arXiv subjects

Jeffrey Negrea

Publications and source records attributed to Jeffrey Negrea.

13 recordsLinked to original sources

arXiv2026

Testing linear combinations of multiple variance components

We test the hypothesis that simulataneous linear contrasts of multiple variance components equal zero in a Gaussian variance components model via a parametric bootstrap. Applications include but are not limited to nested and crossed designs. The main technical contributions are a computationally efficient decomposition of the normalized residual log-likelihood that does not require the variance components to be non-negative or variance design matrices to be positive semi-definite, a modified Newton method for its minimization, and a method for efficient optimization and sampling under the null hypothesis that certain linear combinations of variance components equal zero. A special case of the proposed procedure is a test for multiple variance components simulataneously equalling zero, for which a likelihood ratio test was not previously available. However, the proposed procedure is significantly more general.

stat.ME↗

arXiv2026

A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC

Preconditioning is a common method applied to modify Markov chain Monte Carlo algorithms with the goal of making them more efficient. In practice it is often extremely effective, even when the preconditioner is learned from the chain. We analyse and compare the finite-time computational costs of schemes which learn a preconditioner based on the target covariance or the expected Hessian of the target potential with that of a corresponding scheme that does not use preconditioning. We apply our results to various algorithms including the Unadjusted Langevin Algorithm (ULA) and the proximal sampler for an appropriately regular target, establishing non-asymptotic guarantees for versions of these algorithms that learn and use preconditioners. To do so, we establish non-asymptotic guarantees on the time taken to collect $N$ approximately independent samples from the target for schemes that learn their preconditioners under the assumption that the underlying Markov chain satisfies a contraction in the Wasserstein-2 distance. This approximate independence condition, that we formalize, allows us to bridge the non-asymptotic bounds of modern MCMC theory and classical heuristics of effective sample size and mixing time, and is needed to amortise the costs of learning a preconditioner across the samples it will help to produce.

stat.CO↗

arXiv2025

Bayesian Algorithms for Adversarial Online Learning: from Finite to Infinite Action Spaces

We develop a form Thompson sampling for online learning under full feedback - also known as prediction with expert advice - where the learner's prior is defined over the space of an adversary's future actions, rather than the space of experts. We show regret decomposes into regret the learner expected a priori, plus a prior-robustness-type term we call excess regret. In the classical finite-expert setting, this recovers optimal rates. As an initial step towards practical online learning in settings with a potentially-uncountably-infinite number of experts, we show that Thompson sampling over the $d$-dimensional unit cube, using a certain Gaussian process prior widely-used in the Bayesian optimization literature, has a $\mathcal{O}\Big(β\sqrt{Td\log(1+\sqrt{d}\fracλβ)}\Big)$ rate against a $β$-bounded $λ$-Lipschitz adversary.

cs.LG↗

arXiv2025

Quantitative Error Bounds for Scaling Limits of Stochastic Iterative Algorithms

Stochastic iterative algorithms, including stochastic gradient descent (SGD) and stochastic gradient Langevin dynamics (SGLD), are widely utilized for optimization and sampling in large-scale and high-dimensional problems in machine learning, statistics, and engineering. Numerous works have bounded the parameter error in, and characterized the uncertainty of, these approximations. One common approach has been to use scaling limit analyses to relate the distribution of algorithm sample paths to a continuous-time stochastic process approximation, particularly in asymptotic setups. Focusing on the univariate setting, in this paper, we build on previous work to derive non-asymptotic functional approximation error bounds between the algorithm sample paths and the Ornstein-Uhlenbeck approximation using an infinite-dimensional version of Stein's method of exchangeable pairs. We show that this bound implies weak convergence under modest additional assumptions and leads to a bound on the error of the variance of the iterate averages of the algorithm. Furthermore, we use our main result to construct error bounds in terms of two common metrics: the Lévy-Prokhorov and bounded Wasserstein distances. Our results provide a foundation for developing similar error bounds for the multivariate setting and for more sophisticated stochastic approximation algorithms.

