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Fisher Flow: Information Geometry in Optimization

2 min read Updated:
optimization information-geometry machine-learning

Fisher Flow explores using Fisher information and information geometry for optimization problems, particularly in machine learning.

Information Geometry Primer

Information geometry studies the geometry of probability distributions:

  • Manifold Structure: Probability distributions form a Riemannian manifold
  • Fisher Information Metric: Provides the natural distance metric
  • Geodesics: Shortest paths on the manifold = efficient optimization

Fisher Information Matrix

For parametric distribution p(x|θ), the Fisher Information is:

FIM: I(θ) = E[∇log p(x|θ) ∇log p(x|θ)ᵀ]

This captures:

  • Sensitivity: How much distribution changes with parameters
  • Curvature: Second-order information about likelihood surface
  • Natural Gradient: The true gradient direction in parameter space

Natural Gradient Descent

Standard gradient descent:

  • θₜ₊₁ = θₜ - α ∇L(θₜ)
  • Uses Euclidean geometry
  • Inefficient in parameter space

Natural gradient descent:

  • θₜ₊₁ = θₜ - α I(θₜ)⁻¹ ∇L(θₜ)
  • Uses information geometry
  • Invariant to parametrization

Fisher Flow Dynamics

The paper proposes continuous-time optimization:

dθ/dt = -I(θ)⁻¹ ∇L(θ)

This defines a flow on the parameter manifold with properties:

  • Monotonic Decrease: Loss decreases along flow
  • Geodesic Paths: Flow follows natural geometry
  • Adaptive Step Size: Curvature automatically scales updates

Applications to Deep Learning

Neural Network Training

Benefits:

  • Faster Convergence: Accounts for parameter correlations
  • Better Generalization: Natural geometry reduces overfitting
  • Stable Training: Adaptive scaling prevents instability

Challenges:

  • Computational Cost: Computing I(θ)⁻¹ expensive for large networks
  • Approximations: Use diagonal or block-diagonal FIM
  • Stochastic Setting: Estimate FIM from mini-batches

K-FAC Algorithm

Kronecker-Factored Approximate Curvature:

  • Approximates FIM with Kronecker products
  • O(n) computation vs. O(n³) for exact natural gradient
  • Practical for modern deep networks

Variational Inference

Fisher flow provides natural framework:

  • Variational Distribution: q(z|φ)
  • Objective: KL[q(z|φ) || p(z|x)]
  • Flow: Follow Fisher information geometry

Results in efficient inference with geometric guarantees.

Theoretical Properties

The paper proves:

Convergence:

  • Fisher flow converges to local optima
  • Convergence rate depends on conditioning of FIM

Stability:

  • Flow is stable under perturbations
  • Natural gradient provides implicit regularization

Geometry:

  • Respects underlying probability manifold structure
  • Invariant to reparametrization

Practical Implementation

def fisher_flow_step(params, loss_fn, data):
    # Compute gradient
    grad = compute_gradient(loss_fn, params, data)

    # Estimate Fisher information (diagonal approx)
    fisher = estimate_fisher_diagonal(params, data)

    # Natural gradient step
    natural_grad = grad / (fisher + epsilon)

    # Update parameters
    params = params - learning_rate * natural_grad

    return params

Future Directions

  • Higher-Order Methods: Incorporating 3rd order geometry
  • Adaptive Approximations: Dynamic FIM approximation quality
  • Distributed Optimization: Fisher flow in federated learning
  • Non-Parametric: Extending to infinite-dimensional spaces

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