Optimal Sample Complexity

Streaming principal component analysis (PCA) seeks to recover a leading spectral subspace in a single pass over a data stream. We give a new analysis of the ubiquitous Oja's algorithm [Oja82] for the most general, gap-free variant of this problem, where no eigengap assumptions are made on the underlying mean matrix, complemented by a nearly-matching lower bound. Prior works achieving near-optimal rates for streaming PCA either required gap assumptions [JJK+16, HNWW21], or were limited to rank-one updates [AZL17, Lia23]. Our proof only uses a second moment bound on the individual stochastic updates, bypassing the almost sure bounds needed by prior near-optimal analyses, and the analogous offline matrix Bernstein bound. We also extend our result to a Rayleigh quotient notion of approximate PCA, addressing an open question of [JJK+16]. As our main application, we give gap-free differentially private PCA guarantees for sub-Gaussian data, settling Conjecture 1.1 of [Bro26] up to logarithmic factors.

We study the estimation of a $K$-dimensional simplex from $N$ i.i.d.\ points sampled uniformly from its interior; the observations are convex combinations of $K+1$ unknown prototypes. Existing polynomial-time estimators need cubic per-sample work or $O(NK)$ storage and are impractical at $N\sim 10^6$--$10^8$. We propose DeepMVSA, which re-expresses the minimum-volume principle in neural implicit form: a lightweight coordinate network generates the mixing weights and a triangular LU-type parameterization the dual simplex matrix, reducing the trainable-state memory to $O(K^2)$, independent of $N$, and the cost per data pass to $O(NK^2)$. We prove a non-asymptotic sample-complexity bound of the polynomial-time benchmark order for a localized surrogate estimator; an oracle inequality for every global minimizer of the neural objective, with volume-inflation control and an explicit shrinkage bias; a conditional end-to-end error budget separating statistical, approximation, optimization, and enclosure-residual terms on an explicit envelope event; and two-point lower bounds: at any noise level $σ>0$ fixed independently of $N$, the $N^{-1/2}$ scaling is unimprovable in its $N$-exponent. Experiments with up to $N=10^8$ synthetic observations are consistent with the predicted accuracy and scaling, and feasibility on real scenes of $\sim 10^7$ pixels is demonstrated.

We study the problem of \emph{adversarially robust} PAC learning. In this framework, the learner observes independent samples from an unknown distribution over $\mathcal{X} \times \{0,1\}$, as in classical PAC learning. However, given a perturbation map $\mathcal{U} : \mathcal{X} \to 2^{\mathcal{X}}$ known to the learner, the goal is to output, with high probability, a predictor that correctly classifies \emph{every} perturbation $z \in \mathcal{U}(x)$ of most future examples $(x,y)$ drawn from the same underlying distribution. We determine the \emph{optimal} $\mathcal{U}$-independent sample complexity of this problem in both the realizable and agnostic settings. More specifically, for every concept class $\mathcal{H}$ of $\operatorname{VC}$ dimension $d$, we prove upper bounds of $\mathcal{O} \big( d/ε+ \log(1/δ)/ε\big)$ in the realizable setting and $\mathcal{O} \big( d/ε^2 + \log(1/δ)/ε^2 \big)$ in the agnostic setting, together with an optimal first-order refinement of the latter. These bounds match the corresponding lower bounds for classical PAC learning. Consequently, and perhaps surprisingly, adversarial robustness incurs \emph{no additional} distribution-free statistical cost, uniformly over all perturbation maps. Our bounds improve exponentially on those of [Montasser, Hanneke, and Srebro; COLT '19]. On the technical side, we present short and elementary proofs based on a new algorithmic principle that we call \emph{binomial-bagging}. We believe that binomial-bagging and its analysis may be of independent interest.

Sign-based methods reduce communication costs in distributed environments, but aggregating local signs can introduce bias when data are heterogeneous. As a result, existing sign-based variance reduction methods fail to obtain the optimal convergence rates. In this paper, we solve this problem and obtain optimal rates for both nonconvex stochastic and finite-sum optimization. We first give a counterexample showing that majority voting can fail to approach stationary points even with exact local gradients. Motivated by this limitation, we propose tracking the global gradient at the server through unbiased compression of recursive gradient increments. As a result, we can obtain the convergence rates of $O(\sqrt{d/K}+\sqrt d (a/(nK))^{1/3})$ for the $\ell_1$-norm and $O(\sqrt{a/K}+\sqrt a/(nK)^{1/3})$ for the $\ell_2$-norm. Here, $K$ is the iteration number, $n$ is the number of workers, $d$ is the dimension, and $a=1+ω$, with $ω$ denoting the compressor's relative variance. For finite-sum problems with $M$ components, we combine periodic exact gradient refreshes with compressed component-gradient differences. The resulting total sample complexities are $O(M+d\sqrt{aM}ε^{-2})$ and $O(M+a\sqrt M\ epsilon^{-2})$ for $\ell_1$ and $\ell_2$ gradient norms at most $ε$, matching the corresponding bounds in centralized settings.

In an important recent work, Blanc (2026) gave an algorithm for robustly learning Boolean concept classes with respect to a fixed distribution that outputs a (randomized) classifier achieving the optimal error of $η+ \varepsilon$ where $η$ is the noise rate. In contrast, it is well known that deterministic hypotheses cannot achieve error less than $2η+ \varepsilon.$ Blanc's algorithm is computationally inefficient, and the main problem left open in his work is to find a polynomial-time algorithm given access to an oracle for empirical risk minimization (ERM). In this paper, we resolve this problem and give such an algorithm. Perhaps surprisingly, our techniques make crucial use of various types of no-regret learners. Additionally, we give an efficient algorithm (no ERM oracle required) for robustly learning any function class that admits sandwiching polynomials with respect to hypercontractive distributions. As one consequence, we give the first polynomial-time algorithm for robustly learning a halfspace with respect to Gaussian marginals that achieves error $η+ \varepsilon$ for any constant $\varepsilon$.

We show that logconcave probability measures along the Gaussian cooling path have thin-shell stability, generalizing the thin-shell theorem. This result leads to improved complexity for the fundamental problem of sampling an arbitrary logconcave distribution from a cold start. For (near-)isotropic logconcave distributions, the complexity is nearly $n^{2.5}$, improving the previous bound of $n^{2.75}$, and matching the complexity of the abstract Speedy walk.

