Sample Complexity

Generative models, from diffusion models to large language models, achieve remarkable performance but at a cost in training data orders of magnitude larger than what biological learners require. An alternative paradigm has emerged in which networks are trained to predict their own latent representations of related views or masked regions, as in data2vec and JEPA -- an idea related to predictive-coding accounts of the cortex. Despite strong empirical results, the theoretical understanding of these methods remains limited. Central questions include: by how much does latent prediction actually improve data efficiency? Is there a benefit to stacking such methods into multi-scale hierarchies? We answer both using as data a tractable probabilistic context-free grammar that captures the compositional structure of natural language and images. Such a grammar generates strings of visible tokens by recursively applying production rules along a tree of hidden symbols of depth $L$. For such data, supervised or token-level SSL require a number of samples exponential in $L$ to recover the latent tree; we prove that latent prediction achieves this with a number of samples constant in $L$, up to logarithmic factors. We confirm this bound with (i) a hierarchical clustering algorithm, (ii) an end-to-end neural network whose predictor-clusterer modules predict their own latents at each level via gradient descent, and (iii) the first sample-complexity analysis of data2vec, which we show implicitly performs hierarchical latent prediction. This suggests that explicit stacking such as H-JEPA is largely redundant.

We study the sample complexity of robust binary hypothesis testing under three standard contamination models: $\varepsilon$-additive (Huber), $\varepsilon$-subtractive, and $\varepsilon$-total variation (TV), denoted by $n^*_{\mathrm{Hub}}(\varepsilon)$, $n^*_{\mathrm{Sub}}(\varepsilon)$, and $n^*_{\mathrm{TV}}(\varepsilon)$, respectively. For subtractive contamination, we show that least favourable distributions exist and provide explicit formulas for the same, bringing this model in line with the classical Huber and TV models. Next we show that in all three models, sample complexity may be highly unstable in the contamination parameter $\varepsilon$, increasing by polynomial factors even for $o(\varepsilon)$ perturbations. Similarly, there may be polynomial factor gaps between the sample complexities when $\varepsilon$ is known exactly versus when it is known up to $o(\varepsilon)$ error. Despite the instability of the sample complexity in all models, we show that the sample complexities across models are comparable up to constant-factor rescaling of $\varepsilon$. Specifically, for any fixed $δ_0>0$, the following hold for all distributions $p$ and $q$: (i) $n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(2\varepsilon)$, (ii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((2+δ_0)\varepsilon)$, and (iii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((1+δ_0)\varepsilon)$, and the scaling constants are tight. Finally, we extend our results to adaptive versions of the contamination models.

We prove optimal sampling bounds achieving $(1\pm\varepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms. This includes important functions such as logistic and sigmoid loss, hinge loss, and ReLU loss, as prominent and popular representative examples. In particular, we prove $k^2/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_2/k$ regularization, and $k/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_1/k$ regularization. For $\|\cdot\|_2^2/k$ regularization, the sampling complexity depends mainly on a bounded derivative property: if $|g'(x)|\leq g(x)$, and $g(0)>0$, and $g$ is monotonic or convex, then it admits linear in $k$ sampling complexity; otherwise the general bound is $k^2/\varepsilon^2$. However, if $g(0)=0$, our results indicate that no dimension-free bounds are possible, and even sublinear bounds are ruled out. All upper bounds are complemented by matching lower bounds up to polylogarithmic terms. Moreover, our work relies conceptually and algorithmically on simple uniform or (squared) norm sampling and hereby improves over recent cubic $k^3/\varepsilon^2$ sensitivity sampling bounds of (Alishahi and Phillips, ICML'24). This is achieved by refined arguments involving higher moment bounds and empirical process analyses to avoid overcounting that appears in the de-facto standard VC-dimension and sensitivity framework.

