Gradient_Descent
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A Hybrid Optimization Framework for Grasp Synthesis under Partial Observations
2026-06-16 15:30:00Wenzheng Zhang, Fahira Afzal Maken, Tin Lai, Fabio RamosAbstract
We propose a hybrid grasp synthesis framework that combines a learning-based Energy-Based Model (EBM) with an analytical Iterative Closest Point (ICP) method to generate robust grasps from partially observed point clouds. The learned energy function acts as a prior within a Stein Variational Gradient Descent (SVGD) framework, guiding iterative refinement of grasp configurations. Evaluated on 67 objects with 5,360 grasp attempts, our method achieves an average success rate of 60.9\%, outperforming AnyGrasp (31.1\%) and Grasp Pose Detection (48.4\%) and AS-ICP (56.6\%). These results highlight the strong generalization ability of our approach and demonstrate how combining data-driven learning with geometric optimization addresses the limitations of either strategy in isolation.
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URL
https://arxiv.org/abs/2606.18053
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Conservation Laws for Modern Neural Architectures
2026-06-16 11:44:53Viet-Hoang Tran, Vinh Khanh Bui, Tan Lai Ngoc, Nam Nguyen, Tuan Dam, Tan M. NguyenAbstract
Understanding gradient descent dynamics is key to explaining the success of over-parameterized models, where implicit bias manifests through conservation laws in gradient flow. While such laws are well understood for linear and ReLU networks, they remain largely unexplored for modern architectures. This work develops a unified framework to characterize conservation laws for contemporary models, including feedforward networks with GELU, SiLU, and SwiGLU activations, multihead attention with sinusoidal and rotary positional encodings, and Mixture-of-Experts architectures under diverse gating designs. Our theoretical findings are supported by experiments that validate the predicted invariants.
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URL
https://arxiv.org/abs/2606.17816
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A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks
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We develop a mathematically explicit link between shock-wave theory and the symmetry-quotiented learning dynamics of stochastic gradient descent, drawing on differential geometry, Lie group theory, and fluid mechanics. Specifically, after quotienting parameter symmetries and applying local-entropy coarse-graining, the effective dynamics satisfy a viscous Hamilton--Jacobi equation on the quotient manifold. Moreover, under the assumption that the raw parameter dynamics can be summarized by a gradient field on the quotiented space, the gradient of the coarse-grained loss function obeys a Burgers-type equation, and shock formation can be established rigorously. We apply our theory to multilayer perceptrons, convolutional neural networks, Transformers, and mean-field networks, and show that they obey the Hamilton--Jacobi or Burgers-type equations. We conjecture that this framework also yields practical diagnostics for deep learning. In architectures such as Transformers, raw parameter norms are often distorted by symmetry redundancy and may therefore be misleading, whereas symmetry-corrected quotient observables provide a principled basis for monitoring, forecasting, and controlling training-phase transitions.
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https://arxiv.org/abs/2606.18303
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Theoretical Grounding of Out-Of-Distribution Detection With Reinforcement Learning Optimizer
2026-06-16 03:40:03Salimeh Sekeh, Xin ZhangAbstract
Out-of-distribution (OOD) detection in dynamic open-world environments requires a model to continually adapt to evolving data distributions while generalizing to covariate-shifted inputs and rejecting semantic-shifted OOD examples. Most existing OOD detection methods optimize only the current-step objective and do not explicitly account for how post-deployment environment changes affect future OOD behavior. In this paper, we establish a theoretical grounding for dynamic OOD detection using a reinforcement learning (RL)-guided optimizer that explicitly favors updates that reduce the semantic OOD false positive rate over time. We develop a novel augmented optimizer that uses an RL-guided correction term on top of standard gradient descent (GD) and show its improvement over both future-domain generalization and semantic-OOD rejection. We analyze temporal error decomposition in terms of model-change and environment-change generalization errors and develop a new theoretical framework for comparing the generalization errors under both GD and RL-guided optimizers.