Traditional variance reduction methods (e.g., SPIDER, SARAH, STORM) have been extensively investigated for improving the convergence rates of stochastic optimization. These techniques typically maintain a sequence of estimators for a single function (or gradient) across iterations. However, what if we need to track multiple functions, but can only access stochastic samples of $\mathcal{O}(1)$ functions at each iteration? This scenario arises in an important emerging family of finite-sum coupled compositional optimization (FCCO) problems of the form $\frac{1}{m}\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where each $g_i$ is accessible only through a stochastic oracle. The key challenge is to track $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ over time, where $\mathbf g(\mathbf{w})$ has $m$ blocks but only $\mathcal{O}(1)$ blocks can be probed for their stochastic values at each step. To address this challenge, we propose a novel Multi-block-Single-probe Variance Reduction (MSVR) estimator to efficiently trace $\mathbf g(\mathbf{w})$ under partial block sampling. Building on the MSVR estimator, we develop several algorithms for FCCO problems, achieving improved sample complexities for non-convex, convex, strongly convex, and Polyak-Łojasiewicz (PL) objectives. We further obtain an improved dependence on $m$ when the outer function gradients $\nabla f_i$ are linear. Empirical studies on multi-task deep AUC maximization further demonstrate the superior performance of the proposed estimators.

We revisit Valiant's algorithm (Commun. ACM'84) for learning $n$-variable CNF formulas with clause size $k$ and variable degree $d$ from i.i.d. uniform random solutions in the local lemma regime. For fixed $t\geq1$, under $k\gtrsim(1+1/t)\log d$, Valiant's algorithm achieves total variation error $\varepsilon$ with $\widetilde{O}(n^{\lceil t \rceil}/\varepsilon)$ sample complexity. For $t>1$, we prove a matching lower bound for Valiant's algorithm. At $t=1$ (covering $0<t<1$), we show Valiant's algorithm has optimal sample complexity up to logarithmic factors by an information-theoretic lower bound $\widetildeΩ(n/\varepsilon)$.

Channel simulation has recently emerged as a useful component in machine learning systems where samples from a prescribed probability distribution are to be compressed. Yet, general channel simulation algorithms often suffer from high computational costs, random stopping times or, in the worst case, can require generating an infinite number of shared random samples. We introduce a scheme for both exact and approximate simulation of discrete-to-continuous channels which conversely uses a fixed number of random samples, and therefore has a runtime independent of the channel and the input. Unlike existing channel simulation schemes which generate a sequence of independent samples from a proposal distribution, our approach generates one sample, or alternatively a fixed number of samples, from each potential target distribution. We then apply a latent permutation to the samples before performing sample selection using an exponential race. Our scheme provides a flexible tradeoff between the number of generated samples and the compression rate. Using polar and multilevel coding, we scale our approach to handle long blocklengths in $O(n \log n)$ time in order to benefit from reduced per-symbol overhead. We conclude by demonstrating applications to variable-rate compression with stochastic VQ-VAEs and communication-efficient differentially private distributed mean estimation via exact simulation of the Gaussian mechanism.

We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $\pi(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:\mathbb{R}^d\to\mathbb{R}$ is convex and globally $G$-Lipschitz. For the Moreau-smoothed target $\pi_\lambda$ and the MYULA invariant law $\widehat\pi_{\lambda,h}$, we prove $$\sqrt m\,W_2(\pi_\lambda,\widehat\pi_{\lambda,h}) =O(h)+\widetilde O(h^{3/4})$$ under $0<h(L_f+\lambda^{-1})\le c$, with only logarithmic dependence on $\lambda^{-1}$ in the error coefficients. Combining this estimate with the Moreau approximation bias yields $\widetilde O(\varepsilon^{-4/3})$ iterations to achieve $\sqrt m\,W_2(\mu_N,\pi)\le\varepsilon$, for fixed model parameters and initialization. The proof combines a discrete Poisson corrector with active-trace estimates and a shared-noise bound for the exact--Euler two-point curvature.

We investigate the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in any fixed Euclidean dimension. The potential is $\mu$-strongly convex and $L$-smooth, with an unknown mode in the ball of radius $\mu^{-1/2}$ about the origin. We have access to unbiased stochastic oracles with the variance at most $\sigma^2$. For every $\sigma^2\ge0$ and total variation (TV) accuracy $0<\varepsilon\le1/10$, we prove that the tight complexity of sampling a distribution within $\epsilon$-TV distance from the target distribution is $$N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)+ \frac{\sigma^2}{\mu\epsilon}\right),$$ where $\kappa:=\frac L\mu$ is the condition number. Note that this complexity bound is simultaneously tight for the condition number $\kappa$ and accuracy $\epsilon$. Besides, our tight complexity bound is adaptive to noiseless setting $\sigma=0$, which is $N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)\right)$.

We study KL-regularized contextual bandits under both reward and preference feedback. While existing regret guarantees typically depend on the eluder dimension, we show that simple greedy sampling can achieve polylogarithmic regret without explicit dependence on this complexity measure. For reward feedback, we analyze a greedy algorithm that samples directly from the Gibbs policy induced by the estimated reward. We extend the result to preference feedback under both general preference and Bradley--Terry models, while also sharpening existing dimension-dependent guarantees. Our analysis reveals a trade-off between greedy sampling and upper confidence bound-style exploration: greedy sampling enjoys stronger regret guarantees when KL regularization is sufficiently strong, whereas additional exploration yields sharper bounds as the regularization weakens.

We continue the study of relatively smart learning, introduced by Dughmi and Pour (2026), which asks a supervised learner to compete, marginal by marginal, with every distribution-fixed error guarantee soundly certifiable from unlabeled data. They showed that the One-Inclusion Graph (OIG) learner is relatively smart with a quadratic sample-complexity blowup, and that no relatively smart learner can do better, leaving open whether ERM or another natural or tractable learner achieves comparable guarantees. They also left open whether the blowup can be restricted to unlabeled data. Our firs results shows that ERM---and in fact any proper consistent learner---is relatively smart for binary classification in the distribution-free setting. We show that a small certifiable error with $m$ samples implies a similarly small error on the uniform distribution over a random sample of size $O(m^2)$, yielding a cover of size at most $2^{m+1}$ on that sample. This suffices to control the error of proper consistent learners with $O(m^2)$ samples. We then show that semi-supervised relatively smart learning is information-theoretically possible with a quadratic blowup only in unlabeled sample complexity and no blowup in labeled sample complexity. The learner uses a natural generalization of OIG to a leave-most-out transductive problem, where labels of part of a finite pool are revealed and the remaining labels are predicted. Finally, this label efficiency comes at a cost in simplicity and tractability. If the hypothesis class is accessed only through an agnostic ERM oracle, any semi-supervised relatively smart learner with substantially sub-quadratic labeled-sample blowup requires super-polynomially many oracle calls. This holds even when the marginal is given explicitly, and thus also yields an intractability result for distribution-fixed learning that may be of independent interest.

