Jacob Shkrob Яша Шкроб

We present a version of the vector balancing problem in which each vector may be given a sign and a permutation of its coordinates. We prove that this vector balancing problem and its corresponding prefix problem admit an explicit bound, and we further show that it is asymptotically optimal in the dimension. Our method of proof is purely geometric.

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent \(\frac{2}{d+2}\) obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every \(d \geq 2\), settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if \(\nu\) is a discrete measure and the optimal coupling between \(\mu\) and \(\nu\) transports each point to a nearest atom of \(\nu\), then \(W_p(\mu, \nu) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(\mu, \nu)\) for a universal constant \(C\), where the complexity parameter \(K\) is always at most the number of atoms \(N\) and can be substantially smaller. This complements a similar bound due to Park and Slepčev. An analogous bound holds for the sliced Wasserstein distance based on \(k\)-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on \(K\) in this bound cannot be improved, up to polylogarithmic factors.

Towards understanding the fundamental limits of estimation from data of varied quality, we study the problem of estimating a mean parameter from heteroskedastic Gaussian observations where the variances are unknown and may vary arbitrarily across observations. While a simple linear estimator with known variances attains the smallest mean squared error, estimation without this knowledge is challenging due to the large number of nuisance parameters. We propose a simple and principled approach based on empirical Bayes: model the observations as if they were i.i.d. from a normal scale mixture and compute the profile maximum likelihood estimator (MLE) for the mean, treating the nonparametric mixing distribution as nuisance. Our result shows that this estimator achieves near-optimal error bounds across various heteroskedastic models in the literature. In particular, for the subset-of-signals problem where an unknown subset of observations has small variance, our estimator adaptively achieves the minimax rate for all signal sizes, including the sharp phase transition, without any tuning parameters. One of our key technical steps is a sharper metric entropy bound for normal scale mixtures, obtained via Chebyshev approximations on a transformed polynomial basis. This approach yields an improved polylogarithmic, rather than polynomial, dependence on the variance ratio, which could be of independent interest.

Mechanical ventilation is a key form of life support for patients with pulmonary impairment. Healthcare workers are required to continuously adjust ventilator settings for each patient, a challenging and time consuming task. Hence, it would be beneficial to develop an automated decision support tool to optimize ventilation treatment. We present DeepVent, a Conservative Q-Learning (CQL) based offline Deep Reinforcement Learning (DRL) agent that learns to predict the optimal ventilator parameters for a patient to promote 90 day survival. We design a clinically relevant intermediate reward that encourages continuous improvement of the patient vitals as well as addresses the challenge of sparse reward in RL. We find that DeepVent recommends ventilation parameters within safe ranges, as outlined in recent clinical trials. The CQL algorithm offers additional safety by mitigating the overestimation of the value estimates of out-of-distribution states/actions. We evaluate our agent using Fitted Q Evaluation (FQE) and demonstrate that it outperforms physicians from the MIMIC-III dataset.