Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with thousands to millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order or meta-heuristic methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with millions of parameters. In light of the above challenges, this paper introduces \algoname, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair and comprehensive benchmark, and is open-sourced for the community.

In this paper the authors consider the problem of minimizing a continuous submodular function subject to ordering (isotonic) constraints. They first show that the problem can be solved if we first discretize it (per coordinate, not in [0,1] n), and then solve the resulting discrete optimization problem using convex optimization. The fact that the problem is solvable in polynomial time is of course not surprising, because, as pointed out by the authors in lines 29-36, we can add a penalty to the objective that will implicitly enforce the constraints. However, this can significantly increase the Lipschitz constant of the objective, and that is why the authors take on an alternative approach. First, they prove that seen in the space of measures, the isotonic constraints correspond to dominating inequalities of the CDFs, which I guess is an intuitive result given the results known for the unconstrained case.

Algorithmic fairness in clustering aims to balance the proportions of instances assigned to each cluster with respect to a given sensitive attribute. While recently developed fair clustering algorithms optimize clustering objectives under specific fairness constraints, their inherent complexity or approximation often results in suboptimal clustering utility or numerical instability in practice. To resolve these limitations, we propose a new fair clustering algorithm based on a novel decomposition of the fair $K$-means clustering objective function. The proposed algorithm, called Fair Clustering via Alignment (FCA), operates by alternately (i) finding a joint probability distribution to align the data from different protected groups, and (ii) optimizing cluster centers in the aligned space. A key advantage of FCA is that it theoretically guarantees approximately optimal clustering utility for any given fairness level without complex constraints, thereby enabling high-utility fair clustering in practice. Experiments show that FCA outperforms existing methods by (i) attaining a superior trade-off between fairness level and clustering utility, and (ii) achieving near-perfect fairness without numerical instability.

The analysis of Stochastic Gradient Descent (SGD) often relies on making some assumption on the variance of the stochastic gradients, which is usually not satisfied or difficult to verify in practice. This paper contributes to a recent line of works which attempt to provide guarantees without making any variance assumption, leveraging only the (strong) convexity and smoothness of the loss functions. In this context, we prove new theoretical bounds derived from the monotonicity of a simple Lyapunov energy, improving the current state-of-the-art and extending their validity to larger step-sizes. Our theoretical analysis is backed by a Performance Estimation Problem analysis, which allows us to claim that, empirically, the bias term in our bounds is tight within our framework.

We study offline constrained reinforcement learning (RL) with general function approximation. We aim to learn a policy from a pre-collected dataset that maximizes the expected discounted cumulative reward for a primary reward signal while ensuring that expected discounted returns for multiple auxiliary reward signals are above predefined thresholds. Existing algorithms either require fully exploratory data, are computationally inefficient, or depend on an additional auxiliary function classes to obtain an $ε$-optimal policy with sample complexity $O(ε^{-2})$. In this paper, we propose an oracle-efficient primal-dual algorithm based on a linear programming (LP) formulation, achieving $O(ε^{-2})$ sample complexity under partial data coverage. By introducing a realizability assumption, our approach ensures that all saddle points of the Lagrangian are optimal, removing the need for regularization that complicated prior analyses. Through Lagrangian decomposition, our method extracts policies without requiring knowledge of the data-generating distribution, enhancing practical applicability.

Bisimulation metrics are powerful tools for measuring similarities between stochastic processes, and specifically Markov chains. Recent advances have uncovered that bisimulation metrics are, in fact, optimal-transport distances, which has enabled the development of fast algorithms for computing such metrics with provable accuracy and runtime guarantees. However, these recent methods, as well as all previously known methods, assume full knowledge of the transition dynamics. This is often an impractical assumption in most real-world scenarios, where typically only sample trajectories are available. In this work, we propose a stochastic optimization method that addresses this limitation and estimates bisimulation metrics based on sample access, without requiring explicit transition models. Our approach is derived from a new linear programming (LP) formulation of bisimulation metrics, which we solve using a stochastic primal-dual optimization method. We provide theoretical guarantees on the sample complexity of the algorithm and validate its effectiveness through a series of empirical evaluations.

