Restricted Eigenvalues Beyond Gaussian Width: Threshold Occupancy under Heavy Tails

Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law follows for heavy-tailed designs from a uniform small-ball condition alone. We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form, and the missing obstruction is simultaneous threshold occupancy. A constant-width polyhedral descent cone with fixed small-ball constants has zero empirical RE on every sample path up to half the ambient dimension. More generally, every finite range space admits exact threshold encoding in an arbitrarily narrow spherical cap and a lift to a full polyhedral descent-cone section. For every fixed threshold VC dimension $d$, as $β\downarrow0$, the sharp worst-case sample complexity is $Θ(β^{-1}[d\log(1/β)+\log(1/δ)])$. The separation persists under exact isotropy and all finite moments: on the same constant-width cone, Gaussian measurements succeed with $O(1+\log(1/δ))$ samples, whereas an isotropic heavy-tailed design fails pathwise for $n\lesssim\sqrt{p/\log p}$. Gaussian smoothing yields an everywhere-positive $C^\infty$ density while retaining arbitrarily poor RE. Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.

Discrete Gromov-Wasserstein Duality: Algorithms and Isomorphism Testing

The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.

Two Dimensions Govern Agnostic Multiclass Transductive Learning

In transductive classification, an adversary fixes a labeled population, one label is hidden uniformly, and the learner sees all remaining labels. For binary classes, agnostic transductive and PAC learning have the same minimax rate. Whether this extends to multiclass learning was open, especially for unbounded label spaces where uniform convergence can fail. We resolve the question up to logarithmic factors. For every multiclass class $\mathcal H$ with DS dimension $d_{DS}$ and Natarajan dimension $d_{\mathrm N}$, the optimal agnostic transductive excess error satisfies $\widetildeΘ\left(\frac{d_{DS}}{n}+\sqrt{\frac{d_{\mathrm N}}{n}}\right).$ The result holds for arbitrary label spaces. The two terms are both necessary. A DS pseudo-cube gives the realizable $d_{DS}/n$ obstruction, while a Natarajan cube with repeated points and fair labels gives the agnostic $\sqrt{d_{\mathrm N}/n}$ obstruction. The upper bound uses a random-reservation principle. The learner deliberately ignores a constant fraction of the visible labels, which makes the true test point uniform in a large unseen block. We combine realizable compression, a label-space reduction, and inside-menu agnostic compression across this finite-population split. A new without-replacement multiplicative-weights lemma preserves the fast $d_{DS}/n$ term. Consequently, agnostic multiclass PAC and transductive learning obey the same two-dimension law up to logarithmic factors.

Minimax Optimality of Score-Entropy Discrete Diffusion

Discrete diffusion models have demonstrated strong performance across a range of datasets, including natural language data and graph-structured data. Among many variants, score-entropy discrete diffusion (SEDD) has achieved particularly strong empirical results. In SEDD, new samples are generated by iteratively evaluating a sequence of concrete score functions, which are learned by minimizing a score-entropy loss. While much of the prior theoretical literature on discrete diffusion has focused on the sampling efficiency of SEDD under the assumption of small score estimation error, recent work has begun to investigate the finite-sample properties of score estimation itself. In this work, we take a different route by investigating the fundamental statistical limits of concrete score estimation. We focus on uniform and masking discrete diffusions, two of the most widely adopted discrete diffusion models. We establish a minimax lower bound under the score-entropy loss, and propose an MLE-based thresholding estimator that matches this lower bound up to constant and polylogarithmic factors that depend on neighboring density ratios. We further show that, for any target distribution, this density ratio is naturally controlled under both uniform and masking discrete diffusion models, yielding nearly matching minimax lower and upper bounds for the aggregated score estimation error. Our results imply that, with appropriate initialization and discretization, SEDD can achieve nearly optimal minimax sample complexity, as measured by the KL divergence between the target and generated distributions.

