change laws. For over two decades, the primary existing solution for this problem has employed Vovk's conformal test martingales and associated e-detectors, which control the probability of false alarm (PFA) and the average run length (ARL) respectively. The majority of these works focus on validity, with statistical efficiency usually left for simulations. We develop a comprehensive theory of how conformal p-values behave under non-exchangeable data with a changepoint at an unknown time $T$. We use this to analyze the post-change growth and resulting detection delays and find that the standard existing methods are provably suboptimal for PFA and ARL control, and can lead to delays that are $\Omega(T)$ and $\Omega(\sqrt{\text{ARL}})$ respectively. We construct new conformal e-processes and e-detectors with learned scores and a fixed, distribution-independent betting dictionary. For any distribution pair with positive finite KL divergence, our methods asymptotically achieve $\log T/\mathrm{KL}$ delay for PFA control and $\log\mathrm{ARL}/\mathrm{KL}$ delay (with a sharp leading constant of 1), which are both proven to be minimax optimal in a suitable double-array framework. Simulations illustrate the large empirical benefits of our new approach.
Policy-gradient methods need a scalar reward, yet for many tasks no formula captures what we want, while judging which of two outcomes is better is easy. We study reinforcement learning that receives only such comparisons - no reward function, no reward model - in multi-step tasks. Groups of trajectories start from a shared state, and each action is credited by how its reward-to-go compares with those of other trajectories at the same time step. We show the expected update is policy gradient with reward-to-go replaced by its win rate against the population still running at that time, weighted by the chance an opponent is still running; in deterministic environments the update ranks actions by true value and the optimal policy is stable, whereas under stochastic outcomes it prefers actions that win more often rather than those with higher mean. Label noise near ties or from position bias leaves the expected update unchanged; only noise that also flips clear comparisons halves the signal. A critic trained from the same comparisons removes the state confound without changing the expected update. In a pre-registered study over six environments with ten seeds per claim, comparisons cost no detectable performance relative to scalar returns at the same granularity, learning a reward model from the same comparisons is not better, and ten percent of comparisons suffice when spent on a few time points with all pairs. With real LLM judges, comparisons are 7-10 points more accurate than pointwise scores, and the gap grows with the stronger judge. Trained end-to-end with an LLM judge on a requirement the hand-written reward omits, policies satisfy it in all five seeds but land far less often than with oracle comparisons, because the judge is near chance on the near-tied failures dominating early training. Comparisons do not close the gap to PPO with scalar rewards on HalfCheetah.
The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Specifically, we demonstrate that: (i) When the expected degree $d\geq 2$, the number of negative outliers of the Bethe-Hessian matrix can consistently estimate the number of blocks above the Kesten-Stigum threshold, thus confirming a conjecture from Saade, Krzakala, and Zdeborov\'a (2014) for $d\geq 2$. (ii) For sufficiently large $d$, its eigenvectors can be used to achieve weak recovery. (iii) As $d\to\infty$, we establish the concentration of the locations of its negative outlier eigenvalues, and weak consistency can be achieved via a spectral method based on the Bethe-Hessian matrix.
Conformal prediction gives finite-sample coverage when calibration and test points are exchangeable, but exchangeability can fail when deployment changes the data distribution. We study conformal prediction in performative settings, where the deployed predictor affects the data distribution, and seek prediction sets valid at performative equilibrium: coverage must hold under $D(f^\star)$ for a performatively stable predictor $f^\star$. Under total-variation $L$-sensitivity of the distribution map, we prove a phase transition at $L=1$. If $L \ge 1$, no finite-sample procedure can uniformly guarantee both equilibrium-valid coverage and nontrivial prediction sets. If $L<1$, the minimax excess miscoverage over informative procedures is $\widetilde{\Omega}(1/((1-L)\sqrt{n}))$, implying a sample-complexity inflation of order $1/(1-L)^2$ for fixed excess miscoverage. Our main technical tool is a transfer theorem: any procedure returning a predictor within $\varepsilon$ of equilibrium yields equilibrium-valid coverage up to an additive $L\varepsilon$ term after split-conformal calibration. We instantiate the theorem with two retraining algorithms: a model-free method for discretizable classes and a parametric tail-averaged method that matches the $n$- and $L$-dependence of this rate up to logarithmic factors. Experiments, including semi-synthetic credit scoring, demonstrate the predicted threshold behavior and scaling laws.