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https://arxiv.org/abs/2606.17477
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Towards UAV Image Dehazing: A UAV Atmospheric Scattering Model, Benchmark, and Geometry-Aware Deep Unfolding Network
2026-06-15 08:24:51Wenxuan Fang, Jiangwei Weng, Yu Zheng, Junkai Fan, Guangfa Wang, Xiang Chen, Jian Yang, Jun LiAbstract
In UAV applications, haze significantly obscures distant details and weaken structural information, hindering the recovery of details. Current UAV scenarios still face two key challenges: (i) paired hazy/clean images from the real world are unobtainable, while the classical atmospheric scattering model is inadequate for modeling the spatially non-uniform haze in UAV imagery; (ii) existing dehazing methods struggle to remove the heavy haze accumulated in the upper regions of UAV images. To address these issues, we first propose a UAV Atmospheric Scattering Model (UASM), which explicitly incorporates flight altitude, viewing pitch, and extinction to characterize the non-uniform haze distribution in UAV imaging. Based on UASM, we develop a physics-driven dehazing framework, termed Geometry-aware Proximal Deep Unfolding Network (GP-DUN). Specifically, GP-DUN consists of three key modules: a Latent Geometry Estimator (LGE) that infers transmittance consistent with UAV imaging geometry, a Geometry-aware Gradient Descent Module (GeoGDM) that embeds UASM into the data-fidelity term and performs physics-consistent closed-form updates, and an Pooling-Expert Proximal Mapping Module (PE-PMM) that learns an implicit prior to restore textures and structures beyond the capability of explicit physical modeling. In addition, we further construct UASM-HazeSet, which provides controllable paired synthetic data together with 2,285 real UAV haze images for testing. Extensive experiments show that GP-DUN consistently outperforms existing methods on both UASM-HazeSet and real UAV haze benchmarks.
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URL
https://arxiv.org/abs/2606.16392
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VL2Spike: Spike-driven Distillation from VLMs for Low-Power Visual Perception in Embodied AI
2026-06-14 16:29:53Zinan Liu, Eric Zheng, Soumyaratna Debnath, Hao Shi, Ling Xiao, Lin WangAbstract
Spiking neural networks (SNNs) are brain-inspired, event-driven models that compute with sparse spikes, which enables highly efficient visual perception in resource-constrained embodied AI models. The emergence of Spiking-Transformer models with spike self-attention has substantially improved the learning capacity of pure SNNs. Although SNNs are energy efficient, their performance is still limited by the spike-based architecture and optimization challenges, as standard gradient descent rules cannot be directly applied. Recently, vision-language models (VLMs) have shown rich multi-modal knowledge representation capabilities for visual perception. Thus, it is promising to leverage VLMs for better Spikformer training. To this end, we present VL2Spike, a novel spike-based knowledge distillation (KD) framework that bridges multi-modal knowledge from VLMs with compact Spikformer models. This design enhances the learning capacity of Spikformer models while preserving their energy-efficiency merits, thereby offering a practical pathway toward low-power robotic perception. Our VL2Spike brings two key technical contributions. To align with spiking dynamics, we first propose spatial-temporal visual spike (SVS) distillation, which achieves (1) shared manifold alignment between VLM image features and spike tokens, and (2) warm-started temporal consistency on membrane potentials and spike rates. We then design a novel spike prototype-guided linguistic (SPL) distillation strategy that aligns Spikformer's class prototypes and logits with promptable VLM text embeddings. Extensive experiments show that VL2Spike achieves 6.81% gain across three static datasets with only 15.7% energy consumption. It also exhibits strong generalization capacity on robotic visual place recognition (VPR) with a gain of 6.63%, highlighting its potential for low-power perception in embodied AI.
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URL
https://arxiv.org/abs/2606.15898
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Spokes: Optimizing for Diverse Pretraining Data Selection
2026-06-13 09:28:46Clarence Lee, Yejin Choi, Luke Zettlemoyer, Pang Wei Koh, Hai Leong ChieuAbstract
Diversity plays a critical role in data selection, improving performance under fixed data budgets by reducing redundancy and repetition. However, optimizing for diversity is inherently challenging, as it is a set-level property that depends on interactions between data points rather than individual examples. As a result, existing approaches typically rely on proxies or approximations, which often fail to ensure sufficiently diverse subsets. In this work, we directly optimize diversity by introducing a probabilistic diversification framework based on the G-Vendi score, optimized via exponentiated gradient descent. Our method produces subsets that are substantially more diverse than those obtained via random sampling, achieving a +489 increase in G-Vendi score on a 500k-sample subset. We evaluate our approach on FineWeb and DCLM, where it consistently outperforms existing methods. Notably, SPOKES (diversity-only) improves average downstream performance by +0.4 and +0.5 points over random sampling on DCLM and FineWeb, respectively. More importantly, jointly optimizing for both quality and diversity yields the strongest results: SPOKES achieves gains of +1.5 and +1.4 points on DCLM and FineWeb, outperforming all baselines, including semantic deduplication and quality filtering.