Inverse Kinematics (IK) plays a critical role in robotic motion planning and control. The IK solutions of a robot manipulator could be done by conventional ways such as geometric, algebraic, or Jacobian methods, which have drawbacks. The Artificial Neural Networks (ANNs) have become a promising alternative for approximating IK solutions due to their generalization ability and computational efficiency. This approach basically trains only a few samples of the end effector that are recorded for the solution of the IK problem. However, a fundamental question remains: how many training samples are sufficient to achieve reliable and accurate IK predictions? This study investigates the mathematical framework of relating the size of training datasets and the accuracy of ANN-based IK solvers. Using an articulated robotic manipulator, we generate varying amounts of joint-position pairs to train feedforward neural networks and assess their accuracy, convergence, and generalization capability. The results reveal more training samples than 125 did not contribute to the improvement of the model efficiency that the comparable measure dealing with the approximation accuracy over the sampling size, offering valuable insight into data efficiency. This work provides practical guidance for optimizing the data sizing of ANN solutions, balancing computational cost and model accuracy for real-world robotic applications.

Transfer learning is an essential technique for many machine learning/AI models of complex structures such as large language models and generative AI. The essence of transfer learning is to leverage knowledge from resolved source tasks for a new target task, especially when the sample size $m$ of the training data for the latter is low. In this work, we rigorously analyze the potential benefit of transfer learning in terms of sample efficiency. Specifically, taking an optimal transport viewpoint of transfer learning, we find that when the data dimension $d$ is higher than $3$, the sample complexity for transfer learning is $O(m^{-(α+1)/d})$, with $α$ indicating the smoothness of the data distribution, as opposed to the $O(m^{-p/d})$ sample complexity for direct learning with $p$ indicating the smoothness of the optimal target model. Our finding theoretically supports a better sample efficiency for transfer learning, when the target task is optimizing over a family of not-so-smooth models (i.e., highly complex networks with the possible use of non-smooth activation functions). Using image classification as an example, we numerically demonstrate the sample efficiency for transfer learning, that is, in the data hungry regime, the model performance can be significantly improved by transfer learning.

Neural networks trained with gradient-based methods exhibit a strong simplicity bias: they learn simpler statistical features of their data before moving to more complex features. Previous analyses of this phenomenon have largely focused on settings with (quasi-)isotropic inputs. In this work, we study the simplicity bias from a Fourier perspective, which allows us to include two key features of natural images in the analysis: approximate translation-invariance and power-law spectra. We first show experimentally that simple neural networks trained on image classification tasks first rely on amplitude information -- related to pair-wise correlations between pixels -- before exploiting phase information, which encodes edges and higher-order correlations. In view of this, we introduce a synthetic data model for translation-invariant inputs that allows precise control over amplitudes and phases while remaining tractable. We rigorously establish that for isotropic and high-dimensional inputs, classification based on phase information alone is a genuinely hard task: online stochastic gradient descent (SGD) cannot distinguish the structured inputs from noise within $n \ll N^3$ steps, but needs at least $n \gg N^3 \log^2{N}$ steps. In contrast, we show both experimentally and theoretically that power-law spectra can dramatically accelerate the speed of learning phase information, even if the spectra do not help with classification. Simulations with two-layer networks trained on textures and with deep convolutional networks on ImageNet and CIFAR100 confirm this non-trivial interaction between amplitudes and phases, providing mechanistic insights into how deep neural networks can learn natural image distributions efficiently.

The success of deep learning in high-dimensional settings is often attributed to the presence of low-dimensional structure in real-world data. While standard theoretical models typically assume that this structure lies in the target function, projecting unstructured inputs onto a low-dimensional subspace, data such as images, text or genomic sequences exhibit strong spatial correlations within the input space itself. In this paper, we propose a tractable model to study how these correlations affect the sample complexity of learning with gradient descent on shallow neural networks. Specifically, we consider targets that depend on a small number of latent Boolean variables, and input features grouped into clusters and correlated with the latent variables. Under an identifiability assumption, we show that for a layerwise gradient-descent variant, the sample complexity scales with the number of hidden variables and, when the signal-to-noise ratio is sufficiently high, is independent of the input dimension, up to logarithmic terms. We empirically test our theoretical findings on both synthetic and real data.

Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used method for learning exponential families with continuous variables due to its computational ease when compared against maximum likelihood estimation. However, theoretical understanding of the statistical properties of score matching is still lacking. In this work, we provide a non-asymptotic sample complexity analysis for learning the structure of exponential families of polynomials with score matching. The derived sample bounds show a polynomial dependence on the model dimension. These bounds are the first of its kind, as all prior work has shown only asymptotic bounds on the sample complexity.

We study multiple change point localization under bandit feedback. An unknown piecewise-constant function on a compact interval can be queried sequentially at adaptively chosen inputs, and each query returns a noisy evaluation of the function. The goal is to identify a prescribed number of discontinuities, known as change points, within a target precision $η$ and confidence level $1-δ$, while using as few samples as possible. We propose an adaptive algorithm that first detects intervals likely to contain change points and then refines their locations to precision $η$. We establish non-asymptotic upper bounds on its sample budget, together with corresponding lower bounds. Prior work shows that jump magnitudes alone determine the asymptotic sample complexity as $δ\to 0$. We reveal that this picture is incomplete beyond this regime. We demonstrate, both empirically and theoretically, that for general $δ$ and $η$, the complexity is jointly governed by the jumps and the relative positions of the change points.

Dynamics of interacting systems in engineering, society, and nature often evolve over latent networks that govern which entities can interact. We study the problem of inferring these networks from event-based observations, which arise naturally in finance, seismology, and neuroscience. While there is substantial algorithmic work addressing this important problem, theoretical results are scarce. In this paper we ask the following fundamental question: what is the minimum time that one must observe the dynamics in order to exactly recover the underlying network, as a function of the number $d$ of interacting entities? For a class of stationary Hawkes processes with sparse, weak interactions, we prove that an observation time of order $\log d$ is sufficient and necessary. For the upper bound we construct a two-stage estimator that uses clipped and binned event data for screening, followed by a least-squares refinement, and apply concentration bounds derived from the Poisson cluster representation. For the lower bound we combine Fano's inequality with Jacod's Girsanov formula for point processes on a suitable subclass of networks.

We establish sample complexity results for stochastic optimization over the integers, especially with a view to understand the complexity with respect to the corresponding continuous optimization problem. We show that integer optimization can sometimes require strictly more samples and sometimes strictly smaller number of samples, depending on the structure of the objective and constraints. 1. For Lipschitz objectives over subsets of the $\ell_\infty$ ball, the statistical complexity of general stochastic mixed-integer, nonlinear, nonconvex optimization is exactly the same as stochastic linear optimization with just bound constraints. 2. For Lipschitz objectives over subsets of the $\ell_2$ ball, we show that integer optimization can require strictly smaller sample size compared to the continuous setting in a certain regime. To get to this result, we also establish tight sample complexity results for nonconvex continuous stochastic optimization which, to the best of our knowledge, do not appear in prior work. 3. For strongly convex, smooth objectives, integer optimization has high statistical complexity compared to the continuous setting. In particular, we show that integer optimization requires $Ω(1/ε^2)$ samples to report an $ε$-approximate solution, compared to the well-known $O(1/ε)$ sample complexity from the continuous optimization literature.

This paper presents a parametric solution to piecewise linear regression through the Adaptive Block Gradient Descent (ABGD) algorithm. The heart of the method is the parametrization of piecewise linear functions as the difference of max-affine (DoMA) functions. A non-asymptotic local convergence analysis for ABGD is provided under sub-Gaussian covariate and noise distributions. To initialize ABGD, we adapt a prior algorithm originally developed for the simpler setting of max-affine functions. When suitably initialized, ABGD converges linearly to an $ε$-accurate estimate given $\tilde{\mathcal{O}}(d\max(σ_z/ε,1)^2)$ observations where $σ_z^2$ denotes the noise variance. This implies exact recovery given $\tilde{\mathcal{O}}(d)$ samples in the noiseless case. Also, such a rate is shown to be minimax optimal up to logarithmic factors. Synthetic numerical results corroborate the theoretical guarantees for ABGD. We also observe competitive performance compared to the state-of-the-art methods on real-world datasets.

