Neural Signed Distance Fields (SDFs) provide a differentiable environment representation to readily obtain collision checks and well-defined gradients for robot navigation tasks. However, updating neural SDFs as the scene evolves entails re-training, which is tedious, time consuming, and inefficient, making it unsuitable for robot navigation with limited field-of-view in dynamic environments. Towards this objective, we propose a compositional framework of neural SDFs to solve robot navigation in indoor environments using only an onboard RGB-D sensor. Our framework embodies a dual mode procedure for trajectory optimization, with different modes using complementary methods of modeling collision costs and collision avoidance gradients. The primary stage queries the robot body's SDF, swept along the route to goal, at the obstacle point cloud, enabling swift local optimization of trajectories. The secondary stage infers the visible scene's SDF by aligning and composing the SDF representations of its constituents, providing better informed costs and gradients for trajectory optimization. The dual mode procedure combines the best of both stages, achieving a success rate of 98%, 14.4% higher than baseline with comparable amortized plan time on iGibson 2.0. We also demonstrate its effectiveness in adapting to real-world indoor scenarios.

Bias evaluation is fundamental to trustworthy AI, both in terms of checking data quality and in terms of checking the outputs of AI systems. In testing data quality, for example, one may study a distance of a given dataset, viewed as a distribution, to a given ground-truth reference dataset. However, classical metrics, such as the Total Variation and the Wasserstein distances, are known to have high sample complexities and, therefore, may fail to provide meaningful distinction in many practical scenarios. In this paper, we propose a new notion of distance, the Maximum Subgroup Discrepancy (MSD). In this metric, two distributions are close if, roughly, discrepancies are low for all feature subgroups. While the number of subgroups may be exponential, we show that the sample complexity is linear in the number of features, thus making it feasible for practical applications. Moreover, we provide a practical algorithm for the evaluation of the distance, based on Mixed-integer optimization (MIO). We also note that the proposed distance is easily interpretable, thus providing clearer paths to fixing the biases once they have been identified. It also provides guarantees for all subgroups. Finally, we empirically evaluate, compare with other metrics, and demonstrate the above properties of MSD on real-world datasets.

It is critical to ensure safety for humanoid robots in real-world applications without compromising performance. In this paper, we consider the problem of dexterous safety, featuring limb-level geometry constraints for avoiding both external and self-collisions in cluttered environments. Compared to safety with simplified bounding geometries in sprase environments, dexterous safety produces numerous constraints which often lead to infeasible constraint sets when solving for safe robot control. To address this issue, we propose Projected Safe Set Algorithm (p-SSA), an extension of classical safe control algorithms to multi-constraint cases. p-SSA relaxes conflicting constraints in a principled manner, minimizing safety violations to guarantee feasible robot control. We verify our approach in simulation and on a real Unitree G1 humanoid robot performing complex collision avoidance tasks. Results show that p-SSA enables the humanoid to operate robustly in challenging situations with minimal safety violations and directly generalizes to various tasks with zero parameter tuning.

In modern optimization methods used in deep learning, each update depends on the history of previous iterations, often referred to as memory, and this dependence decays fast as the iterates go further into the past. For example, gradient descent with momentum has exponentially decaying memory through exponentially averaged past gradients. We introduce a general technique for identifying a memoryless algorithm that approximates an optimization algorithm with memory. It is obtained by replacing all past iterates in the update by the current one, and then adding a correction term arising from memory (also a function of the current iterate). This correction term can be interpreted as a perturbation of the loss, and the nature of this perturbation can inform how memory implicitly (anti-)regularizes the optimization dynamics. As an application of our theory, we find that Lion does not have the kind of implicit anti-regularization induced by memory that AdamW does, providing a theory-based explanation for Lion's better generalization performance recently documented.

The application of deep learning for partial differential equation (PDE)-constrained control is gaining increasing attention. However, existing methods rarely consider safety requirements crucial in real-world applications. To address this limitation, we propose Safe Diffusion Models for PDE Control (SafeDiffCon), which introduce the uncertainty quantile as model uncertainty quantification to achieve optimal control under safety constraints through both post-training and inference phases. Firstly, our approach post-trains a pre-trained diffusion model to generate control sequences that better satisfy safety constraints while achieving improved control objectives via a reweighted diffusion loss, which incorporates the uncertainty quantile estimated using conformal prediction. Secondly, during inference, the diffusion model dynamically adjusts both its generation process and parameters through iterative guidance and fine-tuning, conditioned on control targets while simultaneously integrating the estimated uncertainty quantile. We evaluate SafeDiffCon on three control tasks: 1D Burgers' equation, 2D incompressible fluid, and controlled nuclear fusion problem. Results demonstrate that SafeDiffCon is the only method that satisfies all safety constraints, whereas other classical and deep learning baselines fail. Furthermore, while adhering to safety constraints, SafeDiffCon achieves the best control performance.