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URL
https://arxiv.org/abs/2606.15216
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FastMix: Fast Data Mixture Optimization via Gradient Descent
2026-06-12 21:46:57Haoru Tan, Sitong Wu, Yanfeng Chen, Jun Xia, Ruobing Xie, Bin Xia, Xingwu Sun, Xiaojuan QiAbstract
While large and diverse datasets have driven recent advances in large models, identifying the optimal data mixture for pre-training and post-training remains a significant open problem. We address this challenge with FASTMIX, a novel framework that automates data mixture discovery while training only a single proxy model. Instead of relying on predefined heuristics or resource-intensive simulations, FASTMIX jointly optimizes mixture coefficients and model parameters, substantially improving efficiency and scalability over prior approaches. At the core of FASTMIX is a reformulation of mixture selection as a bilevel optimization problem. Under this reformulation, we show that optimizing mixture ratios is mathematically equivalent to assigning per-source loss weights under uniform source sampling. This embeds the mixture coefficients directly into the differentiable iterative optimization objective, enabling efficient, gradient-based optimization of both mixture and model. To solve the optimization problem, FASTMIX implements an approximate iterative optimization procedure, alternating between (i) updating model parameters on data sampled according to current mixture ratios (inner loop) and (ii) updating mixture ratios based on validation feedback (outer loop). Across pre- and post-training, FASTMIX outperforms baselines while drastically reducing search cost. Code (this https URL)
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URL
https://arxiv.org/abs/2606.14971
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Duality for Optimal Multi-Item, Multi-Bidder Auction Design: Revenue Certificates through Deep Learning
2026-06-08 19:42:01Yanchen Jiang, David C. Parkes, Tonghan WangAbstract
Characterizing revenue-optimal auctions for multi-item, multi-bidder settings remains a fundamental open problem, with no known closed-form solution existing beyond restrictive binary-type instances. This has motivated interest in computational approaches to optimal auction design. In this paper, we introduce the first computational framework that directly tackles the dual problem for multi-item, multi-bidder auctions and dominant-strategy incentive compatibility (DSIC), generating certified revenue upper bounds. Our approach parametrizes Lagrange multipliers with a structurally guaranteed strict flow-conservation property using neural networks, enabling efficient optimization over feasible dual solutions via gradient descent. To bridge the gap between discrete computational methods and theoretical guarantees for continuous types, we develop a novel lifting technique that maps dual certificates from coarse discretizations to fine refinements. We prove that lifting gives valid revenue upper bounds for multi-item, multi-bidder auctions with continuous uniform valuations. Furthermore, we give a generalized lifting construction for arbitrary continuous distributions and demonstrate that these lifted duals converge to the revenue of the original continuous problem in the discrete limit. We validate this computational framework for the dual auction design problem by recovering known analytical mechanisms for canonical instances. For multi-item multi-bidder problems, our framework establishes a small gap between the optimal revenue and best-known DSIC mechanisms, providing computational certificates of near-optimality.
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URL
https://arxiv.org/abs/2606.10112
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A Theory on Flow Matching with Neural Networks
2026-06-08 19:16:32Yihan He, Qishuo Yin, Yuan Cao, Jianqing Fan, Han LiuAbstract
In this work, we develop theoretical foundation for flow matching with neural-network-parameterized conditional velocity fields. We establish convergence guarantees for gradient descent in the over-parameterized 2-layered ReLU neural network regime. We derive generalization bounds for the conditional velocity-field matching objective. Building on these results, we provide Wasserstein-distance guarantees for the samples generated by the induced flow. Our analysis is based on generalization bound for multi-task representation learning with unbounded losses, which may be of independent interest beyond flow-based generative modeling. These theoretical results are validated through extensive experiments on both synthetic and real-world image benchmarks.
