The Geometry of Learning¶
You built a Thompson Sampling bandit. It picks rules by sampling from Beta distributions. When a rule catches a real bug, you bump alpha. When it doesn't, you bump beta. Simple.
Then you plot the trajectories and notice something wrong. Two updates that move alpha by exactly 1 feel completely different. One changes everything. The other changes nothing. Your coordinate system is lying to you.
This series introduces the Fisher information metric: the geometry that tells the truth about how much the system is learning, where, and how fast.
The series¶
| Tutorial | What it covers |
|---|---|
| Your Coordinates Are Lying to You | The Fisher information matrix; why Euclidean distance is meaningless for beliefs |
| Paths Through Belief Space | Real bandit trajectories with Fisher-speed coloring; entropy convergence |
| The Shortest Path Is Curved | Christoffel symbols, the geodesic equation, and solving it with scipy |
| Where the Manifold Bends | Gaussian curvature; the GR correspondence; critical learning moments |
| The Update Rule as a Dynamical System | Fixed points, stability, phase portraits, Lyapunov exponents |
| What the System Should Feel | Noether's theorem, conserved quantities, affect signals, the interoception circuit |
What you need¶
- Python with scipy, numpy, matplotlib
bandit_state.jsonlfrom a buildlog Thompson Sampling run (real data)- Comfort with plotting and basic calculus (derivatives, integrals)
- No differential geometry background required (we earn every term)
Depth note¶
Chapters 1-4 are deep: full implementations, real data, runnable code. Chapters 5-6 are sketches: enough to build intuition, write pseudocode, and see the architecture. The full dynamical systems and Noether treatment requires research that's still in progress (see the roadmap).
Where this leads¶
The Fisher layer answers "is the system learning?" The companion series, Causal Reasoning, answers "what leads to what?" Together, they form two halves of a system that knows what it knows, how fast it's learning, and which relationships actually matter.
After both series, a roadmap page maps the horizon: Hamiltonian mechanics on belief space, conservation laws as a sensory apparatus, and the affect signals that close the loop from geometry to action.