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $Θ\left( \frac{dr}{\varepsilon^2} \max\left\{1,\frac r{\sqrt t}\right\} \right)$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt t$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is \emph{highly magnetized}). We use our frameworks to design improved samplers for canonical problems in the study of \emph{Ising models} and \emph{Bayesian sparse linear regression}. Our first main result considers the \emph{Sherrington--Kirkpatrick} (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β>0$, under arbitrary external fields, provided that $k\le c_βd$ for an appropriate constant $c_β$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$β$ limit, our framework permits sampling at field strengths within constant factors of the \emph{Almeida--Thouless line} delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(β)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical \emph{Gaussian spike-and-slab posterior} with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

We study distributed one-dimensional mean estimation under a 1-bit communication constraint. Each agent observes one sample, drawn independently from an unknown distribution, and returns a single bit in response to a query $Q: \mathbb{R}\to\{0,1\}$ chosen by a central learner. The distribution has mean in $[-λ,λ]$ and $k$-th central moment at most $σ^k$, for a fixed $k>1$. The order-optimal two-stage protocol of Lau and Scarlett uses responses from the first batch to choose the second-batch queries, motivating the question of whether this single round of interaction is necessary. We answer this negatively: for every $k>1$, a non-adaptive protocol attains the adaptive 1-bit minimax rate (and concurrent works reached the same conclusion via different strategies). We further determine the minimax sample complexity among non-adaptive 1-bit estimators when every one-set $Q^{-1}(1)$ is restricted to a union of at most $s$ intervals. Relative to unrestricted non-adaptive 1-bit querying, this constraint adds a term of order $(λσ/(s\varepsilon^2))\log(1/δ)$, giving the full tradeoff between sample complexity and interval complexity to within $k$-dependent constant factors. As a corollary, we identify, order-wise, the minimum interval budget needed to retain the unrestricted 1-bit minimax sample rate.

In nonconvex optimization problems arising in geometric machine learning, data augmentation is commonly used to promote invariance by averaging empirical losses over transformations of the data. Computing the fully augmented objective, however, requires access to every element of the transformation group $G$, which may be prohibitively expensive when $G$ is large or accessible only through sampling. We study whether full augmentation can instead be approximated using a small, fixed sample of transformations acquired before optimization and reused thereafter. Under suitable regularity conditions, we show that, with probability at least $1-δ$, gradient descent (GD) on the resulting sparsely augmented objective returns an $\varepsilon$-stationary point of the fully augmented objective using $\mathcal{O}\bigl((\log |G|+\log(1/δ))/\varepsilon^2\bigr)$ group-transformation-oracle queries. By comparison, standard group stochastic gradient descent (group-SGD), which samples a fresh transformation at every iteration, uses $\mathcal{O}(1/\varepsilon^4)$ transformation queries. Therefore, gradient descent with fixed sparse augmentation requires fewer transformation queries than both GD applied to the fully augmented objective and group-SGD. Our proof techniques, which may be of independent interest, establish a uniform approximation of the full group-averaged gradient field by a random group average using spectral properties of group-induced operators and tools from representation theory.

We study the training dynamics of multiclass logistic regression on high-dimensional Gaussian mixture models with a large number of classes and establish precise scaling laws governing the cross-entropy risk under gradient-based optimization. We show that learning proceeds sequentially across classes, from most to least frequent. When the class priors follow a power law distribution, the risk dynamics decompose into three phases: an initial plateau until the first class is learned, a power-law decay regime during which sequential learning occurs, and a final convergence regime. We then analyze how model capacity interacts with optimization under a fixed compute budget. When the effective dimension is restricted via projection onto leading principal components, the risk decomposes into a capacity term (a power law in the retained dimension) and an optimization term (a power law in training time). Optimizing this tradeoff yields a compute-optimal scaling law for logistic regression, with explicit prescriptions for model size and training time as functions of compute. These results extend theoretical scaling laws from linear regression to multiclass classification, while connecting to empirical scaling laws observed in large-scale neural networks.

A system often has to act long before it learns whether the act worked: a recommender sees a click in seconds and a purchase in days. With $K$ actions and a delay of $d$ rounds, the best rate known for this setting is $\widetilde{O}(\sqrt{(K+d)T})$ over $T$ rounds, so a longer menu is always more expensive to learn from. It need not be: if the outcome depends on the action only through the state it produced, then one late outcome informs every action that could have produced the observed state, and the price is set by how many genuinely different states the actions produce rather than by how many actions there are. We measure this using an effective dimension $v_t$ between $1$ and the number of states, and prove $\widetilde{O}(\sqrt{(d+1)V\log K})$ for a rotating algorithm and $\widetilde{O}(\sqrt{V^{-}}+\sqrt{dT})$ for the single-copy algorithm used in practice, for any budget fixed in advance; merging similar states lowers the price further, at an explicit bias. Even when given the exact losses from $d$ rounds ago, no algorithm escapes $Ω(\sqrt{dE\min\{1+\log J,T/d\}})$, where $J$ counts the drifting directions and $E$ bounds how far losses move while the learner waits. On generated data, the state channel cuts regret by up to 79 percent against action-level weighting and, on the funnel family, by 32 to 68 percent against a tuned minimax-optimal method.

The per-instance Jaccard score, or intersection over union (IoU), is standard in multi-label classification and binary segmentation. With $s$ labels, its loss matrix has $2^s$ outcomes and reports. Under the convention $\mathrm{Jac}(\varnothing,\varnothing)=1$, we prove that the Jaccard score, shifted-loss, and ordinary loss matrices are nonsingular and that the loss columns have affine dimension $2^s-1$. The proof combines a finite MinHash Gram representation with Boolean Möbius inversion. For exact calibration, we prove $2^{s-1} \leq \mathrm{CCdim}(L^{\mathrm{Jac}}) \leq 2^s-1$. The lower bound uses a factorially weighted distribution with $2^{s-1}+1$ supported outcomes and Bayes-optimal reports. Consequently, every exactly calibrated convex surrogate requires exponentially many prediction coordinates. We also give two polynomial-dimensional approximation guarantees with explicit regret transfers. A new $F_1$-to-Jaccard transfer turns an existing $(s^2+1)$-dimensional $F_1$ surrogate into a polynomial-time rule with asymptotic Jaccard regret at most $3-2\sqrt{2}$. For any $α>0$ and $0<ρ<1$, a MinHash square-loss surrogate attains Jaccard-regret floor $α$ uniformly over arbitrary conditional label distributions. With probability at least $1-ρ$, the direct construction has dimension $O((s^2+s\log(1/ρ))/α^2)$, while a signed variant has dimension $O((s+\log(1/ρ))/α^2)$. Thus zero-regret calibration requires exponential dimension, whereas every fixed additive regret tolerance admits polynomial prediction dimension.

