notes
Working notes where I elaborate on concepts I'm currently studying. When a proof or technique seems to come out of the blue, I try to reverse-engineer how one might have arrived at it, reconstructing the chain of reasoning from scratch.
On the Contractivity of Stochastic Interpolation Flow
How strong log-concavity of the target controls the Lipschitz constant of the interpolant ODE flow map.
From Behavior Cloning to RL
Tracing the arc from vanilla BC through DAgger and AggreVaTe to approximate policy iteration.
How much of an approximation is PPO?
Relating the discounted return to the unclipped PPO surrogate via state occupancy measures and the performance difference lemma, and clarifying which additional approximations hold in practice.
Vector-Time Flows
Generalizing stochastic interpolants to per-coordinate time schedules, and recovering in-painting as a special case.
Optimality Conditions in Finite and Infinite Dimensions
Deriving necessary and sufficient conditions for local optima, from unconstrained and constrained finite-dimensional problems to infinite-dimensional optimization.
Itô to Epsilon: SDEs, Interpolants, and Tweedie Along the Way
Deriving the diffusion process via Itô calculus, connecting it to Stochastic Interpolants, proving Tweedie's formula for score estimation, and recovering the denoising objective.
Unifying Flow Matching and Stochastic Interpolants
How conditioning on both endpoints extends Flow Matching beyond Gaussian sources and recovers the Stochastic Interpolants objective.
One Gradient to Rule Them All: How the Score Bridges ODEs, SDEs, and Their Reversals
The score function as a toggle between ODE and SDE transports, and why Gaussianization simplifies estimation.
Stochastic Interpolants
Deriving the transport equation for stochastic interpolants using Fourier transforms, and why the Eulerian-Lagrangian toggle fails in this setting.
Eulerian & Lagrangian Views
Toggling between Eulerian and Lagrangian formulations, and the conditions under which this is valid.
Continuous Normalizing Flows
Deriving the Lagrangian and Eulerian views of continuous normalizing flows and the adjoint method for efficient gradient computation.