stat.ML↗

arXiv2024

Concept Algebra for (Score-Based) Text-Controlled Generative Models

This paper concerns the structure of learned representations in text-guided generative models, focusing on score-based models. A key property of such models is that they can compose disparate concepts in a `disentangled' manner. This suggests these models have internal representations that encode concepts in a `disentangled' manner. Here, we focus on the idea that concepts are encoded as subspaces of some representation space. We formalize what this means, show there's a natural choice for the representation, and develop a simple method for identifying the part of the representation corresponding to a given concept. In particular, this allows us to manipulate the concepts expressed by the model through algebraic manipulation of the representation. We demonstrate the idea with examples using Stable Diffusion. Code in https://github.com/zihao12/concept-algebra-code

cs.CL↗

arXiv2023

Tuning Stochastic Gradient Algorithms for Statistical Inference via Large-Sample Asymptotics

The tuning of stochastic gradient algorithms (SGAs) for optimization and sampling is often based on heuristics and trial-and-error rather than generalizable theory. We address this theory--practice gap by characterizing the large-sample statistical asymptotics of SGAs via a joint step-size--sample-size scaling limit. We show that iterate averaging with a large fixed step size is robust to the choice of tuning parameters and asymptotically has covariance proportional to that of the MLE sampling distribution. We also prove a Bernstein--von Mises-like theorem to guide tuning, including for generalized posteriors that are robust to model misspecification. Numerical experiments validate our results and recommendations in realistic finite-sample regimes. Our work lays the foundation for a systematic analysis of other stochastic gradient Markov chain Monte Carlo algorithms for a wide range of models.

stat.CO↗

arXiv2022

Relaxing the I.I.D. Assumption: Adaptively Minimax Optimal Regret via Root-Entropic Regularization

We consider prediction with expert advice when data are generated from distributions varying arbitrarily within an unknown constraint set. This semi-adversarial setting includes (at the extremes) the classical i.i.d. setting, when the unknown constraint set is restricted to be a singleton, and the unconstrained adversarial setting, when the constraint set is the set of all distributions. The Hedge algorithm -- long known to be minimax (rate) optimal in the adversarial regime -- was recently shown to be simultaneously minimax optimal for i.i.d. data. In this work, we propose to relax the i.i.d. assumption by seeking adaptivity at all levels of a natural ordering on constraint sets. We provide matching upper and lower bounds on the minimax regret at all levels, show that Hedge with deterministic learning rates is suboptimal outside of the extremes, and prove that one can adaptively obtain minimax regret at all levels. We achieve this optimal adaptivity using the follow-the-regularized-leader (FTRL) framework, with a novel adaptive regularization scheme that implicitly scales as the square root of the entropy of the current predictive distribution, rather than the entropy of the initial predictive distribution. Finally, we provide novel technical tools to study the statistical performance of FTRL along the semi-adversarial spectrum.

stat.ML↗

arXiv2021

Minimax Optimal Quantile and Semi-Adversarial Regret via Root-Logarithmic Regularizers

Quantile (and, more generally, KL) regret bounds, such as those achieved by NormalHedge (Chaudhuri, Freund, and Hsu 2009) and its variants, relax the goal of competing against the best individual expert to only competing against a majority of experts on adversarial data. More recently, the semi-adversarial paradigm (Bilodeau, Negrea, and Roy 2020) provides an alternative relaxation of adversarial online learning by considering data that may be neither fully adversarial nor stochastic (i.i.d.). We achieve the minimax optimal regret in both paradigms using FTRL with separate, novel, root-logarithmic regularizers, both of which can be interpreted as yielding variants of NormalHedge. We extend existing KL regret upper bounds, which hold uniformly over target distributions, to possibly uncountable expert classes with arbitrary priors; provide the first full-information lower bounds for quantile regret on finite expert classes (which are tight); and provide an adaptively minimax optimal algorithm for the semi-adversarial paradigm that adapts to the true, unknown constraint faster, leading to uniformly improved regret bounds over existing methods.

stat.ML↗

arXiv2021

In Defense of Uniform Convergence: Generalization via derandomization with an application to interpolating predictors