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URL
https://arxiv.org/abs/2606.10089
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Reinforcement Learning for Flow-Matching Policies with Density Transport
2026-06-07 12:28:51Boshu Lei, Kostas Daniilidis, Antonio LoquercioAbstract
We present an online reinforcement learning (RL) algorithm for fine-tuning flow-matching policies in continuous-control problems. Our key insight is to view RL-based policy improvement as a transport of action densities towards regions of high reward, which naturally aligns with the transport formulation of flow matching models. Prior methods either approximate the current or optimal policy distribution or resort to distillation, which introduces biased gradients or sacrifices multimodal modeling capacity. In contrast, our approach for RL with Density Transport, which we name \emph{RLDT}, constructs a transport field from a maximum-entropy RL objective using Stein Variational Gradient Descent (SVGD). Then, it finetunes a pretrained flow matching policy to align with this field. Training with this alignment objective is nontrivial because flow-matching policies generate actions via a multi-step process, making direct gradient-based optimization challenging. To overcome this challenge and stabilize training, we approximate policy actions from intermediate denoising steps via expected-target estimation. This allows the transport-field update to propagate into the network parameters without unstable backpropagation through time. Experimental results demonstrate that RLDT outperforms competitive baselines in reward quality and convergence speed. This performance holds across diverse continuous-control tasks, encompassing both dense and sparse rewards, as well as state- and vision-based long-horizon robot manipulation. The project webpage is \href{this https URL}{this https URL}.
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URL
https://arxiv.org/abs/2606.08602
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Constrained Paraphrase Consistency for LLM Hallucination Detection
2026-06-06 13:14:10Shanshan Lin, Dongsheng Hong, Sibo Ju, Chao Chen, Xi Zhang, Xiangwen LiaoAbstract
Large language models (LLMs) can generate factually inconsistent claims, motivating accurate and scalable hallucination detectors. Prior work largely enlarges training sets via synthesis or new annotations, introducing increasing cost and potential bias while underusing the consistency implied by semantically equivalent paraphrases. We propose Consistency-Constrained Hallucination Detector (CCHD), which formulates training as a constrained optimization problem. The standard cross-entropy on original document-claim pairs is complemented by (i) paraphrase-consistency constraints bounding divergence across paraphrased views, and (ii) label-preservation constraints tying paraphrases to ground truth. We solve the problem by gradient descent-ascent over model parameters and per-view Lagrange multipliers, adding only a few scalar dual variables and no inference-time overhead. With DeBERTa and Flan-T5 backbones, CCHD consistently outperforms strong baselines (FactCG, MiniCheck, and AlignScore) on standard factuality benchmarks, demonstrating its superiority on hallucination detection.
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URL
https://arxiv.org/abs/2606.08158
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Closed-Form Spectral Regularization for Multi-Task Model Merging
2026-06-05 14:00:47Yongxian Wei, Runxi Cheng, Xingxuan Zhang, Li Shen, Chun Yuan, Peng Cui, Dacheng TaoAbstract
Model merging combines several independently fine-tuned experts into a single multi-task model without any training data, reducing the storage, serving, and decentralized-development costs of large foundation models. State-of-the-art merging methods formulate merging as a layer-wise quadratic interference minimization problem. Although this problem admits an exact closed-form pseudoinverse solution, that solution underperforms hundreds of iterations of gradient descent in practice. The iterative loop dominates the cost of the pipeline, yet its effectiveness has remained unexplained. We revisit this regime and show that the iterative solver does not primarily act as an optimizer; rather, it serves as an implicit spectral regularizer for an ill-posed normal equation, where small-eigenvalue directions of the per-layer interference operator amplify proxy noise. Building on this finding, we formalize multi-task model merging as a noisy linear inverse problem and propose a spectral filtering estimator parameterized by a per-direction filter. We instantiate this estimator with SWUDI, a closed-form method that combines a soft exponential filter, which matches the gradient-flow trajectory of iterative descent, with a hard top-K truncation that suppresses noise-amplifying small-eigenvalue directions. Furthermore, we propose SWUDI-A, an adaptive variant that replaces the global rank hyperparameter with per-layer rank rules, further improving robustness across architectures. Both variants share a single symmetric eigendecomposition per linear layer and require no training data or optimizer state. Across four general benchmarks and a multimodal merging benchmark spanning VQA, Geometry, Chart, OCR, Grounding, and modality merging, our proposed spectral solvers match or outperform state-of-the-art merging methods. Crucially, they reduce wall-clock time by 28-72x and peak GPU memory by up to 50%.