Distribution shifts, including covariate shifts, are ubiquitous in real-world applications of machine learning. Empirical evidence suggests that, across a variety of models with and without downstream adaptation, ensembles tend to be more robust to covariate shift than single models. We extend the theory of covariate shift severity from single models to ensembles, providing theoretical justification for the empirical trends. We introduce new discrepancy measures of covariate shift severity to control the risk of partition-based kernel estimators. We then show that with sufficient ensemble size, these models can achieve minimax-optimal risk convergence rates under covariate shift. Our results suggest that modelers can expect to need larger ensemble sizes for more severe covariate shifts.
We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution. Our algorithm achieves classification error $\epsilon$ using $d^{O(k)}2^{O_k(\log^3(1/\epsilon))}$ time and samples, improving over the $d^{O(k^2/\epsilon^2)}$ running time of the polynomial regression approach. To our knowledge, our result is the first to obtain a dimension exponent independent of the accuracy parameter, even for $k=2$. We complement our algorithmic result with an SQ lower bound showing that the complexity dependence of $d^{\Omega(k)}$ is unavoidable. Lifting the input to polynomial features turns the target into an intersection of $k$ halfspaces. This suggests using the first two labeled moments for subspace recovery. In the lifted space, however, these moments can reveal nothing. Our starting point is a Gaussian comparison theorem showing that, for sufficiently regular defining quadratics, the labeled moments needed for subspace recovery are approximately preserved. To handle arbitrary defining quadratics, we establish a new structural result showing that every direction with significant resampling influence can be detected through degree-$O(k)$ moments. These results yield an initial $d^{O(k)}2^{\mathrm{poly}_k(1/\epsilon)}$-time learner using adaptive restrictions. To improve the exponential dependence on $1/\epsilon$, we develop a two-stage dimension-reduction approach that avoids the costly adaptive partitioning. After identifying a subspace containing all directions with large influence, we use generalized regularity and Gaussian comparison to recover the remaining relevant quadratic features through a new derivative-based moment operator. The recovered features support an intersection close to the target. Combining them with robust hinge-loss minimization gives our quasipolynomial-time learner.
We propose a new estimator of low-frequency skewness (monthly) that exploits high-frequency data (5 min) through a direct functional mapping consisting of layers of Convolutional Neural Networks followed by Multi Layer Perceptrons. Monte Carlo experiments involving stochastic volatility models, with and without jumps, show that the estimator reduces finite-sample bias relative to existing realized-skewness estimators and remains accurate under model misspecification. Empirically, our estimator exhibits temporal stability and delivers cross-sectional performance in actual skewness-sorted portfolios. We characterize the finite-sample bias of conventional plug-in skewness estimators and establish risk consistency properties of the proposed NN estimator under standard regularity conditions.
The classical model-based optimal design criteria (D-, T-, L-, C-, and A-optimality) are usually motivated via the Cram\'er--Rao bound as optimizing scalar functions of the Fisher information matrix (FIM). This paper offers a complementary reading in which each criterion appears asymptotically as $N \to \infty$ as the leading-order term of an expected dissimilarity between prior and posterior, with the FIM entering through the Bernstein--von Mises limit. The Kullback--Leibler (KL) divergence of the posterior relative to the prior recovers D-optimality (classically known); a reverse KL of the prior relative to the posterior recovers T-optimality (here defined as maximization of the trace of the FIM, following one common convention) and its weighted-trace generalization; and a first-order approximation of the squared 2-Wasserstein distance recovers L-, C- and A-optimality. The five criteria emerge as instances of a common Bayesian-geometric template, indexed by three choices: the dissimilarity, the prior parametrization, and the approximation scheme used to evaluate it (e.g.\ Taylor expansions of the matrix square root). For the weighted criteria, the Gaussian prior covariance encodes the criterion's weight matrix, so any choice of weights corresponds to a specific prior in the family. The T-, L-, C-, and A-optimality derivations appear novel to the best of my knowledge. A sixth criterion, E-optimality, is recovered not as an independent expected-dissimilarity limit but as a min-max robustification of C-optimality against uncertainty in the modeler's prior direction. The framework is extended to distinguish between a designer's prior over unknown parameters at design time and a modeler's prior used for posterior inference, recovering the classical expected information gain objective as a special case.