Galloping is a common high-speed gait in both animals and quadrupedal robots, yet its energetic characteristics remain insufficiently explored. This study systematically analyzes a large number of possible galloping gaits by categorizing them based on the number of flight phases per stride and the phase relationships between the front and rear legs, following Hildebrand's framework for asymmetrical gaits. Using the A1 quadrupedal robot from Unitree, we model galloping dynamics as a hybrid dynamical system and employ trajectory optimization (TO) to minimize the cost of transport (CoT) across a range of speeds. Our results reveal that rotary and transverse gallop footfall sequences exhibit no fundamental energetic difference, despite variations in body yaw and roll motion. However, the number of flight phases significantly impacts energy efficiency: galloping with no flight phases is optimal at lower speeds, whereas galloping with two flight phases minimizes energy consumption at higher speeds. We validate these findings using a quadratic programming (QP)-based controller, developed in our previous work, in Gazebo simulations. These insights advance the understanding of quadrupedal locomotion energetics and may inform future legged robot designs for adaptive, energy-efficient gait transitions.

Language Models (LMs) are widely used in software engineering for code generation, but they may produce code with errors. Rather than repairing the generated code, an alternative way is to address the underlying failures of models. LM repair offers a lightweight solution to this challenge: it requires minimal data, reduces computational costs, and reduces the side effects. Unlike retraining, LM repair focuses on applying tailored updates to targeted neurons, making it ideal for scenarios with limited resources, high-performance demands, or strict safety requirements. In this paper, we propose \ul{S}emantic \ul{T}argeting for \ul{A}nalytical \ul{R}epair (\textsc{STAR}), a pioneering and novel semantic-based optimization approach for repairing LLMs. \textsc{STAR} realizes main operations in LM repair methods in an optimization process, including locating ``buggy neurons'', solving ``neuron patches'', and patching ``buggy neurons''. Correspondingly, it computes the deltas of weight matrix as the prior information to guide optimization; and attributes the targeted layers and neurons leveraging statistical insights. The neuron patches are computed with a solid semantic-based analytical formula, which directly bridges the changes to logits with the deltas of neurons, by steering latent representations. Compared to the prior work of LM repair (\textsc{MINT}) and optimization methods (\textsc{SGD}), \textsc{STAR} integrates their strengths while mitigating their limitations. \textsc{STAR} supports solving multiple failures together, significantly improving the usefulness. Evaluated on three code generation tasks using popular code LMs, \textsc{STAR} demonstrates superior effectiveness. Additionally, \textsc{STAR} exhibits better efficiency. In terms of side effects, namely the balance between generalization and specificity, \textsc{STAR} outperforms prior work by a significant margin.

We introduce a novel class of regularization functions, called Cauchy-Schwarz (CS) regularizers, which can be designed to induce a wide range of properties in solution vectors of optimization problems. To demonstrate the versatility of CS regularizers, we derive regularization functions that promote discrete-valued vectors, eigenvectors of a given matrix, and orthogonal matrices. The resulting CS regularizers are simple, differentiable, and can be free of spurious stationary points, making them suitable for gradient-based solvers and large-scale optimization problems. In addition, CS regularizers automatically adapt to the appropriate scale, which is, for example, beneficial when discretizing the weights of neural networks. To demonstrate the efficacy of CS regularizers, we provide results for solving underdetermined systems of linear equations and weight quantization in neural networks. Furthermore, we discuss specializations, variations, and generalizations, which lead to an even broader class of new and possibly more powerful regularizers.

Reinforcement learning (RL) has increasingly been applied to solve real-world planning problems, with progress in handling large state spaces and time horizons. However, a key bottleneck in many domains is that RL methods cannot accommodate large, combinatorially structured action spaces. In such settings, even representing the set of feasible actions at a single step may require a complex discrete optimization formulation. We leverage recent advances in embedding trained neural networks into optimization problems to propose SEQUOIA, an RL algorithm that directly optimizes for long-term reward over the feasible action space. Our approach embeds a Q-network into a mixed-integer program to select a combinatorial action in each timestep. Here, we focus on planning over restless bandits, a class of planning problems which capture many real-world examples of sequential decision making. RMAB, a broader class of restless bandits with combinatorial actions that cannot be decoupled across the arms of the restless bandit, requiring direct solving over the joint, exponentially large action space. Our approach significantly outperforms existing methods--which cannot address sequential planning and combinatorial selection simultaneously--by an average of 24.8% on these difficult instances. Reinforcement learning (RL) has made tremendous progress in recent years to solve a wide range of practical problems (Treloar et al., 2020; Marot et al., 2021; Silvestro et al., 2022; Degrave et al., 2022). While successful at dealing with large or infinite state spaces, RL struggles with discrete, combinatorial action spaces. This limitation is pertinent to many real-world sequential decisionmaking problems, where resource constraints frequently lead to combinatorial action spaces (Dulac-Arnold et al., 2020). Consider, for example, a sequential resource allocation problem in which public health workers are dispatched to visit patients. The workers each have a limited daily budget to maximize patient well-being. These requirements give rise to an exponentially large combinatorial action space to optimize over, even when the number of workers and patients is small.

Motivated by the latest research on feasible space monitoring of multiple control barrier functions (CBFs) as well as polytopic collision avoidance, this paper studies the Polytope Volume Monitoring (PVM) problem, whose goal is to design a control law for inputs of nonlinear systems to prevent the volume of some state-dependent polytope from decreasing to zero. Recent studies have explored the idea of applying Chebyshev ball method in optimization theory to solve the case study of PVM; however, the underlying difficulties caused by nonsmoothness have not been addressed. This paper continues the study on this topic, where our main contribution is to establish the relationship between nonsmooth CBF and parametric optimization theory through directional derivatives for the first time, so as to solve PVM problems more conveniently. In detail, inspired by Chebyshev ball approach, a parametric linear program (PLP) based nonsmooth barrier function candidate is established for PVM, and then, sufficient conditions for it to be a nonsmooth CBF are proposed, based on which a quadratic program (QP) based safety filter with guaranteed feasibility is proposed to address PVM problems. Finally, a numerical simulation example is given to show the efficiency of the proposed safety filter.

