Hyperparameter tuning, particularly the selection of an appropriate learning rate in adaptive gradient training methods, remains a challenge. To tackle this challenge, in this paper, we propose a novel parameter-free optimizer, \textsc{AdamG} (Adam with the golden step size), designed to automatically adapt to diverse optimization problems without manual tuning. The core technique underlying \textsc{AdamG} is our golden step size derived for the AdaGrad-Norm algorithm, which is expected to help AdaGrad-Norm preserve the tuning-free convergence and approximate the optimal step size in expectation w.r.t. various optimization scenarios. To better evaluate tuning-free performance, we propose a novel evaluation criterion, \textit{reliability}, to comprehensively assess the efficacy of parameter-free optimizers in addition to classical performance criteria. Empirical results demonstrate that compared with other parameter-free baselines, \textsc{AdamG} achieves superior performance, which is consistently on par with Adam using a manually tuned learning rate across various optimization tasks.

As recommender systems are indispensable in various domains such as job searching and e-commerce, providing equitable recommendations to users with different sensitive attributes becomes an imperative requirement. Prior approaches for enhancing fairness in recommender systems presume the availability of all sensitive attributes, which can be difficult to obtain due to privacy concerns or inadequate means of capturing these attributes. In practice, the efficacy of these approaches is limited, pushing us to investigate ways of promoting fairness with limited sensitive attribute information. Toward this goal, it is important to reconstruct missing sensitive attributes. Nevertheless, reconstruction errors are inevitable due to the complexity of real-world sensitive attribute reconstruction problems and legal regulations. Thus, we pursue fair learning methods that are robust to reconstruction errors. To this end, we propose Distributionally Robust Fair Optimization (DRFO), which minimizes the worst-case unfairness over all potential probability distributions of missing sensitive attributes instead of the reconstructed one to account for the impact of the reconstruction errors. We provide theoretical and empirical evidence to demonstrate that our method can effectively ensure fairness in recommender systems when only limited sensitive attributes are accessible.

We develop a class of mixed-integer formulations for disjunctive constraints intermediate to the big-M and convex hull formulations in terms of relaxation strength. The main idea is to capture the best of both the big-M and convex hull formulations: a computationally light formulation with a tight relaxation. The "P-split" formulations are based on a lifted transformation that splits convex additively separable constraints into P partitions and forms the convex hull of the linearized and partitioned disjunction. The "P-split" formulations are derived for disjunctive constraints with convex constraints within each disjuct, and we generalize the results for the case with nonconvex constraints within the disjuncts. We analyze the continuous relaxation of the P-split formulations and show that, under certain assumptions, the formulations form a hierarchy starting from a big-M equivalent and converging to the convex hull. The goal of the P-split formulations is to form strong approximations of the convex hull through a computationally simpler formulation. We computationally compare the P-split formulations against big-M and convex hull formulations on 344 test instances. The test problems include K-means clustering, semi-supervised clustering, P_ball problems, and optimization over trained ReLU neural networks. The computational results show promising potential of the P-split formulations. For many of the test problems, P-split formulations are solved with a similar number of explored nodes as the convex hull formulation, while reducing the solution time by an order of magnitude and outperforming big-M both in time and number of explored nodes.

The sparse-group lasso performs both variable and group selection, making simultaneous use of the strengths of the lasso and group lasso. It has found widespread use in genetics, a field that regularly involves the analysis of high-dimensional data, due to its sparse-group penalty, which allows it to utilize grouping information. However, the sparse-group lasso can be computationally more expensive than both the lasso and group lasso, due to the added shrinkage complexity, and its additional hyper-parameter that needs tuning. In this paper a novel dual feature reduction method, Dual Feature Reduction (DFR), is presented that uses strong screening rules for the sparse-group lasso and the adaptive sparse-group lasso to reduce their input space before optimization. DFR applies two layers of screening and is based on the dual norms of the sparse-group lasso and adaptive sparse-group lasso. Through synthetic and real numerical studies, it is shown that the proposed feature reduction approach is able to drastically reduce the computational cost in many different scenarios.

The objective of this Philosophiae Doctor (Ph.D) thesis is to propose an efficient approach for optimizing a multidisciplinary black-box model when the optimization problem is constrained and involves a large number of mixed integer design variables (typically 100 variables). The targeted optimization approach, called EGO, is based on a sequential enrichment of an adaptive surrogate model and, in this context, GP surrogate models are one of the most widely used in engineering problems to approximate time-consuming high fidelity models. EGO is a heuristic BO method that performs well in terms of solution quality. However, like any other global optimization method, EGO suffers from the curse of dimensionality, meaning that its performance is satisfactory on lower dimensional problems, but deteriorates as the dimensionality of the optimization search space increases. For realistic aircraft design problems, the typical size of the design variables can even exceed 100 and, thus, trying to solve directly the problems using EGO is ruled out. The latter is especially true when the problems involve both continuous and categorical variables increasing even more the size of the search space. In this Ph.D thesis, effective parameterization tools are investigated, including techniques like partial least squares regression, to significantly reduce the number of design variables. Additionally, Bayesian optimization is adapted to handle discrete variables and high-dimensional spaces in order to reduce the number of evaluations when optimizing innovative aircraft concepts such as the "DRAGON" hybrid airplane to reduce their climate impact.

