Deep neural networks (DNNs) and Kolmogorov-Arnold networks (KANs) are popular methods for function approximation due to their flexibility and expressivity. However, they typically require a large number of trainable parameters to produce a suitable approximation. Beyond making the resulting network less transparent, overparameterization creates a large optimization space, likely producing local minima in training that have quite different generalization errors. In this case, network initialization can have an outsize impact on the model's out-of-sample accuracy. For these reasons, we propose shallow universal polynomial networks (SUPNs). These networks replace all but the last hidden layer with a single layer of polynomials with learnable coefficients, leveraging the strengths of DNNs and polynomials to achieve sufficient expressivity with far fewer parameters. We prove that SUPNs converge at the same rate as the best polynomial approximation of the same degree, and we derive explicit formulas for quasi-optimal SUPN parameters. We complement theory with an extensive suite of numerical experiments involving SUPNs, DNNs, KANs, and polynomial projection in one, two, and ten dimensions, consisting of over 13,000 trained models. On the target functions we numerically studied, for a given number of trainable parameters, the approximation error and variability are often lower for SUPNs than for DNNs and KANs by an order of magnitude. In our examples, SUPNs even outperform polynomial projection on non-smooth functions.

In recent years, dual-arm manipulation has become an area of strong interest in robotics, with end-to-end learning emerging as the predominant strategy for solving bimanual tasks. A critical limitation of such learning-based approaches, however, is their difficulty in generalizing to novel scenarios, especially within cluttered environments. This paper presents an alternative paradigm: a sampling-based optimization framework that utilizes a GPU-accelerated physics simulator as its world model. We demonstrate that this approach can solve complex bimanual manipulation tasks in the presence of static obstacles. Our contribution is a customized Model Predictive Path Integral Control (MPPI) algorithm, \textbf{guided by carefully designed task-specific cost functions,} that uses GPU-accelerated MuJoCo for efficiently evaluating robot-object interaction. We apply this method to solve significantly more challenging versions of tasks from the PerAct$^{2}$ benchmark, such as requiring the point-to-point transfer of a ball through an obstacle course. Furthermore, we establish that our method achieves real-time performance on commodity GPUs and facilitates successful sim-to-real transfer by leveraging unique features within MuJoCo. The paper concludes with a statistical analysis of the sample complexity and robustness, quantifying the performance of our approach. The project website is available at: https://sites.google.com/view/bimanualakslabunitartu .

Fair clustering has gained increasing attention in recent years, especially in applications involving socially sensitive attributes. However, existing fair clustering methods often lack inter-pretability, limiting their applicability in high-stakes scenarios where understanding the rationale behind clustering decisions is essential. In this work, we address this limitation by proposing an interpretable and fair clustering framework, which integrates fairness constraints into the structure of decision trees. Our approach constructs interpretable decision trees that partition the data while ensuring fair treatment across protected groups. To further enhance the practicality of our framework, we also introduce a variant that requires no fairness hyperparameter tuning, achieved through post-pruning a tree constructed without fairness constraints. Extensive experiments on both real-world and synthetic datasets demonstrate that our method not only delivers competitive clustering performance and improved fairness, but also offers additional advantages such as interpretability and the ability to handle multiple sensitive attributes. These strengths enable our method to perform robustly under complex fairness constraints, opening new possibilities for equitable and transparent clustering.

Mixed Integer Linear Programming (MILP) is a fundamental class of NP-hard problems that has garnered significant attention from both academia and industry. The Branch-and-Bound (B\&B) method is the dominant approach for solving MILPs and the branching plays an important role in B\&B methods. Neural-based learning frameworks have recently been developed to enhance branching policies and the efficiency of solving MILPs. However, these methods still struggle with semantic variation across depths, the scarcity of upstream nodes, and the costly collection of strong branching samples. To address these issues, we propose \ours, a Dynamic \underline{\textbf{S}}tratified \underline{\textbf{C}}ontrastive Training Framework for \underline{\textbf{MILP}} Branching. It groups branch-and-bound nodes based on their feature distributions and trains a GCNN-based discriminative model to progressively separate nodes across groups, learning finer-grained node representations throughout the tree. To address data scarcity and imbalance at upstream nodes, we introduce an upstream-augmented MILP derivation procedure that generates both theoretically equivalent and perturbed instances. \ours~effectively models subtle semantic differences between nodes, significantly enhancing branching accuracy and solving efficiency, particularly for upstream nodes. Extensive experiments on standard MILP benchmarks demonstrate that our method enhances branching accuracy, reduces solving time, and generalizes effectively to unseen instances.

Offline data selection and online self-refining generation, which enhance the data quality, are crucial steps in adapting large language models (LLMs) to specific downstream tasks. We tackle offline data selection and online self-refining generations through an optimization perspective. Specifically, bilevel data selection is used for offline data selection with respect to the validation dataset, and we treat online self-refining generation as a model adaptation step of selecting the model trained on current responses that best fits the validation data. Our framework offers a unified understanding of offline data selection and self-refining generation by assigning a learned data weight to each question and response, either explicitly or implicitly. For the first time, we theoretically demonstrate the effectiveness of the bilevel data selection framework and demonstrate its performance gains over unfiltered direct mixing baselines. By combining offline data with validation-weighted online generations, our method enhances fine-tuning performance. Experiments on quality enhancement and safety-aware LLM fine-tuning validate its effectiveness.