This paper introduces TuneNSearch, a hybrid transfer learning and local search approach for addressing different variants of vehicle routing problems (VRP). Recently, multi-task learning has gained much attention for solving VRP variants. However, this adaptability often compromises the performance of the models. To address this challenge, we first pre-train a reinforcement learning model on the multi-depot VRP, followed by a short fine-tuning phase to adapt it to different variants. By leveraging the complexity of the multi-depot VRP, the pre-trained model learns richer node representations and gains more transferable knowledge compared to models trained on simpler routing problems, such as the traveling salesman problem. TuneNSearch employs, in the first stage, a Transformer-based architecture, augmented with a residual edge-graph attention network to capture the impact of edge distances and residual connections between layers. This architecture allows for a more precise capture of graph-structured data, improving the encoding of VRP's features. After inference, our model is also coupled with a second stage composed of a local search algorithm, which yields substantial performance gains with minimal computational overhead added. Results show that TuneNSearch outperforms many existing state-of-the-art models trained for each VRP variant, requiring only one-fifth of the training epochs. Our approach demonstrates strong generalization, achieving high performance across different tasks, distributions and problem sizes, thus addressing a long-standing gap in the literature.

Representation learning has emerged as a crucial focus in machine and deep learning, involving the extraction of meaningful and useful features and patterns from the input data, thereby enhancing the performance of various downstream tasks such as classification, clustering, and prediction. Current mainstream representation learning methods primarily rely on non-linear data mining techniques such as kernel methods and deep neural networks to extract abstract knowledge from complex datasets. However, most of these methods are black-box, lacking transparency and interpretability in the learning process, which constrains their practical utility. To this end, this paper introduces a novel representation learning method grounded in an interpretable fuzzy rule-based model. Specifically, it is built upon the Takagi-Sugeno-Kang fuzzy system (TSK-FS) to initially map input data to a high-dimensional fuzzy feature space through the antecedent part of the TSK-FS. Subsequently, a novel differentiable optimization method is proposed for the consequence part learning which can preserve the model's interpretability and transparency while further exploring the nonlinear relationships within the data. This optimization method retains the essence of traditional optimization, with certain parts of the process parameterized corresponding differentiable modules constructed, and a deep optimization process implemented. Consequently, this method not only enhances the model's performance but also ensures its interpretability. Moreover, a second-order geometry preservation method is introduced to further improve the robustness of the proposed method. Extensive experiments conducted on various benchmark datasets validate the superiority of the proposed method, highlighting its potential for advancing representation learning methodologies.

Diffusion models have demonstrated remarkable performance in modeling complex data priors, catalyzing their widespread adoption in solving various inverse problems. However, the inherently iterative nature of diffusion-based inverse algorithms often requires hundreds to thousands of steps, with performance degradation occurring under fewer steps which limits their practical applicability. While high-order diffusion ODE solvers have been extensively explored for efficient diffusion sampling without observations, their application to inverse problems remains underexplored due to the diverse forms of inverse algorithms and their need for repeated trajectory correction based on observations. To address this gap, we first introduce a canonical form that decomposes existing diffusion-based inverse algorithms into three modules to unify their analysis. Inspired by the linear subspace search strategy in the design of high-order diffusion ODE solvers, we propose the Learnable Linear Extrapolation (LLE) method, a lightweight approach that universally enhances the performance of any diffusion-based inverse algorithm that fits the proposed canonical form. Extensive experiments demonstrate consistent improvements of the proposed LLE method across multiple algorithms and tasks, indicating its potential for more efficient solutions and boosted performance of diffusion-based inverse algorithms with limited steps. Codes for reproducing our experiments are available at https://github.com/weigerzan/LLE_inverse_problem.

To increase the reliability of collaborative robots in performing daily tasks, we require them to be accurate and not only repeatable. However, having a calibrated kinematics model is regrettably a luxury, as available calibration tools are usually more expensive than the robots themselves. With this work, we aim to contribute to the democratization of cobots calibration by providing an inexpensive yet highly effective alternative to existing tools. The proposed minimalist calibration routine relies on a 3D-printable tool as the only physical aid to the calibration process. This two-socket spherical-joint tool kinematically constrains the robot at the end effector while collecting the training set. An optimization routine updates the nominal model to ensure a consistent prediction for each socket and the undistorted mean distance between them. We validated the algorithm on three robotic platforms: Franka, Kuka, and Kinova Cobots. The calibrated models reduce the mean absolute error from the order of 10 mm to 0.2 mm for both Franka and Kuka robots. We provide two additional experimental campaigns with the Franka Robot to render the improvements more tangible. First, we implement Cartesian control with and without the calibrated model and use it to perform a standard peg-in-the-hole task with a tolerance of 0.4 mm between the peg and the hole. Second, we perform a repeated drawing task combining Cartesian control with learning from demonstration. Both tasks consistently failed when the model was not calibrated, while they consistently succeeded after calibration.

Machine learning has been widely applied in many aspects, but training a machine learning model is increasingly difficult. There are more optimization problems named "black-box" where the relationship between model parameters and outcomes is uncertain or complex to trace. Currently, optimizing black-box models that need a large number of query observations and parameters becomes difficult. To overcome the drawbacks of the existing algorithms, in this study, we propose a zeroth-order method that originally came from quantum computing called the parameter-shift rule, which has used a lesser number of parameters than previous methods.