Kullback-Leibler (KL) regularization is widely used in offline decision-making and offers several benefits, motivating recent work on the sample complexity of offline learning with respect to KL-regularized performance metrics. Nevertheless, the exact sample complexity of KL-regularized offline learning remains largely from fully characterized. In this paper, we study this question in the setting of multi-armed bandits (MABs). We provide a sharp analysis of KL-PCB (Zhao et al., 2026), showing that it achieves a sample complexity of $\tilde{O}(ηSAC^{π^*}/ε)$ under large regularization $η= \tilde{O}(ε^{-1})$, and a sample complexity of $\tildeΩ(SAC^{π^*}/ε^2)$ under small regularization $η= \tildeΩ(ε^{-1})$, where $η$ is the regularization parameter, $S$ is the number of contexts, $A$ is the number of arms, $C^{π^*}$ policy coverage coefficient at the optimal policy $π^*$, $ε$ is the desired sub-optimality, and $\tilde{O}$ and $\tildeΩ$ hide all poly-logarithmic factors. We further provide a pair of sharper sample complexity lower bounds, which matches the upper bounds over the entire range of regularization strengths. Overall, our results provide a nearly complete characterization of offline multi-armed bandits with KL regularization.

We characterize the fundamental limits of high-dimensional mean testing under arbitrary truncation, where samples are drawn from the conditional distribution $P(\cdot \mid S)$ for an unknown truncation set $S$ that may hide up to an $\varepsilon$-fraction of the probability mass. For distributions with $p$-th directional moments of magnitude at most $ν_{P,p}$, truncation induces a bias of order $O(ν_{P,p}\varepsilon^{1-1/p})$. This bias creates a sharp information-theoretic detectability floor: when the signal $α$ falls below this threshold, the null and alternative hypotheses are indistinguishable even with infinite data. Above this floor, we prove that a simple second-order test achieving near-optimal sample complexity $n = O\!\left(\frac{\|Σ_P\|}{(α-4ν_{P,p}\varepsilon^{1-1/p})^2}\sqrt{d}\right)$. We further identify a structural escape from this finite-moment bias barrier. Under a directional median regularity assumption, truncation bias improves to linear order $O(\varepsilon)$. This reveals an intermediate regime in which estimation requires $Θ(d)$ samples for uniform recovery, while testing recovers the classical $Θ(\sqrt d)$ rate once truncation bias is eliminated. Together, our results provide a unified framework for mean testing under truncation, connecting finite-moment, sub-Gaussian, and median-regular structural regimes.

Learning curves are a fundamental primitive in supervised learning, describing how an algorithm's performance improves with more data and providing a quantitative measure of its generalization ability. Formally, a learning curve plots the decay of an algorithm's error for a fixed underlying distribution as a function of the number of training samples. Prior work on revenue-maximizing learning algorithms, starting with the seminal work of Cole and Roughgarden [STOC, 2014], adopts a distribution-free perspective, which parallels the PAC learning framework in learning theory. This approach evaluates performance against the hardest possible sequence of valuation distributions, one for each sample size, effectively defining the upper envelope of learning curves over all possible distributions, thus leading to error bounds that do not capture the shape of the learning curves. In this work we initiate the study of learning curves for revenue maximization and provide a near-complete characterization of their rate of decay in the basic setting of a single item and a single buyer. In the absence of any restriction on the valuation distribution, we show that there exists a Bayes-consistent algorithm, meaning that its learning curve converges to zero for any arbitrary valuation distribution as the number of samples $n \to \infty$. However, this convergence must be arbitrarily slow, even if the optimal revenue is finite. In contrast, if the optimal revenue is achieved by a finite price, then the optimal rate of decay is roughly $1/\sqrt{n}$. Finally, for distributions supported on discrete sets of values, we show that learning curves decay almost exponentially fast, a rate unattainable under the PAC framework.

