How Inefficient Is Natural Gradient Descent? From Exact Optimality to Θ( \sqrt{ \log d } ) Divergence
Natural gradient descent (NGD) underlies common methods in ML. For dually flat families, idealized NGD on the forward Kullback--Leibler objective follows the mixture geodesic which is often longer than the shortest Fisher--Rao path. We quantify this overhead by the inefficiency ratio \(R \ge 1\), the Fisher length of the mixture geodesic divided by the Fisher--Rao distance, and bound its supremum over endpoint pairs as a function of the parameter dimension \(d\). A tensor criterion identifies the regime (I) families, with \(R=1\) everywhere: exactly those with quadratic potential or dimension one, such as fixed-covariance Gaussians. For non-quadratic families, we prove two further regimes...
| Authors | Guni Sharon, Alan Kuhnle |
|---|---|
| Primary category | stat.ML |
| Published | 5 October 2026 |
| Last updated at source | 5 October 2026 |
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