Sloan Nietert

Nika Haghtalab, Thodoris Lykouris, Sloan Nietert et al. · berkeley

We study Stackelberg games where a principal repeatedly interacts with a non-myopic long-lived agent, without knowing the agent's payoff function. Although learning in Stackelberg games is well-understood when the agent is myopic, dealing with non-myopic agents poses additional complications. In particular, non-myopic agents may strategize and select actions that are inferior in the present in order to mislead the principal's learning algorithm and obtain better outcomes in the future. We provide a general framework that reduces learning in presence of non-myopic agents to robust bandit optimization in the presence of myopic agents. Through the design and analysis of minimally reactive bandit algorithms, our reduction trades off the statistical efficiency of the principal's learning algorithm against its effectiveness in inducing near-best-responses. We apply this framework to Stackelberg security games (SSGs), pricing with unknown demand curve, general finite Stackelberg games, and strategic classification. In each setting, we characterize the type and impact of misspecifications present in near-best responses and develop a learning algorithm robust to such misspecifications. On the way, we improve the state-of-the-art query complexity of learning in SSGs with $n$ targets from $O(n^3)$ to a near-optimal $\widetilde{O}(n)$ by uncovering a fundamental structural property of these games. The latter result is of independent interest beyond learning with non-myopic agents.

Sloan Nietert, Ritwik Sadhu, Ziv Goldfeld et al. · uw

Sliced Wasserstein distances preserve properties of classic Wasserstein distances while being more scalable for computation and estimation in high dimensions. The goal of this work is to quantify this scalability from three key aspects: (i) empirical convergence rates; (ii) robustness to data contamination; and (iii) efficient computational methods. For empirical convergence, we derive fast rates with explicit dependence of constants on dimension, subject to log-concavity of the population distributions. For robustness, we characterize minimax optimal, dimension-free robust estimation risks, and show an equivalence between robust sliced 1-Wasserstein estimation and robust mean estimation. This enables lifting statistical and algorithmic guarantees available for the latter to the sliced 1-Wasserstein setting. Moving on to computational aspects, we analyze the Monte Carlo estimator for the average-sliced distance, demonstrating that larger dimension can result in faster convergence of the numerical integration error. For the max-sliced distance, we focus on a subgradient-based local optimization algorithm that is frequently used in practice, albeit without formal guarantees, and establish an $O(ε^{-4})$ computational complexity bound for it. Our theory is validated by numerical experiments, which altogether provide a comprehensive quantitative account of the scalability question.

Sloan Nietert, Rachel Cummings, Ziv Goldfeld

We study the problem of robust distribution estimation under the Wasserstein distance, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. Given $n$ samples from an unknown distribution $μ$, of which $\varepsilon n$ are adversarially corrupted, we seek an estimate for $μ$ with minimal Wasserstein error. To address this task, we draw upon two frameworks from OT and robust statistics: partial OT (POT) and minimum distance estimation (MDE). We prove new structural properties for POT and use them to show that MDE under a partial Wasserstein distance achieves the minimax-optimal robust estimation risk in many settings. Along the way, we derive a novel dual form for POT that adds a sup-norm penalty to the classic Kantorovich dual for standard OT. Since the popular Wasserstein generative adversarial network (WGAN) framework implements Wasserstein MDE via Kantorovich duality, our penalized dual enables large-scale generative modeling with contaminated datasets via an elementary modification to WGAN. Numerical experiments demonstrating the efficacy of our approach in mitigating the impact of adversarial corruptions are provided.

Sloan Nietert, Ziv Goldfeld

The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real-world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass. To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.

Thodoris Lykouris, Sloan Nietert, Princewill Okoroafor et al.

We initiate the study of contextual dynamic pricing with a heterogeneous population of buyers, where a seller repeatedly posts prices (over $T$ rounds) that depend on the observable $d$-dimensional context and receives binary purchase feedback. Unlike prior work assuming homogeneous buyer types, in our setting the buyer's valuation type is drawn from an unknown distribution with finite support size $K_{\star}$. We develop a contextual pricing algorithm based on optimistic posterior sampling with regret $\widetilde{O}(K_{\star}\sqrt{dT})$, which we prove to be tight in $d$ and $T$ up to logarithmic terms. Finally, we refine our analysis for the non-contextual pricing case, proposing a variance-aware zooming algorithm that achieves the optimal dependence on $K_{\star}$.