While the optimal sample complexity of binary classification in terms of the VC dimension is well-established, determining the optimal sample complexity of multiclass classification has remained open. The appropriate complexity parameter for multiclass classification is the DS dimension, and despite significant efforts, a gap of $\sqrt{\text{DS}}$ has persisted between the upper and lower bounds on sample complexity. Recent work by Hanneke et al. (2026) shows a novel algebraic characterization of multiclass hypothesis classes in terms of their DS dimension. Building up on this, we show that the maximum hypergraph density of any multiclass hypothesis class is upper-bounded by its DS dimension. This proves a longstanding conjecture of Daniely and Shalev-Shwartz (2014). As a consequence, we determine the optimal dependence of the sample complexity on the DS dimension for multiclass as well as list learning.

Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as $\operatorname{tr}(ρ^t)$, but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order $t\ge 3$, existing protocols show that $\lceil t/2\rceil$ replicas already suffice for polynomial-sample estimation of $\operatorname{tr}(ρ^t)$, yet it has remained open whether one fewer replica must necessarily incur a sample-complexity barrier growing with the dimension. We prove that this is indeed the case in the sample/copy-access model with replica-limited joint measurements: any protocol restricted to $\lceil t/2\rceil-1$ replicas requires dimension-growing sample complexity, while $\lceil t/2\rceil$ replicas suffice by prior work. Thus the exact replica threshold for fixed-order pure moments is $\lceil t/2\rceil$. Equivalently, for fixed-order pure moments, one additional coherent replica is not merely useful but marks the exact threshold between polynomial-sample estimation and a dimension-growing regime in the replica-limited model. We further show that the same threshold law extends to a broad family of observable-weighted moments $\operatorname{tr}(Oρ^t)$, including Pauli observables and other observables with bounded operator norm and macroscopic trace norm. Coherent replica number therefore acts as a genuinely discrete resource for nonlinear quantum-state estimation.

We study the minimax sample complexity of multicalibration in the batch setting. A learner observes $n$ i.i.d. samples from an unknown distribution and must output a (possibly randomized) predictor whose population multicalibration error, measured by Expected Calibration Error (ECE), is at most $\varepsilon$ with respect to a given family of groups. For every fixed $κ> 0$, in the regime $|G|\le \varepsilon^{-κ}$, we prove that $\widetildeΘ(\varepsilon^{-3})$ samples are necessary and sufficient, up to polylogarithmic factors. The lower bound holds even for randomized predictors, and the upper bound is realized by a randomized predictor obtained via an online-to-batch reduction. This separates the sample complexity of multicalibration from that of marginal calibration, which scales as $\widetildeΘ(\varepsilon^{-2})$, and shows that mean-ECE multicalibration is as difficult in the batch setting as it is in the online setting, in contrast to marginal calibration which is strictly more difficult in the online setting. In contrast we observe that for $κ= 0$, the sample complexity of multicalibration remains $\widetildeΘ(\varepsilon^{-2})$ exhibiting a sharp threshold phenomenon. More generally, we establish matching upper and lower bounds, up to polylogarithmic factors, for a weighted $L_p$ multicalibration metric for all $1 \le p \le 2$, with optimal exponent $3/p$. We also extend the lower-bound template to a regular class of elicitable properties, and combine it with the online upper bounds of Hu et al. (2025) to obtain matching bounds for calibrating properties including expectiles and bounded-density quantiles.

