The Halpern iteration for solving monotone inclusion problems has gained increasing interests in recent years due to its simple form and appealing convergence properties. In this paper, we investigate the inexact variants of the scheme in both deterministic and stochastic settings. We conduct extensive convergence analysis and show that by choosing the inexactness tolerances appropriately, the inexact schemes admit an $O(k^{-1})$ convergence rate in terms of the (expected) residue norm. Our results relax the state-of-the-art inexactness conditions employed in the literature while sharing the same competitive convergence properties. We then demonstrate how the proposed methods can be applied for solving two classes of data-driven Wasserstein distributionally robust optimization problems that admit convex-concave min-max optimization reformulations. We highlight its capability of performing inexact computations for distributionally robust learning with stochastic first-order methods.

This paper explores the multifaceted consequences of federated unlearning (FU) with data heterogeneity. We introduce key metrics for FU assessment, concentrating on verification, global stability, and local fairness, and investigate the inherent trade-offs. Furthermore, we formulate the unlearning process with data heterogeneity through an optimization framework. Our key contribution lies in a comprehensive theoretical analysis of the trade-offs in FU and provides insights into data heterogeneity's impacts on FU. Leveraging these insights, we propose FU mechanisms to manage the trade-offs, guiding further development for FU mechanisms. We empirically validate that our FU mechanisms effectively balance trade-offs, confirming insights derived from our theoretical analysis.

Motivated by the advances in conformal prediction (CP), we propose conformal predictive programming (CPP), an approach to solve chance constrained optimization (CCO) problems, i.e., optimization problems with nonlinear constraint functions affected by arbitrary random parameters. CPP utilizes samples from these random parameters along with the quantile lemma -- which is central to CP -- to transform the CCO problem into a deterministic optimization problem. We then present two tractable reformulations of CPP by: (1) writing the quantile as a linear program along with its KKT conditions (CPP-KKT), and (2) using mixed integer programming (CPP-MIP). CPP comes with marginal probabilistic feasibility guarantees for the CCO problem that are conceptually different from existing approaches, e.g., the sample approximation and the scenario approach. While we explore algorithmic similarities with the sample approximation approach, we emphasize that the strength of CPP is that it can easily be extended to incorporate different variants of CP. To illustrate this, we present robust conformal predictive programming to deal with distribution shifts in the uncertain parameters of the CCO problem.

Adapting to a priori unknown noise level is a very important but challenging problem in sequential decision-making as efficient exploration typically requires knowledge of the noise level, which is often loosely specified. We report significant progress in addressing this issue in linear bandits in two respects. First, we propose a novel confidence set that is `semi-adaptive' to the unknown sub-Gaussian parameter $\sigma_*^2$ in the sense that the (normalized) confidence width scales with $\sqrt{d\sigma_*^2 + \sigma_0^2}$ where $d$ is the dimension and $\sigma_0^2$ is the specified sub-Gaussian parameter (known) that can be much larger than $\sigma_*^2$. This is a significant improvement over $\sqrt{d\sigma_0^2}$ of the standard confidence set of Abbasi-Yadkori et al. (2011), especially when $d$ is large. We show that this leads to an improved regret bound in linear bandits. Second, for bounded rewards, we propose a novel variance-adaptive confidence set that has a much improved numerical performance upon prior art. We then apply this confidence set to develop, as we claim, the first practical variance-adaptive linear bandit algorithm via an optimistic approach, which is enabled by our novel regret analysis technique. Both of our confidence sets rely critically on `regret equality' from online learning. Our empirical evaluation in Bayesian optimization tasks shows that our algorithms demonstrate better or comparable performance compared to existing methods.

Identifying correspondences in noisy data is a critically important step in estimation processes. When an informative initial estimation guess is available, the data association challenge is less acute; however, the existence of a high-quality initial guess is rare in most contexts. We explore graph-theoretic formulations for data association, which do not require an initial estimation guess. Existing graph-theoretic approaches optimize over unweighted graphs, discarding important consistency information encoded in weighted edges, and frequently attempt to solve NP-hard problems exactly. In contrast, we formulate a new optimization problem that fully leverages weighted graphs and seeks the densest edge-weighted clique. We introduce two relaxations to this problem: a convex semidefinite relaxation which we find to be empirically tight, and a fast first-order algorithm called CLIPPER which frequently arrives at nearly-optimal solutions in milliseconds. When evaluated on point cloud registration problems, our algorithms remain robust up to at least 95% outliers while existing algorithms begin breaking down at 80% outliers. Code is available at https://mit-acl.github.io/clipper.