Top-k selection is a fundamental computational primitive with applications spanning databases, information retrieval, signal processing, and modern machine learning workloads, including sparse activations and attention pruning. As data sizes grow, existing approaches become inefficient: exact methods incur high memory and compute overhead, while approximate methods often rely on brittle heuristics that degrade under adversarial or heavy-tailed inputs. In this paper, we introduce Prof-K, a fast, scalable, and distribution-agnostic top-k algorithm with probabilistic correctness guarantees. Prof-K performs a single-pass filtering procedure: a small random sample estimates an adaptive threshold, the N input elements are streamed once into a compact buffer, and an exact top-k routine on this buffer recovers the true top-k elements with probability at least 1 - $ε$, where $ε$ > 0 is user specified. We derive high-probability guarantees for correctness and buffer size, together with an approximately optimal sample size that minimizes overhead as a function of N and k. Empirically, Prof-K achieves 1.5x-10x speedups over the highly optimized PyTorch topk and recent RadiK implementations, with the largest gains in the large-scale, small-to-moderate-k regime where prior methods struggle most. Unlike previous approaches, these guarantees hold independently of the input distribution, ensuring robustness to adversarial settings. By relaxing the recall target (e.g., recovering 95% of the true top-k values), Prof-K additionally provides a principled accuracy-speed trade-off. We further demonstrate its impact on training BatchTopK Sparse Autoencoders (SAEs), where top-k selection constitutes a significant portion of the training cost.

\texttt{TensorSketch} by~\cite{pham2013fast,kar2012random} provides efficient sketching algorithms for high-dimensional polynomial kernels $\vec{x}^{\otimes p} \in \R^{d^p}$. \cite{kar2012random} uses dense Johnson-Lindenstrauss (JL)-type projections with computational cost $O(pDd)$, where $D$ denotes the sketch dimension, whereas~\cite{pham2013fast} extends the sparse \texttt{CountSketch}\citep{count_sketch} algorithm, yielding a faster algorithm for high-dimensional sparse inputs with running time $O\big(p(\nnz{\vec{x}} + D \log D)\big)$. However, the variance of both estimators grows exponentially with the polynomial degree $p$, scaling as $3^{p}/D$. Recent work by\cite{pmlr-v206-wacker23a} showed that using complex-valued distribution reduces this dependence to $2^{p}/D$ for the approach of~\cite{kar2012random}. However, their method relies on dense JL-type projections with computational cost $O(pDd)$ and does not extend to the algorithm of~\cite{pham2013fast}. In this work, we introduce a simple variant of \texttt{TensorSketch}\citep{pham2013fast} that achieves the same variance bound as\cite{pmlr-v206-wacker23a}, while retaining its advantage of the input-sparsity running time. We validate our results with supporting experiments on synthetic and real-world datasets.

We prove a sharp lower bound for smooth nonconvex stochastic optimization with uniformly bounded gradient noise. In the $K=1$ fresh-sample model, every randomized adaptive algorithm requires $Ω\left( \frac{ΔL}{ε^2} + \frac{ΔLσ^2}{ε^4} \right)$ queries to find a point with expected gradient norm at most $ε$. This matches the standard upper bound and, to the best of our knowledge, resolves the question raised by [Arjevani et al. 2023] of whether almost-surely bounded oracle error permits a better rate than bounded variance. 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.

Adaptive procedures must work without nuisance information an oracle may use, such as a gradient scale or smoothness index, and robust procedures may have to answer queries whose coordinate and inspection time are chosen only after the data are seen. Such comparisons are meaningful only when the oracle advantage and validity contract are stated explicitly. We formalize nuisance adaptation via a slice-normalized minimax ratio retaining the worst-case instance within each nuisance slice, and separately define the robustness cost of expanding from one preannounced Gaussian query to arbitrary post-hoc inspection. Our main result is a finite-horizon composition law for Gaussian certification: from M independent coordinates, a familywise certifier protecting every coordinate and time up to T pays optimal normalized squared half-width of order log(eM) + log log(e^eT), within the sample-mean-centered rectangular class. Epoch stitching gives the upper bound; independent Gaussian block increments across coordinates and geometric time scales give a matching lower bound, already holding on a geometric checkpoint grid, forcing quantiles of the realized maximum width so selection and stopping taxes add. Two benchmark regimes complete the picture: unknown gradient scale in online convex optimization has constant cost, while pointwise adaptation over nested Holder classes costs order (log n / log log n)^(s1/(2s1+1)). Cast as model monitoring, the law lets an analyst inspect any of M slice metrics at any data-dependent time: the naive fixed-query band's selected coverage degrades sharply, to 0.30 at M=1 and to zero for M>=10, while the epoch-stitched certifier holds familywise coverage at an additive iterated-logarithm width cost. Experiments put both sharp predictions at risk of refutation; both survive.

We study second- and higher-order methods for solving smooth monotone variational inequalities (MVI). Monteiro and Svaiter (SIAM J. Optim., 2012) showed that a second-order method, NPE, converges at the rate of $\mathcal{O}(T^{-1.5})$. For convex-concave minimax optimization, a subset of MVI problems, Chen, Liu, Luo, and Zhang (COLT 2025) recently improved the complexity to $\tilde{\mathcal{O}}( T^{-1.75})$ . However, it is open whether the conjectured complexity for MVI can be improved. In this paper, by using a large-step inexact Halpern iteration, we propose a novel Halpern-NPE method that achieves an even faster rate of $\tilde{\mathcal{O}}(T^{-2})$ for solving MVIs. We also provide the $p$th-order generalization of our method. We first introduce an Anchored Tensor Method (ATM) that achieves the rate of $\mathcal{O}(T^{-(p-1)})$, and then combine it with the Halpern iteration to achieve a faster convergence rate of $\tilde{\mathcal{O}}(T^{-p})$. This improves all prior results for $p \ge 2$ and matches the classical extragradient method for $p=1$.

Probably Approximately Correct (PAC) learning [Val84] is a fundamental learning model that has been extensively investigated. In this model, $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$ is a concept class, and $h^*\in\mathcal{H}$ is the target concept to be learned. Having access to i.i.d. labeled examples from a distribution $\mathcal{D}$ over $\mathcal{X}\times\{0,1\}$, which admits $h^*$ as the best concept in $\mathcal{H}$, the goal is to design a learning algorithm that outputs a hypothesis having low error competitive to $h^{*}$ with high probability. This model was initially studied under the realizable setting, which assumes that $h^*$ has no error. A natural relaxation is to allow label noise, that is, the true label can be flipped with probability $η\in(0,1/2)$. In reality, certain labels might be extremely noisy, especially for those points near the decision boundary. Hence, it is natural to allow very noisy points, though only rarely. This is quantified by a noise model introduced by [MT99] and [Tsy04], now known as Tsybakov noise. For learning general concept classes, [MN06] gave the general upper and lower bounds for error guarantees under Tsybakov noise. However, their upper and lower bounds differ by a logarithmic factor. Resolving this gap has remained a well-known open question for the past twenty years. In this work, we resolve this open question by improving the upper bound to match the best known lower bound, thus establishing the optimal error guarantee for learning under Tsybakov noise. Our learning algorithm operates by adaptively partitioning the instance space into regions, roughly corresponding to different noise levels, and returning a hypothesis in the concept class satisfying a specific error constraint for each region. Our technique shares a conceptual foundation with several recent advances in non-realizable learning, such as [HLZ24] and [Han25].