Sloan Nietert, Ziv Goldfeld, Soroosh Shafiee

Distributionally robust optimization (DRO) is an effective approach for data-driven decision-making in the presence of uncertainty. Geometric uncertainty due to sampling or localized perturbations of data points is captured by Wasserstein DRO (WDRO), which seeks to learn a model that performs uniformly well over a Wasserstein ball centered around the observed data distribution. However, WDRO fails to account for non-geometric perturbations such as adversarial outliers, which can greatly distort the Wasserstein distance measurement and impede the learned model. We address this gap by proposing a novel outlier-robust WDRO framework for decision-making under both geometric (Wasserstein) perturbations and non-geometric (total variation (TV)) contamination that allows an $\varepsilon$-fraction of data to be arbitrarily corrupted. We design an uncertainty set using a certain robust Wasserstein ball that accounts for both perturbation types and derive minimax optimal excess risk bounds for this procedure that explicitly capture the Wasserstein and TV risks. We prove a strong duality result that enables tractable convex reformulations and efficient computation of our outlier-robust WDRO problem. When the loss function depends only on low-dimensional features of the data, we eliminate certain dimension dependencies from the risk bounds that are unavoidable in the general setting. Finally, we present experiments validating our theory on standard regression and classification tasks.

Sloan Nietert, Ziv Goldfeld, Soroosh Shafiee

We consider learning in an adversarial environment, where an $\varepsilon$-fraction of samples from a distribution $P$ are arbitrarily modified (global corruptions) and the remaining perturbations have average magnitude bounded by $ρ$ (local corruptions). Given access to $n$ such corrupted samples, we seek a computationally efficient estimator $\hat{P}_n$ that minimizes the Wasserstein distance $\mathsf{W}_1(\hat{P}_n,P)$. In fact, we attack the fine-grained task of minimizing $\mathsf{W}_1(Π_\# \hat{P}_n, Π_\# P)$ for all orthogonal projections $Π\in \mathbb{R}^{d \times d}$, with performance scaling with $\mathrm{rank}(Π) = k$. This allows us to account simultaneously for mean estimation ($k=1$), distribution estimation ($k=d$), as well as the settings interpolating between these two extremes. We characterize the optimal population-limit risk for this task and then develop an efficient finite-sample algorithm with error bounded by $\sqrt{\varepsilon k} + ρ+ \tilde{O}(d\sqrt{k}n^{-1/(k \lor 2)})$ when $P$ has bounded covariance. This guarantee holds uniformly in $k$ and is minimax optimal up to the sub-optimality of the plug-in estimator when $ρ= \varepsilon = 0$. Our efficient procedure relies on a novel trace norm approximation of an ideal yet intractable 2-Wasserstein projection estimator. We apply this algorithm to robust stochastic optimization, and, in the process, uncover a new method for overcoming the curse of dimensionality in Wasserstein distributionally robust optimization.

Sloan Nietert, Rachel Cummings, Ziv Goldfeld

The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. We propose a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from each contaminated distribution. Under standard moment assumptions, $\mathsf{W}_p^\varepsilon$ is shown to achieve strong robust estimation guarantees under the Huber $\varepsilon$-contamination model. Our formulation of this robust distance amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of $\mathsf{W}_p^\varepsilon$, encompassing robustness guarantees, characterization of optimal perturbations, regularity, duality, and statistical estimation. In particular, by decoupling the optimization variables, we arrive at a simple dual form for $\mathsf{W}_p^\varepsilon$ that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.

Michela Meister, Sloan Nietert

We study an online version of the noisy binary search problem where feedback is generated by a non-stochastic adversary rather than perturbed by random noise. We reframe this as maintaining an accurate estimate for the median of an adversarial sequence of integers, $x_1, x_2, \dots$, in a model where each number $x_t$ can only be accessed through a single threshold query of the form ${1(x_t \leq q_t)}$. In this online comparison feedback model, we explore estimation of general sample statistics, providing robust algorithms for median, CDF, and mean estimation with nearly matching lower bounds. We conclude with several high-dimensional generalizations.

Sloan Nietert, Ziv Goldfeld, Kengo Kato

Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from data, recent work has proposed smoothing out local irregularities in the measured distributions via convolution with a Gaussian kernel. Motivated by the scalability of this framework to high dimensions, we investigate the structural and statistical behavior of the Gaussian-smoothed $p$-Wasserstein distance $\mathsf{W}_p^{(σ)}$, for arbitrary $p\geq 1$. After establishing basic metric and topological properties of $\mathsf{W}_p^{(σ)}$, we explore the asymptotic statistical behavior of $\mathsf{W}_p^{(σ)}(\hatμ_n,μ)$, where $\hatμ_n$ is the empirical distribution of $n$ independent observations from $μ$. We prove that $\mathsf{W}_p^{(σ)}$ enjoys a parametric empirical convergence rate of $n^{-1/2}$, which contrasts the $n^{-1/d}$ rate for unsmoothed $\mathsf{W}_p$ when $d \geq 3$. Our proof relies on controlling $\mathsf{W}_p^{(σ)}$ by a $p$th-order smooth Sobolev distance $\mathsf{d}_p^{(σ)}$ and deriving the limit distribution of $\sqrt{n}\,\mathsf{d}_p^{(σ)}(\hatμ_n,μ)$, for all dimensions $d$. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation using $\mathsf{W}_p^{(σ)}$, with experiments for $p=2$ using a maximum mean discrepancy formulation of $\mathsf{d}_2^{(σ)}$.