Learning to Price with Persuasion

Motivated by modern marketplaces, where the platform or the seller routinely gathers detailed user profiles, we study a novel learning theoretic model that simultaneously involves information and mechanism design. Specifically, we consider the economic setting recently introduced by Bergemann et al. (2022), where in addition to the menu of quality-price pairs, the seller offers information on the value of the match between product quality and buyer's taste via a signaling scheme. We relax the assumption that the seller knows the buyers' belief about the distribution of tastes and study the sample requirements of designing a revenue maximizing scheme. We consider both the batch setting where we have access to data from a set of i.i.d. buyers and an online demand query model where we observe the buyers' behaviors to seller's schemes. Despite the apparent non-convexity of the problem, we also give the first FPTAS to compute a scheme that maximizes the revenue within an arbitrarily small additive loss, which was left open by Bergemann et al. (2022). Overall, this brings a new learning perspective in asymmetric economic settings where buyers and sellers know different types of information.

How Many Samples Are Needed to Determine Causal Direction? Sharp Minimax Bounds for Bivariate LiNGAM

We study how many observations are needed to determine the causal direction between two linearly related variables. Classical LiNGAM theory shows that independent non-Gaussian disturbances identify the direction, but does not quantify the difficulty when the causal effect is weak or the disturbances are nearly Gaussian. Let $β$ bound the absolute structural coefficient from below, let $ν$ measure each standardized disturbance's distance from Gaussianity, and let the disturbance scales lie in $[\underlineσ,\overlineσ]$. We prove the sharp local minimax law \[ N_2^\star(β,ν,δ) \asymp \frac{\log(1/δ)} {d_β^2+β^2ν^2}, \qquad d_β= \left[β^2- \left(1-\frac{\underlineσ^2}{\overlineσ^2}\right)\right]_+. \] Previous theory established population identifiability or assumed a fixed separation between the two directions. By contrast, we establish the sharp sample complexity as a joint function of edge strength, distance from Gaussianity, and scale uncertainty, and characterize when identification comes from non-Gaussian dependence or from covariance alone. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.

Bagging Robustly Learns VC Classes with Linear Sample Complexity

We revisit the problem of learning predictors robust to adversarial examples at test-time. We prove that VC classes are adversarially robustly learnable with sample complexity linear in the VC dimension $d$, providing an exponential improvement over the previous upper bound of Montasser, Hanneke, and Srebro (2019). Remarkably, this result is achieved with a simple improper algorithm that combines the classic heuristic bagging (bootstrap aggregation) of Breiman (1996) with robust empirical risk minimization (RERM). Our algorithm computes RERMs on $O(d^\star)$ independent bootstrap samples and outputs their majority vote, where $d^\star$ denotes the dual VC dimension. We complement this result with a lower bound showing that this is unavoidable: in general, any learner in this oracle model requires $Ω(d^\star)$ calls to an RERM oracle, even when given arbitrarily many training examples.

Tensor-normal maximum likelihood estimation at the operator-norm sample threshold

Let $X_1,\ldots,X_n$ be independent Gaussian tensors in $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_k}$ whose covariance is a Kronecker product of $k$ unknown positive-definite factors, and put $D=\prod_{a=1}^k d_a$ and $d_{\max}=\max_a d_a$. A recent result of Franks et al. (2026) established condition-number-free nonasymptotic guarantees for the tensor-normal maximum likelihood estimator under the sample threshold $nD\gtrsim k^2 d_{\max}^3$. They asked whether the cubic dependence on $d_{\max}$ could be replaced by the operator-norm scale $d_{\max}^2$. We answer this question affirmatively. We prove that, for $t\geq 1$, the maximum likelihood estimator exists uniquely with high probability whenever $nD\geq Ck^2 d_{\max}^2 t^2$, and satisfies $d_{\mathrm{FR}}(\widehatΘ,Θ)\leq Ct\sqrt{k}\,d_{\max}/\sqrt{n}$ and $d_{\mathrm{FR}}(\widehatΘ_a,Θ_a)\leq Ct\sqrt{k d_a}\,d_{\max}/\sqrt{nD}$. For every mode of largest dimension, we also obtain the sharp Thompson bound $d_{\mathrm{op}}(\widehatΘ_a,Θ_a)\leq Ct\,d_{\max}/\sqrt{nD}$. No sparsity, condition-number bound or warm start is assumed. For fixed $k$, the threshold has the information-theoretically optimal dependence on $d_{\max}$, and the displayed rates for the full precision and the largest factor match Gaussian minimax lower bounds up to a factor $\sqrt{k}$. The proof extends a random Gram bound for local group-orbit directions to the full local Lie algebra, transports it to a fixed Thompson ball by exact conjugation, and combines sensitivity of a constrained maximum likelihood estimator with an equivariant Kirszbraun extension and Gaussian concentration. This removes the Frobenius-to-operator loss responsible for the previous extra factor $d_{\max}$ and resolves the explicit open problem posed in the earlier work.