We study policy-based reinforcement learning under the $μ$-resets interaction protocol of Kakade and Langford [KL02]. This interaction protocol enables the learner to sample trajectories from a given exploratory reset distribution $μ$, in addition to the starting distribution. We resolve the question raised by [KLS25] on the role of policy realizability for the sample complexity of this problem. Critically, the dependence on horizon $H$ is governed by the notion of coverage assumed of the reset distribution. Under bounded all-policy concentrability, we show a $\exp(Ω(H))$ sample complexity lower bound; with bounded pushforward concentrability, we show the dependence on horizon is tightly characterized as $\exp(Θ(\sqrt H))$.

Hanneke, Moran, and Waknine \cite{HannekeMoranWaknine2024} asked how the agnostic PAC learning curve of the direct sum $C^r$ depends on the single-instance learning curve $\epsagn(n\mid C)$ and on $r$. We show that the single-instance learning rate does not determine the direct-sum rate. Let $\F$ be the class of the two constant binary functions and let $\G$ consist of the zero function and the identity function. Both classes have agnostic learning curve of order $n^{-1/2}$.

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].

A monotone adversary observes an i.i.d. labeled sample and appends a finite number of further examples of its choice, every one of them labeled correctly by the target hypothesis. The learner sees a uniform shuffle of the combined sample and is scored on the original distribution. Every example is correctly labeled, but the insertions depend on the clean sample, so the combined sample is not exchangeable. Larsen, Pabbaraju, and Shetty, who introduced this model, showed that empirical risk minimization attains expected error $O((d/n)\log(n/d))$ for classes of VC dimension $d$, and that every known optimal learner can be pushed away from the $Θ(d/n)$ rate, optimal for PAC learning. They asked whether the extra logarithm is an artifact of those particular algorithms or an inherent consequence of the lack of exchangeability. We show that this additional cost is inherent beyond VC dimension one. In the worst case over classes of VC dimension $d$ and over known finite insertion budgets, the minimax expected error is $Θ(1/n)$ at $d=1$ and $Θ((d/n)\log(n/d))$ for $d\geq 2$. The same rates hold with Littlestone dimension $d_{\mathrm L}$ in place of $d$, so the clean online-to-batch rate $O(d_{\mathrm L}/n)$ is unattainable as well. Thus, somewhat counterintuitively, adding correctly labeled examples can make learning harder by a logarithmic factor, even for classes that admit finite mistake bounds in online learning. The dimension-one upper bound is achieved by a simple improper learner whose analysis adapts the leave-one-out argument underlying the one-inclusion graph. All of our lower bounds are elementary and come from a single construction: an explicit class and prior on which two target hypothesis, which differ a point of nonnegligible mass, produce the same sample.

Model evaluations may fix all tests before observing any responses or select later tests using earlier responses. We study this choice in a conditional-query model on a finite outcome space $\mathcal{X}$ with $|\mathcal{X}|=N$. We first ask which pairs of distribution classes can be reliably distinguished. We then ask how many additional queries are required to match an adaptive tester when all queried events must be fixed in advance. We show that learnability holds if and only if the two classes have positive separation in their pairwise conditional probabilities. When this separation is zero, the optimal worst-case error is exactly $1/2$ at every finite query budget. For any $T$-query adaptive policy and any $ρ\in (0,1)$, we construct a randomized non-adaptive procedure using $O(N^2(T + \log(1/ρ)))$ pair queries chosen before any response is observed. Its simulated transcript is within $ρ$ in total variation of the adaptive transcript, uniformly over all distributions in the model. We also construct a matching family with constant adaptive query complexity and $Ω_\varepsilon(N^2)$ non-adaptive query complexity. Consequently, the worst-case fixed-error adaptivity gap is $Θ_\varepsilon(N^2)$. Thus interaction can reduce the required number of tests by a quadratic factor, but the apparent exponential branching of an interactive evaluation does not yield an exponential query advantage.

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.

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.

Modern large language models - transformers and diffusion language models - are built around two canonical algorithmic tasks: prediction and generation. We prove unconditional separations between low-depth quantum computation and the corresponding bounded-resource classical language-model architectures in both regimes. Concretely, we exhibit the following: 1. Distributional separation. We give a distribution that is sampleable by $\textsf{QNC}^0$ circuits (i.e., a family of constant-depth quantum circuits consisting of bounded fan-in gates) that no constant-round diffusion language model ($\textsf{DLM}$) with shallow scheduling and denoising can sample within constant distance, even when allowed sublinear chain-of-thought and output-token revision/remasking events, the very features modern $\textsf{DLM}$s rely on. 2. Functional separation. We exhibit a function computable in $\land \circ \textsf{QNC}^0[\log\log n]$ (i.e., a family of O$(\log\log n)$-depth $\textsf{QNC}^0$ circuits, where $n$ is the input length, followed by a single classical $\mathsf{AND}$ gate) such that any constant-depth decoder-only transformer computing the function must be large: it would have to have width $n^{Ω(1)}$. Together, our work initiates the study of quantum advantage in the era of large language models.

In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer, and Tulsiani [RST12]. Concretely, for every fixed $γ\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, $$\lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,κ_{s+k}^{\,1-γ}\right),$$ where $κ_r$ is the restricted condition number at sparsity level $r$. The result holds even on rational instances with $A$ of full column rank. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

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}.

We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $κ\sqrt{d}/\varepsilon$, where $κ$ is the condition number, $d$ is the dimension, and $\varepsilon$ is the target precision: this improves by a factor of $\sqrt{d}/\varepsilon$ over the previous state-of-the-art results.

We study the round complexity of learning a hidden partition $\mathcal{P}$ of an $n$-element universe using PAIR queries: PAIR($x,y$) tells us whether $x$ and $y$ belong to the same part of the partition or not. While it is easy to learn using $n|\mathcal{P}|$ queries using a basic algorithm and this query complexity is optimal, this basic algorithm is highly sequential. Black, Mazumdar, and Saha [COLT 2025] recently gave tight deterministic round/query tradeoffs when the number of parts of $\mathcal{P}$ is known. In particular they prove $Θ(\log\log n)$ rounds are sufficient and necessary to limit the number of queries to $n|\mathcal{P}|$. They leave proving a randomized lower bound as an open direction. We show that randomization dramatically changes the picture. When the number of parts $k = |\mathcal{P}|$ is known, we give a simple 3-round randomized algorithm using $O(nk\log n)$ queries with high probability, and prove that 2 rounds require $Ω(n^{4/3}k^{2/3})$ queries -- the same as deterministic algorithms. We also study a more general setting where the number of parts is unknown. In this case, we give a 4-round randomized algorithm using $O(n|\mathcal P|\log^2 n)$ queries with high probability, and prove that 3-rounds cannot achieve near-optimal query complexity. Furthermore, we show an even bigger separation in this regime between randomized and deterministic algorithms: for the latter, $Θ(\log n/\log\log n)$ rounds are necessary and sufficient to obtain near-optimal query complexity.