Adversarial attacks have become a well-explored domain, frequently serving as evaluation baselines for model robustness. Among these, black-box attacks based on transferability have received significant attention due to their practical applicability in real-world scenarios. Traditional black-box methods have generally focused on improving the optimization framework (e.g., utilizing momentum in MI-FGSM) to enhance transferability, rather than examining the dependency on surrogate white-box model architectures. Recent state-of-the-art approach DiffPGD has demonstrated enhanced transferability by employing diffusion-based adversarial purification models for adaptive attacks. The inductive bias of diffusion-based adversarial purification aligns naturally with the adversarial attack process, where both involving noise addition, reducing dependency on surrogate white-box model selection. However, the denoising process of diffusion models incurs substantial computational costs through chain rule derivation, manifested in excessive VRAM consumption and extended runtime. This progression prompts us to question whether introducing diffusion models is necessary. We hypothesize that a model sharing similar inductive bias to diffusion-based adversarial purification, combined with an appropriate loss function, could achieve comparable or superior transferability while dramatically reducing computational overhead. In this paper, we propose a novel loss function coupled with a unique surrogate model to validate our hypothesis. Our approach leverages the score of the time-dependent classifier from classifier-guided diffusion models, effectively incorporating natural data distribution knowledge into the adversarial optimization process. Experimental results demonstrate significantly improved transferability across diverse model architectures while maintaining robustness against diffusion-based defenses.

Clustering is a fundamental task in both machine learning and data mining. Among various methods, edge-colored clustering (ECC) has emerged as a useful approach for handling categorical data. Given a hypergraph with (hyper)edges labeled by colors, ECC aims to assign vertex colors to minimize the number of edges where the vertex color differs from the edge's color. However, traditional ECC has inherent limitations, as it enforces a nonoverlapping and exhaustive clustering. To tackle these limitations, three versions of ECC have been studied: Local ECC and Global ECC, which allow overlapping clusters, and Robust ECC, which accounts for vertex outliers. For these problems, both linear programming (LP) rounding algorithms and greedy combinatorial algorithms have been proposed. While these LP-rounding algorithms provide high-quality solutions, they demand substantial computation time; the greedy algorithms, on the other hand, run very fast but often compromise solution quality. In this paper, we present an algorithmic framework that combines the strengths of LP with the computational efficiency of combinatorial algorithms. Both experimental and theoretical analyses show that our algorithms efficiently produce high-quality solutions for all three problems: Local, Global, and Robust ECC. We complement our algorithmic contributions with complexity-theoretic inapproximability results and integrality gap bounds, which suggest that significant theoretical improvements are unlikely. Our results also answer two open questions previously raised in the literature.

Routing problems are canonical combinatorial optimization tasks with wide-ranging applications in logistics, transportation, and supply chain management. However, solving these problems becomes significantly more challenging when complex constraints are involved. In this paper, we propose LMask, a novel learning framework that utilizes dynamic masking to generate high-quality feasible solutions for constrained routing problems. LMask introduces the LazyMask decoding method, which lazily refines feasibility masks with the backtracking mechanism. In addition, it employs the refinement intensity embedding to encode the search trace into the model, mitigating representation ambiguities induced by backtracking. To further reduce sampling cost, LMask sets a backtracking budget during decoding, while constraint violations are penalized in the loss function during training to counteract infeasibility caused by this budget. We provide theoretical guarantees for the validity and probabilistic optimality of our approach. Extensive experiments on the traveling salesman problem with time windows (TSPTW) and TSP with draft limits (TSPDL) demonstrate that LMask achieves state-of-the-art feasibility rates and solution quality, outperforming existing neural methods.

We introduce the Best Group Identification problem in a multi-objective multi-armed bandit setting, where an agent interacts with groups of arms with vector-valued rewards. The performance of a group is determined by an efficiency vector which represents the group's best attainable rewards across different dimensions. The objective is to identify the set of optimal groups in the fixed-confidence setting. We investigate two key formulations: group Pareto set identification, where efficiency vectors of optimal groups are Pareto optimal and linear best group identification, where each reward dimension has a known weight and the optimal group maximizes the weighted sum of its efficiency vector's entries. For both settings, we propose elimination-based algorithms, establish upper bounds on their sample complexity, and derive lower bounds that apply to any correct algorithm. Through numerical experiments, we demonstrate the strong empirical performance of the proposed algorithms.