This paper investigates the off-policy evaluation (OPE) problem from a distributional perspective. Rather than focusing solely on the expectation of the total return, as in most existing OPE methods, we aim to estimate the entire return distribution. To this end, we introduce a quantile-based approach for OPE using deep quantile process regression, presenting a novel algorithm called Deep Quantile Process regression-based Off-Policy Evaluation (DQPOPE). We provide new theoretical insights into the deep quantile process regression technique, extending existing approaches that estimate discrete quantiles to estimate a continuous quantile function. A key contribution of our work is the rigorous sample complexity analysis for distributional OPE with deep neural networks, bridging theoretical analysis with practical algorithmic implementations. We show that DQPOPE achieves statistical advantages by estimating the full return distribution using the same sample size required to estimate a single policy value using conventional methods. Empirical studies further show that DQPOPE provides significantly more precise and robust policy value estimates than standard methods, thereby enhancing the practical applicability and effectiveness of distributional reinforcement learning approaches.

Modern large language models generate text autoregressively, producing tokens one at a time. To study the learnability of such systems, Joshi et al. (COLT 2025) introduced a PAC-learning framework for next-token generators, the primitive underlying autoregressive models. In this framework, an unknown next-token generator maps a sequence of tokens to the next token and is iteratively applied for $T$ steps, producing a chain of tokens whose final token constitutes the model's output. The learning task is to learn the input-output mapping induced by this autoregressive process. Depending on the available supervision, training examples may reveal only the final output (End-to-End supervision) or the entire generated chain (Chain-of-Thought supervision). This raises two natural questions: how the sample complexity depends on the generation length $T$, and how much Chain-of-Thought supervision can reduce this dependence. In this work we give a nearly complete answer to both questions by uncovering a taxonomy of how the sample complexity scales with $T$. For End-to-End learning, we show that the landscape is remarkably rich: subject to mild conditions, essentially any growth rate $r(T)$ between constant and linear can arise as the sample complexity, and combined with the linear upper bound of Joshi et al., this yields an essentially complete characterization. In contrast, under Chain-of-Thought supervision we show that the sample complexity is independent of $T$, demonstrating that access to intermediate reasoning steps can eliminate the dependence on the generation length altogether. Our analysis introduces new combinatorial tools, and as corollaries we resolve several open questions posed by Joshi et al. regarding the dependence of learnability on the generation length and the role of Chain-of-Thought supervision.

Replicability requires that algorithmic conclusions remain consistent when rerun on independently drawn data. A central structural question is composition: given $k$ problems each admitting a $ρ$-replicable algorithm with sample complexity $n$, how many samples are needed to solve all jointly while preserving replicability? The naive analysis yields $\widetilde{O}(nk^2)$ samples, and Bun et al. (STOC'23) observed that reductions through differential privacy give an alternative $\widetilde{O}(n^2k)$ bound, leaving open whether the optimal $\widetilde{O}(nk)$ scaling is achievable. We resolve this open problem and, more generally, show that problems with sample complexities $n_1,\ldots,n_k$ can be jointly solved with $\widetilde{O}(\sum_i n_i)$ samples while preserving constant replicability. Our approach converts each replicable algorithm into a perfectly generalizing one, composes them via a privacy-style analysis, and maps back via correlated sampling. This yields the first advanced composition theorem for replicability. En route, we obtain new bounds for the composition of perfectly generalizing algorithms with heterogeneous parameters. As part of our results, we provide a boosting theorem for the success probability of replicable algorithms. For a broad class of problems, the failure probability appears as a separate additive term independent of $ρ$, immediately yielding improved sample complexity bounds for several problems. Finally, we prove an $Ω(nk^2)$ lower bound for adaptive composition, establishing a quadratic separation from the non-adaptive setting. The key technique, which we call the phantom run, yields structural results of independent interest.