{AdaBoost.MH} reduces multi-class classification to a collection of binary subproblems and enjoys the classical boosting-type convergence guarantee under a weak learning condition. A more structured variant, Factorized {AdaBoost.MH}, uses base classifiers of the form $\mathbf{h}(x)=α\mathbf{v} \bm{\varphi}(x)$, where a single binary classifier $\bm{\varphi}$ is shared across all classes and the label dependence is carried by a vote vector $\mathbf{v} \in\{\pm1\}^K$. This factorization is algorithmically attractive and achieves better performance in practice, but its convergence depends on whether one can always choose a vote vector with sufficiently large induced binary weight mass. Previous work resolved this question with a lower bound $\max\{1/n,1/\sqrt{2K}\}$, which still leaves a dimension-dependent slowdown relative to the original {AdaBoost.MH} analysis. In this paper, we sharpen this combinatorial step. For the minimax quantity $\mathfrak{W}_{n,K}$ governing the factorized edge, we prove $\mathfrak{W}_{n,K} = C_{\min\{n+1,K\}}$, where $C_q=1$ for $q=1$, $C_q=q/(3q-4)$ for even $q\ge2$, and $C_q=(q+1)/(3q-1)$ for odd $q\ge2$. Since $C_q\downarrow 1/3$, our bounds show that $\mathfrak{W}_{n,K}=Θ(1)$ uniformly over $n$ and $K$. Consequently, Factorized {AdaBoost.MH} achieves the same boosting-type convergence rate as {AdaBoost.MH} up to a universal constant factor, removing the previously suggested additional dependence on $n$ or $K$ in the number of boosting rounds.

We present HBPI-UCRL, a model-based algorithm for hierarchical reinforcement learning (HRL) that learns high-level and low-level policies in parallel. HBPI-UCRL exploits the fact that a high-level transition corresponds to a multi-step transition at the low level. We introduce two conditions on the low-level dynamics that are sufficient to make parallel HRL learnable. When these conditions hold, we prove that HBPI-UCRL has a polynomial sample complexity in the problem parameters. In the sparse-reward, goal-directed setting, our sample complexity upper bound for HBPI-UCRL is strictly lower than that of its non-hierarchical counterpart, providing theoretical justification for the empirical success of HRL.

Swap-agnostic learning strengthens classical agnostic learning by allowing the comparator to select a different hypothesis on each level set of the learner's predictions. This benchmark captures prediction-dependent postprocessing, but appears to require solving a separate agnostic-learning problem for every possible prediction value. We show that, for proper losses, these prediction-level comparisons can instead be controlled jointly. Our main result is an offline swap-agnostic learner for any fixed proper loss. For a finite hypothesis class $H$ and any fixed smooth proper loss, the excess risk from $m$ i.i.d. samples is $\widetilde{O}((\log |H|/m)^{2/3})$, with a corresponding online swap-regret bound of $\widetilde{O}(T^{1/3}(\log |H|)^{2/3})$. We also give algorithms whose predictions are simultaneously swap-agnostic for entire families of losses. For all proper losses bounded in $[-1,1]$, we obtain online and offline rates of $\widetilde{O}(\sqrt{T\log |H|})$ and $\widetilde{O}(\sqrt{\log |H|/m})$, respectively. For convex, $1$-Lipschitz proper losses, these rates improve to $\widetilde{O}(T^{1/3}(\log |H|)^{2/3})$ online and $\widetilde{O}((\log |H|/m)^{2/3})$ offline. These bounds are tight up to logarithmic factors and improve upon the $\widetilde{O}(T^{2/3}(\log |H|)^{1/3})$ rate implied by the swap-omniprediction guarantee of Luo et al. (2025). Our main technical contribution is a reduction from swap-agnostic learning to a second-order form of multicalibration, obtained via Blackwell approachability with a Bernstein-style variance correction.

Bilevel reinforcement learning (RL) is an important framework within the literature of RL that can be used to formalize various categories of problems, such as meta-learning, hierarchical task decomposition, and reinforcement learning from human feedback (RL-HF). Most of the bilevel RL algorithms are either not scalable because of using hypergradient with Hessian, or they suffer from high sample complexity because of using penalty-based approximation methods. In this work, we propose a hypergradient-based bilevel RL algorithm using the optimality of the Boltzmann policy for the entropy regularized discounted RL objective function. Our proposed algorithm is Hessian-free and obtains an iteration complexity of $O(ε^{-1})$ and state-of-the-art sample complexity of $\tilde{O}(ε^{-2})$ under mild regularity conditions. Further, in our convergence analysis, we are able to remove the assumption of the Polyak-Lojasiewicz (PL) condition on the outer-level objective function present in the prior state-of-the-art sample complexity work.

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).

In this paper, we formulate three communication tasks for empirical optimal transport: distributed coupling sampling, cost-evaluable coupling output, and scalar value-certified sampling. Our main result is a field-code compiler: any communicated transport field approximating an optimal empirical Monge map to error $η$ can be completed by sparse target-cell residuals into an exact-marginal value-certified sampler with scalar certificate $W_1(μ,ν)\leq U\leq W_1(μ,ν)+2Δ$, where $Δ$ is the public target-partition diameter. The certificate accuracy is controlled by $Δ$ alone. The field error $η$ controls residual communication under a cell-margin condition; without a margin, $η$ alone does not bound residuals. We instantiate the compiler via adaptive local-affine and tensor-product spline codes with $d(m+1)^db$ field bits in the spline case, plus residual lists charged separately. For lower bounds, exact Gap-Hamming embeddings prove certified output is hard, including a smooth cell-packing diffeomorphism family requiring $Ω(\varepsilon^{-2d/(d+4)})$ communication for any cost-evaluable, cost-certified, or value-certified protocol. The same gadgets admit zero-communication samplers, formally separating the sampler and certificate-bearing output models. These results identify the transport field as the right communicated object whenever a field code is available, primarily as a residual-sparsity tool.