An Optimal Agnostic PAC Algorithm

Let $H\subseteq\{-1,+1\}^X$ be a class of finite VC dimension $d\ge1$. Writing $L$ for the binary risk and $L^*=\min_{h\in H}L(h)$, we construct a learner achieving the statistically optimal risk bound: from an i.i.d.\ sample of size $n$, for every $0<δ\le 1/2$, with probability at least $1-δ$, \[ L(\widehat h) \le L^*+ 7\cdot10^8\left( \sqrt{\frac{L^*(d+\log(1/δ))}{n}} +\frac{d+\log(1/δ)}{n} \right). \] This settles the sample complexity of agnostic PAC learning up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Györfi, and Lugosi [A Probabilistic Theory of Pattern Recognition, Springer, 1996].

The Sample Complexity of Distributionally Robust PAC Learning under Cressie--Read Divergences

We study distributionally robust PAC learning for the $0$--$1$-loss, where adversarial perturbations of the data distribution are constrained by a Cressie--Read divergence of order $k>1$ and radius $ρ\geq 0$. For hypothesis classes with VC dimension $d$, we establish realizable and agnostic sample-complexity bounds tight up to constant and logarithmic factors, respectively; ordinary empirical risk minimization attains both rates up to logarithmic factors. For target accuracy $\varepsilon\in(0,1)$ and confidence $δ\in(0,1)$, their respective orders are \[ \max\!\left\{\frac{1}{\varepsilon}, \frac{ρ^{\frac 1{k-1}}}{\varepsilon^{k_\star}} \right\}\cdot(d+\log δ^{-1}) \qquad\text{and}\qquad \max\!\left\{\frac{1}{\varepsilon^2}, \frac{ρ^{\frac1{k-1}}}{\varepsilon^{k_\star\vee 2}} \right\}\cdot(d+\log δ^{-1}), \] where $k_\star={k}/{(k-1)}$. For every fixed $ρ>0$, robustness changes the realizable $\varepsilon$-dependence from $\varepsilon^{-1}$ to $\varepsilon^{-k_\star}$ as $\varepsilon\downarrow0$. In the agnostic case, for $1<k<2$, robustness changes the $\varepsilon$-dependence from $\varepsilon^{-2}$ to $\varepsilon^{-k_\star}$, whereas for $k\geq2$ the exponent remains the classical $2$, with nontrivial $ρ$-dependence. Building on the known scalar reduction of robust $0$--$1$ risk to ordinary classification error, our analysis reveals a scale-sensitive interaction between the statistical estimation of classification error and its amplification by robustness, sharply explaining the transition in the agnostic rate. We extend the previously studied $χ^2$-divergence case to every Cressie--Read order $k>1$, close its upper--lower gaps, and recover standard PAC learning rates as $ρ\to0$, unlike previous bounds that fail to interpolate correctly in this limit.

Sample Complexity of Multicalibration for Multilevel Properties

Calibration requires a predictor to be unbiased after conditioning on its own predictions. Multicalibration asks for this guarantee simultaneously across a collection of groups. Many prediction tasks ask for several related features of the same conditional outcome distribution: variance is defined relative to the mean, skewness relative to both mean and variance, and conditional value at risk relative to a quantile. We study multicalibration for a sequence of $k$ properties in which each property is identifiable once the preceding properties are fixed. This framework includes Bayes pairs but does not require the properties to arise from a single loss. For every fixed $k\ge2$, we establish matching upper and lower sample-complexity bounds up to logarithmic factors under regularity conditions. Even with only polylogarithmically many binary groups, achieving multicalibration error $\varepsilon$ requires $\widetildeΩ(\varepsilon^{-(k+2)})$ samples. Conversely, for any finite group family $\mathcal G$, we give a randomized learner using $O(\varepsilon^{-(k+2)}+\varepsilon^{-2}\log|\mathcal G|)$ samples. Thus the sample complexity is $\widetildeΘ(\varepsilon^{-(k+2)})$ for polynomial-size group families. We instantiate the theory for three canonical examples.

