arXiv subjects

Kevin Tian

Publications and source records attributed to Kevin Tian.

At least 19 recordsLinked to original sources

arXiv2026

High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression

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 $\beta>0$, under arbitrary external fields, provided that $k\le c_\beta d$ for an appropriate constant $c_\beta$. 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-$\beta$ 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 $h(\beta)$ 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.

cs.DS↗

arXiv2026

Revisiting the Provable-Auditable Privacy Gap of DP-SGD

Differential privacy (DP) has traditionally been used to provide theoretical upper bounds on an algorithm's stability to changing its training data. In modern private machine learning applications, achieving strong tradeoffs between utility and theoretical privacy is challenging, and thus one may optimistically hope that existing theoretical privacy analyses are loose. Recent work on privacy auditing has adopted a dual viewpoint, instead lower bounding the true privacy of an algorithm by constructing empirical distinguishing events. The auditing literature has thus far yielded a pessimistic outlook on the looseness of theoretical privacy bounds for DP-SGD, the de facto private training method in modern ML, as nearly-matching empirical lower bounds have been achieved under various threat models [NHSBTJCT23, AC24, CBP25]. In this work, we propose the empirical privacy lower bound of an algorithm as a concrete metric to optimize for, complementary to the theoretical upper bound. We give a lightweight defense framework that generically augments optimization methods in the ML pipeline to have significantly-improved empirical privacy on standard benchmarks. Moreover, we show that our framework comes at no theoretical privacy cost when augmenting DP-SGD, unlike previously-proposed defenses against membership inference attacks. We evaluate our defense against a broad range of audit constructions, models, and datasets to demonstrate its flexibility.

cs.LG↗

arXiv2026

Optimal Recalibration of an Online Predictor

We study the problem of recalibrating an online predictor [KE17, OKS24]: given an arbitrary "hint" sequence of forecasts, the learner must output new predictions that are calibrated while incurring small excess error relative to the original forecasts, under a proper loss. We give an online algorithm that achieves $(\varepsilon, \varepsilon^2)$-recalibration for Lipschitz proper losses in $T \approx \varepsilon^{-3}$ rounds, using an imbalanced extension of the recent simultaneous Blackwell approachability reduction framework of [HTY26]. We show that this tradeoff is optimal by proving a matching lower bound for recalibrating against the squared loss. We also prove a companion $\mathcal{K}_2$-recalibration theorem that obtains the same tradeoffs up to a logarithmic factor. As our main application, we show how our recalibration algorithms can be combined with the online refinement method of [FH23] to obtain simultaneous $\varepsilon$-calibration and $\varepsilon^2$-calibeating for smooth proper losses at the same asymptotic rate, improving upon prior works that achieved these properties separately or with a worse $\varepsilon$ dependence. In particular, the $\mathcal{K}_2$ variant answers a question of [CHJL26] on simultaneously achieving near-optimal calibeating and calibration rates. We also derive extensions to settings with multiple hint sequences. Finally, we empirically evaluate our algorithms on a classification dataset undergoing distribution shift.

stat.ML↗

arXiv2026

The Tractability Landscape of Sampling with Inexact Scores

We provide a simple and tight characterization of the types of inexact score oracle access that permit sampling with vanishing total variation bias, for a standard, well-behaved target family. Our main result shows that any weaker error than the sub-Gaussian assumption used by [YW26] rules out the tractability of unbiased sampling. This strengthens the conclusion of [CCSW26] to be algorithm-agnostic, and to hold for a wider range of error assumptions.

stat.ML↗

arXiv2026

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 $\sigma$, we use $\widetilde{O}((n+n^2\sigma^2)/\varepsilon^2)$ 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.