Its simple update, linear scalability with sample size, and compatibility with momentum, mini-batching, and learning-rate heuristics keep it dominant in both industry and academia. Current research continues to refine convergence rates, variance characterizations, and averaging schemes, while engineering efforts focus on hardware-aligned and distributed variants. B. Mini-Batch Stochastic Gradient Descent 1) Background and Development: Batch Gradient Descent (BGD) requires computing the gradient using the entire training dataset at each iteration. As dataset sizes expand to millions or even larger scales, the computational cost of a single iteration becomes extremely high, making it unsuitable for large-scale learning tasks. The convergence of SGD was proven by Robbins and Monro through the stochastic approximation method [1]. SGD uses one sample to update the gradient at each step, resulting in low computational cost but high gradient variance and unstable updates. The mini-batch strategy has gradually become the mainstream in practice, especially with the rise of large-scale machine learning and deep learning. Bottou emphasized the practical value of mini-batches in his research on large-scale learning [5], while systematic monographs and reviews on deep learning have further standardized this approach [6], [7]. Mini-batch SGD achieves an optimal balance between stability, high-frequency updates, and GPU parallel acceleration [2].

With the global energy transition and rapid development of renewable energy, the scheduling optimization challenge for combined power-heat systems under new energy integration and multiple uncertainties has become increasingly prominent. Addressing this challenge, this study proposes an intelligent scheduling method based on the improved Dual-Delay Deep Deterministic Policy Gradient (PVTD3) algorithm. System optimization is achieved by introducing a penalty term for grid power purchase variations. Simulation results demonstrate that under three typical scenarios (10%, 20%, and 30% renewable penetration), the PVTD3 algorithm reduces the system's comprehensive cost by 6.93%, 12.68%, and 13.59% respectively compared to the traditional TD3 algorithm. Concurrently, it reduces the average fluctuation amplitude of grid power purchases by 12.8%. Regarding energy storage management, the PVTD3 algorithm reduces the end-time state values of low-temperature thermal storage tanks by 7.67-17.67 units while maintaining high-temperature tanks within the 3.59-4.25 safety operating range. Multi-scenario comparative validation demonstrates that the proposed algorithm not only excels in economic efficiency and grid stability but also exhibits superior sustainable scheduling capabilities in energy storage device management.

Traveling Salesman Problem (TSP) is a classic NP-hard problem that has garnered significant attention from both academia and industry. While neural-based methods have shown promise for solving TSPs, they still face challenges in scaling to larger instances, particularly in memory constraints associated with global heatmaps, edge weights, or access matrices, as well as in generating high-quality initial solutions and insufficient global guidance for efficiently navigating vast search spaces. To address these challenges, we propose a Hyper Tour Guided Neighborhood Search (HyperNS) method for large-scale TSP instances. Inspired by the ``clustering first, route second" strategy, our approach initially divides the TSP instance into clusters using a sparse heatmap graph and abstracts them as supernodes, followed by the generation of a hyper tour to guide both the initialization and optimization processes. This method reduces the search space by focusing on edges relevant to the hyper tour, leading to more efficient and effective optimization. Experimental results on both synthetic and real-world datasets demonstrate that our approach outperforms existing neural-based methods, particularly in handling larger-scale instances, offering a significant reduction in the gap to the optimal solution.

Accurate state estimation is critical for legged and aerial robots operating in dynamic, uncertain environments. A key challenge lies in specifying process and measurement noise covariances, which are typically unknown or manually tuned. In this work, we introduce a bi-level optimization framework that jointly calibrates covariance matrices and kinematic parameters in an estimator-in-the-loop manner. The upper level treats noise covariances and model parameters as optimization variables, while the lower level executes a full-information estimator. Differentiating through the estimator allows direct optimization of trajectory-level objectives, resulting in accurate and consistent state estimates. We validate our approach on quadrupedal and humanoid robots, demonstrating significantly improved estimation accuracy and uncertainty calibration compared to hand-tuned baselines. Our method unifies state estimation, sensor, and kinematics calibration into a principled, data-driven framework applicable across diverse robotic platforms.

The exponential growth in demand for GPU computing resources has created an urgent need for automated CUDA optimization strategies. While recent advances in LLMs show promise for code generation, current SOTA models achieve low success rates in improving CUDA speed. In this paper, we introduce CUDA-L1, an automated reinforcement learning framework for CUDA optimization that employs a novel contrastive RL algorithm. CUDA-L1 achieves significant performance improvements on the CUDA optimization task: trained on A100, it delivers an average speedup of x3.12 with a median speedup of x1.42 against default baselines over across all 250 CUDA kernels of KernelBench, with peak speedups reaching x120. In addition to the default baseline provided by KernelBench, CUDA-L1 demonstrates x2.77 over Torch Compile, x2.88 over Torch Compile with reduce overhead, x2.81 over CUDA Graph implementations, and remarkably x7.72 over cuDNN libraries. Furthermore, the model also demonstrates portability across different GPU architectures. Beyond these benchmark results, CUDA-L1 demonstrates several properties: it 1) discovers a variety of CUDA optimization techniques and learns to combine them strategically to achieve optimal performance; 2) uncovers fundamental principles of CUDA optimization, such as the multiplicative nature of optimizations; 3) identifies non-obvious performance bottlenecks and rejects seemingly beneficial optimizations that actually harm performance. The capabilities demonstrate that, RL can transform an initially poor-performing LLM into an effective CUDA optimizer through speedup-based reward signals alone, without human expertise or domain knowledge. This paradigm opens possibilities for automated optimization of CUDA operations, and holds promise to substantially promote GPU efficiency and alleviate the rising pressure on GPU computing resources.