Current optimization-based control techniques for humanoid locomotion struggle to adapt step duration and placement simultaneously in dynamic walking gaits due to their reliance on fixed-time discretization, which limits responsiveness to terrain conditions and results in suboptimal performance in challenging environments. In this work, we propose a Gait-Net-augmented implicit kino-dynamic model-predictive control (MPC) to simultaneously optimize step location, step duration, and contact forces for natural variable-frequency locomotion. The proposed method incorporates a Gait-Net-augmented Sequential Convex MPC algorithm to solve multi-linearly constrained variables by iterative quadratic programs. At its core, a lightweight Gait-frequency Network (Gait-Net) determines the preferred step duration in terms of variable MPC sampling times, simplifying step duration optimization to the parameter level. Additionally, it enhances and updates the spatial reference trajectory within each sequential iteration by incorporating local solutions, allowing the projection of kinematic constraints to the design of reference trajectories. We validate the proposed algorithm in high-fidelity simulations and on small-size humanoid hardware, demonstrating its capability for variable-frequency and 3-D discrete terrain locomotion with only a one-step preview of terrain data.

Symbolic regression (SR) is an emerging branch of machine learning focused on discovering simple and interpretable mathematical expressions from data. Although a wide-variety of SR methods have been developed, they often face challenges such as high computational cost, poor scalability with respect to the number of input dimensions, fragility to noise, and an inability to balance accuracy and complexity. This work introduces SyMANTIC, a novel SR algorithm that addresses these challenges. SyMANTIC efficiently identifies (potentially several) low-dimensional descriptors from a large set of candidates (from $\sim 10^5$ to $\sim 10^{10}$ or more) through a unique combination of mutual information-based feature selection, adaptive feature expansion, and recursively applied $\ell_0$-based sparse regression. In addition, it employs an information-theoretic measure to produce an approximate set of Pareto-optimal equations, each offering the best-found accuracy for a given complexity. Furthermore, our open-source implementation of SyMANTIC, built on the PyTorch ecosystem, facilitates easy installation and GPU acceleration. We demonstrate the effectiveness of SyMANTIC across a range of problems, including synthetic examples, scientific benchmarks, real-world material property predictions, and chaotic dynamical system identification from small datasets. Extensive comparisons show that SyMANTIC uncovers similar or more accurate models at a fraction of the cost of existing SR methods.

Solving mixed-integer optimization problems with embedded neural networks with ReLU activation functions is challenging. Big-M coefficients that arise in relaxing binary decisions related to these functions grow exponentially with the number of layers. We survey and propose different approaches to analyze and improve the run time behavior of mixed-integer programming solvers in this context. Among them are clipped variants and regularization techniques applied during training as well as optimization-based bound tightening and a novel scaling for given ReLU networks. We numerically compare these approaches for three benchmark problems from the literature. We use the number of linear regions, the percentage of stable neurons, and overall computational effort as indicators. As a major takeaway we observe and quantify a trade-off between the often desired redundancy of neural network models versus the computational costs for solving related optimization problems.

Optimal transport (OT) is a critical problem in optimization and machine learning, where accuracy and efficiency are paramount. Although entropic regularization and the Sinkhorn algorithm improve scalability, they frequently encounter numerical instability and slow convergence, especially when the regularization parameter is small. In this work, we introduce Proximal Iterations with Sparse Newton and Sinkhorn methods (PINS) to efficiently compute highly accurate solutions for large-scale OT problems. A reduced computational complexity through overall sparsity and global convergence are guaranteed by rigorous theoretical analysis. Our approach offers three key advantages: it achieves accuracy comparable to exact solutions, progressively accelerates each iteration for greater efficiency, and enhances robustness by reducing sensitivity to regularization parameters. Extensive experiments confirm these advantages, demonstrating superior performance compared to related methods.

We introduce a novel alignment method for diffusion models from distribution optimization perspectives while providing rigorous convergence guarantees. We first formulate the problem as a generic regularized loss minimization over probability distributions and directly optimize the distribution using the Dual Averaging method. Next, we enable sampling from the learned distribution by approximating its score function via Doob's $h$-transform technique. The proposed framework is supported by rigorous convergence guarantees and an end-to-end bound on the sampling error, which imply that when the original distribution's score is known accurately, the complexity of sampling from shifted distributions is independent of isoperimetric conditions. This framework is broadly applicable to general distribution optimization problems, including alignment tasks in Reinforcement Learning with Human Feedback (RLHF), Direct Preference Optimization (DPO), and Kahneman-Tversky Optimization (KTO). We empirically validate its performance on synthetic and image datasets using the DPO objective.