cs.LG↗

arXiv2026

On the Curse of Dimensionality in Private Sparse Covariance Estimation and PCA

We study high-dimensional differentially private (DP) covariance estimation in the operator norm, and principal component analysis (PCA), under $k$-row-column sparsity ($k$-RCS) of the covariance matrix. In the non-private setting, it is known that $\mathsf{poly}(k, \log d)$ samples suffice to solve both of these problems. However, the only comparable result known under DP (Wang et al. 2021) requires $\Omega(d)$ samples under standard parameterizations of the problem. We investigate when this curse of dimensionality is inherent for sparse covariance estimation tasks under DP. On the upper bound front, we show that a $\mathsf{poly}(k, \log d)$ sample complexity for PCA is possible under DP, if we also posit sparsity of the leading eigenvector. We complement this result with $\mathsf{poly}(d)$ lower bounds under DP for both sparse covariance estimation and PCA, establishing an exponential gap between the private and non-private variants of these problems when $k = \mathsf{polylog}(d)$. To our knowledge, no such separation has previously been demonstrated for any sparse estimation problems in private high-dimensional statistics. Our techniques are flexible enough that they imply stronger lower bounds even for the well-studied problem of standard DP PCA, without sparsity assumptions.

cs.LG↗

arXiv2026

Combinatorial Sparse PCA Beyond the Spiked Identity Model

Sparse PCA is one of the most well-studied problems in high-dimensional statistics. In this problem, we are given samples from a distribution with covariance $\Sigma$, whose top eigenvector $v \in R^d$ is $s$-sparse. Existing sparse PCA algorithms can be broadly categorized into (1) combinatorial algorithms (e.g., diagonal or elementwise covariance thresholding) and (2) SDP-based algorithms. While combinatorial algorithms are much simpler, they are typically only analyzed under the spiked identity model (where $\Sigma = I_d + \gamma vv^\top$ for some $\gamma > 0$), whereas SDP-based algorithms require no additional assumptions on $\Sigma$. We demonstrate explicit counterexample covariances $\Sigma$ against the success of standard combinatorial algorithms for sparse PCA, when moving beyond the spiked identity model. In light of this discrepancy, we give the first combinatorial method for sparse PCA that provably succeeds for general $\Sigma$ using $s^2 \cdot \mathrm{polylog}(d)$ samples and $d^2 \cdot \mathrm{poly}(s, \log(d))$ time, by providing a global convergence guarantee on a variant of the truncated power method of Yuan and Zhang (2013). We provide a natural generalization of our method to recovering a vector in a sparse leading eigenspace. Finally, we evaluate our method on synthetic and real-world sparse PCA datasets.

stat.ML↗

arXiv2026

Simultaneous Blackwell Approachability and Applications to Multiclass Omniprediction

Omniprediction is a learning problem that requires suboptimality bounds for each of a family of losses $\mathcal{L}$ against a family of comparator predictors $\mathcal{C}$. We initiate the study of omniprediction in a multiclass setting, where the comparator family $\mathcal{C}$ may be infinite. Our main result is an extension of the recent binary omniprediction algorithm of [OKK25] to the multiclass setting, with sample complexity (in statistical settings) or regret horizon (in online settings) $\approx \varepsilon^{-(k+1)}$, for $\varepsilon$-omniprediction in a $k$-class prediction problem. En route to proving this result, we design a framework of potential broader interest for solving Blackwell approachability problems where multiple sets must simultaneously be approached via coupled actions.

cs.DS↗

arXiv2026

Separating Oblivious and Adaptive Models of Variable Selection

Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with $\ell_\infty$ error guarantees. This variant of the problem is motivated by \emph{variable selection} tasks, where the goal is to estimate the support of a $k$-sparse signal in $\mathbb{R}^d$. Our main contribution is a provable separation between the \emph{oblivious} (``for each'') and \emph{adaptive} (``for all'') models of $\ell_\infty$ sparse recovery. We show that under an oblivious model, the optimal $\ell_\infty$ error is attainable in near-linear time with $\approx k\log d$ samples, whereas in an adaptive model, $\gtrsim k^2$ samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard $\ell_2$ setting, where $\approx k \log d$ samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a \emph{partially-adaptive} model, where we show nontrivial variable selection guarantees are possible with $\approx k\log d$ measurements.