Two central challenges in diffusion-based sampling are the theoretical one of understanding their remarkable effectiveness even in high-dimensional settings, and the practical one of designing algorithms with certified performance guarantees. We show that these questions are intimately connected via the \emph{denoising growth complexity} ($\mathsf{DGC}$). It is a geometric measure defined by a log-time weighted integral of the derivative of the denoising mean-squared error along the Gaussian heat flow. We show how the $\mathsf{DGC}$ increments lead to a simple and explicit bound on the KL error of an Euler scheme applied to the stochastic innovations representation. The bound is local along the path: each step is controlled by the corresponding $\mathsf{DGC}$ increment and its relative stepsize. This structure allows us to derive KL sampling guarantees for optimized stepsize schedules, both in a simpler single-block setting and in a more refined $K$-block setting. The $\mathsf{DGC}$ function has a natural martingale structure, which we exploit to develop fully data-certified versions of these algorithms. It also admits information-theoretic upper bounds in terms of covariance, rate distortion, metric entropy, and the Poincar'e constant, thereby recovering and sharpening a range of existing diffusion-sampling guarantees, as well as giving new results. In log heat-time, the fine partition limit is governed by an integral involving the square root of the $\mathsf{DGC}$ density, whereas a single-block schedule depends on its ordinary integral. This comparison precisely characterizes when adaptation to data geometry yields substantial computational gains, including logarithmic-to-constant separations for simple Gaussian mixture models.

We study stochastic composite nonconvex optimization over a compact convex set when gradient samples arrive along a single trajectory of a fixed ergodic Markov chain. Existing single-trajectory variance-reduction theory covers smooth unconstrained objectives; we address the projection-free composite setting using the generalized Frank-Wolfe gap. We propose MC-ALFCG, which combines a momentum conditional-gradient method with coupled capped multilevel Monte Carlo estimation and per-iteration clipping. The deepest nested average uses consecutive states from the same trajectory, yielding conditional bias $O(τ_{\mathrm{mix}}/T)$ uniformly over the starting state, while coupling controls the gradient-difference second moment through the iterate displacement. Clipping enforces the pathwise bounds needed by the adaptive analysis. We reduce the Markovian recursion to its independent-sampling counterpart under $σ^2\mapsto 2ΛG_σ^2$ and $L^2\mapsto 2ΛL^2$, where $Λ=O(τ_{\mathrm{mix}}\log T)$. For positive centered noise, the tuned method achieves expected sample complexity $\widetilde{O}((τ_{\mathrm{mix}}^2G_σ+τ_{\mathrm{mix}}^{5/2}G_σ^2)\varepsilon^{-3}+τ_{\mathrm{mix}}^5\varepsilon^{-2})$. The exactly noiseless specialization achieves $\widetilde{O}(\varepsilon^{-2})$ with mixing-time-free constants, while a mixing-time-oblivious variant achieves $\widetilde{O}(τ_{\mathrm{mix}}^6\varepsilon^{-3}+τ_{\mathrm{mix}}^3\varepsilon^{-2})$. All guarantees are in expectation under a fixed transition kernel. Controlled numerical studies examine dependence sensitivity, a nonconvex composite instance, and clipping behavior.

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 $\widetilde Θ(n^α/ε^2)$.

We study streaming federated learning with limited client memory, where newly generated training data incur time-varying sampling costs and must be selectively admitted and retained over time. We consider a joint server-side admission and client-side memory-management framework with the objective of minimizing the cumulative excess population risk under a sampling-cost budget and buffer constraints. We first derive a learning-error bound that explicitly captures the effects of instantaneous training sample size, distinct-sample growth, and reuse imbalance through a characterization of the effective sample size. Through a surrogate penalty obtained from this bound, we develop an Active-Constraint Drift-Plus-Penalty (ACDPP) policy that combines a structured client-side $K$-step retention rule with a server-side online admission rule and a time-varying rectangular admission region. We further present a sequence of comparison arguments, via an auxiliary constant-admission policy, that connects the ACDPP learning bound to a costless oracle benchmark. This yields explicit guarantees in terms of sublinear regret and sampling-cost violation, while the buffer-occupancy violation is controlled through offline selection of the retention horizon. Experiments on multiple datasets demonstrate that the proposed policy remains close to the oracle benchmark while satisfying the sampling-cost and buffer constraints.

Recent years have witnessed increasing interests in tackling heteroscedastic noise in bandits and reinforcement learning. In these works, the cumulative variance of the noise $Λ= \sum_{t=1}^T σ_t^2$, where $σ_t^2$ is the variance of the noise at round $t$, is used to characterize the statistical complexity of the problem, yielding \emph{simple regret} bounds of order $\tilde{\cal{O}}(d \sqrt{Λ/ T^2})$ for $d$-dimensional linear bandits with heteroscedastic noise. However, with a closer look, $Λ$ remains the same order even if the noise is close to zero at half of the rounds, which indicates that the $Λ$-dependence is not optimal. In this paper, we revisit the stochastic linear bandit problem with heteroscedastic noise, where the action set is prefixed throughout the learning process. We propose a novel variance-adaptive algorithm \texttt{VAEE} (Variance-Aware Exploration with Elimination) for large action set, which actively explores actions that maximizes the information gain among a candidate set of actions that are not eliminated. With the active-exploration strategy, we show that \texttt{VAEE} achieves a \emph{simple regret} with a nearly \emph{harmonic-mean} dependent rate. For finitely many actions, we propose a variance-aware variant of G-optimal design based exploration, which achieves a simple regret with sharper dependence on $d$. We also establish a nearly matching lower bound for the fixed action set setting indicating that \emph{harmonic-mean} dependent rate is unavoidable. To the best of our knowledge, this is the first work that breaks the $\sqrtΛ$ barrier for stochastic linear bandits with heteroscedastic noise.

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.

A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including integer linear programming (ILP). It is known that, by applying gradient-based optimization methods to the suboptimality loss, the inverse optimization of ILPs can be solved exactly within finitely many oracle iterations, and that the required number of iterations is bounded as $T=O(1/γ(\ell_{\mathrm{sub}})^2)$ in terms of a problem-dependent geometric constant $γ(\ell_{\mathrm{sub}})$. However, no means of bounding $γ(\ell_{\mathrm{sub}})$ from below as a function of the problem size has been available, and hence the number of iterations could not be given as an explicit function of the problem size. We therefore give, when the forward problem is an integer linear program (ILP), the number of iterations sufficient for projected subgradient descent applied to the suboptimality loss to achieve exact consistency with the observed data, as a fully explicit function of the number of samples, the dimension of the features, the ranges of the features, and the structure of the constraint coefficient matrix, up to polynomial factors in the basic constants (the diameter of the weight set, the step-size parameter, and the Lipschitz constant of the suboptimality loss).