Interaction Is Not Necessary for Order-Optimal 1-Bit Mean Estimation

This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-λ,λ]$ and absolute $k$-th central moment at most $σ^k$, where $k>1$ is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy $ε$ and confidence $1-δ$, its sample complexity scales as \[ \log\fracλσ + \begin{cases} (σ/ε)^2\log(1/δ), & k>2,\\ (σ/ε)^2\log(σ/ε)\log(1/δ), & k=2,\\ (σ/ε)^{k/(k-1)}\log(1/δ), & 1<k<2, \end{cases} \] up to constants depending only on $k$. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries \citep[Open Problem~1]{lau2026open}.

Tight Sample Complexity for Low-Rank Adaptation: Matching Bounds and Rank Selection

Low-Rank Adaptation (LoRA) has become the standard mechanism for fine-tuning large pretrained models, yet its statistical properties remain only partially understood. Existing generalization results provide upper bounds of the form O~(sqrt(rd/n)) or O~(rd/n), but a matching lower bound is missing, and the question of how to choose the LoRA rank r has no formal answer. Both gaps are closed here. A local Rademacher argument establishes an upper bound of O~(rd/n) on the excess risk of the empirical risk minimizer over rank-r LoRA, whenever the target adaptation has rank at most r. A matching minimax lower bound of Omega(rd/n) is then proved via a Fano-type packing of the rank-r subspace of R^{d x d}; the bound applies to any estimator whose output lies in the rank-r LoRA class. Combining the two yields a rank-selection dichotomy. For the constrained empirical risk minimizer, the optimal rank equals the intrinsic rank r*, and over-ranking strictly hurts. For adaptive estimators of the nuclear-norm-then-truncate type, over-ranking is harmless and the rate saturates at Theta~(r* d / n) regardless of r. Taken together, the three results characterize the statistical complexity of LoRA fine-tuning within the well-specified locally quadratic regime, and identify the empirically observed over-parameterization penalty as a property of unregularized empirical risk minimization rather than of the LoRA class itself. Predictions of the theory are verified on a synthetic trace-regression benchmark and on real LoRA fine-tuning across three (model, task) configurations covering DistilBERT and RoBERTa on SST-2 and MRPC. All configurations exhibit the predicted U-shape in validation loss, with two showing statistically significant loss inflation at large ranks (paired permutation p = 0.016).

Actions Have Consequences: Detecting Outcome Performativity using Intervention Testing

In many domains such as Palliative Care, Credit Assignment and Recommender Systems, predictions may causally influence the outcomes they predict. This phenomena is known as Outcome Performativity. This paper formalises an approach for detecting Outcome Performativity using prediction intervention called Outcome Performativity A/B Detection (OPAB). OPAB enables the detection of Outcome Performativity by assessing the dissimilarity in outcome distributions produced by different predictions groups (interventions). If that dissimilarity is significant, Outcome Performativity is detected. We derive sample complexity bounds for OPAB under various Outcome Performative assumption classes which we empirically validate. Results show that detecting Outcome Performativity using OPAB is achievable in numerous cases. Results also show the presence of regions of indistinguishability which describe settings where the allotted number of interventions are insufficient for detecting Outcome Performativity. The results of which have broader practical implications for the detectability of Outcome Performativity in settings where samples are scarce, cost-prohibitive or potentially unethical to obtain. The paper concludes with a case study on the efficacy of OPAB on the Open Bandits dataset, and provides directions for future work.

Learning Distributions from Multiple Data Providers

Motivated by learning from heterogeneous and overlapping data providers, we study a stylized model of distribution learning from restricted conditional samples. The goal is to learn an unknown distribution $p$ on a finite domain $[n]$. The learner is given a fixed family of queryable sets $\mathscr{S} \subseteq 2^{[n]}$, and each query to $S \in \mathscr{S}$ returns an independent sample from the conditional distribution $p(\cdot \mid S)$. Learnability is governed by the co-occurrence graph associated with $\mathscr{S}$: two domain elements are adjacent if they appear together in some queryable set. Pointwise consistency is achievable when this graph is connected on the target support. PAC learning requires more: it is possible when the co-occurrence graph is complete. The optimal sample complexity of PAC learning ranges from nearly linear to quadratic. Every query family with complete co-occurrence graph admits sample complexity $\widetilde O(n^2/ε^2)$, and this bound is tight in the worst case. On the other hand, if $[n]$ is queryable then ordinary sampling improves the bound to $Θ(n/ε^2)$, and this cannot be improved further even if every set is queryable. More generally, we identify hierarchical comparabilityas a sufficient structural condition on $\mathscr S$ under which the optimal complexity is nearly linear, $\widetilde Θ(n/ε^2)$, with pairwise query families as a canonical example. Finally, the full range of polynomial rates between linear and quadratic is attainable: for every $α\in (1,2)$, there exists a query family with optimal PAC rate $\widetilde Θ(n^α/ε^2)$.

