Regularized nonnegative low-rank approximations such as sparse Nonnegative Matrix Factorization or sparse Nonnegative Tucker Decomposition are an important branch of dimensionality reduction models with enhanced interpretability. However, from a practical perspective, the choice of regularizers and regularization coefficients, as well as the design of efficient algorithms, is challenging because of the multifactor nature of these models and the lack of theory to back these choices. This paper aims at improving upon these issues. By studying a more general model called the Homogeneous Regularized Scale-Invariant, we prove that the scale-invariance inherent to low-rank approximation models causes an implicit regularization with both unexpected beneficial and detrimental effects. This observation allows to better understand the effect of regularization functions in low-rank approximation models, to guide the choice of the regularization hyperparameters, and to design balancing strategies to enhance the convergence speed of dedicated optimization algorithms. Some of these results were already known but restricted to specific instances of regularized low-rank approximations. We also derive a generic Majorization Minimization algorithm that handles many regularized nonnegative low-rank approximations, with convergence guarantees. We showcase our contributions on sparse Nonnegative Matrix Factorization, ridge-regularized Canonical Polyadic decomposition and sparse Nonnegative Tucker Decomposition.

Diffusion models have demonstrated great potential in generating high-quality content for images, natural language, protein domains, etc. However, how to perform user-preferred targeted generation via diffusion models with only black-box target scores of users remains challenging. To address this issue, we first formulate the fine-tuning of the targeted reserve-time stochastic differential equation (SDE) associated with a pre-trained diffusion model as a sequential black-box optimization problem. Furthermore, we propose a novel covariance-adaptive sequential optimization algorithm to optimize cumulative black-box scores under unknown transition dynamics. Theoretically, we prove a $O(\frac{d^2}{\sqrt{T}})$ convergence rate for cumulative convex functions without smooth and strongly convex assumptions. Empirically, experiments on both numerical test problems and target-guided 3D-molecule generation tasks show the superior performance of our method in achieving better target scores.

Unbalanced optimal transport (UOT) has recently gained much attention due to its flexible framework for handling un-normalized measures and its robustness properties. In this work, we explore learning (structured) sparse transport plans in the UOT setting, i.e., transport plans have an upper bound on the number of non-sparse entries in each column (structured sparse pattern) or in the whole plan (general sparse pattern). We propose novel sparsity-constrained UOT formulations building on the recently explored maximum mean discrepancy based UOT. We show that the proposed optimization problem is equivalent to the maximization of a weakly submodular function over a uniform matroid or a partition matroid. We develop efficient gradient-based discrete greedy algorithms and provide the corresponding theoretical guarantees. Empirically, we observe that our proposed greedy algorithms select a diverse support set and we illustrate the efficacy of the proposed approach in various applications.

Ensuring the safety of Reinforcement Learning (RL) is crucial for its deployment in real-world applications. Nevertheless, managing the trade-off between reward and safety during exploration presents a significant challenge. Improving reward performance through policy adjustments may adversely affect safety performance. In this study, we aim to address this conflicting relation by leveraging the theory of gradient manipulation. Initially, we analyze the conflict between reward and safety gradients. Subsequently, we tackle the balance between reward and safety optimization by proposing a soft switching policy optimization method, for which we provide convergence analysis. Based on our theoretical examination, we provide a safe RL framework to overcome the aforementioned challenge, and we develop a Safety-MuJoCo Benchmark to assess the performance of safe RL algorithms. Finally, we evaluate the effectiveness of our method on the Safety-MuJoCo Benchmark and a popular safe RL benchmark, Omnisafe. Experimental results demonstrate that our algorithms outperform several state-of-the-art baselines in terms of balancing reward and safety optimization.

This paper presents a trajectory optimization and control approach for the guidance of an orbital four-arm robot in extravehicular activities. The robot operates near the target spacecraft, enabling its arm's end-effectors to reach the spacecraft's surface. Connections to the target spacecraft can be established by the arms through specific footholds (docking devices). The trajectory optimization allows the robot path planning by computing the docking positions on the target spacecraft surface, along with their timing, the arm trajectories, the six degrees of freedom body motion, and the necessary contact forces during docking. In addition, the paper introduces a controller designed to track the planned trajectories derived from the solution of the nonlinear programming problem. A weighted controller formulated as a convex optimization problem is proposed. The controller is defined as the optimization of an objective function that allows the system to perform a set of tasks simultaneously. Simulation results show the application of the trajectory optimization and control approaches to an on-orbit servicing scenario.

