Week 7: Projection, Natural Gradient, and Optimal Protocols

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Neil D. Lawrence

Abstract:

Maximum entropy as an m-projection onto a constraint manifold, and natural gradient descent as steepest descent in the Fisher metric. Geodesics of that metric are Crooks’ minimum-dissipation protocols. One computed geodesic on a two-parameter exponential family is enough.

Worksheet 3 is due at the start of this session. Quiz 3 occupies the first ten minutes (Fisher metric, natural gradient, dual flatness). Worksheet 4 is released; due 1 December.

This Session

Time plan (120 minutes)

Information, entropy and intelligence course notebook setup

We install some bespoke code for creating and saving plots as well as loading data sets.

import importlib.util

cmd = install_command('pods')

%system {cmd}

cmd = install_command('mlai')

%system {cmd}

MaxEnt as Projection

MaxEnt is the \(m\)-projection of a reference distribution (often uniform) onto the constraint surface written in moment coordinates \(\eta\). Week 6: \(\mathrm{KL}\) is non-negative even when differential entropy is not — the same \(\min\mathrm{KL}(q\|r)\) recipe applies to Jaynes’ die (discrete) and the Gaussian (continuous). On an exponential family the constraint surface is a straight line in natural parameters \(\theta\).

Figure: MaxEnt die as \(m\)-projection of the uniform reference.

Natural Gradient

Natural gradient ascent \(\theta\leftarrow\theta+\eta F^{-1}\nabla L\) is steepest ascent in the Fisher metric. It removes arbitrary parameterisation dependence. The same \(F\) appears in Crooks’ thermodynamic length.

Figure: Vanilla versus natural gradient paths for Gaussian MLE — Worksheet 3 core task.

Geodesics as Optimal Protocols

The geodesic is the prescription for minimum-dissipation protocols under Crooks’ bound — last week’s near-equilibrium expansion of the fluctuation theorem, not an independent axiom. Natural gradient is the local form of the same instruction. A straight line in \((\mu,\sigma^2)\) is not generally a geodesic, but it gives a first length estimate.

Figure: Worksheet 3 uses the straight line; higher marks compare a second path.

GAIST on Minimum-Dissipation Protocols

(Welling et al., 2026) has a theory of minimum-dissipation protocols. It is not this week’s theory. Sections 22.3–22.4 minimise dissipated work in the Benamou–Brenier formulation of optimal transport. The geodesic that saturates the bound lives in the Wasserstein metric, and the bound is \(\Sigma \ge W_2^2/(T\tau)\). Counterdiabatic driving (Section 22.4) prescribes a path of densities and solves for the control that the system must follow.

We are still on the equilibrium manifold, with the Fisher metric. Today’s geodesic is Crooks’ minimum-dissipation protocol — justified last week as the near-equilibrium expansion of the fluctuation theorem, packaged as Fisher–Rao length. Today’s local instruction is the natural gradient \(F^{-1}\nabla L\). GAIST does not treat m-projection, dual flatness, or the natural gradient. Stay with Amari this week. Wasserstein and the Schrödinger bridge arrive in week 8, where the three geometries must be kept apart.

Define This Week

Named, Not Yet Answered

An alternating \(m\)-projection onto two constraint sets is how Sinkhorn enforces two prescribed marginals. Name it; do not compute it. Week 8.

This Week’s Pair

No-go: you cannot beat \(\mathcal{L}^2/\tau\). Prescription: descend by the natural gradient; travel by the geodesic.

After This Lecture

Quiz 3 at the start of lecture 7 (24 November): Fisher metric, natural gradient, dual flatness, on a new parameterised family.

Further Reading

• Sections 22.3–22.4 (Wasserstein geodesics, not Fisher–Rao) of Welling et al. (2026)

• Chapters 3–4 of Amari (2016)

Thanks!

For more information on these subjects and more you might want to check the following resources.

• company: Trent AI
• book: The Atomic Human
• twitter: @lawrennd
• podcast: The Talking Machines
• newspaper: Guardian Profile Page
• blog: http://inverseprobability.com

References