Local Regularization Does Not Characterize Multiclass PAC Learnability

Local regularization assigns each hypothesis a test-point-dependent score and predicts with a minimum-score hypothesis consistent with the sample. Asilis et al. asked whether this principle characterizes multiclass PAC learnability. We give a negative answer. There is a countable class of Daniely--Shalev-Shwartz dimension at most two with realizable PAC sample complexity \[ O\!\left(\frac{1}{\varepsilon}\log\frac{1}δ\right), \] that no local regularizer learns. Hypotheses are edges of complete graphs and instances are tournaments. At a test tournament, the scores fix an edge ranking while the training sample independently removes competitors. Cyclic triangles force enough inversions that surviving competitors produce constant population error at arbitrarily large sample sizes.

Efficient Learning of Truncated Boolean Product Distributions: Influence to the Rescue

Learning the natural parameters $z \in \mathbb{R}^n$ of discrete distributions $μ_z$ from independent samples constrained to a subset $S \subseteq \{0,1\}^n$ is a foundational challenge in high-dimensional statistics. Existing methods for efficiently estimating truncated Boolean product distributions, notably the work of [Fotakis et al' COLT'20, Algorithmica '22], require either strong local connectivity assumptions on $S$ -- a property denoted fatness -- or stringent anti-concentration assumptions and necessitate the total mass of the truncation set to be a constant with respect to $n$. Moreover, the results in [Fotakis et al' COLT'20, Algorithmica '22] suffer from sample complexities that scale as $Ω(2^n)$ if the mass of $S$ is exponentially small in $n$. In this work, we circumvent these limitations by analyzing the geometry of $S$ under the measure $μ_z$. We refine the existing parameter estimation guarantees under the fatness assumption, improving the prior sample complexity to $O( \log n / ε^2)$ for $\ell_\infty$-recovery, matching the untruncated minimax rate. We further generalize fatness using the notion of influence utilized in the analysis of Boolean functions and provide sufficient conditions for efficient inference. Notably, unlike previous work, our method does not require sampling at arbitrary parameterizations of the model. Lastly, we establish a theoretical lower bound demonstrating the sample complexity exhibits an intrinsic exponential dependence on the width of the model and the minimum distance between elements in the set.

Total Variation Distance Estimation in Autoregressive Models

Modern LLM deployments use a number of implementation choices and inference optimizations (e.g., batching, custom kernels, and quantization) on top of fixed weights, so two engines serving "the same model" can produce meaningfully different distributions. We study the problem of estimating the total variation (TV) distance between two length-$n$ autoregressive distributions to additive error $\varepsilon$, under three access models. (1) Under sample access, we use $\widetilde{O}(n^2 K/\varepsilon^2)$ queries, where $K$ is the maximum support of the next-token distribution. This improves upon the $\widetilde{O}(n^3 m/\varepsilon^5)$-query estimator of Meel et al. (2025), where $m \geq K$ is the total size of the token alphabet. (2) Under logit access, we use $O(n/\varepsilon^2)$ queries, and this is tight. (3) Under noisy logit access, we smoothly interpolate between the above two guarantees: if probability values are given to relative error $σ$, we use $\widetilde{O}((n+n^2σ^2)/\varepsilon^2)$ queries. We complement our theoretical results with an empirical evaluation of our algorithms, for example measuring the distance between SGLang and vLLM serving identical weights. Our experiments highlight the robustness and practicality of estimating the total variation distance, which remains estimable where the KL divergence is infinite. Our code is available at https://github.com/XunZhiyang/llm-tv-estimation.