Efficient numerical optimization methods can improve performance and reduce the environmental impact of computing in many applications. This work presents a proof-of-concept study combining primitive state representations and agent-environment interactions as first-order optimizers in the setting of budget-limited optimization. Through reinforcement learning (RL) over a set of training instances of an optimization problem class, optimal policies for sequential update selection of algorithmic iteration steps are approximated in generally formulated low-dimensional partial state representations that consider aspects of progress and resource use. For the investigated case studies, deployment of the trained agents to unseen instances of the quadratic optimization problem classes outperformed conventional optimal algorithms with optimized hyperparameters. The results show that elementary RL methods combined with succinct partial state representations can be used as heuristics to manage complexity in RL-based optimization, paving the way for agentic optimization approaches.

Multi-objective formulations have been widely used to solve single-objective optimization problems. The initial study carried out by Knowles et al. [8] for the H-IFF and the traveling salesperson problem shows that such formulations can significantly reduce the number of local optima in the search space and uses the term multi-objectivization for such approaches. Using multi-objective formulations to solve constrained single-objective optimization problems by evolutionary multi-objective optimization using the constraint as an additional objective has shown to be highly beneficial for a wide range of problems [4,9,12]. Using the constraint as an additional objective for such problems allows simple evolutionary multi-objective algorithms such as GSEMO mimic a greedy behaviour and as a consequence allows us to achieve theoretically best possible performance guarantees for a wide range of constrained submodular optimization problems [17-19]. Such approaches have been widely studied recently under the term Pareto optimization in the artificial intelligence and machine learning literature [22]. In the context of problems with stochastic constraints, it has recently been shown that 3-objective formulations where the given constraint is relaxed into a third objective lead to better performance than 2-objective formulations that optimize the expected value and variance of the given stochastic components under the given constraint [14, 15].

Optimizing discrete black-box functions is key in several domains, e.g. protein engineering and drug design. Due to the lack of gradient information and the need for sample efficiency, Bayesian optimization is an ideal candidate for these tasks. Several methods for high-dimensional continuous and categorical Bayesian optimization have been proposed recently. However, our survey of the field reveals highly heterogeneous experimental set-ups across methods and technical barriers for the replicability and application of published algorithms to real-world tasks. To address these issues, we develop a unified framework to test a vast array of high-dimensional Bayesian optimization methods and a collection of standardized black-box functions representing real-world application domains in chemistry and biology. These two components of the benchmark are each supported by flexible, scalable, and easily extendable software libraries (poli and poli-baselines), allowing practitioners to readily incorporate new optimization objectives or discrete optimizers.

We study the problem of online multi-group learning, a learning model in which an online learner must simultaneously achieve small prediction regret on a large collection of (possibly overlapping) subsequences corresponding to a family of groups. Groups are subsets of the context space, and in fairness applications, they may correspond to subpopulations defined by expressive functions of demographic attributes. In contrast to previous work on this learning model, we consider scenarios in which the family of groups is too large to explicitly enumerate, and hence we seek algorithms that only access groups via an optimization oracle. In this paper, we design such oracle-efficient algorithms with sublinear regret under a variety of settings, including: (i) the i.i.d. setting, (ii) the adversarial setting with smoothed context distributions, and (iii) the adversarial transductive setting.

We study the implicit bias of optimization in robust empirical risk minimization (robust ERM) and its connection with robust generalization. In classification settings under adversarial perturbations with linear models, we study what type of regularization should ideally be applied for a given perturbation set to improve (robust) generalization. We then show that the implicit bias of optimization in robust ERM can significantly affect the robustness of the model and identify two ways this can happen; either through the optimization algorithm or the architecture. We verify our predictions in simulations with synthetic data and experimentally study the importance of implicit bias in robust ERM with deep neural networks.