Independent Component Analysis (ICA) is a fundamental unsupervised learning technique foruncovering latent structure in data by separating mixed signals into their independent sources. While substantial progress has been made in establishing asymptotic identifiability guarantees for nonlinear ICA, the finite-sample statistical properties of learning algorithms remain poorly understood. This gap poses significant challenges for practitioners who must determine appropriate sample sizes for reliable source recovery. This paper presents a comprehensive finite-sample analysis of nonlinear ICA with neural network encoders, providing the first complete characterization with matching upper and lower bounds. Our theoretical development introduces three key technical contributions. First, we establish a direct relationship between excess risk and identification error that bypasses parameter-space arguments, thereby avoiding the rate degradation that would otherwise yield suboptimal scaling. Second, we prove matching information-theoretic lower bounds that confirm the optimality of our sample complexity results. Third, we extend our analysis to practical SGD optimization, showing that the same sample efficiency can be achieved with finite-iteration gradient descent under standard landscape assumptions. We validate our theoretical predictions through carefully designed simulation experiments. This gap points toward valuable future research on finite-sample behavior of neural network training and highlights the importance of our validated scaling laws for dimension and diversity.

We propose an active learning algorithm for linear system identification with optimal centered noise excitation. Notably, our algorithm, based on ordinary least squares and semidefinite programming, attains the minimal sample complexity while allowing for efficient computation of an estimate of a system matrix. More specifically, we first establish lower bounds of the sample complexity for any active learning algorithm to attain the prescribed accuracy and confidence levels. Next, we derive a sample complexity upper bound of the proposed algorithm, which matches the lower bound for any algorithm up to universal factors. Our tight bounds are easy to interpret and explicitly show their dependence on the system parameters such as the state dimension.

We study compressed sensing when the sampling vectors are chosen from the rows of a unitary matrix. In the literature, these sampling vectors are typically chosen randomly; the use of randomness has enabled major empirical and theoretical advances in the field. However, in practice there are often certain crucial sampling vectors, in which case practitioners will depart from the theory and sample such rows deterministically. In this work, we derive an optimized sampling scheme for Bernoulli selectors which naturally combines random and deterministic selection of rows, thus rigorously deciding which rows should be sampled deterministically. This sampling scheme provides measurable improvements in image compressed sensing for both generative and sparse priors when compared to with-replacement and without-replacement sampling schemes, as we show with theoretical results and numerical experiments. Additionally, our theoretical guarantees feature improved sample complexity bounds compared to previous works, and novel denoising guarantees in this setting.

Generating samples from a continuous probability density is a central algorithmic problem across statistics, engineering, and the sciences. For high-dimensional settings, Hamiltonian Monte Carlo (HMC) is the default algorithm across mainstream software packages. However, despite the extensive line of work on HMC and its widespread empirical success, it remains unclear how many iterations of HMC are required as a function of the dimension $d$. On one hand, a variety of results show that Metropolized HMC converges in $O(d^{1/4})$ iterations from a warm start close to stationarity. On the other hand, Metropolized HMC is significantly slower without a warm start, e.g., requiring $Ω(d^{1/2})$ iterations even for simple target distributions such as isotropic Gaussians. Finding a warm start is therefore the computational bottleneck for HMC. We resolve this issue for the well-studied setting of sampling from a probability distribution satisfying strong log-concavity (or isoperimetry) and third-order derivative bounds. We prove that non-Metropolized HMC generates a warm start in $\tilde{O}(d^{1/4})$ iterations, after which we can exploit the warm start using Metropolized HMC. Our final complexity of $\tilde{O}(d^{1/4})$ is the fastest algorithm for high-accuracy sampling under these assumptions, improving over the prior best of $\tilde{O}(d^{1/2})$. This closes the long line of work on the dimensional complexity of MHMC for such settings, and also provides a simple warm-start prescription for practical implementations.

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Our analysis considers trajectory-dependent regressors and allows marginally stable dynamics with polynomial mean-square state growth. We prove that the diameter of the feasible parameter set shrinks with sample complexity $\widetilde{\mathcal O}(1/ε)$ where $ε$ is the estimation error. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.