Online linear programming plays an important role in both revenue management and resource allocation, and recent research has focused on developing efficient first-order online learning algorithms. Despite the empirical success of first-order methods, they typically achieve a regret no better than $\mathcal{O}(\sqrt{T})$, which is suboptimal compared to the $\mathcal{O}(\log T)$ bound guaranteed by the state-of-the-art linear programming (LP)-based online algorithms. This paper establishes several important facts about online linear programming, which unveils the challenge for first-order-method-based online algorithms to achieve beyond $\mathcal{O}(\sqrt{T})$ regret. To address the challenge, we introduce a new algorithmic framework that decouples learning from decision-making. More importantly, for the first time, we show that first-order methods can attain regret $\mathcal{O}(T^{1/3})$ with this new framework. Lastly, we conduct numerical experiments to validate our theoretical findings.

Optimal allocation of agricultural water in the event of droughts is an important global problem. In addressing this problem, many aspects, including the welfare of farmers, the economy, and the environment, must be considered. Under this backdrop, our work focuses on several resource-matching problems accounting for agents with multi-crop portfolios, geographic constraints, and fairness. First, we address a matching problem where the goal is to maximize a welfare function in two-sided markets where buyers' requirements and sellers' supplies are represented by value functions that assign prices (or costs) to specified volumes of water. For the setting where the value functions satisfy certain monotonicity properties, we present an efficient algorithm that maximizes a social welfare function. When there are minimum water requirement constraints, we present a randomized algorithm which ensures that the constraints are satisfied in expectation. For a single seller--multiple buyers setting with fairness constraints, we design an efficient algorithm that maximizes the minimum level of satisfaction of any buyer. We also present computational complexity results that highlight the limits on the generalizability of our results. We evaluate the algorithms developed in our work with experiments on both real-world and synthetic data sets with respect to drought severity, value functions, and seniority of agents.

This paper addresses the challenge of optimizing meta-parameters (i.e., hyperparameters) in machine learning algorithms, a critical factor influencing training efficiency and model performance. Moving away from the computationally expensive traditional meta-parameter search methods, we introduce MetaOptimize framework that dynamically adjusts meta-parameters, particularly step sizes (also known as learning rates), during training. More specifically, MetaOptimize can wrap around any first-order optimization algorithm, tuning step sizes on the fly to minimize a specific form of regret that accounts for long-term effect of step sizes on training, through a discounted sum of future losses. We also introduce low complexity variants of MetaOptimize that, in conjunction with its adaptability to multiple optimization algorithms, demonstrate performance competitive to those of best hand-crafted learning rate schedules across various machine learning applications.

We consider the problem of finding stationary points in Bilevel optimization when the lower-level problem is unconstrained and strongly convex. The problem has been extensively studied in recent years; the main technical challenge is to keep track of lower-level solutions $y^*(x)$ in response to the changes in the upper-level variables $x$. Subsequently, all existing approaches tie their analyses to a genie algorithm that knows lower-level solutions and, therefore, need not query any points far from them. We consider a dual question to such approaches: suppose we have an oracle, which we call $y^*$-aware, that returns an $O(\epsilon)$-estimate of the lower-level solution, in addition to first-order gradient estimators {\it locally unbiased} within the $\Theta(\epsilon)$-ball around $y^*(x)$. We study the complexity of finding stationary points with such an $y^*$-aware oracle: we propose a simple first-order method that converges to an $\epsilon$ stationary point using $O(\epsilon^{-6}), O(\epsilon^{-4})$ access to first-order $y^*$-aware oracles. Our upper bounds also apply to standard unbiased first-order oracles, improving the best-known complexity of first-order methods by $O(\epsilon)$ with minimal assumptions. We then provide the matching $\Omega(\epsilon^{-6})$, $\Omega(\epsilon^{-4})$ lower bounds without and with an additional smoothness assumption on $y^*$-aware oracles, respectively. Our results imply that any approach that simulates an algorithm with an $y^*$-aware oracle must suffer the same lower bounds.

In this study, we propose a new method for constructing UCB-type algorithms for stochastic multi-armed bandits based on general convex optimization methods with an inexact oracle. We derive the regret bounds corresponding to the convergence rates of the optimization methods. We propose a new algorithm Clipped-SGD-UCB and show, both theoretically and empirically, that in the case of symmetric noise in the reward, we can achieve an $O(\log T\sqrt{KT\log T})$ regret bound instead of $O\left (T^{\frac{1}{1+\alpha}} K^{\frac{\alpha}{1+\alpha}} \right)$ for the case when the reward distribution satisfies $\mathbb{E}_{X \in D}[|X|^{1+\alpha}] \leq \sigma^{1+\alpha}$ ($\alpha \in (0, 1])$, i.e. perform better than it is assumed by the general lower bound for bandits with heavy-tails. Moreover, the same bound holds even when the reward distribution does not have the expectation, that is, when $\alpha<0$.