diff --git a/lecture_6/.gitignore b/lecture_6/.gitignore deleted file mode 100644 index 5ad7551fe653514b4e5b44f00060a99bc9c2dd86..0000000000000000000000000000000000000000 --- a/lecture_6/.gitignore +++ /dev/null @@ -1,4 +0,0 @@ -__pycache__/ -.venv/ -outputs/ -*.pt diff --git a/lecture_6/EQUATION_READINGS.md b/lecture_6/EQUATION_READINGS.md deleted file mode 100644 index 7b5d48725913ae72fe253bdb0d0d337abd3c79b7..0000000000000000000000000000000000000000 --- a/lecture_6/EQUATION_READINGS.md +++ /dev/null @@ -1,3765 +0,0 @@ -# Lecture 6 · Equation readings - -The presentation reveals the equation, then its spoken reading, then intuition. This guide reproduces the literal readings; the teaching notes retain the surrounding derivations and explanations. - -## We can see the difficulty in a one-dimensional flow - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u004_r0) - -Start at one and let the velocity equal the current position. - -$$ -\dot x_t=x_t,\qquad x_0=1 -$$ - -The time derivative of x equals x itself, and the initial value at time zero is one. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u004_r1) - -Separate variables and integrate from zero to t. - -$$ -\int_1^{x_t}\frac{dz}{z}=\int_0^t d\tau\quad\Longrightarrow\quad\log x_t=t -$$ - -Integrating one over z from one to the current state equals integrating time from zero to t, so the logarithm of the current state equals t. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u004_r2) - -Exponentiating gives the exact destination. - -$$ -x_t=e^t,\qquad x_1=e\approx2.71828 -$$ - -The state at time t is e raised to t; at time one, the exact destination is approximately 2.71828. - -## Using the initial velocity once misses that acceleration - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u005_r0) - -One Euler step uses the velocity at the starting point. - -$$ -x_1^{(1)}=1+1(1)=2 -$$ - -The one-step Euler approximation adds the interval length one times the initial velocity one to the starting value one, giving two. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u005_r1) - -Two steps update the velocity after the first move. - -$$ -x_{1/2}=1+\tfrac12(1)=1.5,\qquad x_1^{(2)}=1.5+\tfrac12(1.5)=2.25 -$$ - -The first half step reaches 1.5; the second adds half of the updated velocity 1.5, giving 2.25. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u005_r2) - -With N equal steps, the approximation approaches the exact result. - -$$ -x_1^{(N)}=(1+1/N)^N\longrightarrow e -$$ - -With N equal steps, the endpoint is one plus one over N, raised to N; this converges to e as the number of steps grows. - -## First, recall how flow matching specifies the motion - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u008_r0) - -Pair a source sample with a data sample and interpolate. - -$$ -I_t=(1-t)X_0+tX_1,\qquad (X_0,X_1)\sim\rho -$$ - -The interpolant is one minus t times the source sample plus t times the data sample, with the endpoint pair drawn from the coupling ρ. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u008_r1) - -Average the conditional velocity at the observed state. - -$$ -b_t(x)=\mathbb E[X_1-X_0\mid I_t=x] -$$ - -The velocity at state x and time t is the conditional average of the endpoint displacement, given that the interpolant currently equals x. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u008_r2) - -The resulting ODE follows the interpolant marginals. - -$$ -\dot x_t=b_t(x_t),\qquad x_t\sim p_t=\operatorname{Law}(I_t) -$$ - -The ODE moves with velocity b at its current state, and its time-t distribution equals the law of the interpolant, under the stated regularity assumptions. - -## The continuity equation explains that agreement - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u009_r0) - -Differentiate the expectation of a smooth test function. - -$$ -\frac{d}{dt}\mathbb E[\varphi(I_t)]=\mathbb E[\nabla\varphi(I_t)\cdot\dot I_t] -$$ - -The time derivative of the average test-function value equals the average dot product of its spatial gradient and the interpolant velocity. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u009_r1) - -Condition on the current interpolant state. - -$$ -=\mathbb E[\nabla\varphi(I_t)\cdot\mathbb E[\dot I_t\mid I_t]] -$$ - -Conditioning on the current interpolant replaces its sample-specific velocity by the conditional mean velocity inside the same expected dot product. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u009_r2) - -Integrating by parts identifies the probability evolution. - -$$ -\partial_t p_t+\nabla\cdot(p_t b_t)=0 -$$ - -The time derivative of the density plus the divergence of density times velocity is zero; probability changes locally through incoming and outgoing flow. - -## A flow map records where that ODE arrives - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u010_r0) - -Start the trajectory at x at time s. - -$$ -F_{s,t}(x)=x_t,\qquad x_s=x,\qquad \dot x_\tau=b_\tau(x_\tau) -$$ - -The map F from s to t returns the state reached at t by the ODE whose state at s is x and whose velocity is b. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u010_r1) - -Integrate its velocity along the trajectory. - -$$ -F_{s,t}(x)=x+\int_s^t b_\tau(F_{s,\tau}(x))\,d\tau -$$ - -The destination equals the starting point plus the integral, from s to t, of the velocity evaluated along that same trajectory. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u010_r2) - -A zero-length interval leaves the state unchanged. - -$$ -F_{s,s}(x)=x -$$ - -The map from time s to that same time s returns x unchanged. - -## The average velocity turns this into one residual update - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u011_r0) - -Divide the total displacement by the interval length. - -$$ -v_{s,t}(x)=\frac{F_{s,t}(x)-x}{t-s},\qquad s0 -$$ - -The time-t distribution is Gaussian with mean μ at t and variance σ at t squared, where its standard deviation is strictly positive. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u024_r1) - -Preserve the standardized coordinate of each sample. - -$$ -F_{s,t}(x)=\mu_t+\frac{\sigma_t}{\sigma_s}(x-\mu_s) -$$ - -The map subtracts the source mean, scales by the target-to-source standard-deviation ratio, and adds the target mean. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u024_r2) - -Choose a numerical starting point. - -$$ -\mu_s=0,\ \sigma_s=1,\ \mu_t=2,\ \sigma_t=3,\ x=1\ \Longrightarrow\ F_{s,t}(1)=5 -$$ - -With source mean zero and scale one, target mean two and scale three, the source point one maps to five. - -## The diagonal loss retains the flow matching target - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u027_r0) - -Use the two-time model with equal time inputs. - -$$ -\mathcal L_{\rm diag}=\mathbb E\|v_\theta(t,t,I_t)-\dot I_t\|^2 -$$ - -The diagonal loss averages the squared difference between the model’s equal-time velocity at the interpolant and the interpolant’s actual velocity. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u027_r1) - -Write the conditional mean velocity plus a zero-mean residual. - -$$ -\dot I_t=b_t(I_t)+\eta,\qquad\mathbb E[\eta\mid I_t,t]=0 -$$ - -The interpolant velocity is its conditional mean b plus a residual η whose average is zero when the current state and time are fixed. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u027_r2) - -The cross term vanishes after conditioning. - -$$ -\mathcal L_{\rm diag}=\mathbb E\|v_\theta(t,t,I_t)-b_t(I_t)\|^2+\mathbb E\|\eta\|^2 -$$ - -The loss splits into the squared error against the conditional mean and the residual variance; only the first term depends on the model. - -## The Lagrangian loss compares two endpoint velocities - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u028_r0) - -Differentiate the learned finite-interval map. - -$$ -A_\theta=\partial_tF_\theta(s,t,x) -$$ - -A is the derivative of the learned map with respect to its destination time, with the start time and starting state held fixed. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u028_r1) - -Evaluate the model’s local velocity at its predicted endpoint. - -$$ -B_\theta=v_\theta(t,t,F_\theta(s,t,x)) -$$ - -B is the model’s diagonal velocity at time t, evaluated at its own predicted destination. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u028_r2) - -Penalize their disagreement. - -$$ -\mathcal L_{\rm LSD}=\mathbb E\|A_\theta-B_\theta\|^2 -$$ - -The Lagrangian self-distillation loss averages the squared difference between the map’s endpoint derivative A and its endpoint velocity prediction B. - -## The Eulerian loss checks whether the destination changes - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u029_r0) - -Follow the starting state with the local diagonal velocity. - -$$ -D_sF_\theta=\partial_sF_\theta+J_xF_\theta\,v_\theta(s,s,x) -$$ - -The derivative along the moving start equals the partial start-time derivative plus the map’s spatial Jacobian applied to the model’s diagonal starting velocity. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u029_r1) - -The exact destination is constant along that change. - -$$ -\mathcal L_{\rm ESD}=\mathbb E\|D_sF_\theta\|^2 -$$ - -The Eulerian self-distillation loss averages the squared norm of that derivative along the moving start, penalizing changes in the predicted destination. - -## The composition loss compares the two routes - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u030_r0) - -Compute the direct destination. - -$$ -y_{\rm direct}=F_\theta(s,t,x) -$$ - -The direct destination is the output of the learned map from s to t applied to the starting point x. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u030_r1) - -Compute the destination through an intermediate time. - -$$ -y_{\rm split}=F_\theta(u,t,F_\theta(s,u,x)) -$$ - -The split destination first applies the map from s to u, then applies the map from u to t to that intermediate output. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u030_r2) - -Match the two destinations. - -$$ -\mathcal L_{\rm PSD}=\mathbb E\|y_{\rm direct}-y_{\rm split}\|^2 -$$ - -The progressive self-distillation loss averages the squared distance between the direct destination and the destination obtained through the intermediate time. - -## A model that always stays still exposes that missing condition - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u031_r0) - -Consider the identity map at every pair of times. - -$$ -\widehat F_{s,t}(x)=x -$$ - -The candidate map returns the input x for every choice of starting and ending times. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u031_r1) - -It satisfies composition automatically. - -$$ -\widehat F_{u,t}(\widehat F_{s,u}(x))=x=\widehat F_{s,t}(x) -$$ - -Composing two identity maps still returns x, exactly matching the direct identity map. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u031_r2) - -Its tangent is wrong whenever the desired velocity is nonzero. - -$$ -\widehat v_{t,t}(x)=0\ne b_t(x) -$$ - -This map’s diagonal velocity is zero, so it disagrees with the desired local velocity wherever b is nonzero. - -## Detaching the target determines which branch learns - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u032_r0) - -Build a target from the two shorter intervals. - -$$ -y_{\rm target}=\operatorname{sg}[F_\theta(u,t,F_\theta(s,u,x))] -$$ - -The target is the two-interval composed prediction inside stop-gradient, so its value is used without differentiating through its construction. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u032_r1) - -Train the direct prediction against that fixed target for this update. - -$$ -\ell=\|F_\theta(s,t,x)-y_{\rm target}\|^2 -$$ - -The loss is the squared distance between the direct map prediction and that fixed target for the current update. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u032_r2) - -Only the direct branch contributes to the gradient. - -$$ -\nabla_\theta\ell=2J_\theta F_\theta(s,t,x)^\top(F_\theta(s,t,x)-y_{\rm target}) -$$ - -The parameter gradient is twice the direct map’s parameter Jacobian transpose applied to its prediction error; the detached target contributes no derivative. - -## Small velocity errors accumulate along the generated path - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u033_r0) - -Let the learned path have residual r relative to a Lipschitz drift. - -$$ -y_t=\widehat F_{0,t}(x),\quad r_t=\dot y_t-b_t(y_t),\quad e_t=y_t-x_t -$$ - -The learned path y has residual r equal to its time derivative minus the desired drift at y, and state error e equal to y minus the exact path. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u033_r1) - -A drift Lipschitz constant L controls error growth. - -$$ -\frac{d}{dt}\|e_t\|\leq L\|e_t\|+\|r_t\|,\qquad e_0=0 -$$ - -The growth rate of the error norm is at most the Lipschitz constant L times the error norm plus the residual norm, starting from zero error. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u033_r2) - -Grönwall bounds the terminal discrepancy. - -$$ -\|e_1\|\leq\int_0^1 e^{L(1-\tau)}\|r_\tau\|\,d\tau -$$ - -The endpoint error is at most the integral of each residual norm multiplied by its remaining-time amplification factor, e raised to L times one minus τ. - -## Coupling both samplers turns this into a distributional bound - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u034_r0) - -Use the same initial noise for both paths. - -$$ -W_2^2(\widehat p_1,p_1)\leq\mathbb E\|\widehat F_{0,1}(X_0)-F_{0,1}(X_0)\|^2 -$$ - -The squared Wasserstein distance between terminal laws is at most the expected squared endpoint difference when both maps use the same initial random sample. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u034_r1) - -Apply Cauchy–Schwarz to the residual integral. - -$$ -\mathbb E\|e_1\|^2\leq\left(\int_0^1e^{2L(1-\tau)}d\tau\right)\mathbb E\int_0^1\|r_\tau\|^2d\tau -$$ - -The mean squared endpoint error is bounded by the integral of the squared exponential amplification times the expected time-integrated squared residual. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u034_r2) - -Evaluate the deterministic factor for L > 0. - -$$ -W_2^2(\widehat p_1,p_1)\leq\frac{e^{2L}-1}{2L}\,\mathbb E\int_0^1\|r_\tau\|^2d\tau -$$ - -For positive L, the squared Wasserstein error is bounded by the expected integrated squared residual multiplied by e to the two L minus one, divided by two L. - -## We can test the training logic on a flow with a known answer - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u037_r0) - -Train on the local ODE and check the full-interval destination. - -$$ -\dot x=x,\qquad F_\theta(0,1,x)=x+v_\theta(0,1,x)\approx e x -$$ - -For the ODE whose velocity equals x, the learned full-interval map adds its predicted average velocity to x and should approximate e times x. - -## A consistency model always points to the same endpoint - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u042_r0) - -Fix the destination time at one. - -$$ -C_t(x)=F_{t,1}(x),\qquad C_1(x)=x -$$ - -The consistency function C at time t is the flow map from t to the fixed endpoint one; at time one, it returns its input. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u042_r1) - -Points on the same trajectory have the same final prediction. - -$$ -C_s(x)=C_t(F_{s,t}(x)) -$$ - -The final prediction made at s equals the final prediction made at t after transporting the state from s to t. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u042_r2) - -Differentiating along the trajectory gives a transport equation. - -$$ -\partial_t C_t(x)+J_xC_t(x)b_t(x)=0 -$$ - -The time derivative of C plus its spatial Jacobian applied to the current velocity is zero, expressing constancy along the trajectory. - -## Two steps of length h define an average for length 2h - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u044_r0) - -Take a first short step from x. - -$$ -x_{t+h}=x+h\,a_\theta(x,t,h) -$$ - -The first short step adds h times the model’s average velocity for the interval starting at x and t. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u044_r1) - -Take the second short step from the updated state. - -$$ -x_{t+2h}=x_{t+h}+h\,a_\theta(x_{t+h},t+h,h) -$$ - -The second short step adds h times a new average velocity evaluated at the updated state and the later starting time t plus h. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u044_r2) - -Divide the combined displacement by 2h. - -$$ -a_{2h}^{\rm target}=\tfrac12\left[a_\theta(x,t,h)+a_\theta(x_{t+h},t+h,h)\right] -$$ - -The target average velocity for the interval of length two h is the arithmetic mean of the two short-interval average velocities. - -## For the exponential flow, the short-step averages are different - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u046_r0) - -Use exact half-interval averages starting from one. - -$$ -a_1=\frac{e^{1/2}-1}{1/2}=1.29744 -$$ - -The first half-interval average is e to the one half minus one, divided by one half, approximately 1.29744. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u046_r1) - -The second interval starts at e to the one half. - -$$ -a_2=\frac{e-e^{1/2}}{1/2}=2.13912 -$$ - -The second half-interval average is e minus e to the one half, divided by one half, approximately 2.13912. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u046_r2) - -Their mean gives the full-interval average. - -$$ -\tfrac12(a_1+a_2)=1.71828=e-1 -$$ - -Half the sum of these two averages is 1.71828, which equals the full-interval displacement e minus one. - -## Write the average along the interval from r to t - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u049_r0) - -Average the instantaneous field along a trajectory. - -$$ -u(z_t,r,t)=\frac{1}{t-r}\int_r^t v(z_\tau,\tau)\,d\tau -$$ - -The MeanFlow average u is the integral of instantaneous velocity along the trajectory from r to t, divided by t minus r. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u049_r1) - -The integral is the forward displacement. - -$$ -(t-r)u(z_t,r,t)=z_t-z_r -$$ - -The interval length times u equals the later state z at t minus the earlier state z at r. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u049_r2) - -Generate toward the earlier data time. - -$$ -z_r=z_t-(t-r)u(z_t,r,t) -$$ - -To recover the earlier state, subtract the interval length times the predicted average velocity from the later state. - -## Differentiating the interval identity gives the MeanFlow target - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u050_r0) - -Differentiate while the later state follows its trajectory. - -$$ -\frac{d}{dt}[(t-r)u(z_t,r,t)]=v(z_t,t) -$$ - -The total derivative of interval length times u, while the later state follows the trajectory, equals the instantaneous velocity at that later state. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u050_r1) - -Use the product and chain rules. - -$$ -u+(t-r)(\partial_tu+J_zu\,v)=v -$$ - -The product and chain rules give u plus the interval length times the sum of its partial time derivative and spatial Jacobian applied to v. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u050_r2) - -Isolate the average velocity. - -$$ -u=v-(t-r)(\partial_tu+J_zu\,v) -$$ - -The average velocity equals the instantaneous velocity minus the interval length times that total derivative correction. - -## Encoding defines the target distribution for the latent map - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u055_r0) - -Encode data into a latent representation. - -$$ -Y\sim p_{\rm data},\qquad Z_1=E(Y),\qquad p_1^Z=E_\#p_{\rm data} -$$ - -A data sample Y is encoded into the latent Z at time one, and the latent target law is the pushforward of the data law through the encoder E. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u055_r1) - -Learn a flow map from latent noise to encoded data. - -$$ -Z_0\sim p_0^Z,\qquad Z_1^{\rm gen}=F^Z_{0,1}(Z_0) -$$ - -A latent source sample is drawn from its noise law and transformed by the latent flow map into a generated endpoint latent. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u055_r2) - -Decode the generated latent. - -$$ -Y^{\rm gen}=D(Z_1^{\rm gen}),\qquad p_{\rm gen}=D_\#(F^Z_{0,1})_\#p_0^Z -$$ - -Decoding the generated latent produces Y; its law is obtained by pushing the noise law through the latent map and then through the decoder D. - -## Decoder sensitivity affects the visible generation error - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u057_r0) - -Assume the decoder is L-sub-D Lipschitz. - -$$ -\|D(z)-D(z')\|\leq L_D\|z-z'\| -$$ - -The distance between two decoded outputs is at most the decoder’s Lipschitz constant times the distance between their latent inputs. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u057_r1) - -Push a coupling of the latent distributions through the decoder. - -$$ -W_2(D_\#\widehat p^Z,D_\#p_1^Z)\leq L_DW_2(\widehat p^Z,p_1^Z) -$$ - -The Wasserstein distance between decoded latent laws is at most that same constant times the Wasserstein distance between the latent laws. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u057_r2) - -Include the encoder–decoder reconstruction discrepancy. - -$$ -W_2(p_{\rm gen},p_{\rm data})\leq L_DW_2(\widehat p^Z,p_1^Z)+W_2((D\circ E)_\#p_{\rm data},p_{\rm data}) -$$ - -The data-space error is bounded by amplified latent distribution error plus the Wasserstein discrepancy introduced by encoding and then decoding real data. - -## A simple decoder shows why latent error can be amplified - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u058_r0) - -Choose a one-dimensional decoder. - -$$ -D(z)=3z+2,\qquad L_D=3 -$$ - -The decoder multiplies z by three and adds two, so its Lipschitz constant is three. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u058_r1) - -Compare a target latent with a perturbed prediction. - -$$ -z^*=1,\quad\widehat z=1.1,\quad|\widehat z-z^*|=0.1 -$$ - -The target latent is one and the predicted latent is 1.1, giving an absolute latent error of 0.1. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u058_r2) - -The visible error is three times as large. - -$$ -D(z^*)=5,\quad D(\widehat z)=5.3,\quad|5.3-5|=0.3 -$$ - -The two decoded values are five and 5.3, so the absolute output error is 0.3. - -## Two equal source atoms cannot produce every target law - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u061_r0) - -Start from two atoms of equal probability. - -$$ -p_0(a)=p_0(b)=\tfrac12 -$$ - -The source distribution assigns probability one half to each of the two atoms a and b. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u061_r1) - -A deterministic map assigns each atom to one destination. - -$$ -p_1(A)=\tfrac12\mathbf1\{F(a)=A\}+\tfrac12\mathbf1\{F(b)=A\} -$$ - -The probability of destination A is half the indicator that a maps to A plus half the indicator that b maps to A. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u061_r2) - -Only three destination probabilities are possible. - -$$ -p_1(A)\in\{0,\tfrac12,1\},\qquad p_1(A)=0.3\ \text{is impossible} -$$ - -A deterministic map can therefore assign A only probability zero, one half, or one; probability 0.3 is unavailable from these two source atoms. - -## Represent each token as a vertex, then add continuous noise - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u063_r0) - -Embed a length-L sequence using V-dimensional one-hot vectors. - -$$ -X_1\in\{e_1,\ldots,e_V\}^{L}\subset\mathbb R^{L\times V} -$$ - -The clean sequence consists of L one-hot vectors chosen from V vocabulary vertices, embedded in a continuous space with L times V coordinates. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u063_r1) - -Use Gaussian noise in the same ambient coordinates. - -$$ -X_0\sim\mathcal N(0,I),\qquad I_t=(1-t)X_0+tX_1 -$$ - -The source is standard Gaussian noise, and the noisy sequence interpolates between that noise and the clean one-hot sequence with weights one minus t and t. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u063_r2) - -The denoiser predicts token probabilities from the entire noisy sequence. - -$$ -D_t(x)=\mathbb E[X_1\mid I_t=x]\in(\Delta^{V-1})^L -$$ - -The denoiser is the conditional mean clean sequence given the noisy state, returning one categorical probability vector per position. - -## The posterior mean gives the marginal velocity - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u065_r0) - -Solve the linear interpolant for its noise endpoint. - -$$ -X_0=\frac{x-tX_1}{1-t}\quad\text{when }I_t=x -$$ - -When the interpolant equals x, its noise endpoint is x minus t times the clean endpoint, divided by one minus t. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u065_r1) - -Substitute into the endpoint displacement. - -$$ -X_1-X_0=\frac{X_1-x}{1-t} -$$ - -Substituting that expression makes the clean-minus-noise displacement equal to clean endpoint minus x, divided by one minus t. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u065_r2) - -Average over the conditional clean endpoint. - -$$ -b_t(x)=\frac{D_t(x)-x}{1-t},\qquad t<1 -$$ - -The marginal velocity is the denoiser’s conditional mean minus the current state, divided by the remaining time one minus t. - -## Cross-entropy learns that posterior mean one token at a time - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u066_r0) - -Train each position against its observed clean token. - -$$ -\mathcal L_{\rm CE}=\mathbb E\left[-\sum_{\ell=1}^{L}\log D_\theta^{\ell,Y_\ell}(t,I_t)\right] -$$ - -The cross-entropy loss averages the negative sum, over sequence positions, of the log probability assigned to each observed clean token. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u066_r1) - -At a fixed noisy state, write its true token marginal as q. - -$$ -\mathbb E[-\log D^Y\mid I_t=x]=H(q)+D_{\rm KL}(q\|D) -$$ - -At a fixed noisy state, expected negative log probability equals the true token distribution’s entropy plus its KL divergence to the predicted distribution. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u066_r2) - -The minimum occurs at the conditional categorical distribution. - -$$ -D^{\ell,k}_*(t,x)=\Pr(Y_\ell=k\mid I_t=x) -$$ - -The minimizing probability for category k at position ℓ is the conditional probability that the clean token there is k, given the noisy sequence. - -## Independent draws can destroy a simple two-word relationship - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u068_r0) - -Suppose only two phrases occur, each with equal probability. - -$$ -p(\text{New York})=p(\text{San Diego})=\tfrac12 -$$ - -The joint distribution assigns probability one half to New York and one half to San Diego. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u068_r1) - -At complete uncertainty, each position has two equally likely words. - -$$ -p_1(\text{New})=p_1(\text{San})=p_2(\text{York})=p_2(\text{Diego})=\tfrac12 -$$ - -The first-position words New and San and the second-position words York and Diego each have marginal probability one half. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u068_r2) - -Independent token draws also produce two invalid combinations. - -$$ -p(\text{New Diego})+p(\text{San York})=\tfrac14+\tfrac14=\tfrac12 -$$ - -Independent draws assign one quarter to New Diego and one quarter to San York, so the total probability of an invalid phrase is one half. - -## The finite map can also be written using an endpoint prediction - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u069_r0) - -Define a two-time mean denoiser from the interval average. - -$$ -\psi_{s,t}(x)=x+(1-s)v_{s,t}(x) -$$ - -The two-time mean denoiser ψ equals the current state plus the remaining source-time interval, one minus s, times the average velocity. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u069_r1) - -Substitute into the residual map. - -$$ -F_{s,t}(x)=x+\frac{t-s}{1-s}(\psi_{s,t}(x)-x) -$$ - -The map adds ψ minus x to x, scaled by the elapsed interval divided by the time remaining at the start. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u069_r2) - -Separate the two coefficients. - -$$ -F_{s,t}(x)=\frac{1-t}{1-s}x+\frac{t-s}{1-s}\psi_{s,t}(x) -$$ - -The destination is a weighted sum of x and ψ, with weights one minus t over one minus s and t minus s over one minus s. - -## Why does the two-time prediction stay on the simplex? - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u070_r0) - -Use the ODE and an integrating factor. - -$$ -\frac{d}{du}\left[\frac{F_{s,u}(x)}{1-u}\right]=\frac{D_u(F_{s,u}(x))}{(1-u)^2} -$$ - -The derivative of the current flow state divided by its remaining time equals the denoiser at that state divided by remaining time squared. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u070_r1) - -Integrate from s to t, with t < 1. - -$$ -F_{s,t}(x)=\frac{1-t}{1-s}x+(1-t)\int_s^t\frac{D_u(F_{s,u}(x))}{(1-u)^2}\,du -$$ - -Integrating gives a scaled starting state plus one minus t times the integral of denoiser values weighted by inverse remaining time squared. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u070_r2) - -Compare with the mean-denoiser parameterization. - -$$ -\psi_{s,t}(x)=\int_s^t\frac{(1-s)(1-t)}{(t-s)(1-u)^2}\,D_u(F_{s,u}(x))\,du -$$ - -The two-time denoiser is the integral of along-path denoisers weighted by the displayed factor involving s, t, and the intermediate time u. - -## The weights integrate to one - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u071_r0) - -Write the nonnegative weighting function. - -$$ -w(u)=\frac{(1-s)(1-t)}{(t-s)(1-u)^2}\geq0 -$$ - -The weight at intermediate time u is the product of one minus s and one minus t, divided by t minus s and one minus u squared; it is nonnegative. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u071_r1) - -Integrate it explicitly. - -$$ -\int_s^tw(u)du=\frac{(1-s)(1-t)}{t-s}\left(\frac1{1-t}-\frac1{1-s}\right) -$$ - -The integral of the weight equals its constant prefactor times the difference between the inverse remaining times at t and s. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u071_r2) - -Cancel the interval factors. - -$$ -\int_s^tw(u)du=1\quad\Longrightarrow\quad\psi_{s,t}(x)\in\Delta^{V-1} -$$ - -The weights integrate to one, so averaging categorical probability vectors with them keeps ψ inside the probability simplex. - -## We can calculate one step in these coordinates - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u072_r0) - -Choose an ambient state and a simplex-valued prediction. - -$$ -s=0.25,\ t=0.75,\quad x=(-0.2,0.6,1.1),\quad\psi=(0.1,0.7,0.2) -$$ - -The interval runs from 0.25 to 0.75; the ambient state is minus 0.2, 0.6, 1.1, and the denoiser predicts probabilities 0.1, 0.7, 0.2. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u072_r1) - -Compute the interpolation weight. - -$$ -\frac{t-s}{1-s}=\frac{0.5}{0.75}=\frac23 -$$ - -The elapsed interval 0.5 divided by the remaining starting interval 0.75 gives interpolation weight two thirds. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u072_r2) - -Apply the finite map. - -$$ -F_{s,t}(x)=\tfrac13x+\tfrac23\psi=(0,\,0.66667,\,0.5) -$$ - -The destination is one third of the ambient state plus two thirds of the denoiser, giving coordinates zero, 0.66667, and 0.5. - -## Composition induces a weighted consistency rule for denoisers - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u073_r0) - -Compose the two affine map expressions. - -$$ -F_{s,t}(x)=F_{u,t}(F_{s,u}(x)) -$$ - -The map from s to t equals the map from s to u followed by the map from u to t. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u073_r1) - -Match their coefficients after subtracting the shared x term. - -$$ -\psi_{s,t}(x)=\gamma\psi_{s,u}(x)+(1-\gamma)\psi_{u,t}(F_{s,u}(x)) -$$ - -The direct denoiser is γ times the first-interval denoiser plus one minus γ times the second-interval denoiser evaluated at the intermediate destination. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u073_r2) - -The coefficient depends on all three times. - -$$ -\gamma=\frac{(1-t)(u-s)}{(1-u)(t-s)},\qquad0\leq\gamma\leq1 -$$ - -The weight γ is one minus t times u minus s, divided by one minus u times t minus s, and lies between zero and one. - -## The same two shorter predictions receive unequal weights - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u074_r0) - -Choose s = 0, u = 0.5, and t = 0.75. - -$$ -\gamma=\frac{(0.25)(0.5)}{(0.5)(0.75)}=\frac13 -$$ - -For these three times, γ is 0.25 times 0.5 divided by 0.5 times 0.75, giving one third. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u074_r1) - -Suppose the two predictions are different probability vectors. - -$$ -\psi_{s,u}=(0.8,0.2),\qquad\psi_{u,t}=(0.2,0.8) -$$ - -The first short-interval denoiser predicts 0.8 and 0.2, while the second predicts 0.2 and 0.8. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u074_r2) - -Their consistency target is still normalized. - -$$ -\psi_{s,t}^{\rm target}=\tfrac13(0.8,0.2)+\tfrac23(0.2,0.8)=(0.4,0.6) -$$ - -The direct target takes one third of the first prediction and two thirds of the second, yielding the normalized vector 0.4, 0.6. - -## A KL loss matches this probability-valued target - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u075_r0) - -Detach the composed denoiser target. - -$$ -q=\operatorname{sg}[\gamma\psi_\theta(s,u,x)+(1-\gamma)\psi_\theta(u,t,F_\theta(s,u,x))] -$$ - -The target q is the γ-weighted mixture of the two composed denoiser predictions, enclosed in stop-gradient to freeze the target during the update. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u075_r1) - -Match it with the direct prediction at each position. - -$$ -\mathcal L_{\rm SG}=\mathbb E\sum_{\ell=1}^L D_{\rm KL}(q^\ell\|\psi_\theta^\ell(s,t,x)) -$$ - -The consistency loss averages the sum, over positions, of KL divergence from the detached target q to the direct two-time denoiser prediction. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u075_r2) - -Retain the diagonal cross-entropy anchor. - -$$ -\mathcal L=\mathcal L_{\rm CE,diag}+\lambda\mathcal L_{\rm SG} -$$ - -The total loss adds diagonal cross-entropy to λ times the semigroup consistency loss. - -## The decoding error defines a more evenly spaced clock - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u077_r0) - -Measure the probability of decoding the wrong token. - -$$ -P_e(t)=\Pr[\arg\max I_t\ne\arg\max X_1] -$$ - -The decoding error probability is the chance that the largest coordinate of the noisy interpolant identifies a different token from the clean endpoint. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u077_r1) - -Normalize between random guessing and perfect recovery. - -$$ -\tau(t)=1-\frac{V}{V-1}P_e(t) -$$ - -The new clock τ is one minus the error probability multiplied by V over V minus one, normalizing random guessing and perfect recovery. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u077_r2) - -Sample a value in the new clock and invert the schedule. - -$$ -q\sim\mathcal U(0,1),\qquad t=\tau^{-1}(q) -$$ - -Draw q uniformly between zero and one, then use the inverse clock to convert q into the original interpolation time t. - -## Let’s implement the probability-preserving map update - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u079_r0) - -The code should move the state using a valid categorical prediction. - -$$ -F_{s,t}(x)=(1-h)x+h\psi_{s,t}(x),\qquad h=\frac{t-s}{1-s},\quad\psi_{s,t}\in\Delta^{V-1} -$$ - -The map mixes the current state with a simplex-valued denoiser using weight h equal to the elapsed interval divided by the remaining starting time. - -## Their map has the same affine structure - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u085_r0) - -Use the paper’s endpoint prediction π. - -$$ -F_{s,t}(x)=x+\eta_{s,t}(\pi_{s,t}(x)-x),\qquad\eta_{s,t}=\frac{t-s}{1-s} -$$ - -The categorical flow map adds the endpoint prediction π minus the current state, scaled by η, the elapsed interval divided by the remaining starting time. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u085_r1) - -The diagonal prediction supplies the velocity. - -$$ -b_t(x)=\frac{\pi_{t,t}(x)-x}{1-t} -$$ - -The diagonal endpoint prediction minus x, divided by one minus t, gives the instantaneous velocity. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u085_r2) - -Multiply the Lagrangian identity by the remaining time. - -$$ -(1-t)\partial_tF_{s,t}(x)=\pi_{t,t}(F_{s,t}(x))-F_{s,t}(x) -$$ - -The remaining time times the map’s end-time derivative equals the diagonal endpoint prediction at the destination minus that destination. - -## Differentiate the map to isolate the two sources of residual - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u086_r0) - -The time derivative acts on both the coefficient and the prediction. - -$$ -\partial_tF_{s,t}=\frac{\pi_{s,t}-x}{1-s}+\eta_{s,t}\partial_t\pi_{s,t} -$$ - -The end-time derivative is π minus x divided by one minus s, plus η times the end-time derivative of π. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u086_r1) - -Define the scaled Lagrangian residual. - -$$ -r=(1-t)\partial_tF_{s,t}-\pi_{t,t}(F_{s,t})+F_{s,t} -$$ - -The residual r is one minus t times the map’s derivative, minus the diagonal endpoint prediction at the destination, plus the destination itself. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u086_r2) - -Collect terms. - -$$ -r=\pi_{s,t}-\pi_{t,t}(F_{s,t})+(1-t)\eta_{s,t}\partial_t\pi_{s,t} -$$ - -Collecting terms leaves the difference between the two endpoint predictions plus one minus t times η times the derivative of the finite-interval prediction. - -## KL divergence can bound the endpoint mismatch - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u087_r0) - -Divide by remaining time and use the squared-sum inequality. - -$$ -\left\|\frac r{1-t}\right\|^2\leq\frac{2\|\pi_{s,t}-\pi_{t,t}(F)\|^2}{(1-t)^2}+2\eta_{s,t}^2\|\partial_t\pi_{s,t}\|^2 -$$ - -The squared residual divided by remaining time squared is at most twice the scaled squared endpoint mismatch plus twice η squared times the squared prediction derivative. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u087_r1) - -Pinsker bounds squared Euclidean discrepancy by KL. - -$$ -\|p-q\|_2^2\leq\|p-q\|_1^2\leq2D_{\rm KL}(p\|q) -$$ - -The squared Euclidean distance between probability vectors is at most their squared L1 distance, which Pinsker bounds by twice their KL divergence. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u087_r2) - -Apply it to the endpoint term. - -$$ -\mathbb E\left\|\frac r{1-t}\right\|^2\leq4\mathcal L_{\rm EC}+2\mathcal L_{\rm TD} -$$ - -The expected squared scaled residual is therefore at most four times the endpoint-consistency loss plus twice the time-derivative loss. - -## The two penalties measure different failures - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u088_r0) - -Suppose the current endpoint predictions agree. - -$$ -\pi_{s,t}=\pi_{t,t}(F_{s,t})=(0.4,0.6) -$$ - -The finite-interval endpoint prediction and the diagonal prediction at its destination both equal the probability vector 0.4, 0.6. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u088_r1) - -Allow the finite-interval prediction to vary with t. - -$$ -t=0.5,\ \eta=0.5,\quad\partial_t\pi_{s,t}=(0.2,-0.2) -$$ - -At time 0.5 with interpolation weight 0.5, the endpoint prediction changes with derivative 0.2, minus 0.2. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u088_r2) - -A nonzero residual remains. - -$$ -r=(0.5)(0.5)(0.2,-0.2)=(0.05,-0.05),\qquad\|r\|^2=0.005 -$$ - -The derivative correction produces residual 0.05, minus 0.05, whose squared norm is 0.005 despite agreement between the endpoint predictions. - -## The Lagrangian identity becomes a relation between denoisers - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u094_r0) - -Use the same affine map and mean denoiser ψ. - -$$ -F_{s,t}=\frac{1-t}{1-s}x+\frac{t-s}{1-s}\psi_{s,t} -$$ - -The destination combines x and the mean denoiser ψ with weights one minus t over one minus s and t minus s over one minus s. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u094_r1) - -Substitute into the Lagrangian equation and collect terms. - -$$ -\psi_{s,t}+c_{s,t}\partial_t\psi_{s,t}=\psi_{t,t}(F_{s,t}) -$$ - -The finite-interval denoiser plus c times its end-time derivative equals the diagonal denoiser evaluated at the predicted destination. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u094_r2) - -The derivative coefficient is determined by the times. - -$$ -c_{s,t}=\frac{(t-s)(1-t)}{1-s} -$$ - -The derivative coefficient c is the elapsed interval times the remaining end-time interval, divided by the remaining starting interval. - -## A raw derivative correction can leave the simplex - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u095_r0) - -A tentative teacher subtracts a derivative from a probability vector. - -$$ -q_{\rm raw}=\psi_{t,t}(F)-c_{s,t}\partial_t\psi_{s,t} -$$ - -The raw teacher subtracts c times the finite-interval denoiser’s end-time derivative from the diagonal denoiser at the destination. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u095_r1) - -Consider a two-category numerical example. - -$$ -\psi_{t,t}(F)=(0.1,0.9),\quad c=0.2,\quad\partial_t\psi=(1,-1) -$$ - -The diagonal prediction is 0.1, 0.9, the coefficient is 0.2, and the derivative is one, minus one. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u095_r2) - -The first coordinate becomes negative. - -$$ -q_{\rm raw}=(-0.1,1.1) -$$ - -The corrected vector is minus 0.1, 1.1, so its first coordinate is negative even though the coordinates still sum to one. - -## The softmax derivative gives a multiplicative correction - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u096_r0) - -Write the prediction using logits z. - -$$ -\psi_k=\frac{e^{z_k}}{\sum_j e^{z_j}} -$$ - -The probability of category k is the exponential of its logit divided by the sum of exponentiated logits over all categories. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u096_r1) - -Differentiate and subtract the probability-weighted logit derivative. - -$$ -\partial_t\psi_k=\psi_k\left(\partial_tz_k-\sum_j\psi_j\partial_tz_j\right)=\psi_k\delta_k -$$ - -The derivative of probability k equals that probability times its logit derivative minus the probability-weighted average logit derivative; this centered difference is δ. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u096_r2) - -The Lagrangian identity becomes coordinate-wise multiplication. - -$$ -\psi_k(1+c\delta_k)=\psi_{t,t,k}(F) -$$ - -For each category, the finite-interval probability multiplied by one plus c times δ equals the diagonal probability at the destination. - -## Taking logs produces a probability-valued Lagrangian teacher - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u097_r0) - -Assume each correction denominator is positive. - -$$ -a_k=1+c\delta_k>0 -$$ - -For each category k, the correction denominator a is one plus c times δ and must be strictly positive. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u097_r1) - -Solve the coordinate relation and normalize. - -$$ -\psi_k\propto\frac{\psi_{t,t,k}(F)}{a_k} -$$ - -The finite-interval probability is proportional to the diagonal destination probability divided by its correction denominator, followed by normalization. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u097_r2) - -Express the normalized result using softmax. - -$$ -T^{\rm LSD}=\operatorname{softmax}\!\left(z_{t,t}(F)-\log(1+c\delta)\right) -$$ - -The Lagrangian teacher applies softmax to the diagonal destination logits minus the coordinate-wise logarithm of one plus c times δ. - -## The Eulerian teacher uses a derivative along the starting flow - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u098_r0) - -Differentiate logits with respect to the moving start. - -$$ -D_sz=\partial_sz+J_xz\,b_s,\qquad\delta=D_sz-\langle\psi,D_sz\rangle\mathbf1 -$$ - -The start-flow derivative of the logits adds their partial start-time derivative to their spatial Jacobian times b; subtracting its probability-weighted mean gives δ. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u098_r1) - -The probability identity determines its coefficient. - -$$ -D_s\psi=\kappa(\psi-\psi_{s,s}),\qquad\kappa=\frac{1-t}{(1-s)(t-s)} -$$ - -The start-flow derivative of ψ equals κ times the difference between the finite-interval prediction and its starting diagonal prediction, with κ given by the time ratio. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u098_r2) - -Solve in logits on the positive-denominator domain. - -$$ -T^{\rm ESD}=\operatorname{softmax}\!\left(z_{s,s}-\log(\mathbf1-\kappa^{-1}\delta)\right) -$$ - -The Eulerian teacher is softmax of the starting diagonal logits minus the logarithm of one minus δ divided by κ, where every denominator is positive. - -## A common scalar can be removed before taking softmax - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u099_r0) - -Write the Eulerian denominator without an explicit inverse ratio. - -$$ -1-\kappa^{-1}\delta=\frac{(1-t)\mathbf1-(1-s)(t-s)\delta}{1-t} -$$ - -One minus δ divided by κ equals one minus t times the all-ones vector, minus one minus s times t minus s times δ, all divided by one minus t. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u099_r1) - -The denominator contributes the same logit shift to every category. - -$$ -\operatorname{softmax}(z+c\mathbf1)=\operatorname{softmax}(z) -$$ - -Adding the same scalar c to every logit leaves its softmax probabilities unchanged. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u099_r2) - -Use the equivalent target for t < 1. - -$$ -T^{\rm ESD}=\operatorname{softmax}\!\left(z_{s,s}-\log[(1-t)\mathbf1-(1-s)(t-s)\delta]\right) -$$ - -The equivalent Eulerian teacher subtracts the logarithm of the unscaled positive correction vector from the starting diagonal logits before applying softmax. - -## The KL gradient has a simple interpretation in logit space - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u100_r0) - -Let p be the detached teacher and q the student softmax. - -$$ -\ell=D_{\rm KL}(p\|q),\qquad q=\operatorname{softmax}(z) -$$ - -The loss is KL divergence from the detached teacher p to the student q, where q is softmax of the trainable logits z. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u100_r1) - -Differentiate the student cross-entropy term. - -$$ -\frac{\partial\ell}{\partial z_k}=q_k-p_k -$$ - -The derivative with respect to logit k is the student probability q at k minus the teacher probability p at k. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u100_r2) - -A detached adaptive weight rescales this mismatch. - -$$ -w=\operatorname{sg}[(\|q-p\|^2+c_0)^{-r}],\qquad\nabla_z(w\ell)=w(q-p) -$$ - -The detached weight is a negative power of squared probability mismatch plus c zero; the weighted logit gradient is that fixed weight times q minus p. - -## The desired distribution reweights data by its reward - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u104_r0) - -Let r assign a scalar reward to a clean sample z. - -$$ -p_1^r(z)=\frac{p_1(z)e^{r(z)}}{Z},\qquad Z=\mathbb E_{p_1}[e^{r(Z_1)}] -$$ - -The tilted endpoint law multiplies the original probability by exponential reward and divides by Z, the original-law average of exponential reward. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u104_r1) - -At a noisy state, average the exponentiated future reward. - -$$ -h_t(x)=\mathbb E[e^{r(Z_1)}\mid X_t=x],\qquad V_t(x)=\log h_t(x) -$$ - -The function h at x and t averages exponentiated endpoint reward conditional on the current state x; the value V is the logarithm of h. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u104_r2) - -The intermediate tilted marginal follows from Bayes’ rule. - -$$ -p_t^r(x)=\frac{p_t(x)h_t(x)}{Z} -$$ - -The tilted intermediate density is the original intermediate density multiplied by its conditional reward average h and divided by the same normalizer Z. - -## A small posterior makes the exponential weighting explicit - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u105_r0) - -Suppose two clean endpoints are equally likely. - -$$ -p(z_A\mid x)=p(z_B\mid x)=\tfrac12,\qquad r_A=0,\quad r_B=\log4 -$$ - -The two clean endpoints each have conditional probability one half, with rewards zero and log four. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u105_r1) - -Average exponentiated reward before taking the logarithm. - -$$ -h=\tfrac12(1)+\tfrac12(4)=2.5,\qquad V=\log2.5\approx0.91629 -$$ - -The conditional exponential-reward average is half of one plus half of four, giving 2.5; its logarithm is approximately 0.91629. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u105_r2) - -The reward-tilted posterior probabilities are different. - -$$ -p^r(z_A\mid x)=\frac{0.5}{2.5}=0.2,\qquad p^r(z_B\mid x)=\frac2{2.5}=0.8 -$$ - -Dividing the reward-weighted masses 0.5 and two by 2.5 gives tilted posterior probabilities 0.2 and 0.8. - -## Even the denoiser mean can hide the available outcomes - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u106_r0) - -Let the clean posterior place mass at minus one and plus one. - -$$ -\Pr(Z_1=-1\mid x)=\Pr(Z_1=1\mid x)=\tfrac12 -$$ - -The clean posterior places probability one half at minus one and probability one half at plus one. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u106_r1) - -Choose a reward that favors magnitude. - -$$ -D_t(x)=\mathbb E[Z_1\mid x]=0,\qquad r(z)=z^2 -$$ - -The denoiser’s conditional mean is zero, while the reward of a candidate z is its square. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u106_r2) - -Compare mean-based lookahead with the true value. - -$$ -r(D_t(x))=0,\qquad\log\mathbb E[e^{Z_1^2}\mid x]=1 -$$ - -Reward at the denoiser mean is zero, but the logarithm of the posterior average of exponentiated squared endpoint values is one. - -## Two clocks distinguish generation from posterior sampling - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u109_r0) - -Fix the outer noisy observation and its time. - -$$ -(x,t)\quad\text{is held fixed} -$$ - -The outer noisy state x and its outer time t are fixed conditioning information throughout this inner sampling calculation. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u109_r1) - -An inner flow acts on a separate state and inner times a, b. - -$$ -G_{a,b}(\bar x\mid x,t) -$$ - -The conditional map G moves a separate inner state from inner time a to inner time b while conditioning on the fixed outer observation x and t. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u109_r2) - -Fresh inner noise gives a clean conditional sample. - -$$ -\epsilon\sim\mathcal N(0,I),\qquad G_{0,1}(\epsilon\mid x,t)\sim p_{1\mid t}(\cdot\mid x) -$$ - -Drawing fresh Gaussian inner noise and applying G from zero to one produces a sample from the clean endpoint distribution conditional on the outer observation. - -## Several posterior samples estimate the value function - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u110_r0) - -Draw K independent inner noise samples. - -$$ -z_k=G_{0,1}(\epsilon_k\mid x,t),\qquad\epsilon_k\overset{\rm iid}{\sim}\mathcal N(0,I) -$$ - -Each candidate z is the conditional map applied to an independent standard Gaussian inner-noise sample, using the same outer state and time. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u110_r1) - -Estimate the logarithm of the conditional reward average. - -$$ -\widehat V_t(x)=\log\left(\frac1K\sum_{k=1}^K e^{r(z_k)}\right) -$$ - -The estimated value is the logarithm of the arithmetic average of the K exponentiated candidate rewards. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u110_r2) - -Differentiate through the posterior sampler. - -$$ -\nabla_x\widehat V_t=\sum_k\operatorname{softmax}(r(z_{1:K}))_k\,\nabla_x r(z_k) -$$ - -The value gradient is a weighted sum of candidate reward gradients, with weights given by softmax of their rewards and derivatives passing through the conditional sampler. - -## The value gradient changes the transport velocity - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u111_r0) - -Write the noisy state using signal and noise schedules. - -$$ -X_t=\alpha_tZ_1+\sigma_t\epsilon -$$ - -The noisy state equals the clean endpoint multiplied by signal scale α plus standard Gaussian noise multiplied by noise scale σ. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u111_r1) - -Define the schedule-dependent guidance coefficient. - -$$ -c_t=\sigma_t^2\frac{\dot\alpha_t}{\alpha_t}-\dot\sigma_t\sigma_t -$$ - -The guidance coefficient is noise variance times the signal’s relative time derivative, minus noise scale times its own time derivative. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u111_r2) - -The reward-tilted probability-flow velocity adds the value gradient. - -$$ -b_t^r(x)=b_t(x)+c_t\nabla_xV_t(x) -$$ - -The reward-tilted probability-flow velocity equals the original velocity plus that schedule coefficient times the spatial gradient of the value function. - -## Two noisy observations add their Gaussian precisions - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u113_r0) - -Condition on outer x and inner x-bar with independent noises. - -$$ -x=\alpha_tz+\sigma_t\epsilon,\qquad\bar x=\alpha_az+\sigma_a\bar\epsilon -$$ - -The outer and inner observations are separately scaled versions of the same clean endpoint z, each corrupted by its own independent Gaussian noise. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u113_r1) - -Collect the quadratic coefficient and the linear coefficient in z. - -$$ -\Lambda=\frac{\alpha_t^2}{\sigma_t^2}+\frac{\alpha_a^2}{\sigma_a^2},\qquad m=\Lambda^{-1}\left(\frac{\alpha_tx}{\sigma_t^2}+\frac{\alpha_a\bar x}{\sigma_a^2}\right) -$$ - -The precision Λ sums the two squared signal-to-noise ratios; m is the sum of their signal-weighted, inverse-variance observations divided by Λ. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u113_r2) - -Complete the square in the product likelihood. - -$$ -p(x,\bar x\mid z)\propto\exp\!\left[-\tfrac12\Lambda\|z-m\|^2\right] -$$ - -The joint observation likelihood, as a function of z, is proportional to the exponential of minus one half Λ times the squared distance from z to m. - -## An effective noise level turns this into a denoiser query - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u114_r0) - -Choose an effective time with matching noise-to-signal ratio. - -$$ -\frac{\sigma_{t_*}^2}{\alpha_{t_*}^2}=\Lambda^{-1} -$$ - -The effective time is chosen so that its noise variance divided by squared signal scale equals the inverse combined precision. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u114_r1) - -Rescale the effective observation into the original path coordinates. - -$$ -\widetilde x=\alpha_{t_*}m -$$ - -The effective observation is the combined likelihood center m multiplied by the signal scale at that effective time. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u114_r2) - -The original denoiser supplies the joint posterior mean. - -$$ -\mathbb E[Z_1\mid x,\bar x]=D_{t_*}(\widetilde x) -$$ - -The conditional mean clean endpoint given both observations equals the original denoiser evaluated at the effective time and effective observation. - -## That posterior mean defines the inner conditional velocity - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u115_r0) - -Differentiate the inner interpolant while holding outer x and t fixed. - -$$ -\dot{\bar X}_a=\dot\alpha_a Z_1+\dot\sigma_a\bar\epsilon -$$ - -The inner interpolant’s time derivative is the signal-scale derivative times the clean endpoint plus the noise-scale derivative times the fixed inner noise. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u115_r1) - -Eliminate the inner noise using its observed state. - -$$ -\bar\epsilon=(\bar x-\alpha_a Z_1)/\sigma_a -$$ - -The inner noise is the observed inner state minus its signal-scaled clean endpoint, divided by the inner noise scale. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u115_r2) - -Average over the posterior conditioned on both observations. - -$$ -\bar b_a(\bar x\mid x,t)=\frac{\dot\sigma_a}{\sigma_a}\bar x+\left(\dot\alpha_a-\frac{\dot\sigma_a\alpha_a}{\sigma_a}\right)D_{t_*}(\widetilde x) -$$ - -The inner conditional velocity combines the inner state times the relative noise-scale derivative with the joint posterior mean times the remaining signal derivative coefficient. - -## The posterior map follows its own composition rule - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u116_r0) - -Keep the same outer observation throughout the inner trajectory. - -$$ -G_{a,c}(\bar x\mid x,t)=G_{b,c}(G_{a,b}(\bar x\mid x,t)\mid x,t) -$$ - -The conditional map from a to c equals the map from a to b followed by b to c, using the same outer observation in both maps. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u116_r1) - -Anchor the diagonal to the conditional velocity. - -$$ -\partial_bG_{a,b}(\bar x\mid x,t)\big|_{b=a}=\bar b_a(\bar x\mid x,t) -$$ - -The end-time derivative on the inner time diagonal equals the conditional inner velocity at the starting inner state. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u116_r2) - -Distill the inner Lagrangian identity. - -$$ -\mathcal L=\mathbb E\|\partial_bG_{a,b}-\bar b_b(G_{a,b}\mid x,t)\|^2 -$$ - -The loss averages the squared difference between the inner map’s end-time derivative and the conditional velocity at its predicted endpoint. - -## Re-noising defines a proposal from an earlier outer time - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u118_r0) - -Choose an earlier, noisier time q < t. - -$$ -x_q=kx+\sqrt{\sigma_q^2-k^2\sigma_t^2}\,\epsilon,\qquad k=\alpha_q/\alpha_t -$$ - -The earlier noisy proposal scales x by the signal ratio k and adds Gaussian noise with variance equal to σ at q squared minus k squared times σ at t squared. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u118_r1) - -Map each perturbed state to a clean endpoint. - -$$ -z=F_{q,1}(x_q) -$$ - -The proposed earlier state is mapped from time q to a clean endpoint using the outer flow map. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u118_r2) - -The endpoint law depends on the re-noising proposal. - -$$ -z\sim Q(\cdot\mid x),\qquad Q(\cdot\mid x)\ \text{need not equal}\ p_{1\mid t}(\cdot\mid x) -$$ - -That endpoint has the proposal law Q conditional on x, which need not be the true clean posterior conditional on x at time t. - -## A recovery term enforces compatibility with the observation - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u119_r0) - -Include the likelihood of the current noisy observation. - -$$ -r_{\rm rec}(x,z)=r(z)-\frac{\|x-\alpha_tz\|^2}{2\sigma_t^2} -$$ - -The recovery reward is endpoint reward minus the squared observation mismatch divided by twice the observation-noise variance. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u119_r1) - -Use the score to recover the change in the earlier marginal density. - -$$ -\Gamma=(x_q-x)^\top\int_0^1\nabla\log p_q(x+u(x_q-x))\,du -$$ - -The density correction Γ is the proposal displacement dotted with the integral of the earlier-time score along the straight segment connecting the two states. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u119_r2) - -The exact-design log weight includes the proposal correction. - -$$ -\ell=r_{\rm rec}+\Gamma+\tfrac12\|\epsilon\|^2 -$$ - -The corrected log weight adds recovery reward, the density correction Γ, and one half the squared norm of the Gaussian proposal noise. - -## The gradient correction also depends on the noisy-state scores - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u120_r0) - -Define the additional score difference. - -$$ -\Delta s=\frac{\alpha_q}{\alpha_t}\nabla\log p_q(x_q)-\nabla\log p_t(x) -$$ - -The score correction is the earlier-time score at the proposal multiplied by the signal-scale ratio, minus the current-time score at x. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u120_r1) - -Normalize the corrected log weights. - -$$ -w_k=\operatorname{softmax}(\ell_{1:K})_k -$$ - -The candidate weights are the softmax of their corrected log weights, so they are positive and sum to one. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u120_r2) - -Combine recovery gradients with the score correction. - -$$ -\widehat{\nabla V_t}(x)=\sum_kw_k\left[\nabla_xr_{\rm rec}(x,z_k)+\Delta s_k\right] -$$ - -The estimated value gradient averages each candidate’s recovery-reward gradient plus its score correction using those normalized weights. - -## Both observations share the same clean endpoint - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u126_r0) - -Draw a clean example and two independent noise samples. - -$$ -Z_1\sim p_1,\qquad Z_0,\bar Z_0\overset{\rm iid}{\sim}p_0 -$$ - -Draw one clean endpoint from the data law and two independent source samples from the noise law. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u126_r1) - -Construct the outer observation. - -$$ -I_t=\alpha_tZ_0+\beta_tZ_1 -$$ - -The outer interpolant combines its noise sample with the clean endpoint using the outer noise coefficient α and signal coefficient β. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u126_r2) - -Construct the inner interpolant using the same clean example. - -$$ -\bar I_s=\alpha_s\bar Z_0+\beta_sZ_1 -$$ - -The inner interpolant combines its separate noise sample with the same clean endpoint using the inner-time coefficients α and β. - -## Conditional flow matching now needs no solved trajectories - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u127_r0) - -Differentiate the inner interpolant. - -$$ -\dot{\bar I}_s=\dot\alpha_s\bar Z_0+\dot\beta_sZ_1 -$$ - -The inner interpolant’s derivative is the derivative of α times the inner noise sample plus the derivative of β times the shared clean endpoint. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u127_r1) - -Train its diagonal velocity conditioned on the outer observation. - -$$ -\mathcal L_{\rm diag}=\mathbb E\|v_\theta(s,s,\bar I_s;t,I_t)-\dot{\bar I}_s\|^2 -$$ - -The diagonal loss averages squared error between the conditional equal-time velocity prediction and that directly computed inner interpolant derivative. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u127_r2) - -The conditional mean is the minimizing velocity. - -$$ -\bar b_s(\bar x;t,x)=\mathbb E[\dot{\bar I}_s\mid\bar I_s=\bar x,I_t=x] -$$ - -The minimizing conditional velocity is the mean inner interpolant derivative given both the inner state and the outer observation. - -## Finite inner maps reuse the same consistency identities - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u128_r0) - -Parameterize the inner destination with an average velocity. - -$$ -G_{s,u}(\bar x;t,x)=\bar x+(u-s)v_\theta(s,u,\bar x;t,x) -$$ - -The inner map adds the inner interval length times its learned average velocity to the inner state while conditioning on outer time t and state x. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u128_r1) - -Compose inner intervals while keeping the outer context fixed. - -$$ -G_{s,v}(\bar x;t,x)=G_{u,v}(G_{s,u}(\bar x;t,x);t,x) -$$ - -The inner map from s to v equals the map from s to u followed by u to v, with the outer conditioning unchanged. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u128_r2) - -At the final inner time, generate a conditional clean endpoint. - -$$ -G_{0,1}(\bar Z_0;t,x)\sim p_{1\mid t}(\cdot\mid x) -$$ - -Applying the inner map from zero to one to a fresh source sample produces the clean endpoint law conditional on the outer observation. - -## The unconditional generator is included at the noise endpoint - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u130_r0) - -At t = 0, the outer observation is independent source noise. - -$$ -I_0=Z_0,\qquad Z_0\perp Z_1 -$$ - -At outer time zero, the observation is the source noise Z zero, which is independent of the clean endpoint Z one. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u130_r1) - -Conditioning on it does not change the clean distribution. - -$$ -p_{1\mid0}(z\mid x_0)=p_1(z) -$$ - -The clean endpoint law conditioned on a time-zero observation is therefore just the unconditional clean data law. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u130_r2) - -The conditional family therefore contains an unconditional flow. - -$$ -G_{0,1}(\bar Z_0;0,x_0)\sim p_1 -$$ - -Running the inner map with outer time zero consequently produces an unconditional data sample, even though the model accepts an outer observation. - -## Reward weighting gives us another guidance estimator - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u131_r0) - -Define the tilted posterior mean. - -$$ -D_t^r(x)=\frac{\mathbb E[Z_1e^{r(Z_1)}\mid I_t=x]}{\mathbb E[e^{r(Z_1)}\mid I_t=x]} -$$ - -The reward-tilted denoiser is the conditional average of clean endpoint times exponential reward divided by the conditional average of exponential reward. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u131_r1) - -Differentiate the Gaussian likelihood of the outer observation. - -$$ -\nabla_xV_t(x)=\frac{\beta_t}{\alpha_t^2}\left(D_t^r(x)-D_t(x)\right) -$$ - -The value gradient equals the signal coefficient β divided by noise coefficient α squared, multiplied by tilted posterior mean minus ordinary posterior mean. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u131_r2) - -Insert the tilted mean directly into the velocity. - -$$ -b_t^r(x)=\frac{\dot\alpha_t}{\alpha_t}x+\left(\dot\beta_t-\frac{\dot\alpha_t\beta_t}{\alpha_t}\right)D_t^r(x) -$$ - -The tilted velocity combines x times the relative noise derivative with the tilted denoiser times the signal derivative corrected for that relative noise change. - -## Our two-outcome example gives that tilted mean immediately - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u132_r0) - -Use clean endpoints minus one and plus one with equal posterior mass. - -$$ -z_A=-1,\quad z_B=1,\quad e^{r_A}=1,\quad e^{r_B}=4 -$$ - -The two endpoint values are minus one and plus one, with exponentiated reward weights one and four. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u132_r1) - -The untilted mean is zero and the tilted mean favors plus one. - -$$ -D=0,\qquad D^r=0.2(-1)+0.8(1)=0.6 -$$ - -The ordinary mean is zero, while the tilted mean weights minus one by 0.2 and plus one by 0.8, giving 0.6. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u132_r2) - -Choose α = β = 0.5 to calculate the value gradient. - -$$ -\nabla_xV=\frac{0.5}{0.5^2}(0.6-0)=1.2 -$$ - -With signal and noise coefficients both 0.5, the value gradient is 0.5 divided by 0.5 squared times the mean difference 0.6, giving 1.2. - -## Weighted samples estimate the tilted mean - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u133_r0) - -Draw clean candidates from the conditional map. - -$$ -z_k=G_{0,1}(\epsilon_k;t,x) -$$ - -Each clean candidate z is generated by applying the inner conditional map to an independent noise input at the fixed outer context. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u133_r1) - -Normalize the exponentiated rewards. - -$$ -w_k=\frac{e^{r(z_k)}}{\sum_je^{r(z_j)}} -$$ - -The weight of candidate k is its exponentiated reward divided by the sum of all candidates’ exponentiated rewards. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u133_r2) - -Average the candidates using these weights. - -$$ -\widehat D_t^r(x)=\sum_kw_kz_k -$$ - -The estimated tilted denoiser is the sum of candidate endpoints multiplied by these normalized reward weights. - -## Adding diffusion requires a drift correction - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u135_r0) - -Start from a probability-flow velocity b and density p. - -$$ -\partial_tp=-\nabla\cdot(bp) -$$ - -The density’s time derivative is minus the divergence of density times probability-flow velocity b. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u135_r1) - -Add scalar diffusion g and a score correction to the drift. - -$$ -dX_t=\left[b_t(X_t)+\tfrac12g_t^2\nabla\log p_t(X_t)\right]dt+g_t\,dW_t -$$ - -The SDE adds Brownian noise with scale g and uses drift b plus one half g squared times the score of the current density. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u135_r2) - -The score term cancels the new diffusion term in Fokker–Planck. - -$$ --\nabla\cdot\left(\tfrac12g_t^2p_t\nabla\log p_t\right)+\tfrac12g_t^2\Delta p_t=0 -$$ - -The negative divergence of the added score-weighted probability flux exactly cancels one half g squared times the density Laplacian. - -## The Doob transform determines stochastic guidance - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u136_r0) - -A reward tilt adds the value gradient to the SDE drift. - -$$ -f_t^r=f_t+g_t^2\nabla V_t -$$ - -The Doob-transformed SDE drift equals its original drift plus diffusion variance g squared times the gradient of the value function. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u136_r1) - -The tilted score also changes. - -$$ -\nabla\log p_t^r=\nabla\log p_t+\nabla V_t -$$ - -The tilted density’s score equals the original score plus the value gradient. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u136_r2) - -Subtract half the diffusion-weighted score to obtain probability flow. - -$$ -b_t^r=b_t+\tfrac12g_t^2\nabla V_t -$$ - -Subtracting half the diffusion-weighted tilted score from the tilted SDE drift leaves probability-flow velocity b plus one half g squared times the value gradient. - -## Fine-tuning can avoid a normalized Monte Carlo ratio - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u137_r0) - -Write the desired change in velocity as d and the reward weight as w. - -$$ -d=b_{\rm student}-b_{\rm teacher},\qquad w=e^{r(G_{0,1}(\epsilon;t,x))},\qquad a=\tfrac12g_t^2 -$$ - -The drift difference d is student minus teacher; w is exponentiated reward of a conditional sample, and a is one half the diffusion variance. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u137_r1) - -The desired drift relation can be written without dividing by E[w]. - -$$ -\mathbb E[w]d=a\,\mathbb E[\nabla_xw] -$$ - -The desired relation multiplies d by the expected reward weight and sets it equal to a times the expected spatial gradient of that weight. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u137_r2) - -Equivalently, its estimating equation has zero mean. - -$$ -\mathbb E[w\,d-a\nabla_xw]=0 -$$ - -Equivalently, the expected residual w times d minus a times the weight gradient is zero. - -## The detached surrogate has exactly that expected gradient - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u138_r0) - -Treat the posterior sampler and teacher as frozen. - -$$ -R=d+(w-1)\operatorname{sg}(d)-a\nabla_xw -$$ - -The surrogate residual is d plus w minus one times a detached copy of d, minus a times the spatial gradient of w. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u138_r1) - -Use the squared surrogate residual. - -$$ -\mathcal L_{\rm FT}=\mathbb E\|R\|^2 -$$ - -The fine-tuning loss is the expected squared norm of that surrogate residual R. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u138_r2) - -Differentiate with respect to the trainable drift difference. - -$$ -\nabla_d\mathcal L_{\rm FT}=2\mathbb E[w\,d-a\nabla_xw] -$$ - -Differentiating with respect to the trainable drift difference gives twice the expected weighted drift residual, because the extra copy of d is detached. - -## Factor the finite move into expansion and transport - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u146_r0) - -At time s, the active state has dimension d(s). - -$$ -x_s\in\mathbb R^{d(s)},\qquad d(t)\geq d(s) -$$ - -The active state at time s lies in a space of dimension d at s, and the later dimension is at least as large. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u146_r1) - -Inject conditional noise into the additional coordinates. - -$$ -E_{s,t}(x_s,\epsilon)\in\mathbb R^{d(t)} -$$ - -The expansion map takes the current state and a fresh noise input and returns a state in the later, possibly larger-dimensional space. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u146_r2) - -Transport the augmented state to the later time. - -$$ -\Phi_{s,t}(x_s,\epsilon)=X_{s,t}(E_{s,t}(x_s,\epsilon)) -$$ - -The full finite move first applies expansion E and then applies transport X to the augmented state. - -## A two-coordinate example makes the change of space explicit - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u147_r0) - -Begin with a single observed coordinate. - -$$ -x_s=(2)\in\mathbb R -$$ - -The starting state contains the single coordinate two and lies on the real line. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u147_r1) - -Append one noise coordinate. - -$$ -\epsilon=-0.4,\qquad E_{s,t}(x_s,\epsilon)=(2,-0.4)\in\mathbb R^2 -$$ - -Appending the noise coordinate minus 0.4 gives the two-dimensional augmented state two, minus 0.4. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u147_r2) - -Apply a transport in the enlarged space. - -$$ -X_{s,t}(z_1,z_2)=(z_1+0.5,z_2+0.8)\quad\Longrightarrow\quad\Phi=(2.5,0.4) -$$ - -The transport adds 0.5 to the first coordinate and 0.8 to the second, producing the destination 2.5, 0.4. - -## Each coordinate needs a clock that begins at its birth - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u148_r0) - -Let coordinate i enter the state at global time b-i. - -$$ -b_i\in[0,1),\qquad i\text{ is inactive when }ts)=\frac{a(t)-a(s)}{1-a(s)} -$$ - -Given that insertion has not occurred by s, its probability of occurring by t is the cumulative probability increase divided by the remaining probability at s. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u161_r2) - -Denote the fraction of remaining positions inserted over the interval. - -$$ -\rho_{s,t}=\frac{a(t)-a(s)}{1-a(s)} -$$ - -The interval insertion fraction ρ is a at t minus a at s, divided by one minus a at s. - -## The expected gap count scales with that insertion probability - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u162_r0) - -Let g-i be the number of clean positions still missing from gap i. - -$$ -m_i(s,x_s)=\mathbb E[g_i(s)\mid X_s=x_s] -$$ - -The predicted remaining gap count m is the conditional mean of the clean positions still missing from that gap, given the current state. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u162_r1) - -Each missing position is inserted with probability rho. - -$$ -I_{s,t}(x_s)[i]=\rho_{s,t}\,m_i(s,x_s) -$$ - -The expected inserted count in gap i is the interval insertion probability ρ multiplied by that gap’s predicted remaining count. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u162_r2) - -In the infinitesimal limit, recover the birth hazard. - -$$ -\lambda_t=\frac{\dot a(t)}{1-a(t)} -$$ - -The instantaneous birth hazard is the schedule derivative divided by the probability that insertion has not yet occurred. - -## A concrete gap example gives two expected insertions - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u163_r0) - -Use a linear insertion schedule and three expected missing positions. - -$$ -a(t)=t,\qquad s=0.25,\quad t=0.75,\quad m_i=3 -$$ - -The cumulative insertion schedule equals time, the interval runs from 0.25 to 0.75, and the predicted remaining gap count is three. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u163_r1) - -Calculate the conditional insertion probability. - -$$ -\rho_{s,t}=\frac{0.75-0.25}{1-0.25}=\frac23 -$$ - -The conditional insertion probability is 0.75 minus 0.25 divided by one minus 0.25, giving two thirds. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u163_r2) - -Multiply by the expected remaining count. - -$$ -I_{s,t}[i]=\frac23(3)=2 -$$ - -Multiplying insertion probability two thirds by remaining count three gives an expected inserted count of two. - -## The count loss learns a real-valued insertion mean - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u164_r0) - -Use the nonnegative count divergence for target a and predicted mean b. - -$$ -\phi(a,b)=b-a+a\log(a/b),\qquad b>0 -$$ - -The count divergence is predicted mean b minus target count a plus a times the logarithm of a divided by b, with b strictly positive. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u164_r1) - -Differentiate the conditional expected loss. - -$$ -\partial_b\mathbb E[\phi(A,b)\mid x]=1-\frac{\mathbb E[A\mid x]}b -$$ - -The derivative of its conditional expected loss is one minus the conditional expected target count divided by the predicted mean. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u164_r2) - -Set the derivative to zero. - -$$ -b^*(x)=\mathbb E[A\mid x] -$$ - -The minimizing prediction b equals the conditional mean of the target count given x. - -## The sampler turns that mean into a bounded count - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u165_r0) - -Let B be the remaining token budget and μ the predicted gap mean. - -$$ -0\leq\mu\leq B,\qquad p=\mu/B -$$ - -The predicted mean μ lies between zero and the remaining budget B, and the binomial success probability is μ divided by B. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u165_r1) - -Use the paper’s capped binomial count proposal. - -$$ -L_i\sim\operatorname{Binomial}(B,p),\qquad\mathbb E[L_i]=\mu -$$ - -The proposed inserted count is binomial with B trials and success probability p, so its expectation equals μ. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u165_r2) - -For B = 4 and μ = 2, the variance is one. - -$$ -p=\tfrac12,\qquad\operatorname{Var}(L_i)=4(\tfrac12)(\tfrac12)=1 -$$ - -With budget four and mean two, the success probability is one half and the count variance is four times one half times one half, giving one. - -## Each token uses its own local interpolation weight - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u166_r0) - -For coordinate i, start from its current local time tau-s. - -$$ -\eta_i=\frac{\tau_i(t)-\tau_i(s)}{1-\tau_i(s)} -$$ - -Coordinate i uses interpolation weight equal to its local-time advance divided by its remaining local time at the start. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u166_r1) - -Apply the categorical mean-denoiser update coordinate-wise. - -$$ -F^i=(1-\eta_i)z^i+\eta_i\psi^i -$$ - -The coordinate destination is one minus η times its current value plus η times its categorical denoiser prediction. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u166_r2) - -A newly inserted token starts from local time zero. - -$$ -\tau_i(s)=0\quad\Longrightarrow\quad F^i=(1-\tau_i(t))z^i+\tau_i(t)\psi^i -$$ - -For a newly inserted coordinate with starting local time zero, η equals its destination local time, giving the displayed noise-to-denoiser interpolation. - -## Let’s first implement one insertion interval - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u167_r0) - -Expand the state and preserve the matching coordinate clocks. - -$$ -(x_s,\{b_i\})\xrightarrow{E_{s,t}}(z,\{b_i^{\rm expanded}\}),\qquad\tau_i(t)=\frac{t-b_i}{1-b_i} -$$ - -The expansion carries both the state and its coordinate birth times into the larger space; each coordinate’s local time is then computed from its own birth time. - -## The SDE solution depends on an entire driving path - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u176_r0) - -Consider the additive-noise Itô SDE. - -$$ -dX_t=f(t,X_t)\,dt+g(t)\,dW_t -$$ - -The state increment consists of drift f at the current time and state times dt, plus time-dependent noise scale g times a Brownian increment. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u176_r1) - -Write its integral form from a starting state x. - -$$ -X_t=x+\int_s^t f(u,X_u)\,du+\int_s^tg(u)\,dW_u -$$ - -The endpoint equals the starting state plus the ordinary integral of drift along the path and the Itô integral of noise scale against Brownian motion. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u176_r2) - -The Itô map takes the initial state and that noise realization. - -$$ -\Psi_{s,t}(x,W_{[s,t]})=X_t -$$ - -The Itô map Ψ takes the starting state and the entire Brownian path segment from s to t and returns the resulting SDE endpoint. - -## Strong and weak error ask different questions - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u177_r0) - -Strong error compares solutions driven by the same Brownian path. - -$$ -\mathbb E\|\widehat\Psi(x,W)-\Psi(x,W)\|^2 -$$ - -The strong squared error averages the squared distance between learned and exact endpoints driven by the same starting state and Brownian path. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u177_r1) - -Weak error compares expectations of test functions. - -$$ -\left|\mathbb E[\varphi(\widehat X_t)]-\mathbb E[\varphi(X_t)]\right| -$$ - -The weak error is the absolute difference between the expected test-function values under the learned and exact terminal distributions. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u177_r2) - -Equal marginal laws can coexist with a poor pathwise coupling. - -$$ -\widehat X_t\overset d=X_t\quad\not\Longrightarrow\quad\widehat X_t=X_t\ \text{for the same }W -$$ - -Equality in distribution of the two endpoints does not imply that they agree when both are evaluated using the same Brownian realization. - -## A constant-coefficient SDE gives an exact first example - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u178_r0) - -Choose constant drift and diffusion. - -$$ -dX_t=a\,dt+\sigma\,dW_t -$$ - -The SDE has constant drift a and constant noise scale σ multiplying Brownian motion. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u178_r1) - -Integrate both terms over the interval. - -$$ -\Psi_{s,t}(x,W)=x+a(t-s)+\sigma(W_t-W_s) -$$ - -The exact endpoint is x plus a times the elapsed interval plus σ times the Brownian increment from s to t. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u178_r2) - -Only the total Brownian increment is needed in this special case. - -$$ -\Delta W_{s,t}=W_t-W_s\sim\mathcal N(0,t-s) -$$ - -The total Brownian increment is the difference W at t minus W at s, normally distributed with mean zero and variance t minus s. - -## Two half steps agree when their noise increments add - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u179_r0) - -Choose x = 1, a = 0.5, sigma = 0.8, and two half intervals. - -$$ -\Delta W_L=0.2,\qquad\Delta W_R=-0.1,\qquad\Delta W=0.1 -$$ - -The left and right Brownian increments are 0.2 and minus 0.1, so the whole-interval increment is their sum, 0.1. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u179_r1) - -The direct move uses the summed increment. - -$$ -X_1=1+0.5(1)+0.8(0.1)=1.58 -$$ - -The direct endpoint adds drift 0.5 and noise contribution 0.8 times 0.1 to the starting value one, giving 1.58. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u179_r2) - -The split move reaches the same endpoint. - -$$ -X_{1/2}=1+0.25+0.16=1.41,\qquad X_1=1.41+0.25-0.08=1.58 -$$ - -The first half reaches 1.41; the second adds drift 0.25 and noise minus 0.08, reaching the same endpoint 1.58. - -## Independent coarse noise breaks that agreement - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u180_r0) - -Keep the fine increments from the preceding example. - -$$ -\Delta W_L+\Delta W_R=0.1 -$$ - -The two fine-interval Brownian increments sum to 0.1, fixing the whole-interval increment for that same path. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u180_r1) - -Independently draw a coarse increment equal to minus 0.3. - -$$ -\widetilde{\Delta W}=-0.3 -$$ - -The independently drawn coarse increment is minus 0.3, which differs from the fine increments’ sum. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u180_r2) - -The coarse endpoint is now different. - -$$ -\widetilde X_1=1+0.5+0.8(-0.3)=1.26\ne1.58 -$$ - -Using that independent increment gives endpoint 1.26, differing from the coupled fine-path endpoint 1.58. - -## A changing drift can require more path information - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u181_r0) - -Consider the Ornstein–Uhlenbeck SDE. - -$$ -dX_t=-\lambda X_t\,dt+\sigma\,dW_t -$$ - -The Ornstein–Uhlenbeck drift is minus λ times the current state, with constant noise scale σ multiplying Brownian motion. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u181_r1) - -An integrating factor gives the exact solution. - -$$ -X_t=e^{-\lambda(t-s)}x+\sigma\int_s^t e^{-\lambda(t-u)}\,dW_u -$$ - -The exact solution exponentially damps the starting state and adds a Brownian integral whose later increments receive larger exponential weights. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u181_r2) - -The marginal variance has a closed form. - -$$ -\operatorname{Var}(X_t\mid X_s=x)=\frac{\sigma^2}{2\lambda}(1-e^{-2\lambda(t-s)}) -$$ - -The conditional variance is σ squared divided by two λ, multiplied by one minus e raised to minus two λ times the interval length. - -## The weighted-noise variance follows from Itô isometry - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u182_r0) - -Let J be the stochastic integral in the OU solution. - -$$ -\mathbb E[J^2]=\sigma^2\int_s^t e^{-2\lambda(t-u)}\,du -$$ - -The second moment of the noise contribution J equals σ squared times the integral of the squared exponential weighting function. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u182_r1) - -Evaluate the ordinary integral. - -$$ -\sigma^2\int_0^{t-s}e^{-2\lambda r}dr=\frac{\sigma^2}{2\lambda}(1-e^{-2\lambda(t-s)}) -$$ - -Evaluating this integral gives σ squared over two λ times one minus the exponential of minus two λ times the elapsed interval. - -## Shifted Legendre polynomials define the path coefficients - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u184_r0) - -Normalize time within an interval of length h. - -$$ -h=t-s,\qquad q=(u-s)/h\in[0,1] -$$ - -The interval length h is t minus s, and the normalized inner time q is u minus s divided by h, lying between zero and one. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u184_r1) - -The first two shifted Legendre polynomials are simple. - -$$ -\widetilde P_0(q)=1,\qquad\widetilde P_1(q)=2q-1 -$$ - -The zeroth shifted Legendre polynomial is one, and the first is two q minus one. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u184_r2) - -Project Brownian increments onto each polynomial. - -$$ -I^{(n)}_{s,t}=\int_s^t\widetilde P_n\!\left(\frac{u-s}{h}\right)dW_u -$$ - -Coefficient n is the Itô integral of the nth shifted Legendre polynomial, evaluated at normalized interval time, against Brownian increments. - -## Orthogonality gives independent Gaussian coefficients - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u185_r0) - -Apply Itô isometry to a pair of deterministic integrands. - -$$ -\mathbb E[I^{(n)}I^{(m)}]=h\int_0^1\widetilde P_n(q)\widetilde P_m(q)dq -$$ - -The expected product of coefficients n and m equals h times the integral, from zero to one, of the product of their shifted Legendre polynomials. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u185_r1) - -Use the orthogonality of shifted Legendre polynomials. - -$$ -\mathbb E[I^{(n)}I^{(m)}]=\frac{h}{2n+1}\,\mathbf1\{n=m\} -$$ - -Orthogonality makes that covariance zero for different indices and h divided by two n plus one when the indices agree. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u185_r2) - -Joint Gaussianity turns zero covariance into independence. - -$$ -I^{(n)}_{s,t}\overset{\rm independent}{\sim}\mathcal N\!\left(0,\frac{h}{2n+1}\right) -$$ - -The coefficients are independent zero-mean Gaussians, with the nth coefficient having variance h divided by two n plus one. - -## Integrating the basis reconstructs a polynomial noise path - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u186_r0) - -Keep N coefficients on the interval. - -$$ -W^{(N)}_{u,v}=\sum_{n=0}^{N-1}\frac{2n+1}{h}I^{(n)}_{s,t}\int_u^v\widetilde P_n\!\left(\frac{r-s}{h}\right)dr -$$ - -The reconstructed noise increment sums each retained coefficient times its integrated polynomial basis, scaled by two n plus one divided by h. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u186_r1) - -One coefficient produces a straight interpolation of the increment. - -$$ -W^{(1)}_{s,s+qh}=qI^{(0)} -$$ - -With only the zeroth coefficient, the reconstructed path at fractional time q is q times the total Brownian increment. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u186_r2) - -Two coefficients add an interior curvature term. - -$$ -W^{(2)}_{s,s+qh}=qI^{(0)}+3I^{(1)}(q^2-q) -$$ - -With two coefficients, the path adds three times the first-order coefficient times q squared minus q to that straight interpolation. - -## The next coefficient changes the interior of the path - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u187_r0) - -Choose a unit interval and two coefficients. - -$$ -I^{(0)}=0.1,\qquad I^{(1)}=0.2 -$$ - -On the unit interval, choose total increment 0.1 and first-order path coefficient 0.2. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u187_r1) - -Evaluate the two-coefficient path at its midpoint. - -$$ -W^{(2)}_{0,1/2}=0.5(0.1)+3(0.2)(0.25-0.5)=-0.1 -$$ - -At the midpoint, the linear term is 0.05 and the curvature term is minus 0.15, so the reconstructed path value is minus 0.1. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u187_r2) - -At the endpoint, the correction vanishes. - -$$ -W^{(2)}_{0,1}=0.1+3(0.2)(1-1)=0.1 -$$ - -At the endpoint q equals one, the curvature factor is zero, so the path still ends at the total increment 0.1. - -## The zeroth coefficient combines by adding increments - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u189_r0) - -Split the interval at u. - -$$ -[s,t]=[s,u]\cup[u,t] -$$ - -The full interval from s to t is split into the left interval from s to u and the right interval from u to t. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u189_r1) - -Split the stochastic integral of the constant basis function. - -$$ -I^{(0)}_{s,t}=\int_s^u dW+\int_u^t dW -$$ - -The zeroth coefficient is the sum of the Brownian integrals over the left and right intervals. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u189_r2) - -The coarse coefficient is the sum of the fine coefficients. - -$$ -I^{(0)}_{s,t}=I^{(0)}_{s,u}+I^{(0)}_{u,t} -$$ - -The whole-interval zeroth coefficient equals the left zeroth coefficient plus the right zeroth coefficient. - -## The first-order coefficient also has an exact composition rule - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u190_r0) - -Let the left and right lengths be h-L and h-R. - -$$ -h_L=u-s,\qquad h_R=t-u,\qquad h=h_L+h_R -$$ - -The left length is u minus s, the right length is t minus u, and the total length is their sum. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u190_r1) - -Restrict the global linear basis to each local interval. - -$$ -\widetilde P_1^{\rm global}=\frac{h_L}{h}\widetilde P_1^L-\frac{h_R}{h}\ \text{on L},\qquad\frac{h_R}{h}\widetilde P_1^R+\frac{h_L}{h}\ \text{on R} -$$ - -On each half, the global linear polynomial is its local polynomial scaled by that half’s length fraction, with the displayed constant offset. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u190_r2) - -Integrate those expressions against the same Brownian path. - -$$ -I^{(1)}_{s,t}=\frac{h_L}{h}I_L^{(1)}+\frac{h_R}{h}I_R^{(1)}-\frac{h_R}{h}I_L^{(0)}+\frac{h_L}{h}I_R^{(0)} -$$ - -The coarse first-order coefficient combines length-weighted fine first-order coefficients, subtracts the right fraction of the left increment, and adds the left fraction of the right increment. - -## We can calculate the coefficients for two equal halves - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u191_r0) - -Use the previously chosen fine increments and two first-order terms. - -$$ -I_L^{(0)}=0.2,\quad I_R^{(0)}=-0.1,\quad I_L^{(1)}=0.04,\quad I_R^{(1)}=-0.02 -$$ - -The fine zeroth coefficients are 0.2 and minus 0.1, while their first-order coefficients are 0.04 and minus 0.02. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u191_r1) - -The coarse increment remains their sum. - -$$ -I^{(0)}=0.2-0.1=0.1 -$$ - -Adding the zeroth coefficients gives the coarse total increment 0.1. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u191_r2) - -Combine the first-order coefficients with the equal-half rule. - -$$ -I^{(1)}=\tfrac12(0.04-0.02-0.2-0.1)=-0.14 -$$ - -The equal-half formula takes half of 0.04 minus 0.02 minus 0.2 minus 0.1, giving the coarse first-order coefficient minus 0.14. - -## The composition rule also preserves the correct variance - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u192_r0) - -On half intervals, increment variance is one half and first-order variance is one sixth. - -$$ -\operatorname{Var}(I_L^{(0)})=\operatorname{Var}(I_R^{(0)})=\tfrac12,\quad\operatorname{Var}(I_L^{(1)})=\operatorname{Var}(I_R^{(1)})=\tfrac16 -$$ - -Each half-interval zeroth coefficient has variance one half, and each first-order coefficient has variance one sixth. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u192_r1) - -The four coefficients are independent. - -$$ -\operatorname{Var}(I^{(1)})=\tfrac14\left(\tfrac16+\tfrac16+\tfrac12+\tfrac12\right) -$$ - -Because the four terms are independent and have coefficients of magnitude one half, the coarse variance is one quarter of the sum of their variances. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u192_r2) - -This equals the unit-interval first-order variance. - -$$ -\operatorname{Var}(I^{(1)})=\tfrac13 -$$ - -The resulting coarse first-order variance is one third, matching the first-order coefficient on a unit interval. - -## The map learns interval drift and diffusion terms - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u193_r0) - -Use the paper’s Euler–Maruyama-like parameterization. - -$$ -\Psi_{s,t}(x,W)=x+(t-s)f_{s,t}(x,W)+g_{s,t}(W)(W_t-W_s) -$$ - -The learned stochastic map adds interval length times its interval drift and the Brownian increment multiplied by its interval diffusion coefficient to x. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u193_r1) - -On the diagonal, the coefficients recover the SDE. - -$$ -f_{t,t}(x)=f(t,x),\qquad g_{t,t}=g(t) -$$ - -On the time diagonal, the map’s drift and diffusion coefficients equal the instantaneous coefficients of the target SDE. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u193_r2) - -In practice, pass retained path coefficients to the networks. - -$$ -W\longrightarrow(I^{(0)},\ldots,I^{(N-1)}) -$$ - -In the implementation, the noise path is represented by its retained coefficients from index zero through N minus one. - -## Pathwise composition uses the same noise realization - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u194_r0) - -Predict the whole interval directly. - -$$ -y=\Psi_{s,t}(x,W_{[s,t]}) -$$ - -The direct destination y is the stochastic map over the full interval using that interval’s Brownian path. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u194_r1) - -Predict the first part, then continue with the remaining noise. - -$$ -\widetilde y=\Psi_{u,t}(\Psi_{s,u}(x,W_{[s,u]}),W_{[u,t]}) -$$ - -The split destination first uses the left Brownian segment, then continues from that result using the right segment. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u194_r2) - -The exact solution satisfies the stochastic composition identity. - -$$ -\Psi_{s,t}(x,W)=\Psi_{u,t}(\Psi_{s,u}(x,W_L),W_R) -$$ - -The exact stochastic map equals this composition when the two segments belong to the same whole-interval Brownian realization. - -## These conditions identify the strong solution - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u195_r0) - -Compose over a partition and separate the diagonal terms. - -$$ -\widehat X_t-x=\sum_i f(t_i,\widehat X_{t_i})h_i+\sum_i g(t_i)\Delta W_i+R_f+R_g -$$ - -The composed endpoint displacement is the sum of diagonal drift times interval lengths and diagonal diffusion times Brownian increments, plus drift and diffusion remainders. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u195_r1) - -The time-Lipschitz bounds force both remainders to vanish. - -$$ -|R_f|\leq L_f(t-s)|\pi|,\qquad |R_g|\leq L_g(t-s)\max_i|\Delta W_i| -$$ - -The drift remainder is bounded by a constant times total duration and maximum step size; the diffusion remainder is bounded by a constant times duration and the largest Brownian increment. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u195_r2) - -The remaining sums converge to the defining SDE integrals. - -$$ -\widehat X_t=x+\int_s^t f(u,\widehat X_u)\,du+\int_s^t g(u)\,dW_u -$$ - -As the partition is refined, the endpoint equals x plus the drift integral and the Itô noise integral, recovering the defining SDE. - -## Training anchors the SDE and enforces composition - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u196_r0) - -Match the instantaneous SDE coefficients on the diagonal. - -$$ -\mathcal L_{f,g}=\mathbb E[\|\widehat f_{t,t}(X_t)-f(t,X_t)\|^2+\|\widehat g_{t,t}-g(t)\|^2] -$$ - -The diagonal loss averages the sum of squared errors in the learned instantaneous drift and diffusion coefficients against their target SDE values. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u196_r1) - -Use coupled noise representations in the off-diagonal loss. - -$$ -\mathcal L_D=\mathbb E\|\widehat\Psi_{s,t}(x,W)-\widehat\Psi_{u,t}(\widehat\Psi_{s,u}(x,W_L),W_R)\|^2 -$$ - -The composition loss averages the squared difference between the direct stochastic map and its two-interval composition, using coupled representations of the same noise. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u196_r2) - -Combine both requirements. - -$$ -\mathcal L_{\rm SSFM}=\mathcal L_{f,g}+\mathcal L_D -$$ - -The strong stochastic flow-map loss is the sum of the diagonal coefficient loss and the coupled composition loss. - -## Let’s implement the coupled noise inputs first - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u197_r0) - -Construct the coarse noise from its two fine intervals. - -$$ -I^{(0)}=I_L^{(0)}+I_R^{(0)},\qquad I^{(1)}=\tfrac12(I_L^{(1)}+I_R^{(1)}-I_L^{(0)}+I_R^{(0)}) -$$ - -The coarse zeroth coefficient adds the fine increments; the first-order coefficient adds the two fine first-order terms, subtracts the left increment, adds the right increment, and halves that total. - -## Local dynamics and interval consistency remain central - -[Equation reading 1](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u208_r0) - -The diagonal fixes the intended local motion. - -$$ -v_{t,t}(x)=b_t(x),\qquad F_{t,t}(x)=x -$$ - -The diagonal average velocity equals the intended instantaneous velocity, and the zero-length flow map returns its input unchanged. - -[Equation reading 2](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u208_r1) - -Composition connects the finite intervals. - -$$ -F_{s,t}=F_{u,t}\circ F_{s,u} -$$ - -The full finite map equals the first interval’s map followed by the second interval’s map. - -[Equation reading 3](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u208_r2) - -For a strong stochastic map, retain the same driving noise. - -$$ -\Psi_{s,t}(x,W)=\Psi_{u,t}(\Psi_{s,u}(x,W_L),W_R) -$$ - -The stochastic composition uses the left and right pieces of the same driving path to reproduce the full-interval map. diff --git a/lecture_6/MATHEMATICS.md b/lecture_6/MATHEMATICS.md deleted file mode 100644 index cbdaee8a45f85595852f273a875e3f87d460d7ef..0000000000000000000000000000000000000000 --- a/lecture_6/MATHEMATICS.md +++ /dev/null @@ -1,225 +0,0 @@ -# From the lecture equations to the running code - -The [slide code map](SLIDE_CODE_MAP.md) supplies the corresponding slide links. -Unless stated otherwise, time zero denotes Gaussian noise and time one denotes -data. All expectations below are population quantities; the training code uses -finite minibatches and learned approximations. - -## We first learn the motion at one time - -For a noise sample $X_0$ and a data sample $X_1$, construct - -$$I_t=(1-t)X_0+tX_1,\qquad \dot I_t=X_1-X_0.$$ - -Here $I_t$ is the sampled interpolant and $t$ is its progress from noise to data. -Regressing the displacement against the noisy state learns - -$$b_t(x)=\mathbb E[X_1-X_0\mid I_t=x].$$ - -`interpolate` constructs the pair; `diagonal_loss` performs the regression. -The conditional-mean field transports the same one-time marginals as the -interpolant. Individual interpolant lines generally differ from its ODE paths. - -## Now predict the destination over an interval - -$$F_{s,t}(x)=x+(t-s)v_{s,t}(x).$$ - -$s$ is the start, $t$ is the arrival time, and $v_{s,t}$ is the average velocity -along that ODE trajectory. `finite_map` uses this residual parameterization, -which gives $F_{s,s}(x)=x$ exactly. The diagonal average $v_{s,s}$ must still be -trained to equal $b_s$. - -For $\dot x=x$, the exact map is $e^{t-s}x$. From $x=1$ over a unit interval, -the average velocity is $e-1\approx1.71828$. The residual update reaches -$2.71828$, while one Euler step using the initial velocity reaches only $2$. -`numerical_examples.py` trains this scalar map and reports its actual error. - -## The ODE supplies three finite-interval identities - -$$\partial_t F_{s,t}(x)=b_t(F_{s,t}(x)),$$ - -$$\partial_sF_{s,t}(x)+J_xF_{s,t}(x)b_s(x)=0,$$ - -$$F_{s,t}(x)=F_{u,t}(F_{s,u}(x)).$$ - -The first changes the destination time. The second advances the start along the -same path, leaving the destination fixed. The third splits the interval at an -intermediate time $u$. `lagrangian_residual` and `eulerian_residual` use JVPs; -`semigroup_loss` evaluates the direct and split paths. - -The identity map satisfies composition for every interval but has zero velocity. -This is why composition needs the diagonal anchor. A detached target also -changes the optimization: only the student branch receives its gradient. - -## Shortcut and MeanFlow choose particular targets - -Shortcut learns a velocity for a longer interval from two shorter intervals: - -$$v_{s,s+2d}(x)\approx\operatorname{sg}\left[ -\tfrac12 v_{s,s+d}(x)+\tfrac12 v_{s+d,s+2d}(x+d\,v_{s,s+d}(x))\right].$$ - -$d$ is the half-step length. The second prediction receives the moved state. -`shortcut_loss` samples dyadic lengths and builds that detached target. - -MeanFlow uses the opposite clock. Its $z_t=(1-t)X_{\rm data}+tX_{\rm noise}$ -has data at zero and noise at one. For $r\leq t$, - -$$z_r=z_t-(t-r)u(z_t,r,t),$$ - -$$u=v-(t-r)(\partial_tu+J_zu\,v).$$ - -$v=X_{\rm noise}-X_{\rm data}$ is the conditional training velocity. -`meanflow_loss` evaluates the JVP in direction $(v,0,1)$ for inputs $(z,r,t)$, -then detaches the complete right-hand target. The sampler retains the minus sign. - -## A latent representation changes the coordinates - -For encoder $E$, decoder $D$, and data distribution $p$, fit a map to $E_\#p$. -Generation is $D(F_{0,1}(Z_0))$. The code first trains an autoencoder and freezes -it; a stored mean and standard deviation normalize its latent coordinates. -`train_latent` saves the encoder, decoder, normalization, and map together. - -For an $L_D$-Lipschitz decoder, the distributional error is bounded by latent -transport error multiplied by $L_D$, plus reconstruction error. The code -reports reconstruction MSE separately; it does not assert a measured global -Lipschitz constant or equate that MSE with a Wasserstein distance. - -## Categorical maps constrain their predictions - -For one-hot endpoints, the posterior mean denoiser $D_s(x)$ lies on the -probability simplex. The noisy state $x$ may lie outside it. The finite map is - -$$F_{s,t}(x)=\frac{1-t}{1-s}x+\frac{t-s}{1-s}\psi_{s,t}(x),$$ - -where $\psi_{s,t}$ is the two-time softmax prediction. `categorical_map` implements -this expression. At $s=.25,t=.75$, state $(-.2,.6,1.1)$ and prediction -$(.1,.7,.2)$ produce $(0,2/3,.5)$; the intermediate state is not normalized. - -Composition induces - -$$q=\gamma\psi_{s,u}(x)+(1-\gamma)\psi_{u,t}(F_{s,u}(x)),\qquad -\gamma=\frac{(1-t)(u-s)}{(1-u)(t-s)}.$$ - -At $(s,u,t)=(0,.5,.75)$, the weight is $1/3$. Combining predictions $(.8,.2)$ -and $(.2,.8)$ gives $(.4,.6)$. `composition_target` builds this probability-valued -target; a detached KL objective has logit gradient $p_{\rm student}-q$. - -The decoding clock is $\tau(t)=1-\frac{V}{V-1}P_e(t)$, with vocabulary size $V$ -and single-token corruption error $P_e$. `DecodingClock` numerically inverts it. -Endpoint enforcement prevents floating-point accuracy saturation from stopping -the sampler before physical time one. - -## Categorical and discrete consistency constrain different residuals - -Categorical ECLD combines endpoint agreement with the time derivative of the -finite denoiser. With $\eta=(t-s)/(1-s)$, the scaled Lagrangian residual is - -$$r=\psi_{s,t}-\psi_{t,t}(F_{s,t})+(1-t)\eta\partial_t\psi_{s,t}.$$ - -The endpoint term can vanish while the derivative term remains nonzero. The -implementation includes both and follows the released code's finite weights. - -For Discrete Flow Maps, write $\psi=\operatorname{softmax}(z)$ and center the -arrival-time logit derivative: - -$$\delta_k=\partial_tz_k-\sum_j\psi_j\partial_tz_j,\qquad -c=\frac{(t-s)(1-t)}{1-s}.$$ - -The Lagrangian teacher is - -$$T_{\rm LSD}=\operatorname{softmax}\left[z_{t,t}(F_{s,t})-\log(1+c\delta)\right].$$ - -For the Eulerian teacher, replace the derivative with -$D_sz=\partial_sz+J_xz\,b_s$, center it in the same way, and use - -$$T_{\rm ESD}=\operatorname{softmax}\left[z_{s,s}- -\log\left(1-\frac{(1-s)(t-s)}{1-t}\delta\right)\right].$$ - -`corrected_logit_teacher` computes both versions. Log arguments must be positive; -the code records its stabilization frequency so that violations remain visible. - -## Posterior maps need a second noise draw - -Meta training constructs - -$$I_t=(1-t)X_0+tX_1,\qquad \bar I_s=(1-s)\bar X_0+sX_1,$$ - -with independent $X_0,\bar X_0$. The outer observation $(t,I_t)$ stays fixed -while the inner map advances from $s$ to its target time. At that fixed context, -the shared $X_1$ has exactly the desired posterior distribution. - -Diamond distillation obtains the corresponding conditional velocity through -GLASS. For two independent observations, the likelihood precisions add: - -$$\Lambda=\frac{s^2}{(1-s)^2}+\frac{t^2}{(1-t)^2},\qquad -S=\Lambda^{-1}\left[\frac{s\bar x}{(1-s)^2}+\frac{tx}{(1-t)^2}\right].$$ - -The equivalent linear-interpolant time is -$t^*=\sqrt\Lambda/(1+\sqrt\Lambda)$. Evaluating the original denoiser at -$(t^*,t^*S)$ supplies the posterior mean conditioned on both observations. -`glass_denoiser` is checked against an independent Gaussian conditioning formula. - -## Posterior samples make reward averages computable - -For reward $r$ and conditional samples $Z_k$, - -$$\widehat V_t(x)=\log\left(\frac1K\sum_{k=1}^K e^{r(Z_k)}\right),\qquad -\widehat D_t^r(x)=\sum_k\operatorname{softmax}(r(Z))_k Z_k.$$ - -`posterior_value` differentiates through the learned conditional samples. -`guided_samples` uses $(\widehat D_t^r(x)-x)/(1-t)$ as the guided velocity. -The ratio is biased at finite $K$, even with an exact conditional sampler. - -For Meta fine-tuning, let $d=b_{\rm student}-b_{\rm base}$, $w=e^{r(Z)}$, and -$a=g_t^2/2$. The detached surrogate is - -$$\ell=\left\|d+(w-1)\operatorname{sg}(d)-a\operatorname{sg}(\nabla_xw)\right\|^2.$$ - -Its expected gradient in $d$ is $2\mathbb E[wd-a\nabla_xw]$. Squaring $wd-a\nabla w$ -without that stop-gradient placement introduces an extra weight. The test suite -checks this distinction directly. For the linear path, $a=(1-t)/t$; the optional -training demonstration therefore excludes the singular initial time. - -## Expansion gives each new token its own clock - -A token born at $b_i$ has local time - -$$\tau_i(t)=\max\left(0,\frac{t-b_i}{1-b_i}\right),\qquad -\frac{d\tau_i}{dt}=\frac1{1-b_i}\quad(t>b_i).$$ - -`local_clock` and `local_map` preserve this information during transport. -`insert_tokens` keeps token order, inserted noise, and birth metadata aligned. -The linear insertion CDF gives conditional birth fraction -$\rho_{s,t}=(t-s)/(1-s)$. Predicted remaining gap means are converted into -interval means using this factor. - -The count divergence is - -$$\phi(a,b)=b-a+a\log(a/b),\qquad b>0,$$ - -where $a$ is a realized count and $b$ is the predicted mean. Its population -minimum is $b=\mathbb E[a]$. The zero-count branch is implemented with a finite -autodiff expression. Sampling uses bounded binomial proposals and reports -counts removed by global budget capping. - -## Strong maps must retain the same driving path - -For an interval of length $h$, the first two Brownian integrals satisfy - -$$I_0\sim\mathcal N(0,h),\qquad I_1\sim\mathcal N(0,h/3),\qquad I_0\perp I_1.$$ - -For left/right subintervals of lengths $h_L,h_R$, Chen composition is - -$$I_0=I_0^L+I_0^R,$$ - -$$I_1=\frac{h_LI_1^L+h_RI_1^R-h_RI_0^L+h_LI_0^R}{h_L+h_R}.$$ - -With equal halves, $(I_0^L,I_1^L)=(.2,.04)$ and -$(I_0^R,I_1^R)=(-.1,-.02)$ combine into $(.1,-.14)$. -`chen_two` and `aggregate_tree` ensure the direct and split maps refer to that -same history. Independent coarse noise would compare different sample paths. - -The SSFM objective combines a short stochastic matching step with same-noise -composition. The reported strong error compares the learned result to a fine -Euler-Maruyama trajectory using the same increments. Terminal variance checks -the marginal separately; matching variance cannot establish pathwise accuracy. diff --git a/lecture_6/README.md b/lecture_6/README.md deleted file mode 100644 index 491dcaa529c3335e98f8b3c2851667739b345123..0000000000000000000000000000000000000000 --- a/lecture_6/README.md +++ /dev/null @@ -1,225 +0,0 @@ -# CIS 6270 - Lecture 6 - Flow Maps - -Course hub: [ChatterjeeLab/CIS6270](https://huggingface.co/ChatterjeeLab/CIS6270). -Slides: [Lecture 6 - Flow Maps](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit). -Previous lecture: [Discrete Flow Matching](../lecture_5/README.md). - -Lecture 5 learned how a distribution moves locally. Here we learn the motion over -an entire time interval. The examples progress from continuous flow maps to -latent representations, categorical text, posterior maps, expanding states, -and maps driven by a shared Brownian path. - -This folder contains **16 complete training and generation examples**, with -checkpoint loading, data, mathematical checks, and saved execution results. -The code uses small PyTorch networks and inspectable datasets. Each method's -objective and sampler are implemented here; the large image and language -benchmark runs from the papers require their original architectures, datasets, -and training budgets. [SOURCES.md](SOURCES.md) records the exact papers, inspected -author-code revisions, and differences from those implementations. - -## Train a velocity, then learn a finite map - -Use Python 3.11 or later. The minimal requirements retain the course's PyTorch -2.9.1 baseline. All examples run on CPU; `--device cuda` selects an available GPU. -The data require no downloads. - -From the course repository root: - -```bash -cd lecture_6 -python -m venv .venv -source .venv/bin/activate -python -m pip install -r requirements.txt - -python run.py --method flow-matching --sample-steps 32 --out outputs/flow-matching -python run.py --method self-distill --out outputs/self-distill -python run.py --mode sample --out outputs/self-distill --sample-steps 1 -``` - -The first command fits the conditional-mean velocity and integrates it with -Heun's method. The second trains a two-time map using a diagonal velocity loss -and a detached composition target. The third loads its checkpoint and generates -in one map evaluation. The baseline uses two velocity evaluations per Heun step; -finite maps use one evaluation per step. - -Training writes `checkpoint.pt`, `config.json`, `losses.json`, `report.json`, and -`samples.txt`. A teacher or autoencoder stage also writes `teacher_losses.json`. -Generation-only mode reads model architecture and vocabulary from the checkpoint -and writes `resampled.txt` and `sample_report.json`, preserving training records. -The checkpoint contains the inference state, not optimizer state for resuming -training. `--mode train` saves training records without generating samples. - -## Run every method in the same structure as Lectures 4 and 5 - -```bash -python run_all.py --quick -python run_all.py -python -m unittest discover -s tests -v -python numerical_examples.py --output outputs/numerical -``` - -`--quick` performs 20 optimization steps per method and verifies execution and -checkpoint reloading. The regular command uses 1,000 steps per training stage. -Both run every method and require seeded generated samples to agree exactly -after checkpoint reload. The quick run measures execution correctness; inspect -the longer run's losses and sample metrics when discussing learned behavior. - -| Method | Training | Generation | -| --- | --- | --- | -| `flow-matching` | Conditional displacement regression on a linear interpolant | Heun integration of the learned diagonal velocity | -| `fmm-lagrangian` | Train a velocity teacher, then match the map's arrival-time derivative to the teacher | Residual finite-time maps | -| `fmm-eulerian` | Train a velocity teacher, then match the start-time directional derivative to zero | Residual finite-time maps | -| `self-distill` | Diagonal flow matching and EMA two-subinterval composition targets | One or more direct map updates | -| `consistency` | Endpoint consistency along short Heun-solved teacher intervals, with the diagonal anchor | One endpoint prediction | -| `shortcut` | Conditional flow matching and two-half-step average-velocity targets on dyadic intervals | One or more shortcuts | -| `meanflow` | Backward average-velocity target with a directional JVP and detached adaptive weights | Backward updates from noise at time one to data at zero | -| `latent` | Train a 3D-to-2D autoencoder, freeze it, standardize its encodings, then train a flow map | Latent map followed by the saved decoder | -| `fmlm` | Diagonal token CE and weighted progressive denoiser consistency | Warped-time affine categorical maps, followed by argmax | -| `categorical` | Diagonal CE and endpoint consistency plus temporal-derivative energy | The same categorical map parameterization | -| `discrete-lsd` | Diagonal CE and Lagrangian logit-corrected KL teacher | Categorical finite-time maps | -| `discrete-esd` | Diagonal CE and Eulerian directional-JVP logit teacher | Categorical finite-time maps | -| `diamond` | Distill a GLASS posterior velocity built from an analytic Gaussian-mixture denoiser | Conditional posterior samples, differentiable value estimates, and reward steering | -| `meta` | Independent inner and outer noise with a shared data endpoint, diagonal regression, and conditional composition | Posterior samples and reward-weighted posterior-mean steering | -| `expanding` | Local-clock token CE, compatible-canvas composition, and remaining/interval gap-count losses | Learned binomial gap insertion, budget capping, and local-clock transport | -| `ssfm` | OU drift anchoring, small-step stochastic matching, and same-noise composition | Strong maps using consistently aggregated Brownian coefficients | - -Each method also has its own entry point: - -```bash -python examples/fmm_lagrangian.py -python examples/fmm_eulerian.py -python examples/consistency.py -python examples/shortcut.py -python examples/meanflow.py --sample-steps 1 -python examples/latent.py --teacher-steps 2000 -python examples/fmlm.py --sample-steps 1 -python examples/categorical.py -python examples/discrete_lsd.py -python examples/discrete_esd.py -python examples/diamond.py --posterior-steps 4 --particles 64 -python examples/meta.py --finetune-steps 300 -python examples/expanding.py --sample-steps 4 -python examples/ssfm.py --sample-steps 4 -python examples/ssfm.py --ssfm-target paper --lr 0.0001 --train-steps 2000 --out outputs/ssfm-paper -``` - -The common options are `--train-steps`, `--teacher-steps`, `--batch-size`, -`--width`, `--lr`, `--seed`, `--samples`, `--sample-steps`, and `--out`. -`--teacher-steps` controls the teacher or autoencoder stage where one exists. -Run `python run.py --help` for the remaining options. The `consistency` example -always makes one endpoint prediction. SSFM requires a power-of-two sampling -step count so its finite Brownian tree supports every requested partition. - -## Change one component at a time - -`lecture_core.py` exposes the functions used in the slide walkthroughs. -`common.py` contains networks, data handling, optimization, and reference -distributions. Each method family has a separate implementation file. - -| File | Responsibility | -| --- | --- | -| `continuous.py` | Diagonal, Lagrangian, Eulerian, semigroup, Shortcut, MeanFlow, and latent training | -| `categorical.py` | Affine probability-valued maps, inverse decoding clock, ECLD, and logit teachers | -| `posterior.py` | GLASS fusion, conditional map training, posterior values, importance correction, and reward fine-tuning | -| `expanding.py` | Birth times, compaction, gap labels, count losses, and expansion/transport sampling | -| `stochastic.py` | Brownian integrals, Chen composition, OU training, and strong-error evaluation | -| `numerical_examples.py` | The lecture's numerical examples, including the trained exponential-flow example | -| `SLIDE_CODE_MAP.md` | Stable links from every code slide and major method to the implementation | -| `MATHEMATICS.md` | Equations, variable definitions, and the meaning of each loss | -| `verified_examples/` | Actual seeded training logs, generated samples, and verification reports | - -The small sequence MLP receives the entire sequence. Its per-position output -can therefore depend on every input position. The variable-length network also -receives padding masks and birth-time metadata. Replacing these backbones with -transformers does not require changing the displayed map identities. - -## Data and evaluation - -The continuous examples use four 2D Gaussians with known centers and variance. -Reports include nearest-center distance, mode fractions, mode entropy, and exact -target log density. These metrics describe different aspects of the samples; -high mode entropy alone does not establish sample fidelity. - -`data/phrases.txt` contains correlated four-token phrases such as `red circle -moves left`. `data/variable_text.txt` contains two-, four-, and five-token -versions. Reports record token entropy, unique fraction, training-support -fraction, and lengths. These tiny-data statistics are not LM1B/OWT perplexity or -large-language-model evaluations. The split is a seeded 80/20 split of examples; -the small grammar intentionally appears in both splits. - -Custom fixed-length text can be supplied with `--data my_phrases.txt`. -The expanding example also accepts variable-length lines, up to 32 tokens. -Continuous methods accept `--data points.csv`, with two finite numeric columns -and no header. Diamond, Meta, and SSFM retain their specified reference -distributions so their analytic diagnostics remain valid. See [data/README.md](data/README.md). - -## Numerical and theorem boundaries - -- **Continuous maps.** The diagonal is exactly the identity as a map, while its - predicted average velocity is trained by flow matching. Interpolant samples - are not individual ODE trajectories. Diagonal regression has an irreducible - conditional-variance floor, so raw training loss need not approach zero. -- **Teacher distillation.** The finite-map teachers are learned in a separate - first stage. This retains teacher approximation error. The fixed MLP, bounded - interval sampling, and loss weights are classroom choices. -- **Consistency and Shortcut.** Endpoint consistency is a compact - flow-matching-clock teaching implementation. Shortcut uses dyadic interval - lengths, a continuous start-time sample, EMA targets, and a separate diagonal - loss. The authors' large-image code has additional schedules and clipping. -- **MeanFlow.** Data are at time zero and noise at one. The directional JVP uses - the conditional training velocity, and the entire corrected target is - detached. Seventy-five percent of training examples use equal times. Uniform - time sampling replaces the author's default logit-normal schedule. -- **Latent maps.** This section demonstrates a learned representation and - decoder. It does not claim a separate canonical paper named Latent Flow Maps. - Reconstruction error and latent transport error are reported separately. -- **Categorical maps.** Only the predicted denoiser lies on the simplex during - the trajectory. The final step reaches time one exactly; argmax is an explicit - decoding approximation for an imperfect model. We use Gaussian quadrature - for the decoding clock. Categorical ECLD uses the released code's bounded - endpoint-plus-time-energy objective, with an EMA endpoint target. Detached KL - and CE have the same student gradient; they log different scalar values. -- **Discrete logit teachers.** The logarithmic correction requires positive - denominators. The implementation clamps them at 0.05 and logs the affected - fraction. This changes the teacher away from the exact solution; softmax - alone cannot repair an invalid logarithm. Differential objectives stay below - terminal time 0.97. No unreported gradient surgery is applied. -- **Diamond and Meta.** The two noises are independent and share a clean - endpoint. Diamond uses an exact, evaluable mixture denoiser as its GLASS - teacher; Meta learns directly from samples. Posterior diagnostics compare - both means and variances. Finite importance estimates and reward-weighted - posterior means have self-normalization bias. The weighted-Diamond companion - uses an explicit full-support Gaussian proposal with a known density. -- **Meta fine-tuning.** `--finetune-steps` runs and saves the detached surrogate - from Equation 43. It uses a bounded coordinate reward and the analytic base - mixture drift. Training is restricted to outer times 0.05 through 0.85; - extending the learned drift outside that interval is extrapolation. -- **Expanding maps.** The data use a linear birth CDF. The network predicts - remaining gap means and interval means through the conditional insertion - factor. Binomial proposals and global budget capping approximate a joint count - law; a mean alone does not identify that law. A finite sampling jump introduces - new coordinates at its start, then transports them. Training compares routes - on a shared destination canvas with compatible noise and local clocks. - These are explicit finite-step choices, not a proof of exact conditional-law - recovery by the fitted network. Empty generated sequences remain visible in - the reports; the sampler does not silently replace them. -- **Strong stochastic maps.** The example is an additive-noise OU SDE with a - known diffusion coefficient. It uses the first two Legendre integrals and a - learned average drift. Default `official-code` targets detach an EMA split - prediction; `--ssfm-target paper` detaches the direct prediction, as in the - lecture's pseudocode. Both reuse exactly the same Brownian history. The - reference is a fine Euler-Maruyama solve on that history, so its residual - discretization error is retained. No claim is made for multiplicative noise - or exact pathwise recovery with only two coefficients. The optional paper - orientation remained inaccurate in the recorded experiments, including the - lower-learning-rate recipe above; see the explicit sensitivity results in - `verified_examples/README.md`. Use the default released-code orientation for - the successful OU demonstration. - -## Read the saved results with the implementation - -[verified_examples/README.md](verified_examples/README.md) records the executed -environment, commands, and metrics. Checkpoints are generated locally and are -excluded from this folder's Git history, matching Lectures 4 and 5. The saved -text/JSON records let students inspect a complete run before training their own -models. The [slide code map](SLIDE_CODE_MAP.md) connects that run to the lecture. diff --git a/lecture_6/SLIDE_CODE_MAP.md b/lecture_6/SLIDE_CODE_MAP.md deleted file mode 100644 index 237d6a66847ddddc7a05407cac10692d59adef71..0000000000000000000000000000000000000000 --- a/lecture_6/SLIDE_CODE_MAP.md +++ /dev/null @@ -1,36 +0,0 @@ -# Lecture 6 slide-to-code map - -The updated deck adds spoken equation readings before intuition; see [Equation readings](EQUATION_READINGS.md). These links use stable native slide identifiers. Each code slide connects to the same functions used by the training runner. - -| Code walkthrough | Slide | Implementation | -| --- | --- | --- | -| The map takes the state and both times as inputs | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u038_b0) | `common.finite_map`, `continuous.diagonal_loss`, `continuous.semigroup_loss`, `numerical_examples.train_scalar` | -| The two losses connect local motion to a longer move | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u039_b0) | `common.finite_map`, `continuous.diagonal_loss`, `continuous.semigroup_loss`, `numerical_examples.train_scalar` | -| A JVP computes that correction without forming a Jacobian | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u051_b0) | `continuous.meanflow_loss` | -| Softmax constrains the prediction before the map is applied | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u080_b0) | `categorical.categorical_map`, `categorical.composition_target`, `categorical.probability_kl` | -| The target combines probabilities from two different states | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u081_b0) | `categorical.categorical_map`, `categorical.composition_target`, `categorical.probability_kl` | -| We can differentiate through the posterior samples | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u122_b0) | `posterior.posterior_samples`, `posterior.posterior_value` | -| The stop-gradient placement is essential in this fine-tuning loss | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u139_b0) | `posterior.fine_tune_surrogate`, `posterior.train_reward_drift` | -| Keep the count, noise, and clock updates together | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u168_b0) | `expanding.bounded_counts`, `expanding.insert_tokens`, `expanding.local_clock` | -| Sampling and combining two coefficients is only a few lines | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u198_b0) | `stochastic.sample_coefficients`, `stochastic.chen_two`, `stochastic.train_stochastic` | -| The consistency update compares those two routes | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u199_b0) | `stochastic.sample_coefficients`, `stochastic.chen_two`, `stochastic.train_stochastic` | - -## Method sections - -| Lecture section | First slide | Code | -| --- | --- | --- | -| From velocities to finite motion | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u001_b0) | [continuous.py](continuous.py) | -| Learning the identities | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u025_b0) | [continuous.py](continuous.py) | -| Related finite-step formulations | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u041_b0) | [continuous.py](continuous.py) | -| MeanFlow and the direction of time | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u047_b0) | [continuous.py](continuous.py) | -| Flow maps in learned latent spaces | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u054_b0) | [posterior.py](posterior.py) | -| Flow Map Language Models | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u060_b0) | [categorical.py](categorical.py) | -| Categorical Flow Maps | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u083_b0) | [categorical.py](categorical.py) | -| Discrete Flow Maps and logit consistency | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u093_b0) | [categorical.py](categorical.py) | -| Diamond Maps and posterior lookahead | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u103_b0) | [posterior.py](posterior.py) | -| Meta Flow Maps | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u125_b0) | [posterior.py](posterior.py) | -| Expanding Flow Maps | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u143_b0) | [expanding.py](expanding.py) | -| Strong Stochastic Flow Maps | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u173_b0) | [stochastic.py](stochastic.py) | -| Connecting the formulations | [Open](https://docs.google.com/presentation/d/1wfLAazqYveMoyvdy0y5ILeionp7eUjcPgxu-5QFlWwI/edit#slide=id.fm6_u206_b0) | [stochastic.py](stochastic.py) | - -The exponential-flow code slides use the exact scalar drift `b(x)=x`; the generative runners use learned fields on a Gaussian mixture. The scalar training experiment remains in `numerical_examples.py`. The SSFM consistency code slide corresponds to `--ssfm-target paper`; the default follows the released code orientation, described in `SOURCES.md`. diff --git a/lecture_6/SOURCES.md b/lecture_6/SOURCES.md deleted file mode 100644 index 52719e3c4ebbd039a317108909c6e7088bc72cdc..0000000000000000000000000000000000000000 --- a/lecture_6/SOURCES.md +++ /dev/null @@ -1,97 +0,0 @@ -# Primary papers and author-code checks - -The teaching implementations were written for this course. Upstream training files were inspected to check objectives and conventions; their code is not vendored here. Small networks, synthetic data, and explicit numerical choices make the methods runnable on CPU. Inspection date: 2026-09-13. - -## Flow Map Matching - -[Paper](https://arxiv.org/abs/2406.07507v2). Class methods: `fmm-lagrangian, fmm-eulerian`. - -[Inspected author source](https://github.com/nmboffi/flow-maps/blob/2f115a07fa9073553193e4b265dfc303827af2b0/py/common/losses.py), revision `2f115a07fa90`. - -Frozen learned velocity teachers; residual-map JVPs; small MLP and bounded time range. - -## How to build a consistency model: Learning flow maps via self-distillation - -[Paper](https://arxiv.org/abs/2505.18825v2). Class methods: `self-distill`. - -[Inspected author source](https://github.com/nmboffi/flow-maps/blob/2f115a07fa9073553193e4b265dfc303827af2b0/py/common/losses.py), revision `2f115a07fa90`. - -Diagonal regression plus an EMA progressive target, normalized by a floored interval length. Learned uncertainty weights are omitted. - -## One Step Diffusion via Shortcut Models - -[Paper](https://arxiv.org/abs/2410.12557v3). Class methods: `shortcut`. - -[Inspected author source](https://github.com/kvfrans/shortcut-models/blob/601004348667094e1b71f30942199759412d4432/targets_shortcut.py), revision `601004348667`. - -Two half-step average target and dyadic intervals retained. Continuous start-time sampling and small MLP replace image-specific schedules and clipping. - -## Mean Flows for One-step Generative Modeling - -[Paper](https://arxiv.org/abs/2505.13447). Class methods: `meanflow`. - -[Inspected author