Convergence Properties of Natural Gradient Descent for Minimizing KL Divergence (2504.19259v1)

Abstract: The Kullback-Leibler (KL) divergence plays a central role in probabilistic machine learning, where it commonly serves as the canonical loss function. Optimization in such settings is often performed over the probability simplex, where the choice of parameterization significantly impacts convergence. In this work, we study the problem of minimizing the KL divergence and analyze the behavior of gradient-based optimization algorithms under two dual coordinate systems within the framework of information geometry$-$ the exponential family ($\theta$ coordinates) and the mixture family ($\eta$ coordinates). We compare Euclidean gradient descent (GD) in these coordinates with the coordinate-invariant natural gradient descent (NGD), where the natural gradient is a Riemannian gradient that incorporates the intrinsic geometry of the parameter space. In continuous time, we prove that the convergence rates of GD in the $\theta$ and $\eta$ coordinates provide lower and upper bounds, respectively, on the convergence rate of NGD. Moreover, under affine reparameterizations of the dual coordinates, the convergence rates of GD in $\eta$ and $\theta$ coordinates can be scaled to $2c$ and $\frac{2}{c}$, respectively, for any $c>0$, while NGD maintains a fixed convergence rate of $2$, remaining invariant to such transformations and sandwiched between them. Although this suggests that NGD may not exhibit uniformly superior convergence in continuous time, we demonstrate that its advantages become pronounced in discrete time, where it achieves faster convergence and greater robustness to noise, outperforming GD. Our analysis hinges on bounding the spectrum and condition number of the Hessian of the KL divergence at the optimum, which coincides with the Fisher information matrix.

• The paper critically examines the convergence properties of Natural Gradient Descent (NGD) for minimizing KL divergence, comparing it to Euclidean Gradient Descent (GD) in continuous and discrete time settings.
• Key findings show that NGD's continuous-time convergence rate falls between GD in the \(\theta\) and \(\eta\) coordinate systems and remains invariant under affine reparameterizations, unlike GD rates.
• The study demonstrates that NGD achieves superior convergence speed and noise robustness compared to GD in the discrete-time setting, attributing this to effective loss landscape conditioning.

Convergence Properties of Natural Gradient Descent for Minimizing KL Divergence

The paper by Datar and Ay critically examines the convergence dynamics of natural gradient descent (NGD) when applied to the task of minimizing the Kullback-Leibler (KL) divergence within the context of information geometry. KL divergence is a fundamental loss function in probabilistic machine learning, and optimizing this metric is frequently essential in various applications, including the training of probabilistic models over the probability simplex. This research juxtaposes the behavior of NGD with that of Euclidean gradient descent (GD) in both the exponential family (using $\theta$ coordinates) and mixture family (using $\eta$ coordinates) to understand the advantages and limitations of different methodologies in continuous and discrete-time frameworks.

Key Findings and Claims

1. Continuous-Time Convergence:
   • The paper establishes that, in a continuous-time setting, the convergence rates of GD in the $\theta$ and $\eta$ coordinates provide bounds for NGD. Specifically, (a) GD using $\theta$ coordinates historically benchmarks a lower convergence rate than NGD, while (b) GD using $\eta$ coordinates outpaces NGD, suggesting that NGD does not uniformly outperform in this setting.
2. Affine Reparameterization Impact:
   • Under affine reparameterizations of dual coordinates, the convergence rates for GD in $\eta$ and $\theta$ can be scaled by factors of $2c$ and $\frac{2}{c}$, whereas NGD maintains a fixed convergence rate of 2. This factorization illustrates NGD’s invariance and reliability, positioning its rate between GD in the respective coordinate systems.
3. Discrete-Time Dynamics:
   • The NGD method shows notable advantages when transitioning to discrete time. It achieves pronounced convergence speed and demonstrates robustness against noise, outperforming traditional GD. This superiority is attributed to optimal loss landscape conditioning, essentially implying that NGD's updates are akin to minimization with a condition number equivalent to 1.
4. Theoretical Implications:
   • The superiority of NGD in discrete time is theoretically linked to its effective conditioning properties, akin to loss minimization scenarios where the Hessian condition number is optimized. This particular aspect distinguishes NGD in terms of practical implementation scaling.

Implications and Future Directions

This paper's quantitative analysis contributes vital understandings to the theoretical and practical discourse on gradient-based optimization methods for minimizing information-theoretic divergences. The results imply that while NGD does not guarantee superior continuous-time performance across all coordinate frameworks, its discrete-time performance makes it an attractive choice for learning algorithms in noisy environments or where exact trajectory tracking is crucial.

Furthermore, the preservation of convergence characteristics under affine reparameterizations suggests a broader resilience and applicability of NGD across diverse machine learning contexts, potentially informing the parameterization strategies in architectural designs like neural networks. Encouraging future research could explore extending these results into more complex landscapes, including high-dimensional problem spaces or non-Euclidean geometries pertinent to over-parameterized models or those adhering to different statistical families.

By providing a deeper insight into the nuanced interplay of gradient methodologies, the paper lays groundwork for both theorists and practitioners to design more robust optimization algorithms that leverage specific strengths of NGD, helping to advance progress in fields such as Bayesian inference, neural networks, and reinforcement learning.