Large Language Models (LLMs) have demonstrated impressive capabilities across numerous fields, presenting an opportunity to revolutionize optimization problem-solving, a crucial, ubiquitous, and complex domain. This paper explores the proficiency of LLMs in handling Sequential Optimization Problems (SOPs). We introduce WorldGen, a dynamic framework for generating unseen SOPs with controllable complexities, to evaluate LLM performance. Our initial observations reveal that while LLMs perform well on simple SOPs, their performance significantly degrades with increased complexity. Motivated by this, we revisit philosophical hypotheses on reasoning to enhance LLM performance. Inspired by the influential framework of Hegelian Dialectics, we propose ACE, demonstrating how the performance of LLMs in SOP contexts can be significantly improved without any retraining or further fine-tuning.

Deep generative models have recently shown success in solving complex engineering design problems where models predict solutions that address the design requirements specified as input. However, there remains a challenge in aligning such models for effective design exploration. For many design problems, finding a solution that meets all the requirements is infeasible. In such a case, engineers prefer to obtain a set of Pareto optimal solutions with respect to those requirements, but uniform sampling of generative models may not yield a useful Pareto front. To address this gap, we introduce a new framework for Pareto-front design exploration with simulation fine-tuned generative models. First, the framework adopts preference alignment methods developed for Large Language Models (LLMs) and showcases the first application in fine-tuning a generative model for engineering design. The important distinction here is that we use a simulator instead of humans to provide accurate and scalable feedback. Next, we propose epsilon-sampling, inspired by the epsilon-constraint method used for Pareto-front generation with classical optimization algorithms, to construct a high-quality Pareto front with the fine-tuned models. Our framework, named e-SimFT, is shown to produce better-quality Pareto fronts than existing multi-objective alignment methods.

A fundamental question in data-driven decision making is how to quantify the uncertainty of predictions in ways that can usefully inform downstream action. This interface between prediction uncertainty and decision-making is especially important in risk-sensitive domains, such as medicine. In this paper, we develop decision-theoretic foundations that connect uncertainty quantification using prediction sets with risk-averse decision-making. Specifically, we answer three fundamental questions: (1) What is the correct notion of uncertainty quantification for risk-averse decision makers? We prove that prediction sets are optimal for decision makers who wish to optimize their value at risk. (2) What is the optimal policy that a risk averse decision maker should use to map prediction sets to actions? We show that a simple max-min decision policy is optimal for risk-averse decision makers. Finally, (3) How can we derive prediction sets that are optimal for such decision makers? We provide an exact characterization in the population regime and a distribution free finite-sample construction. Answering these questions naturally leads to an algorithm, Risk-Averse Calibration (RAC), which follows a provably optimal design for deriving action policies from predictions. RAC is designed to be both practical-capable of leveraging the quality of predictions in a black-box manner to enhance downstream utility-and safe-adhering to a user-defined risk threshold and optimizing the corresponding risk quantile of the user's downstream utility. Finally, we experimentally demonstrate the significant advantages of RAC in applications such as medical diagnosis and recommendation systems. Specifically, we show that RAC achieves a substantially improved trade-off between safety and utility, offering higher utility compared to existing methods while maintaining the safety guarantee.

Signed networks, where edges are labeled as positive or negative to indicate friendly or antagonistic interactions, offer a natural framework for studying polarization, trust, and conflict in social systems. Detecting meaningful group structures in these networks is crucial for understanding online discourse, political division, and trust dynamics. A key challenge is to identify groups that are cohesive internally yet antagonistic externally, while allowing for neutral or unaligned vertices. In this paper, we address this problem by identifying $k$ polarized communities that are large, dense, and balanced in size. We develop an approach based on Frank-Wolfe optimization, leading to a local search procedure with provable convergence guarantees. Our method is both scalable and efficient, outperforming state-of-the-art baselines in solution quality while remaining competitive in terms of computational efficiency.

Recently, message-passing graph neural networks (MPNNs) have shown potential for solving combinatorial and continuous optimization problems due to their ability to capture variable-constraint interactions. While existing approaches leverage MPNNs to approximate solutions or warm-start traditional solvers, they often lack guarantees for feasibility, particularly in convex optimization settings. Here, we propose an iterative MPNN framework to solve convex optimization problems with provable feasibility guarantees. First, we demonstrate that MPNNs can provably simulate standard interior-point methods for solving quadratic problems with linear constraints, covering relevant problems such as SVMs. Secondly, to ensure feasibility, we introduce a variant that starts from a feasible point and iteratively restricts the search within the feasible region. Experimental results show that our approach outperforms existing neural baselines in solution quality and feasibility, generalizes well to unseen problem sizes, and, in some cases, achieves faster solution times than state-of-the-art solvers such as Gurobi.