Self-Balancing Sequential Sampling: Fast Convergence with Controlled Predictability

Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.

Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design

We consider the parameter estimation problem in logistic regression with Gaussian design: the estimation of a fixed unknown parameter $θ^*\in \mathbb{R}^d$ ($\|θ^*\|_2\ge 1$) from $n$ i.i.d. samples $\{(x_i,y_i)\}_{i=1}^n$, where $x_i\sim N(0,I_d)$ and $y_i|x_i \sim {\rm Bernoulli}(1/(1+\exp(-x_i^\top θ^*)))$. Our main aim is to characterize the finite-sample estimation performance and convergence behavior of gradient descent (GD) on the maximum likelihood objective (i.e., the logistic loss). Under small $O(1)$ stepsize and $0$ initialization, we show that GD linearly converges to a small neighborhood of $θ^*$ achieving an $\ell_2$ error of order $O(\sqrt{\|θ^*\|_2^5d/n})$. This substantially goes beyond existing theoretical results that lack non-asymptotic estimation error rate and exhibit much slower parameter convergence. We also establish a faster local linear convergence to the same statistical error under a large $Θ(\|θ^*\|_2)$ stepsize. The main technical component is to show that the gradient of the logistic loss satisfies a certain approximate invertibility condition (AIC). To that end, we uniformly control the deviation of the gradient from its population counterpart by covering and peeling arguments, and then show that the population GD is a contraction by a delicate analysis based on the eigenvalues of population Hessian matrices. Finally, we build upon the recent work Matsumoto and Mazumdar (2025) and devise a novel efficient estimator that attains a sharper rate in high dimensions. This indicates that the existing non-asymptotic guarantees exhibit sub-optimal dependence on $\|θ^*\|_2$, and that in many regimes $Θ(\sqrt{\|θ^*\|_2d/n})$ is the tight estimation error rate. Numerical examples are provided to corroborate our theoretical results.

A unified complexity bound for logconcave sampling

We give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincaré constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).