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URL
https://arxiv.org/abs/2606.07289
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Generalization in Deep Neural Networks: Minimax Rates for Gradient Methods
2026-06-04 23:31:52Junyu Zhou, Puyu Wang, Yunwen Lei, Marius Kloft, Yiming YingAbstract
Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory. While recent advances, particularly works under the Neural Tangent Kernel (NTK) regime, have shed light on the behavior of shallow architectures, the statistical generalization properties of deep neural networks (DNNs), especially in regression tasks, remain far less understood. In this paper, we make significant progress toward closing this gap by providing a comprehensive generalization analysis of DNNs trained using gradient-based methods. First, we establish, for the first time, a crucial connection between the learning dynamics of a DNN with smooth activation functions trained via gradient-based methods and those of kernel methods, showing that gradient-based methods on over-parameterized DNNs can fully inherit the favorable learning dynamics of their kernel counterparts. Building on this connection and the well-established optimality of kernel methods, we derive the first known minimax-optimal rates for the excess population risk of both gradient descent (GD) and stochastic gradient descent (SGD), under the assumption that network width scales polynomially with the sample size. Our results demonstrate that, with sufficient width, DNNs trained by GD or SGD can achieve generalization performance comparable to kernel-based methods.
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URL
https://arxiv.org/abs/2606.06772
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Optimal Rates for Generalization of Gradient Descent Methods with Deep Neural Networks
2026-06-04 23:04:49Junyu Zhou, Puyu Wang, Yunwen Lei, Yiming Ying, Ding-Xuan ZhouAbstract
Recent progress has been made in understanding the statistical generalization performance of gradient descent methods for overparameterized neural networks within the neural tangent kernel (NTK) regime. However, most of the existing work on regression problems is limited to shallow network architectures, leaving a notable gap in the theory of deep neural networks. This paper addresses this gap by presenting a comprehensive generalization analysis for deep ReLU networks trained using gradient descent (GD) and stochastic gradient descent (SGD). Specifically, we establish the first known minimax-optimal rates of excess population risk for both GD and SGD with deep ReLU networks, under the assumption that the network width scales polynomially with respect to the network depth and training sample size. Our results demonstrate that with sufficient width, gradient descent methods for deep ReLU networks can achieve optimal generalization rates on par with kernel methods.
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URL
https://arxiv.org/abs/2606.06764
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PC Layer: Polynomial Weight Preconditioning for Improving LLM Pre-Training
2026-06-04 17:55:11Senmiao Wang, Tiantian Fang, Haoran Zhang, Yushun Zhang, Kunxiang Zhao, Alex Schwing, Ruoyu SunAbstract
We propose a preconditioning (PC) layer, a weight parameterization via polynomial preconditioner that ensures stable weight conditioning throughout LLM training. The PC module reshapes the singular-value spectrum of weight matrices via low-degree polynomial preconditioning. After training, the preconditioned weights can be merged back into the original architecture, incurring no inference overhead. We demonstrate the advantage of the proposed PC layer over standard transformers in Llama-1B pre-training, for both the AdamW and Muon optimizers. Theoretically, we justify this spectrum-control principle by proving that uniformly bounding each layer's singular values ensures geometric convergence of gradient descent to global minima, for certain deep linear networks. Our code is available at this https URL.
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URL
https://arxiv.org/abs/2606.06470
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Cross-Epoch Adaptive Rollout Optimization for RL Post-Training
2026-06-04 02:27:51Yiming Zong, Yige Wang, Jiashuo JiangAbstract
LLM post-training often relies on reinforcement learning methods that sample multiple rollouts per prompt, yet most existing approaches use a fixed rollout budget for every prompt, despite large differences in the training signal different prompts provide. In this paper, we study adaptive rollout allocation under a fixed global budget and formulate the problem as online resource allocation with prompt-level diminishing returns. Our method, CERO, maintains a Beta posterior over each prompt's success probability and uses the posterior expected Bernoulli variance as a Bayesian estimate of the value of additional rollouts. We use this estimate to construct a concave, saturating utility over cumulative allocations, yielding an objective in which decisions across prompts and epochs are coupled by the global budget. Since the resulting objective is temporally nonseparable, we derive a Fenchel-dual reformulation and update both prompt-level and budget-level dual variables via projected online gradient descent. Under fixed prompt utilities, we prove an $O(\sqrt{K})$ regret bound against the offline allocation benchmark. Experiments on mathematical-reasoning problems show that CERO consistently outperforms GRPO across multiple open-weight LLMs and benchmarks, demonstrating that adaptive rollout budgeting can improve sample efficiency.