math.ST↗

arXiv2026

Functional Stochastic Localization

Eldan's stochastic localization is a probabilistic construction that has proved instrumental to modern breakthroughs in high-dimensional geometry and the design of sampling algorithms. Motivated by sampling under non-Euclidean geometries and the mirror descent algorithm in optimization, we develop a functional generalization of Eldan's process that replaces Gaussian regularization with regularization by any positive integer multiple of a log-Laplace transform. We further give a mixing time bound on the Markov chain induced by our localization process, which holds if our target distribution satisfies a functional Poincar\'e inequality. Finally, we apply our framework to differentially private convex optimization in $\ell_p$ norms for $p \in [1, 2)$, where we improve state-of-the-art query complexities in a zeroth-order model.

math.PR↗

arXiv2025

Perspectives on Stochastic Localization

We survey different perspectives on the stochastic localization process of Eldan, a powerful construction that has had many exciting recent applications in high-dimensional probability and algorithm design. Unlike prior surveys on this topic, our focus is on giving a self-contained presentation of all known alternative constructions of Eldan's stochastic localization, with an emphasis on connections between different constructions. Our hope is that by collecting these perspectives, some of which had primarily arisen within a particular community (e.g., probability theory, theoretical computer science, information theory, or machine learning), we can broaden the accessibility of stochastic localization, and ease its future use.

math.PR↗

arXiv2025

The Power of Iterative Filtering for Supervised Learning with (Heavy) Contamination

Inspired by recent work on learning with distribution shift, we give a general outlier removal algorithm called iterative polynomial filtering and show a number of striking applications for supervised learning with contamination: (1) We show that any function class that can be approximated by low-degree polynomials with respect to a hypercontractive distribution can be efficiently learned under bounded contamination (also known as nasty noise). This is a surprising resolution to a longstanding gap between the complexity of agnostic learning and learning with contamination, as it was widely believed that low-degree approximators only implied tolerance to label noise. In particular, it implies the first efficient algorithm for learning halfspaces with $\eta$-bounded contamination up to error $2\eta+\epsilon$ with respect to the Gaussian distribution. (2) For any function class that admits the (stronger) notion of sandwiching approximators, we obtain near-optimal learning guarantees even with respect to heavy additive contamination, where far more than $1/2$ of the training set may be added adversarially. Prior related work held only for regression and in a list-decodable setting. (3) We obtain the first efficient algorithms for tolerant testable learning of functions of halfspaces with respect to any fixed log-concave distribution. Even the non-tolerant case for a single halfspace in this setting had remained open. These results significantly advance our understanding of efficient supervised learning under contamination, a setting that has been much less studied than its unsupervised counterpart.

cs.LG↗

arXiv2025

Private Geometric Median in Nearly-Linear Time

Estimating the geometric median of a dataset is a robust counterpart to mean estimation, and is a fundamental problem in computational geometry. Recently, [HSU24] gave an $(\varepsilon, \delta)$-differentially private algorithm obtaining an $\alpha$-multiplicative approximation to the geometric median objective, $\frac 1 n \sum_{i \in [n]} \|\cdot - \mathbf{x}_i\|$, given a dataset $\mathcal{D} := \{\mathbf{x}_i\}_{i \in [n]} \subset \mathbb{R}^d$. Their algorithm requires $n \gtrsim \sqrt d \cdot \frac 1 {\alpha\varepsilon}$ samples, which they prove is information-theoretically optimal. This result is surprising because its error scales with the \emph{effective radius} of $\mathcal{D}$ (i.e., of a ball capturing most points), rather than the worst-case radius. We give an improved algorithm that obtains the same approximation quality, also using $n \gtrsim \sqrt d \cdot \frac 1 {\alpha\epsilon}$ samples, but in time $\widetilde{O}(nd + \frac d {\alpha^2})$. Our runtime is nearly-linear, plus the cost of the cheapest non-private first-order method due to [CLM+16]. To achieve our results, we use subsampling and geometric aggregation tools inspired by FriendlyCore [TCK+22] to speed up the "warm start" component of the [HSU24] algorithm, combined with a careful custom analysis of DP-SGD's sensitivity for the geometric median objective.