We propose to study the generalization error of a learned predictor $\hat h$ in terms of that of a surrogate (potentially randomized) predictor that is coupled to $\hat h$ and designed to trade empirical risk for control of generalization error. In the case where $\hat h$ interpolates the data, it is interesting to consider theoretical surrogate classifiers that are partially derandomized or rerandomized, e.g., fit to the training data but with modified label noise. We also show that replacing $\hat h$ by its conditional distribution with respect to an arbitrary $σ$-field is a convenient way to derandomize. We study two examples, inspired by the work of Nagarajan and Kolter (2019) and Bartlett et al. (2019), where the learned classifier $\hat h$ interpolates the training data with high probability, has small risk, and, yet, does not belong to a nonrandom class with a tight uniform bound on two-sided generalization error. At the same time, we bound the risk of $\hat h$ in terms of surrogates constructed by conditioning and denoising, respectively, and shown to belong to nonrandom classes with uniformly small generalization error.

cs.LG↗

arXiv2020

Sharpened Generalization Bounds based on Conditional Mutual Information and an Application to Noisy, Iterative Algorithms

The information-theoretic framework of Russo and J. Zou (2016) and Xu and Raginsky (2017) provides bounds on the generalization error of a learning algorithm in terms of the mutual information between the algorithm's output and the training sample. In this work, we study the proposal, by Steinke and Zakynthinou (2020), to reason about the generalization error of a learning algorithm by introducing a super sample that contains the training sample as a random subset and computing mutual information conditional on the super sample. We first show that these new bounds based on the conditional mutual information are tighter than those based on the unconditional mutual information. We then introduce yet tighter bounds, building on the "individual sample" idea of Bu, S. Zou, and Veeravalli (2019) and the "data dependent" ideas of Negrea et al. (2019), using disintegrated mutual information. Finally, we apply these bounds to the study of Langevin dynamics algorithm, showing that conditioning on the super sample allows us to exploit information in the optimization trajectory to obtain tighter bounds based on hypothesis tests.

stat.ML↗

arXiv2020

Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates

In this work, we improve upon the stepwise analysis of noisy iterative learning algorithms initiated by Pensia, Jog, and Loh (2018) and recently extended by Bu, Zou, and Veeravalli (2019). Our main contributions are significantly improved mutual information bounds for Stochastic Gradient Langevin Dynamics via data-dependent estimates. Our approach is based on the variational characterization of mutual information and the use of data-dependent priors that forecast the mini-batch gradient based on a subset of the training samples. Our approach is broadly applicable within the information-theoretic framework of Russo and Zou (2015) and Xu and Raginsky (2017). Our bound can be tied to a measure of flatness of the empirical risk surface. As compared with other bounds that depend on the squared norms of gradients, empirical investigations show that the terms in our bounds are orders of magnitude smaller.

stat.ML↗

arXiv2019

Optimal Scaling and Shaping of Random Walk Metropolis via Diffusion Limits of Block-I.I.D. Targets

This work extends Roberts et al. (1997) by considering limits of Random Walk Metropolis (RWM) applied to block IID target distributions, with corresponding block-independent proposals. The extension verifies the robustness of the optimal scaling heuristic, to tune the acceptance rate to $\approx0.234$, for any choice of proposal shaping. We upgrade the form of weak convergence from a finite-dimensional subprocess to the infinite dimensional process. We show that the optimal shaping (in terms of the decay of autocorrelations of linear functions) is the variance of the target distribution. We show that this choice coincides with the optimal shaping in terms of spectral gaps in special cases where they can be computed. Lastly, we provide some negative guarantees, showing that RWM performance degrades with higher-order dependence. In such cases, no tuning of RWM will yield performance comparable to an IID target.

math.PR↗

arXiv2017

Approximations of Geometrically Ergodic Reversible Markov Chains

A common tool in the practice of Markov Chain Monte Carlo is to use approximating transition kernels to speed up computation when the desired kernel is slow to evaluate or intractable. A limited set of quantitative tools exist to assess the relative accuracy and efficiency of such approximations. We derive a set of tools for such analysis based on the Hilbert space generated by the stationary distribution we intend to sample, $L_2(\pi)$. Our results apply to approximations of reversible chains which are geometrically ergodic, as is typically the case for applications to Markov Chain Monte Carlo. The focus of our work is on determining whether the approximating kernel will preserve the geometric ergodicity of the exact chain, and whether the approximating stationary distribution will be close to the original stationary distribution. For reversible chains, our results extend the results of Johndrow et al. [18] from the uniformly ergodic case to the geometrically ergodic case, under some additional regularity conditions. We then apply our results to a number of approximate MCMC algorithms.

math.PR↗