Mixing-Free and Signal-Optimal Learning of Gaussian Graphical Models from Glauber Dynamics

Gaussian graphical model selection is usually studied under independent sampling, but in many applications the data arise as a single trajectory of a dependent stochastic process. We study exact recovery of the graph from one trajectory of random-scan Gaussian Glauber dynamics. Existing techniques for this problem either inherit the mixing time of the chain, which can be super-polynomial in the dimension $p$ without strong assumptions, or are suboptimal in the minimum normalized edge strength $κ$. We propose two algorithms that are mixing-free and attain the $κ^{-2}$ dependence of the information-theoretic lower bounds. Both instantiate a shared dueling-neighborhood search meta-algorithm with a local statistic built directly from the update sequence. For every fixed precision matrix and deterministic initialization, the first algorithm fits a least-squares regression at the updates of each node and has pointwise recovery horizon $\widetilde O(pd^{2}/κ^{2})$, where $d$ is the maximum degree. Its horizon depends logarithmically on a local conditioning quantity and on the initialization potential. The second algorithm is based on counting occurences of a specific update pattern and requires $\widetilde O(pd^{4}/κ^{2})$ updates, with no dependence on any condition number. The central technical challenge is that both statistics are built from dependent, non-stationary observations. Our analysis tackles this by demonstrating how to extract fresh Gaussian innovations from the update sequence, which yields mixing-free control of appropriate quantities. Neither the algorithms nor their analyses invoke stationarity, a spectral gap, or mixing conditions.

Tight Sample Bounds for Renyi and Min-Entropy Estimation

Estimating entropy from samples is fundamental in information theory and property testing. Shannon entropy measures average uncertainty and can be estimated to constant additive accuracy over a $k$-symbol alphabet using $Θ(k/\log k)$ samples. Min-entropy depends only on the most likely symbol. Both are special cases of order-$α$ R'{e}nyi entropy, $H_α$. We characterize the sample complexity of estimating min-entropy and R'{e}nyi entropy for $k$ and integer $α>1$; our lower bounds also hold for noninteger $α\ge1.001$. We prove that min-entropy estimation to constant additive accuracy has sample complexity $Θ(k\log k)$. The upper bound uses the largest empirical frequency and concentration via dyadic grouping. The matching lower bound hides a slightly heavier symbol at a uniformly random location. Thus, min-entropy requires $Θ(\log^2 k)$ more samples than Shannon entropy and corrects a previously stated $Θ(k/\log k)$ characterization. For every integer $2\leα\le c_0\log k$, we prove the matching fixed-accuracy bound $Θ_{c_0}(αk^{1-1/α})$. Previous results gave $Ω_α(k^{1-1/α})$ for fixed integer $α>1$ and $O_{c_0}(α^2k^{1-1/α})$ for all integer $α>1$. Our upper bound analyzes an unbiased falling-factorial estimator based on $α$-way collisions, while a hidden-heavy-coordinate construction gives the matching lower bound and shows that the factor $α$ is unavoidable. For every real $1.001\leα\le c_0\log k$, we prove the uniform lower bound $Ω_{c_0}(αk^{1-1/α})$. Finally, since $0\le H_α(p)-H_\infty(p)\le\log k/(α-1)$, min-entropy uniformly approximates $H_α$ when $α$ is a sufficiently large multiple of $\log k$. Combining this reduction with our min-entropy bounds gives $Θ_\varepsilon(k\log k)$ sample complexity in the high-order regime.

Semi-Supervised Conditional Generative Learning through Stochastic Interpolation and Sufficient Representations

Conditional generative modeling remains a challenging problem in semi-supervised settings where labeled data is scarce but unlabeled samples are abundant. To effectively leverage structural information embedded within the unlabeled dataset and compensate for sparse conditioning signals, we propose a semi-supervised framework combining conditional stochastic interpolation with low-dimensional latent representations. RepG decomposes generation into two stages: label-dependent latent sampling and high-dimensional reconstruction. This isolates the supervised learning of conditional dependencies to a low-dimensional space, requiring few labels while utilizing the abundant unlabeled data purely for reconstruction. Theoretically, we establish an error decomposition showing that the Kullback-Leibler divergence of RepG comprises stage-wise estimation errors and a structural bias quantified by conditional mutual information. For deep neural network estimators, we derive non-asymptotic convergence rates proving that RepG significantly improves sample complexity. By confining the supervised estimation burden to the low intrinsic dimension of the latent representation, RepG achieves a strictly faster convergence rate. Complemented by a minimax lower bound, our theoretical results demonstrate that this method effectively mitigates the curse of dimensionality inherent in direct ambient-space generative modeling.