Standard Direction of Arrival (DOA) estimation methods are typically derived based on the Gaussian noise assumption, making them highly sensitive to outliers. Therefore, in the presence of impulsive noise, the performance of these methods may significantly deteriorate. In this paper, we model impulsive noise as Gaussian noise mixed with sparse outliers. By exploiting their statistical differences, we propose a novel DOA estimation method based on sparse signal recovery (SSR). Furthermore, to address the issue of grid mismatch, we utilize an alternating optimization approach that relies on the estimated outlier matrix and the on-grid DOA estimates to obtain the off-grid DOA estimates. Simulation results demonstrate that the proposed method exhibits robustness against large outliers.

In human-AI interaction, a prominent goal is to attain human's desirable outcome with the assistance of AI agents, which can be ideally delineated as a problem of seeking the optimal Nash Equilibrium that matches the human's desirable outcome. However, reaching the outcome is usually challenging due to the existence of multiple Nash Equilibria that are related to the assisting task but do not correspond to the human's desirable outcome. To tackle this issue, we employ a theoretical framework called structural causal game (SCG) to formalize the human-AI interactive process. Furthermore, we introduce a strategy referred to as pre-policy intervention on the SCG to steer AI agents towards attaining the human's desirable outcome. In more detail, a pre-policy is learned as a generalized intervention to guide the agents' policy selection, under a transparent and interpretable procedure determined by the SCG. To make the framework practical, we propose a reinforcement learning-like algorithm to search out this pre-policy. The proposed algorithm is tested in both gridworld environments and realistic dialogue scenarios with large language models, demonstrating its adaptability in a broader class of problems and potential effectiveness in real-world situations.

We study nonconvex zeroth-order optimization (ZOO) in a high-dimensional space $\mathbb R^d$ for functions with approximately $s$-sparse gradients. To reduce the dependence on the dimensionality $d$ in the query complexity, high-dimensional ZOO methods seek to leverage gradient sparsity to design gradient estimators. The previous best method needs $O\big(s\log\frac ds\big)$ queries per step to achieve $O\big(\frac1T\big)$ rate of convergence w.r.t. the number T of steps. In this paper, we propose *Gradient Compressed Sensing* (GraCe), a query-efficient and accurate estimator for sparse gradients that uses only $O\big(s\log\log\frac ds\big)$ queries per step and still achieves $O\big(\frac1T\big)$ rate of convergence. To our best knowledge, we are the first to achieve a *double-logarithmic* dependence on $d$ in the query complexity under weaker assumptions. Our proposed GraCe generalizes the Indyk--Price--Woodruff (IPW) algorithm in compressed sensing from linear measurements to nonlinear functions. Furthermore, since the IPW algorithm is purely theoretical due to its impractically large constant, we improve the IPW algorithm via our *dependent random partition* technique together with our corresponding novel analysis and successfully reduce the constant by a factor of nearly 4300. Our GraCe is not only theoretically query-efficient but also achieves strong empirical performance. We benchmark our GraCe against 12 existing ZOO methods with 10000-dimensional functions and demonstrate that GraCe significantly outperforms existing methods.

Racial and other demographic imputation is necessary for many applications, especially in auditing disparities and outreach targeting in political campaigns. The canonical approach is to construct continuous predictions -- e.g., based on name and geography -- and then to $\textit{discretize}$ the predictions by selecting the most likely class (argmax). We study how this practice produces $\textit{discretization bias}$. In particular, we show that argmax labeling, as used by a prominent commercial voter file vendor to impute race/ethnicity, results in a substantial under-count of African-American voters, e.g., by 28.2% points in North Carolina. This bias can have substantial implications in downstream tasks that use such labels. We then introduce a $\textit{joint optimization}$ approach -- and a tractable $\textit{data-driven thresholding}$ heuristic -- that can eliminate this bias, with negligible individual-level accuracy loss. Finally, we theoretically analyze discretization bias, show that calibrated continuous models are insufficient to eliminate it, and that an approach such as ours is necessary. Broadly, we warn researchers and practitioners against discretizing continuous demographic predictions without considering downstream consequences.

Better fuel efficiency leads to better financial security as well as a cleaner environment. We propose a novel approach for improving fuel efficiency in unstructured and unregulated traffic environments. Existing intelligent transportation solutions for improving fuel efficiency, however, apply only to traffic intersections with sparse traffic or traffic where drivers obey the regulations, or both. We propose GameOpt+, a novel hybrid approach for cooperative intersection control in dynamic, multi-lane, unsignalized intersections. GameOpt+ is a hybrid solution that combines an auction mechanism and an optimization-based trajectory planner. It generates a priority entrance sequence for each agent and computes velocity controls in real-time, taking less than 10 milliseconds even in high-density traffic with over 10,000 vehicles per hour. Compared to fully optimization-based methods, it operates 100 times faster while ensuring fairness, safety, and efficiency. Tested on the SUMO simulator, our algorithm improves throughput by at least 25%, reduces the time to reach the goal by at least 70%, and decreases fuel consumption by 50% compared to auction-based and signaled approaches using traffic lights and stop signs. GameOpt+ is also unaffected by unbalanced traffic inflows, whereas some of the other baselines encountered a decrease in performance in unbalanced traffic inflow environments.