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URL
https://arxiv.org/abs/2606.05606
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Gradient descent at the Edge of Stability: free energy model and kinetic description of the two-layer network
2026-06-03 18:11:33Antonin Chodron de CourcelAbstract
We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the loss and the sharpness. We propose a continuous-time effective model that tracks the evolution of the average trajectory coupled with the time-averaged covariance of its fast oscillations. Our analysis reveals that the natural quantity to monitor in such unstable regimes is an effective free energy, which combines the original risk functional with a curvature-related "entropic" term. Our model allows us to track the envelope of the oscillations even in situations where its dynamics evolve on similar timescales as the averaged weights. Otherwise stated, we can track the spikes that occur during the training of some neural network architectures. For wide two-layer neural networks optimized under stable non-vanishing oscillations, we derive a mean-field limit that results in a novel kinetic equation describing the joint distribution of weights and their fluctuations. We show that this equation can be interpreted as a Wasserstein-2 gradient flow of a macroscopic free energy. Finally, we provide numerical evidence on matrix factorization and deep learning tasks (CIFAR-10) to demonstrate the model's accuracy in capturing the envelope of the oscillations and the predictive power of the effective free energy.
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URL
https://arxiv.org/abs/2606.05326
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Trace-Mediated Peak Bias: Bridging Temporal Credit Assignment and Cognitive Heuristics in Deep Reinforcement Learning
2026-06-03 11:19:29Viktor Vesel\'y, Aleksandar Todorov, Erwan Escudie, Matthia SabatelliAbstract
Temporal credit assignment is central to both biological and artificial intelligence, yet its interaction with non-linear function approximation is poorly understood. We identify a systematic failure mode in deep reinforcement learning (RL) termed Trace-Mediated Peak Bias (TMPB). At intermediate eligibility trace depths, agents irrationally prefer trajectories with high-magnitude reward ``peaks'' over alternatives with higher cumulative returns. This provides a mechanistic account of the Peak-End Rule: a human memory bias where experiences are judged by their most intense moments rather than integrated utility. We show that TMPB emerges because traces amplify distal Temporal Difference errors into ``gradient shocks'' that fixed-step-size Stochastic Gradient Descent cannot normalize, leading to global overestimation. Conversely, adaptive optimizers mitigate this pathology via second-moment normalization. Our results suggest that human-like saliency distortions may emerge naturally from the mathematical constraints of credit assignment in distributed systems, and that adaptive optimization is a theoretical necessity for rational value estimation.
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URL
https://arxiv.org/abs/2606.04735
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An Ensembled Latent Factor Model via Differential Evolution and Gradient Descent Optimization
2026-06-03 03:36:27Rui Zhang, Jinhang Liu, Wenbo ZhangAbstract
High-dimensional and incomplete (HDI) data are prevalent in many real-world big data scenarios. Latent factor models serve as a common representation learning approach, capable of uncovering informative latent factors from such data. Nevertheless, most existing latent factor models rely solely on gradient descent for optimization, which may lead to insufficient and biased representations, particularly when dealing with heterogeneous HDI data. Thus, this study proposes an Ensembled Latent Factor Model via Differential Evolution and Gradient Descent Optimization (ELFM-DEGDO) with two-fold designed: 1) two diverse latent factor models are independently modeled via differential evolution and gradient descent optimization, respectively, and 2) the two diverse latent factor models are combined via a customized self-adaptive weighting mechanism to effectively fuse their strengths. By leveraging the complementary advantages of both optimization paradigms, ELFM-DEGDO is able to produce more comprehensive and less biased representations for HDI data. Three HDI datasets are tested to show that ELFM-DEGDO consistently performs better than related several latent factor models.
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URL
https://arxiv.org/abs/2606.04408
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