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Foundation Lab

Natural Gradient & Riemannian Optimization

Natural gradient is coordinate-invariant—it gives the same update regardless of parameterization

Concept 56 of 100OptimizationPhase 10
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#56Natural GradOptimization
key equation
∇~L=F(θ)−1∇θL\tilde{\nabla} L = F(\theta)^{-1} \nabla_\theta L
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Concept 56 of 100Natural GradOptimization / Phase 10: Mathematical foundations & information geometry
Current question

The gradient is a covector, not a vector—the metric turns it into a direction of steepest descent

∇~L=F(θ)−1∇θL\tilde{\nabla} L = F(\theta)^{-1} \nabla_\theta L
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Why It Matters for Modern Models

  • Natural gradient is coordinate-invariant—it gives the same update regardless of parameterization
  • TRPO and PPO are approximations to natural gradient updates for policy optimization
  • Adam can be viewed as a diagonal approximation to natural gradient with adaptive preconditioning

What Tutorials Skip

What is still poorly explained in textbooks and papers:

  • The gradient is a covector, not a vector—the metric turns it into a direction of steepest descent
  • Euclidean gradient depends on how you parameterize; natural gradient depends only on the distributions
  • Natural gradient avoids plateaus faster because it accounts for local curvature in distribution space

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Core Math (Optional Deep Dive)

If you want intuition first, start with the key equation and cited sources. Come back here for the full walkthrough.

Key Equation
∇~L=F(θ)−1∇θL\tilde{\nabla} L = F(\theta)^{-1} \nabla_\theta L

Natural gradient uses the Fisher metric instead of Euclidean:

∇~L=F(θ)−1∇θL\tilde{\nabla} L = F(\theta)^{-1} \nabla_\theta L

Update rule:

θt+1=θt−ηF(θt)−1∇L\theta_{t+1} = \theta_t - \eta F(\theta_t)^{-1} \nabla L

Variational characterization (why it's "natural"):

δ∗=arg⁡min⁡δ⟨∇L,δ⟩s.t.KL(pθ∥pθ+δ)≤ϵ\delta^* = \arg\min_\delta \langle \nabla L, \delta \rangle \quad \text{s.t.} \quad \text{KL}(p_\theta \| p_{\theta+\delta}) \leq \epsilon
⇒δ∗∝F−1∇L\Rightarrow \delta^* \propto F^{-1} \nabla L

Canonical Papers

Natural Gradient Works Efficiently in Learning

Amari1998Neural Computation
Read paper →

Trust Region Policy Optimization

Schulman et al.2015ICML
Read paper →

Connections

Prerequisites

ℐFisher Info∇Adam

Enables

🪞Bregman

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