Optimal Top-$k$ Identification from Pairwise Comparisons

We study the active learning problem of fixed-confidence top-$k$ identification from noisy pairwise comparisons. In this problem, an algorithm sequentially chooses pairs of items to compare, observes the outcomes, and stops when it can return the set of top-$k$ items with error probability at most $δ$. The objective is to design such a $δ$-correct procedure that minimizes the expected number of comparisons (the sample complexity). This problem falls within the broader literature on fixed-confidence pure exploration in bandit models, where a common target is asymptotic optimality: the algorithm's expected sample complexity matches the information theoretic lower bound as $δ\to 0$. Asymptotically optimal procedures have been developed for a range of fixed-confidence pure-exploration problems, however to the best of our knowledge, for top-$1$, or more generally top-$k$ identification from pairwise comparisons under latent utility models an asymptotically optimal algorithm has not been established. In this setting, we develop such an algorithm. We characterize the structure of the lower bound and formulate it as a saddle-point problem. This structure enables a computationally efficient primal-dual procedure that learns the asymptotically optimal comparison allocation online. We then construct an adaptive comparison-allocation algorithm that tracks the allocation learned by the primal-dual procedure and prove it is asymptotically optimal.

Near-Optimal Learning of Gaussian Sobolev Operators

A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time. Whereas smooth operators can be approximated efficiently, i.e., with at least algebraic convergence in the amount of training data, learning finitely regular operators is known to be less efficient. The reason is an intrinsic curse of sample complexity, which allows only subalgebraic sample complexity rates. This fact makes it all the more important to develop algorithms which provably achieve these rates. In this work, we present a fully data-driven algorithm, termed Hermite-PCA approximation, for learning Gaussian Sobolev operators with near-optimal sample complexity. It employs principal component analysis and weighted least-squares methods and is therefore computationally efficient. Moreover, it is spectral, in the sense that it achieves faster (and near-optimal) convergence the higher the Sobolev regularity. We provide a full error analysis of this algorithm, taking into account all sources of error, along with numerical experiments that verify our theoretical results and empirically confirm the efficacy of Hermite-PCA approximation for learning Sobolev operators.

Best-Arm Identification with Generative Proxy

Best-arm identification is a canonical model for data-driven decision-making, but in many applications each reward observation is costly. Motivated by the growing availability of cheap predictions from machine learning and large language models, we study fixed-confidence best-arm identification in which each costly reward pull is paired with a cheap but correlated proxy score. The marginal mean of the proxy can be estimated offline and is treated as known, whereas its correlation $ρ$ with the reward, which governs how much the proxy helps, is unknown and must be learned online in pair with real rewards. We show that a control-variate adjustment turns this model into a heteroscedastic identification problem whose oracle sample complexity improves by residual variance $1-ρ^2$. The central difficulty is that the correlation must be learned from the same costly samples that identification consumes online, and that a plug-in estimate of the residual variance is anti-conservative and can compromise correctness. We propose PROBE (PRoxy OLS for Best-arm Exploration), a phase-elimination algorithm that directly maintains an upper certificate on the residual variance with an ordinary least squares fit, whose exact chi-square law keeps the certificate valid regardless of the unknown correlation. We prove that PROBE is $δ$-PAC and attains the known-correlation oracle sample complexity up to a constant multiplicative factor and a constant additive calibration cost. The guarantee extends to the $(ε,δ)$-PAC setting under minimal changes to the algorithm. Numerical experiments on synthetic instances and on an auto-loan pricing replay with large language model and tabular proxies confirm that the sample savings of PROBE scale with the strength of the reward-proxy correlation, exactly as the theory predicts.