cs.DS↗

arXiv2025

Radial Isotropic Position via an Implicit Newton's Method

Placing a dataset $A = \{\mathbf{a}_i\}_{i \in [n]} \subset \mathbb{R}^d$ in radial isotropic position, i.e., finding an invertible $\mathbf{R} \in \mathbb{R}^{d \times d}$ such that the unit vectors $\{(\mathbf{R} \mathbf{a}_i) \|\mathbf{R} \mathbf{a}_i\|_2^{-1}\}_{i \in [n]}$ are in isotropic position, is a powerful tool with applications in functional analysis, communication complexity, coding theory, and the design of learning algorithms. When the transformed dataset has a second moment matrix within a $\exp(\pm \epsilon)$ factor of a multiple of $\mathbf{I}_d$, we call $\mathbf{R}$ an $\epsilon$-approximate Forster transform. We give a faster algorithm for computing approximate Forster transforms, based on optimizing an objective defined by Barthe [Barthe98]. When the transform has a polynomially-bounded aspect ratio, our algorithm uses $O(nd^{\omega - 1}(\frac n \epsilon)^{o(1)})$ time to output an $\epsilon$-approximate Forster transform with high probability, when one exists. This is almost the natural limit of this approach, as even evaluating Barthe's objective takes $O(nd^{\omega - 1})$ time. Previously, the state-of-the-art runtime in this regime was based on cutting-plane methods, and scaled at least as $\approx n^3 + n^2 d^{\omega - 1}$. We also provide explicit estimates on the aspect ratio in the smoothed analysis setting, and show that our algorithm similarly improves upon those in the literature. To obtain our results, we develop a subroutine of potential broader interest: a reduction from almost-linear time sparsification of graph Laplacians to the ability to support almost-linear time matrix-vector products. We combine this tool with new stability bounds on Barthe's objective to implicitly implement a box-constrained Newton's method [CMTV17, ALOW17].

cs.DS↗

arXiv2025

Spike-and-Slab Posterior Sampling in High Dimensions

Posterior sampling with the spike-and-slab prior [MB88], a popular multimodal distribution used to model uncertainty in variable selection, is considered the theoretical gold standard method for Bayesian sparse linear regression [CPS09, Roc18]. However, designing provable algorithms for performing this sampling task is notoriously challenging. Existing posterior samplers for Bayesian sparse variable selection tasks either require strong assumptions about the signal-to-noise ratio (SNR) [YWJ16], only work when the measurement count grows at least linearly in the dimension [MW24], or rely on heuristic approximations to the posterior. We give the first provable algorithms for spike-and-slab posterior sampling that apply for any SNR, and use a measurement count sublinear in the problem dimension. Concretely, assume we are given a measurement matrix $\mathbf{X} \in \mathbb{R}^{n\times d}$ and noisy observations $\mathbf{y} = \mathbf{X}\mathbf{\theta}^\star + \mathbf{\xi}$ of a signal $\mathbf{\theta}^\star$ drawn from a spike-and-slab prior $\pi$ with a Gaussian diffuse density and expected sparsity k, where $\mathbf{\xi} \sim \mathcal{N}(\mathbb{0}_n, \sigma^2\mathbf{I}_n)$. We give a polynomial-time high-accuracy sampler for the posterior $\pi(\cdot \mid \mathbf{X}, \mathbf{y})$, for any SNR $\sigma^{-1}$ > 0, as long as $n \geq k^3 \cdot \text{polylog}(d)$ and $X$ is drawn from a matrix ensemble satisfying the restricted isometry property. We further give a sampler that runs in near-linear time $\approx nd$ in the same setting, as long as $n \geq k^5 \cdot \text{polylog}(d)$. To demonstrate the flexibility of our framework, we extend our result to spike-and-slab posterior sampling with Laplace diffuse densities, achieving similar guarantees when $\sigma = O(\frac{1}{k})$ is bounded.

