Natural Gradient Descent: Empirical Validation of Local Efficiency and Coordinate Invariance
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2025
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| _version_ | 1866901851891302400 |
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| author | HIDEKI |
| author_facet | HIDEKI |
| contents | <p>Abstract</p> <p>Natural Gradient descent is a second-order optimization method that incorporates the Fisher information matrix to account for the geometry of probability distributions. While its theoretical foundation in information geometry is well-established, empirical validation of its key properties—particularly coordinate invariance—has been limited. We present three complementary experiments that validate Natural Gradient's fundamental characteristics in softmax regression: (1) local efficiency advantage under matched Kullback-Leibler divergence, (2) performance under strict KL-controlled learning, and (3) coordinate invariance under reparameterization. Our results demonstrate that Natural Gradient achieves 1.98% superior loss reduction per step, maintains performance parity with SGD under exact KL budgeting in linear models, and exhibits machine-precision coordinate invariance—showing 1.9 billion-fold better consistency than SGD across parameter rescalings. We introduce a KL-controlled learning protocol enabling fair comparison of optimization methods and validate it through extensive experimentation. These findings provide the first systematic empirical confirmation of Natural Gradient's coordinate invariance property in a controlled setting and establish benchmarks for evaluating geometric optimization methods.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17996325 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Natural Gradient Descent: Empirical Validation of Local Efficiency and Coordinate Invariance HIDEKI Natural Gradient Coordinate Invariance Fisher Information Matrix Information Geometry Optimization KL Divergence Control Second-Order Methods Reparameterization Invariance Machine Learning Softmax Regression <p>Abstract</p> <p>Natural Gradient descent is a second-order optimization method that incorporates the Fisher information matrix to account for the geometry of probability distributions. While its theoretical foundation in information geometry is well-established, empirical validation of its key properties—particularly coordinate invariance—has been limited. We present three complementary experiments that validate Natural Gradient's fundamental characteristics in softmax regression: (1) local efficiency advantage under matched Kullback-Leibler divergence, (2) performance under strict KL-controlled learning, and (3) coordinate invariance under reparameterization. Our results demonstrate that Natural Gradient achieves 1.98% superior loss reduction per step, maintains performance parity with SGD under exact KL budgeting in linear models, and exhibits machine-precision coordinate invariance—showing 1.9 billion-fold better consistency than SGD across parameter rescalings. We introduce a KL-controlled learning protocol enabling fair comparison of optimization methods and validate it through extensive experimentation. These findings provide the first systematic empirical confirmation of Natural Gradient's coordinate invariance property in a controlled setting and establish benchmarks for evaluating geometric optimization methods.</p> |
| title | Natural Gradient Descent: Empirical Validation of Local Efficiency and Coordinate Invariance |
| topic | Natural Gradient Coordinate Invariance Fisher Information Matrix Information Geometry Optimization KL Divergence Control Second-Order Methods Reparameterization Invariance Machine Learning Softmax Regression |
| url | https://doi.org/10.5281/zenodo.17996325 |