We study machine unlearning: the removal of memorized training data from a trained model. Specifically, we investigate the algorithmic complexity of certified unlearning from an optimization perspective. We formalize the goal of an unlearning algorithm as simultaneously achieving certified unlearning and optimization accuracy. Utilizing the notion of uniformly convex regularizers, we prove new bounds on the distance between initial and unlearned models using a novel substitute for generalization error. Thus we theoretically demonstrate that if the removed data is well-predicted by the unlearned model, the corresponding optimization problem is simple. Furthermore, we develop a new second-order unlearning algorithm with an anisotropic Gaussian mechanism and state-of-the-art global convergence. We prove fast rates for our method in achieving certified unlearning for linear models with quasi-self-concordant losses. As a direct application, our theory covers unlearning for logistic and exponential regressions and shows a provable benefit of utilizing second-order information compared to first-order unlearning methods.

We consider the problem of constructing a $K$-class classifier from the combination of $O(\log K)$ simple binary classifiers -- this is a natural paradigm to construct a sophisticated classifier in a distributed manner with each agent performing a relatively straightforward task. We study the fundamental performance limits of such a classifier when the corresponding binary classifiers are hyperplanes. For a stylized Gaussian setting where the $K$ class centers are independent Gaussian points in $\mathbb R^d$ and the observations are corrupted by Gaussian noise, we derive explicit performance bounds across several decoding and dimensional regimes. Extensive simulation experiments provide strong empirical validation of the presented theoretical results.

Reachability analysis is central to safety-critical control, robotics, and neural network verification, but classical computational methods, such as Hamilton--Jacobi reachability and set propagation, scale poorly with state dimension. Sampling-based methods have emerged as a promising alternative, often providing finite-sample guarantees that bound the probability-mass left uncovered. However, an explicit account of how the geometry of the initial set, the dynamics, and the sampling law affect the accuracy of the estimator is not fully available in the literature. We study this by casting sampling-based reachable-set recovery as geometric support estimation over a family of problems specified by an initial set, its dynamics, and a sampling law. First, we identify two regularity properties, positive reach of the initial set's complement and Lipschitz continuity of the dynamics, that together make recovery well-posed: a probability-mass coverage guarantee can be upgraded to accuracy $r$ in Hausdorff distance. Second, we bound the resulting sample complexity: recovery is achievable with $\tilde{\mathcal{O}}\big((e^{3LT}/r)^n\big)$ samples, exponential in both the state dimension and the time horizon. Third, we show that neither can be removed: an minimax lower bound of $Ω\big((e^{LT}/r)^n\big)$ holds for every estimator, so the exponential dependence on dimension and the degradation over the horizon are both intrinsic, not artifacts of a particular method. Experiments on nonlinear systems confirm that adversarial sampling improves constants but not the scaling.

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.

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.

Motivated by the challenge of testing distributions over high-dimensional or continuous domains, we study distribution testing with respect to bounded classes of distinguishers. A representative task is to use samples from an unknown distribution $P$ over a very large domain to decide between two cases: $P = P_{\mathsf{ref}}$ for a fixed reference distribution $P_{\mathsf{ref}}$, or there exists a distinguisher $f$ in a bounded class $\mathcal{F}$ which witnesses the separation $|\mathbf{E}_P[f] - \mathbf{E}_{P_{\mathsf{ref}}}[f]| > ε$. This is the task of identity testing with respect to fooling distance, a name inspired by the conceptual connection with pseudorandomness. (Formally, our model instantiates integral probability metrics from Boolean classes of bounded expressivity.) We show that testing with respect to fooling distance is not only a natural computational problem that admits sample-efficient algorithms even in high-dimensional settings, but also one that reveals and underlies connections between three seemingly unrelated areas of study: testable learning, verification of learning algorithms, and testing of structured distributions (whose "$\mathcal{A}_k$-testing" model our framework extends). These connections yield new results for all of these models, including: 1. Testable proper learners using membership queries for halfspaces and decision trees. 2. A lower bound for testable PAC verification in terms of Rademacher complexity, and a distribution-free verification protocol for disjoint unions of $k$ multidimensional rectangles. 3. Identity testers (with respect to total variation distance) for decision tree distributions and distributions with low-degree polynomial densities, over Boolean and continuous hypercube domains.

Following Goldwasser, Rothblum, Shafer, and Yehudayoff, who defined a framework for interactive proofs of learning [ITCS'21], we initiate the study of non-interactive proofs of learning. We define and study a new notion: Publicly-Verifiable Certificates of Statistical Validity (pvCSVs), which allow for public, distributionally-robust certification that the result of a learning algorithm is valid. In a pvCSV, a learner publishes a hypothesis $h$ and corresponding certificate $π$; then, any user, who holds a user-specific distribution, can read the pair $(h,π)$ and determine efficiently whether the hypothesis is valid according to the user-specific distribution. We construct pvCSVs in the context of Adaptive Statistical Query (SQ) Algorithms. To certify SQ algorithms that makes $k$ adaptive queries, we construct pvCSVs where the sample complexity scales with $O(\log k)$, whereas the sample complexity of the best learning algorithms scale with $\tilde{O}(\sqrt{k})$. More generally, we study proof systems for learning in the SQ model, demonstrating the model's strengths as well as its limitations.

Inspired by interior-point methods (IPM) for structured convex optimization, Kannan and Narayanan introduced the Dikin walk for sampling uniformly from polytopes in 2009. As in IPMs, the Dikin walk is affine-invariant, and its convergence is governed by the barrier geometry used to define its local proposal. They showed that the Dikin walk with the logarithmic barrier for a polytope in $\mathbb{R}^{d}$ with $m$ linear inequalities mixes in $md$ iterations. In 2017, Chen, Dwivedi, Wainwright, and Yu improved this to $d^{2.5}$ using a Lewis-weight barrier, and conjectured that the correct mixing time should be $d^{2}$. We make progress toward this conjecture by improving the previous $d^{2.5}$-mixing bound. For exponential sampling over a polytope, we prove that the Dikin walk with a scaled Lee--Sidford metric mixes from a warm start in $d^{2.25}$ iterations. This also yields an improved cold-start complexity via a known annealing framework. The main technical ingredient is improved average self-concordance of the Lee--Sidford metric, which gives high acceptance probability for the Metropolis filter along a random Dikin proposal. While previous analyses were effectively limited to second-order control due to technical difficulties, we develop a principled higher-order analysis. The proof combines a selective higher-order expansion of recursive bottleneck terms, a moving orthonormal-frame calculus for higher derivatives of the Lewis weights, and Wiener-chaos decompositions via multiple stochastic integrals to control the resulting Gaussian polynomials.

A theoretical understanding of Transformers is crucial to better understand the capacities and limitations of large language models (LLMs). There is much work analyzing the expressivity of attention-based models. By proposing handcrafted weights or using computational complexity arguments, a large amount of past theoretical works have sought to characterize which tasks are and which are not in the hypothesis class of Transformer models. However, little work investigates the learnability of such solutions. In this work, we make progress towards this goal. Inspired by recent loss landscape analysis work, we propose preliminary sample complexity bounds for learning C-RASP constructions with Transformers.