stat.ML↗

arXiv2025

More of the Same: Persistent Representational Harms Under Increased Representation

To recognize and mitigate the harms of generative AI systems, it is crucial to consider whether and how different societal groups are represented by these systems. A critical gap emerges when naively measuring or improving who is represented, as this does not consider how people are represented. In this work, we develop GAS(P), an evaluation methodology for surfacing distribution-level group representational biases in generated text, tackling the setting where groups are unprompted (i.e., groups are not specified in the input to generative systems). We apply this novel methodology to investigate gendered representations in occupations across state-of-the-art large language models. We show that, even though the gender distribution when models are prompted to generate biographies leads to a large representation of women, even representational biases persist in how different genders are represented. Our evaluation methodology reveals that there are statistically significant distribution-level differences in the word choice used to describe biographies and personas of different genders across occupations, and we show that many of these differences are associated with representational harms and stereotypes. Our empirical findings caution that naively increasing (unprompted) representation may inadvertently proliferate representational biases, and our proposed evaluation methodology enables systematic and rigorous measurement of the problem.

cs.CL↗

arXiv2025

Learning Noisy Halfspaces with a Margin: Massart is No Harder than Random

We study the problem of PAC learning $\gamma$-margin halfspaces with Massart noise. We propose a simple proper learning algorithm, the Perspectron, that has sample complexity $\widetilde{O}((\epsilon\gamma)^{-2})$ and achieves classification error at most $\eta+\epsilon$ where $\eta$ is the Massart noise rate. Prior works [DGT19,CKMY20] came with worse sample complexity guarantees (in both $\epsilon$ and $\gamma$) or could only handle random classification noise [DDK+23,KIT+23] -- a much milder noise assumption. We also show that our results extend to the more challenging setting of learning generalized linear models with a known link function under Massart noise, achieving a similar sample complexity to the halfspace case. This significantly improves upon the prior state-of-the-art in this setting due to [CKMY20], who introduced this model.

cs.LG↗

arXiv2024

Omnipredicting Single-Index Models with Multi-Index Models

Recent work on supervised learning [GKR+22] defined the notion of omnipredictors, i.e., predictor functions $p$ over features that are simultaneously competitive for minimizing a family of loss functions $\mathcal{L}$ against a comparator class $\mathcal{C}$. Omniprediction requires approximating the Bayes-optimal predictor beyond the loss minimization paradigm, and has generated significant interest in the learning theory community. However, even for basic settings such as agnostically learning single-index models (SIMs), existing omnipredictor constructions require impractically-large sample complexities and runtimes, and output complex, highly-improper hypotheses. Our main contribution is a new, simple construction of omnipredictors for SIMs. We give a learner outputting an omnipredictor that is $\varepsilon$-competitive on any matching loss induced by a monotone, Lipschitz link function, when the comparator class is bounded linear predictors. Our algorithm requires $\approx \varepsilon^{-4}$ samples and runs in nearly-linear time, and its sample complexity improves to $\approx \varepsilon^{-2}$ if link functions are bi-Lipschitz. This significantly improves upon the only prior known construction, due to [HJKRR18, GHK+23], which used $\gtrsim \varepsilon^{-10}$ samples. We achieve our construction via a new, sharp analysis of the classical Isotron algorithm [KS09, KKKS11] in the challenging agnostic learning setting, of potential independent interest. Previously, Isotron was known to properly learn SIMs in the realizable setting, as well as constant-factor competitive hypotheses under the squared loss [ZWDD24]. As they are based on Isotron, our omnipredictors are multi-index models with $\approx \varepsilon^{-2}$ prediction heads, bringing us closer to the tantalizing goal of proper omniprediction for general loss families and comparators.

cs.LG↗