We introduce the problem of best arm identification (BAI) with post-action context, a new BAI problem in a stochastic multi-armed bandit environment and the fixed-confidence setting. The problem addresses the scenarios in which the learner receives a $\textit{post-action context}$ in addition to the reward after playing each action. This post-action context provides additional information that can significantly facilitate the decision process. We analyze two different types of the post-action context: (i) $\textit{non-separator}$, where the reward depends on both the action and the context, and (ii) $\textit{separator}$, where the reward depends solely on the context. For both cases, we derive instance-dependent lower bounds on the sample complexity and propose algorithms that asymptotically achieve the optimal sample complexity. For the non-separator setting, we do so by demonstrating that the Track-and-Stop algorithm can be extended to this setting. For the separator setting, we propose a novel sampling rule called $\textit{G-tracking}$, which uses the geometry of the context space to directly track the contexts rather than the actions. Finally, our empirical results showcase the advantage of our approaches compared to the state of the art.

In Group Trip Planning (GTP) Query Problem, we are given a city road network where a number of Points of Interest (PoI) have been marked with their respective categories (e.g., Cafeteria, Park, Movie Theater, etc.). A group of agents want to visit one PoI from every category from their respective starting location and once finished, they want to reach their respective destinations. This problem asks which PoI from every category should be chosen so that the aggregated travel cost of the group is minimized. This problem has been studied extensively in the last decade, and several solution approaches have been proposed. However, to the best of our knowledge, none of the existing studies have considered the different modalities of the journey, which makes the problem more practical. To bridge this gap, we introduce and study the GTP Query Problem with Multimodal Journey in this paper. Along with the other inputs of the GTP Query Problem, we are also given the different modalities of the journey that are available and their respective cost. Now, the problem is not only to select the PoIs from respective categories but also to select the modality of the journey. For this problem, we have proposed an efficient solution approach, which has been analyzed to understand their time and space requirements. A large number of experiments have been conducted using real-life datasets and the results have been reported. From the results, we observe that the PoIs and modality of journey recommended by the proposed solution approach lead to much less time and cost than the baseline methods.

We consider the problem of differentially private (DP) convex empirical risk minimization (ERM). While the standard DP-SGD algorithm is theoretically well-established, practical implementations often rely on shuffled gradient methods that traverse the training data sequentially rather than sampling with replacement in each iteration. Despite their widespread use, the theoretical privacy-accuracy trade-offs of private shuffled gradient methods (\textit{DP-ShuffleG}) remain poorly understood, leading to a gap between theory and practice. In this work, we leverage privacy amplification by iteration (PABI) and a novel application of Stein's lemma to provide the first empirical excess risk bound of \textit{DP-ShuffleG}. Our result shows that data shuffling results in worse empirical excess risk for \textit{DP-ShuffleG} compared to DP-SGD. To address this limitation, we propose \textit{Interleaved-ShuffleG}, a hybrid approach that integrates public data samples in private optimization. By alternating optimization steps that use private and public samples, \textit{Interleaved-ShuffleG} effectively reduces empirical excess risk. Our analysis introduces a new optimization framework with surrogate objectives, adaptive noise injection, and a dissimilarity metric, which can be of independent interest. Our experiments on diverse datasets and tasks demonstrate the superiority of \textit{Interleaved-ShuffleG} over several baselines.

The organization of latent token representations plays a crucial role in determining the stability, generalization, and contextual consistency of language models, yet conventional approaches to embedding refinement often rely on parameter modifications that introduce additional computational overhead. A hierarchical alignment method was introduced to restructure token embeddings without altering core model weights, ensuring that representational distributions maintained coherence across different linguistic contexts. Experimental evaluations demonstrated improvements in rare token retrieval, adversarial robustness, and long-range dependency tracking, highlighting the advantages of hierarchical structuring in mitigating inconsistencies in latent space organization. The comparative analysis against conventional fine-tuning and embedding perturbation methods revealed that hierarchical restructuring maintained computational efficiency while achieving measurable gains in representation quality. Structural refinements introduced through the alignment process resulted in improved contextual stability across varied linguistic tasks, reducing inconsistencies in token proximity relationships and enhancing interpretability in language generation. A detailed computational assessment confirmed that the realignment process introduced minimal inference overhead, ensuring that representational improvements did not compromise model efficiency. The findings reinforced the broader significance of structured representation learning, illustrating that hierarchical embedding modifications could serve as an effective strategy for refining latent space distributions while preserving pre-learned semantic associations.

Most existing methods for motion planning of mobile robots involve generating collision-free trajectories. However, these methods focusing solely on contact avoidance may limit the robots' locomotion and can not be applied to tasks where contact is inevitable or intentional. To address these issues, we propose a novel contact-aware motion planning (CAMP) paradigm for robotic systems. Our approach incorporates contact between robots and movable objects as complementarity constraints in optimization-based trajectory planning. By leveraging augmented Lagrangian methods (ALMs), we efficiently solve the optimization problem with complementarity constraints, producing spatial-temporal optimal trajectories of the robots. Simulations demonstrate that, compared to the state-of-the-art method, our proposed CAMP method expands the reachable space of mobile robots, resulting in a significant improvement in the success rate of two types of fundamental tasks: navigation among movable objects (NAMO) and rearrangement of movable objects (RAMO). Real-world experiments show that the trajectories generated by our proposed method are feasible and quickly deployed in different tasks.