We study the basic task of mean estimation in the presence of mean-shift contamination. In the mean-shift contamination model, an adversary is allowed to replace a small constant fraction of the clean samples by samples drawn from arbitrarily shifted versions of the base distribution. Prior work characterized the sample complexity of this task for the special cases of the Gaussian and Laplace distributions. Specifically, it was shown that consistent estimation is possible in these cases, a property that is provably impossible in Huber's contamination model. An open question posed in earlier work was to determine the sample complexity of mean estimation in the mean-shift contamination model for general base distributions. In this work, we study and essentially resolve this open question. Specifically, we show that, under mild spectral conditions on the characteristic function of the (potentially multivariate) base distribution, there exists a sample-efficient algorithm that estimates the target mean to any desired accuracy. We complement our upper bound with a qualitatively matching sample complexity lower bound. Our techniques make critical use of Fourier analysis, and in particular introduce the notion of a Fourier witness as an essential ingredient of our upper and lower bounds.

In this paper, we consider the problem of replicable realizable PAC learning. We construct a particularly hard learning problem and show a sample complexity lower bound with a close to $(\log|H|)^{3/2}$ dependence on the size of the hypothesis class $H$. Our proof uses several novel techniques and works by defining a particular Cayley graph associated with $H$ and analyzing a suitable random walk on this graph by examining the spectral properties of its adjacency matrix. Furthermore, we show an almost matching upper bound for the lower bound instance, meaning if a stronger lower bound exists, one would have to consider a different instance of the problem.

Diffusion models over discrete spaces have recently shown striking empirical success, yet their theoretical foundations remain incomplete. In this paper, we study the sampling efficiency of score-based discrete diffusion models under a continuous-time Markov chain (CTMC) formulation, with a focus on $τ$-leaping-based samplers. We establish sharp convergence guarantees for attaining $\varepsilon$ accuracy in Kullback-Leibler (KL) divergence for both uniform and masking noising processes. For uniform discrete diffusion, we show that the $τ$-leaping algorithm achieves an iteration complexity of order $\tilde O(d/\varepsilon)$, with $d$ the ambient dimension of the target distribution, eliminating linear dependence on the vocabulary size $S$ and improving existing bounds by a factor of $d$; moreover, we establish a matching algorithmic lower bound showing that linear dependence on the ambient dimension is unavoidable in general. For masking discrete diffusion, we introduce a modified $τ$-leaping sampler whose convergence rate is governed by an intrinsic information-theoretic quantity, termed the effective total correlation, which is bounded by $d \log S$ but can be sublinear or even constant for structured data. As a consequence, the sampler provably adapts to low-dimensional structure without prior knowledge or algorithmic modification, yielding sublinear convergence rates for various practical examples (such as hidden Markov models, image data, and random graphs). Our analysis requires no boundedness or smoothness assumptions on the score estimator beyond control of the score entropy loss.

We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown $d$-dimensional quantum state $ρ$ and a known set of observables $\{O_i\}_{i=1}^m$, the goal is to estimate expectation values $\{\mathrm{tr}(O_iρ)\}_{i=1}^m$ to accuracy $ε$ in $L_p$-norm, using possibly adaptive measurements that act on $O(\mathrm{polylog}(d))$ number of copies of $ρ$ at a time. We focus on the regime where $ε$ is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as $\tildeΘ(Γ_p/ε^2)$, where $Γ_p$ is a function of $\{O_i\}_{i=1}^m$ defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with $L_\infty$-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form $\sum_{i=1}^m α_i O_i$ with $\|α\|_q = 1$ (where $q$ is dual to $p$) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of $Θ(Γ^{\mathrm{ob}}_p/ε^2)$. We then show $\tildeΘ(Γ_p/ε^2)$ is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, $Γ^{\mathrm{ob}}_\infty = Γ_\infty$. In both cases, allowing $c$-copy measurements improves the sample complexity by at most $Ω(1/c)$. Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.