发表机构
Indian Institute of Technology Bhubaneswar; Narula Institute of Technology; The University of Tokyo（印度理工学院布巴内斯瓦尔分校; 纳鲁拉理工学院; 东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出HAMLET，一个将参数优化视为耗散动力系统演化的几何能量框架，通过哈密顿动力学与瑞利耗散建模，在CIFAR-10和MNIST基准上验证了其有效性与优越性能。
AI 中文摘要
机器学习中的优化通常通过迭代规则来表达，这些规则利用损失景观的信息更新模型参数。在本工作中，我们研究了一种替代视角，将参数优化视为耗散动力系统的演化。模型参数被视为广义坐标，损失充当势能，正定度量定义参数空间的局部动能几何。从变分公式出发，我们推导了相应的哈密顿动力学、由位置相关度量产生的几何力，以及与度量相容的瑞利耗散定律。所得连续系统满足单调能量耗散关系，而其离散形式允许直接研究曲率对优化轨迹的影响。我们使用一个受控的CIFAR-10图像重建问题来说明该框架，该问题的最优解在解析上已知。当使用图像Hessian作为度量时，二次模态动力学的曲率依赖性被消除。然后，我们将同一图像重新参数化为矩阵乘积态，产生一个真正位置相关的度量和一个非零几何力。庞加莱回归映射提供了对所得收缩动力学的互补相空间视图。这些例子确立了HAMLET作为一个几何的、基于能量的框架，用于研究优化作为参数空间中的耗散运动。在五个种子的MNIST MLP基准测试中，HAMLET在三个评估优化器中达到了最高的平均测试准确率（97.95%）和最低的平均测试负对数似然（0.0696）。
英文摘要
Optimization in machine learning is usually expressed through iterative rules that update model parameters using information from the loss landscape. In this work, we study an alternative viewpoint in which parameter optimization is treated as the evolution of a dissipative dynamical system. The model parameters are regarded as generalized coordinates, the loss acts as a potential energy, and a positive-definite metric defines the local kinetic geometry of parameter space. Starting from a variational formulation, we derive the corresponding Hamiltonian dynamics, the geometric force generated by a position-dependent metric, and a metric-compatible Rayleigh dissipation law. The resulting continuous system satisfies a monotonic energy-dissipation relation, while its discrete form allows the influence of curvature on the optimization trajectory to be studied directly. We illustrate the framework using a controlled CIFAR-10 image-reconstruction problem for which the optimum is known analytically. With the image Hessian used as the metric, the curvature dependence of the quadratic modal dynamics is removed. We then reparameterize the same image as a matrix product state, producing a genuinely position-dependent metric and a nonzero geometric force. Poincaré return maps provide a complementary phase-space view of the resulting contraction dynamics. These examples establish HAMLET as a geometric, energy-based framework for studying optimization as dissipative motion in parameter space. On a five-seed MNIST MLP benchmark, HAMLET attains the highest mean test accuracy (97.95\%) and the lowest mean test negative log-likelihood (0.0696) among the three evaluated optimizers.