A Hierarchy of Policy Learning Problems

Policy learning has received substantial attention with the goal of learning policies from observational data for decision-making. A majority of work in this space has focused on developing algorithms for computing policies that minimize regret compared to the optimal policy. However, in many practical settings, there is insufficient data to obtain low regret. As a result, recent work has shifted attention to alternative objectives, most notably, studying whether it is possible to learn an improving policy that statistically significantly outperforms baseline policies. We argue that there is substantial merit in studying a broader range of policy learning problems. When there is insufficient data to learn an improving policy, there may still be useful questions that can be answered. To this end, we provide a mathematical framework for studying the relationships between policy learning problems. We formalize three problems within our framework: beyond the optimal policy problem and the improving policy problem, we also propose the policy existence problem, which aims to determine if an improving policy exists. Within our framework, we show that the policy existence problem reduces to the improving policy problem, which in turn reduces to the optimal policy problem; these reductions prove that each problem is at least as easy as the next one (in sample complexity). A key question remains: is this hardness strict? We provide partial answers. First, the gap between the optimal policy and improving policy problems is strict. For the improving policy and policy existence problems, we prove that a sublinear polynomial gap exists under natural conditions on improving policy learning algorithms. Thus, we may be able to answer questions about the existence of an improving policy even when we cannot find one. These results highlight the value in studying a broader range of policy learning problems.

Sample Complexities of Estimating Gumbel--Max Watermark Proportions with and without Reduction to Pivotal Statistics

Watermarking promises statistical traceability of large language model (LLM) uses, but real documents rarely arrive as purely human-written or purely LLM-generated. This motivates a quantitative question beyond detection: what proportion of a document is generated from a pre-specified watermarked LLM? We study this watermark proportion estimation problem under the Gumbel--max watermarking mechanism, treating the next-token prediction distributions as unknown and arbitrary nuisance parameters subject to a non-degeneracy condition. We compare two observation regimes: in the full observation regime, the estimator observes the pseudorandom vector and the selected token at each position; in the more prevalent setting of pivotal reduction, it observes only a scalar pivot, which follows a one-dimensional Uniform--Beta mixture distribution. Under pivotal reduction, we develop a Laguerre-polynomial estimator and establish a matching information-theoretic lower bound for the sample complexity. For full observation, we introduce an event-counting estimator and show a matching lower bound, yielding a substantially smaller sample complexity. As our results imply, although reducing to pivotal statistics is an elegant and prevalent choice, it is not always sample-efficient for estimating the proportion of watermarks.

The Fundamental Limits of Valid Transport Map Estimation

Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.

Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS

This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

Fast algorithms for learning a Gaussian under halfspace truncation with optimal sample complexity

We study the fundamental problem of learning a high-dimensional Gaussian truncated to an unknown halfspace. Lee, Mehrotra and Zampetakis (FOCS'24) recently obtained the first polynomial time algorithm for this problem, but their resulting sample and time complexity bounds are not optimal. Under non-trivial truncation, for any target accuracy $\varepsilon > 0$ and dimension $d$ we give an efficient algorithm that uses $n = \tilde{O}(d^2/\varepsilon^2)$ samples and learns the underlying Gaussian to error $\varepsilon$ in total variation distance. Our algorithm is also fast: its runtime is dominated by the cost of computing the empirical covariance matrix. Both our sample and time complexity are optimal in terms of $d$ and $\varepsilon$ even without truncation: in this regard, we can learn a Gaussian under halfspace truncation for free. The key ingredient behind our result is a novel reinterpretation of the low-degree moments of the truncated Gaussian in terms of a relative truncation parameter. This relative truncation parameter uniquely determines the parameters of the untruncated Gaussian and enables direct parameter recovery. This reinterpretation allows us to circumvent the time intensive projected stochastic gradient descent procedure that is widely used in learning under truncation.

Sample complexity of unbalanced entropic OT

Optimal transport (OT) has become a central language for comparing probability measures, but exact balanced OT is often both too rigid for data with missing, created, or destroyed mass and subject to unfavorable high-dimensional sample complexity. Entropic regularization and unbalanced relaxations address these limitations in complementary ways. Entropy smooths the geometry, improves statistical behavior, and enables fast Sinkhorn-type algorithms, while unbalanced marginal penalties replace hard conservation constraints by divergence terms adapted to noisy empirical data. This paper studies the sample complexity of entropic unbalanced OT at the level of the optimal coupling, rather than only the scalar transport value. We develop a translation-invariant dual formulation, prove compactness and strong convexity properties for the intrinsic dual variables, and convert these geometric estimates into high-probability finite-sample bounds for empirical couplings. The results clarify why regularization is a practical necessity in machine learning applications: it softens the curse of dimensionality, reduces the number of samples needed for stable transport estimation, and keeps the resulting estimators compatible with scalable Sinkhorn-type solvers.