Lessons

What differentiable rendering is actually used for, how it relates to NeRF and Gaussian splatting, and the four failure modes that make inverse rendering harder than the optimisation loop suggests.

Correct gradients are useless if computing them exhausts memory. Radiative backpropagation, path replay backpropagation's constant-memory trick, and why differentiable renderers needed their own compiler.

Differentiating a renderer is easy until geometry moves. Why silhouettes break naive automatic differentiation, and the three families of solutions: edge sampling, reparameterization, and warped-area methods.

Before you can differentiate a renderer you have to see it as mathematics: the rendering equation, why it has no closed-form solution, and how Monte Carlo path tracing turns light transport into an estimation problem.

What happens when a prediction model leaves the benchmark: adversarial fragility, distribution shift, the feedback loop where your own robot changes the behaviour it predicts, and how planners consume uncertainty safely.

Displacement error is the field's default metric and it rewards the wrong behaviour: hedged average predictions, physically impossible trajectories, and best-of-many reporting that flatters diverse nonsense. What to measure instead.

How the field learned to represent people influencing each other: the physics-inspired force model, the Social LSTM pooling layer that replaced hand-designed rules with learned ones, and the attention and graph architectures that followed.

Why forecasting human motion is not a physics problem: the multimodality that makes a single correct answer impossible, the social conventions people navigate by, and the joint prediction problem where everyone is predicting everyone else.

Where this machinery earns its place: low-rank matrix completion, synchronization, and PCA. Plus the two ideas that explain why non-convex manifold problems are often solved to global optimality anyway.

How to move along a curved space without solving differential equations: retractions as cheap approximations to geodesics, vector transport, and the Riemannian versions of gradient descent, conjugate gradients, and trust regions.

Building the machinery: the linear space of allowed directions at a point, the inner product that gives it geometry, and why the Riemannian gradient is the ambient gradient projected rather than a new derivative.

Rotations, subspaces, low-rank matrices and covariances are not vectors in flat space. Why treating those constraints as penalties or projections wastes structure, and what it means to say the search space is a manifold.

What model updates actually leak, the two mechanisms that turn data minimisation into a quantified guarantee, poisoning by malicious clients, and what the Apertus project shows about openness at the other end of the spectrum.

What happens when nobody coordinates: averaging by talking only to neighbours, why the graph's spectral gap sets the convergence rate, and how compressing messages by orders of magnitude still converges.

Why local training on non-identical data pulls clients apart, how averaging their updates produces a model that suits nobody, and the control-variate fix that corrects the drift.

The federated setting: why hospitals, phones, and banks cannot pool their data, what changes when the training loop crosses a network, and the FedAvg algorithm that made the idea practical.

Taking the framework beyond solvable toy models: sharp transitions in real learning, why the loss landscape of a neural network is not the fractured one theory warns about, and what the physics lens genuinely explains.

Belief propagation, the cavity method, and approximate message passing: how physics-derived algorithms achieve the best performance any efficient method can, and how state evolution predicts their behaviour exactly before you run them.

The central result of the field: problems split into three regimes as data increases, and the middle one contains enough information to solve them while no efficient algorithm can. Community detection and planted clique make it concrete.

Why statistical physics has anything to say about algorithms: planted models with known ground truth, the large-system limit where randomness stops fluctuating, and the Bayes-optimal benchmark that makes hardness measurable.

Competition law was built to catch people agreeing in rooms. What happens when the coordination has no agreement, no communication, and no human intent, and how regulators are responding anyway.

What happened to margins when German petrol stations adopted pricing software, and the substantial body of research arguing the simulation results are fragile, parameter-dependent, and may not survive contact with real markets.

The simulation result that started an entire literature: independent Q-learning agents converging on supra-competitive prices with punishment strategies, nobody having programmed them to, and what the mechanism actually is.

From revenue management to reinforcement learning: the mechanics of software that sets prices, the distinction between following rules and learning a policy, and why that distinction turns out to be the one that matters.