source](https://github.com/Gsunshine/meanflow/blob/d70cb55d298ee03c53bf6da67bec281082e4e2d9/meanflow.py), revision `d70cb55d298e`. - -Backward clock, conditional-velocity JVP, detached target, adaptive loss, and 75% diagonal proportion retained. Uniform time sampling; no class guidance. - -## Flow Map Language Models: One-step Language Modeling via Continuous Denoising - -[Paper](https://arxiv.org/abs/2602.16813v3). Class methods: `fmlm`. - -[Inspected author source](https://github.com/david3684/flm/blob/a1918d5164e5038e37d0b7a4fb2010ce75b863b3/algo.py), revision `a1918d5164e5`. - -PSD denoiser target and Gaussian decoding clock retained. Independent clean-data diagonal supervision; EMA targets; quadrature lookup and MLP replace full language architecture. - -## Categorical Flow Maps - -[Paper](https://arxiv.org/abs/2602.12233v1). Class methods: `categorical`. - -[Inspected author source](https://github.com/olsdavis/semicat/blob/558602a0fa722514e4a6012f5c46a8ae178b3068/semicat/models/semicat.py), revision `558602a0fa72`. - -Released ECLD endpoint CE plus sum-of-squares time energy. Detached-target KL has identical student gradient. EMA target replaces the current network target. - -## Discrete Flow Maps - -[Paper](https://arxiv.org/abs/2604.09784v1). Class methods: `discrete-lsd, discrete-esd`. - -Derived from the paper. The linked project page has a placeholder Code link, so no author implementation was verified. Positive log denominators are explicitly clamped and counted; no gradient surgery. - -## Diamond Maps: Stochastic Flow Maps - -[Paper](https://arxiv.org/abs/2602.05993). Class methods: `diamond`. - -[Inspected author source](https://github.com/PeterHolderrieth/diamond_maps/blob/d30f65c75a169a2ed624f146b2770aeef882543a/posterior_diamond_maps/py/common/losses.py), revision `d30f65c75a16`. - -GLASS conditional velocity and Lagrangian posterior distillation. Exact Gaussian-mixture denoiser replaces a pretrained image teacher. Weighted posterior recovery uses an explicit Gaussian proposal with a known density. - -## Meta Flow Maps enable scalable reward alignment - -[Paper](https://arxiv.org/abs/2601.14430v2). Class methods: `meta`. - -[Inspected author source](https://github.com/adh1s/mfm/blob/53c0f60db695cad88bbace8fb26469614e9e7d7d/src/mfm/losses/losses.py), revision `53c0f60db695`. - -Data training with independent inner/outer noises and a shared endpoint; conditional semigroup objective. Optional Equation 43 fine-tuning also checked against src/mfm/losses/finetune.py. - -## Expanding Flow Maps - -[Paper](https://arxiv.org/abs/2607.21585v1). Class methods: `expanding`. - -[Inspected author source](https://github.com/sophtang/ExpandingFlowMaps/blob/4defc1ff168d12526d85b551bea78150b9aec605/README.md), revision `4defc1ff168d`. - -Paper Algorithms 2-4 and local-clock equations. The inspected repository contains README and artwork only. Classroom implementation includes a shared lifted canvas, remaining/interval count heads, explicit local clocks, and budget-capped binomial insertions. - -## Strong Stochastic Flow Maps - -[Paper](https://arxiv.org/abs/2606.01086v1). Class methods: `ssfm`. - -[Inspected author source](https://github.com/sammccallum/ssfm/blob/24b563620a8ee61c683d791fe802b56668f98f9b/ssfm/losses.py), revision `24b563620a8e`. - -First two Legendre integrals, Chen composition, known diffusion, small-step matching. The released loss detaches an EMA split target; paper Algorithm 1 detaches the direct target. Both orientations are selectable. OU replaces image/molecular experiments; no learned uncertainty weights. - -## Supporting lecture demonstrations - -`flow-matching` supplies the local velocity baseline. `consistency` demonstrates endpoint consistency on a learned flow teacher using the forward clock and a residual endpoint parameterization; it is not a reproduction of the original EDM consistency training system. `latent` demonstrates the encoder/map/decoder construction and does not attribute it to a separate paper called Latent Flow Maps. `numerical_examples.py` retains the lecture’s scalar ODE experiment and exact arithmetic checks. - -## Version and result scope - -Paper links retain the lecture versions when supplied. Author repositories may have advanced after those paper versions. The source revision above records exactly what was inspected. The lecture’s paper figures and benchmark values remain in the slide deck; the files under `verified_examples/` contain newly executed course examples, with their own data, compute settings, and metrics. No reported FID, generative perplexity, or biological benchmark is reproduced by these CPU teaching runs. diff --git a/lecture_6/categorical.py b/lecture_6/categorical.py deleted file mode 100644 index fd0ee80622a9347c6a5af65df6d884ce6929fa61..0000000000000000000000000000000000000000 --- a/lecture_6/categorical.py +++ /dev/null @@ -1,145 +0,0 @@ -"""Simplex-valued mean denoisers for Flow Map Language Models and discrete maps. - -The state is Gaussian-noised one-hot text and can be outside the simplex. -Only the softmax endpoint prediction is a probability vector. -""" -import math - -import torch -from torch.nn import functional as F - -from common import SequenceNet, draw_batch, ema_copy, interpolate, optimize, time_like - - -def categorical_map(model, x, s, t): - s, t = time_like(s, x), time_like(t, x) - probability = model(x, s, t).softmax(-1) - h = ((t - s) / (1 - s)).unsqueeze(-1) - return x + h * (probability - x), probability - - -def composition_target(model, x, s, u, t): - mid, p1 = categorical_map(model, x, s, u) - _, p2 = categorical_map(model, mid, u, t) - gamma = ((1 - t) * (u - s) / ((1 - u) * (t - s))).unsqueeze(-1) - return gamma * p1 + (1 - gamma) * p2 - - -def probability_kl(logits, target): - return F.kl_div(logits.log_softmax(-1), target, reduction='none').sum(-1).mean() - - -class DecodingClock: - """Inverse of the FMLM decoding-accuracy clock for isotropic Gaussian noise. - - P(correct) = E_Z[Phi(Z + t/(1-t))^(V-1)]. Gaussian quadrature gives a - deterministic lookup. Endpoint values are exact. No empirical vocabulary - frequency estimate is mixed into this corruption-only clock. - """ - def __init__(self, vocab, device='cpu'): - import numpy as np - nodes, weights = np.polynomial.hermite.hermgauss(64) - self.t = torch.linspace(0, .999, 1001, device=device) - z = torch.as_tensor(nodes * math.sqrt(2), device=device, dtype=torch.float64) - w = torch.as_tensor(weights / math.sqrt(math.pi), device=device, dtype=torch.float64) - snr = (self.t / (1 - self.t)).double() - cdf = .5 * (1 + torch.erf((z[None] + snr[:, None]) / math.sqrt(2))) - accuracy = (cdf.pow(vocab - 1) * w).sum(-1) - tau = ((vocab * accuracy - 1) / (vocab - 1)).clamp(0, 1).float() - self.tau = torch.cummax(tau, 0).values - self.tau[0], self.tau[-1], self.t[-1] = 0., 1., 1. - - def inverse(self, tau): - indices = torch.searchsorted(self.tau, tau.contiguous()).clamp(1, len(self.tau)-1) - low, high = self.tau[indices-1], self.tau[indices] - alpha = (tau-low) / (high-low).clamp_min(1e-7) - result = self.t[indices-1] + alpha * (self.t[indices]-self.t[indices-1]) - # Finite precision can saturate the lookup before t=1. Enforce the - # mathematical endpoint so generation actually reaches the simplex. - return torch.where(tau>=1,torch.ones_like(result),torch.where(tau<=0,torch.zeros_like(result),result)) - - -def corrected_logit_teacher(model, teacher, x, s, t, kind, floor=.05): - """Lagrangian/Eulerian logit teachers and explicit domain stabilization. - - At an exact solution the denominators are positive. Unconverged networks - can violate that domain. We clamp to floor and report the changed fraction. - These clamped off-solution targets are an explicit numerical approximation. - """ - if kind == 'discrete-lsd': - logits, dz = torch.func.jvp(lambda end: model(x, s, end), - (t,), (torch.ones_like(t),)) - p = logits.softmax(-1) - delta = dz - (p * dz).sum(-1, keepdim=True) - coefficient = ((t-s) * (1-t) / (1-s)).unsqueeze(-1) - denominator = 1 + coefficient * delta - with torch.no_grad(): - y, _ = categorical_map(teacher, x, s, t) - base_logits = teacher(y, t, t) - else: - with torch.no_grad(): - p0 = teacher(x, s, s).softmax(-1) - velocity = (p0 - x) / (1-s).unsqueeze(-1) - base_logits = teacher(x, s, s) - logits, dz = torch.func.jvp(lambda z, start: model(z, start, t), - (x, s), (velocity, torch.ones_like(s))) - p = logits.softmax(-1) - delta = dz - (p * dz).sum(-1, keepdim=True) - coefficient = ((1-s)*(t-s)/(1-t)).unsqueeze(-1) - denominator = 1 - coefficient * delta - target = (base_logits - denominator.clamp_min(floor).log()).softmax(-1).detach() - return logits, target, (denominator.detach() < floor).float().mean() - - -def train_categorical(method, ids, vocab, args): - model = SequenceNet(ids.shape[1], len(vocab), args.width).to(ids.device) - teacher = ema_copy(model) - clock = DecodingClock(len(vocab), ids.device) - - def objective(step): - batch = draw_batch(ids, args.batch_size) - clean = F.one_hot(batch, len(vocab)).float() - ordered = clock.inverse(torch.rand(len(batch), 2, device=ids.device)).sort(-1).values - # Stay away from singular coefficients in differential teacher targets. - s, t = (.97 * ordered).split(1, -1) - t = torch.maximum(t, s + 1e-4) - x, _ = interpolate(clean, s) - diagonal = F.cross_entropy(model(x, s, s).flatten(0, 1), batch.flatten()) - logits = model(x, s, t) - extra = {} - if method == 'fmlm': - with torch.no_grad(): - target = composition_target(teacher, x, s, (s+t)/2, t) - finite = probability_kl(logits, target) - elif method == 'categorical': - y, _ = categorical_map(model, x, s, t) - with torch.no_grad(): - endpoint = teacher(y.detach(), t, t).softmax(-1) - ec = probability_kl(logits, endpoint) - _, dt = torch.func.jvp(lambda end: model(x, s, end).softmax(-1), - (t,), (torch.ones_like(t),)) - eta = ((t-s)/(1-s)).unsqueeze(-1) - temporal = (eta * dt).square().flatten(1).sum(-1).mean() - # Released ECLD implementation: CE + sum-of-squares time energy. - # Detached-target KL has the same student gradient as its CE term. - finite = ec + temporal - extra = {'endpoint_kl': ec, 'temporal_derivative': temporal} - else: - logits, target, fraction = corrected_logit_teacher(model, teacher, x, s, t, method) - finite = probability_kl(logits, target) - extra = {'denominator_clamp_fraction': fraction} - weight = min(1., (step + 1) / max(1, args.train_steps // 4)) - return diagonal + weight * finite, {'diagonal_ce': diagonal, 'finite': finite, **extra} - - logs = optimize(model, objective, args.train_steps, args.lr, teacher) - return model, {'model': model.state_dict(), 'length': ids.shape[1], 'vocab': vocab}, logs - - -@torch.no_grad() -def sample_categorical(model, count, steps, device): - clock = DecodingClock(model.vocab, device) - grid = clock.inverse(torch.linspace(0, 1, steps+1, device=device)) - x = torch.randn(count, model.length, model.vocab, device=device) - for s, t in zip(grid[:-1], grid[1:]): - x, _ = categorical_map(model, x, s, t) - return x.argmax(-1) diff --git a/lecture_6/common.py b/lecture_6/common.py deleted file mode 100644 index 59650c512f0e0006e32c873ac69148d784646030..0000000000000000000000000000000000000000 --- a/lecture_6/common.py +++ /dev/null @@ -1,207 +0,0 @@ -"""Shared networks, data, and optimization for CIS 6270 Lecture 6. - -The small MLPs keep time derivatives and all training steps visible. Sequence -networks receive the whole padded sequence, so predictions can depend on other -positions. No pretrained checkpoint or dataset download is required. -""" -import copy -import json -import math -import random -from pathlib import Path - -import numpy as np -import torch -from torch import nn -from torch.nn import functional as F - - -def seed_all(seed, threads=1): - random.seed(seed) - np.random.seed(seed) - torch.manual_seed(seed) - torch.set_num_threads(threads) - - -def mlp(inputs, outputs, width): - return nn.Sequential(nn.Linear(inputs, width), nn.SiLU(), - nn.Linear(width, width), nn.SiLU(), - nn.Linear(width, outputs)) - - -def time_like(t, x): - """One scalar time per batch member, shape [B,1].""" - t = torch.as_tensor(t, dtype=x.dtype, device=x.device) - if t.numel() == 1: - return t.expand(len(x), 1) - return t.reshape(len(x), 1) - - -class MapNet(nn.Module): - def __init__(self, dim=2, width=64, context_dim=0): - super().__init__() - self.net = mlp(dim + 2 + context_dim, dim, width) - - def forward(self, x, s, t, context=None): - inputs = [x, time_like(s, x), time_like(t, x)] - if context is not None: - inputs.append(context) - return self.net(torch.cat(inputs, -1)) - - -class SequenceNet(nn.Module): - def __init__(self, length, vocab, width=64): - super().__init__() - self.length, self.vocab = length, vocab - self.net = mlp(length * vocab + 2, length * vocab, width) - - def forward(self, x, s, t): - inputs = torch.cat([x.flatten(1), time_like(s, x), time_like(t, x)], -1) - return self.net(inputs).reshape(-1, self.length, self.vocab) - - -def finite_map(model, x, s, t, context=None): - """F(s,t,x) = x + (t-s) v(s,t,x), including exact F(s,s,x)=x.""" - s, t = time_like(s, x), time_like(t, x) - return x + (t - s) * model(x, s, t, context) - - -def ordered_times(x, ceiling=.98): - times = ceiling * torch.rand(len(x), 2, device=x.device, dtype=x.dtype) - s, t = times.sort(-1).values.split(1, -1) - return s, t, (s + t) / 2 - - -def draw_batch(data, count): - return data[torch.randint(len(data), (count,), device=data.device)] - - -def interpolate(data, time, noise=None): - noise = torch.randn_like(data) if noise is None else noise - t = time.reshape(len(data), *([1] * (data.ndim - 1))) - return (1 - t) * noise + t * data, data - noise - - -def ema_copy(model): - result = copy.deepcopy(model).eval() - result.requires_grad_(False) - return result - - -@torch.no_grad() -def update_ema(ema, model, decay=.99): - for p, q in zip(ema.parameters(), model.parameters()): - p.lerp_(q, 1 - decay) - - -def optimize(model, objective, steps, lr=1e-3, ema=None): - optimizer = torch.optim.Adam(model.parameters(), lr=lr) - logs = [] - for step in range(steps): - loss, details = objective(step) - if not torch.isfinite(loss): - raise FloatingPointError(f'Nonfinite loss at step {step}: {details}') - optimizer.zero_grad(set_to_none=True) - loss.backward() - norm = torch.nn.utils.clip_grad_norm_(model.parameters(), 10.) - if not torch.isfinite(norm): - raise FloatingPointError(f'Nonfinite gradient at step {step}') - optimizer.step() - if ema is not None: - update_ema(ema, model) - logs.append({'step': step, 'loss': float(loss.detach()), - **{k: float(torch.as_tensor(v).detach()) for k, v in details.items()}}) - return logs - - -CENTERS = torch.tensor([[-1.5, -1.5], [-1.5, 1.5], [1.5, -1.5], [1.5, 1.5]]) -DATA_STD = .22 - - -def mixture_data(n, generator=None): - ids = torch.randint(4, (n,), generator=generator) - return CENTERS[ids] + DATA_STD * torch.randn(n, 2, generator=generator) - - -def mixture_posterior(x, t): - """Exact p(X1 | (1-t)X0+tX1=x) for the four Gaussian mixture. - - Return component probabilities, conditional means, scalar variances. - This analytic oracle is used for diagnostics and the GLASS example. - """ - t = time_like(t, x) - a = 1 - t - variance = a.square() + (t * DATA_STD).square() - centers = CENTERS.to(x) - delta = x[:, None, :] - t[:, None, :] * centers - logits = -delta.square().sum(-1) / (2 * variance) - prob = logits.softmax(-1) - gain = t * DATA_STD**2 / variance - means = centers + gain[:, None, :] * delta - posterior_variance = DATA_STD**2 * a.square() / variance - return prob, means, posterior_variance - - -def exact_denoiser(x, t): - p, means, _ = mixture_posterior(x, t) - return (p[..., None] * means).sum(1) - - -def exact_velocity(x, t): - t = time_like(t, x) - # Direct conditional velocity avoids cancellation at t=1. - a = 1 - t - variance = a.square() + (t * DATA_STD).square() - centers = CENTERS.to(x) - delta = x[:, None] - t[:, None] * centers - weights = (-delta.square().sum(-1) / (2 * variance)).softmax(-1) - component_velocity = centers + ((t * DATA_STD**2 - a) / variance)[:, None] * delta - return (weights[..., None] * component_velocity).sum(1) - - -def mixture_metrics(samples): - x = samples.detach().cpu() - distances = (x[:, None] - CENTERS).square().sum(-1) - counts = torch.bincount(distances.argmin(-1), minlength=4).float() - p = counts / len(x) - logp = torch.logsumexp(-distances / (2 * DATA_STD**2), -1) - logp -= math.log(4 * 2 * math.pi * DATA_STD**2) - return {'finite_samples': bool(torch.isfinite(x).all()), - 'mean_distance_to_center': float(distances.min(-1).values.sqrt().mean()), - 'mode_fractions': p.tolist(), 'mode_entropy': float(-(p * p.clamp_min(1e-12).log()).sum()), - 'mean_target_log_density': float(logp.mean())} - - -def load_text(path, variable=False): - lines = [x.strip().split() for x in Path(path).read_text().splitlines() if x.strip()] - if len(lines) < 2: - raise ValueError('Text data need at least two nonempty lines.') - vocab = sorted(set(word for line in lines for word in line)) - lengths = torch.tensor([len(x) for x in lines]) - if not variable and len(set(lengths.tolist())) != 1: - raise ValueError('Fixed-length methods require equal tokens per line; use expanding otherwise.') - if lengths.max() > 32: - raise ValueError('Teaching MLP supports up to 32 positions.') - ids = {word: i for i, word in enumerate(vocab)} - encoded = torch.zeros(len(lines), int(lengths.max()), dtype=torch.long) - for i, line in enumerate(lines): - encoded[i, :len(line)] = torch.tensor([ids[word] for word in line]) - return encoded, lengths, vocab - - -def text_metrics(ids, vocab, lengths=None, reference=None): - lengths = [ids.shape[1]] * len(ids) if lengths is None else lengths.tolist() - texts = [' '.join(vocab[j] for j in row[:length]) for row, length in zip(ids.tolist(), lengths)] - flat = [j for row, length in zip(ids.tolist(), lengths) for j in row[:length]] - counts = torch.bincount(torch.tensor(flat, dtype=torch.long), minlength=len(vocab)).float() - p = counts / counts.sum().clamp_min(1) - metrics = {'unique_fraction': len(set(texts)) / len(texts), - 'token_entropy': float(-(p * p.clamp_min(1e-12).log()).sum()), - 'mean_length': sum(lengths) / len(lengths), 'empty_fraction': lengths.count(0) / len(lengths)} - if reference is not None: - metrics['training_support_fraction'] = sum(x in reference for x in texts) / len(texts) - return texts, metrics - - -def write_json(path, value): - Path(path).write_text(json.dumps(value, indent=2, allow_nan=False) + '\n') diff --git a/lecture_6/continuous.py b/lecture_6/continuous.py deleted file mode 100644 index e0e3b49aa4819dc65946ea73c2ee926385f23648..0000000000000000000000000000000000000000 --- a/lecture_6/continuous.py +++ /dev/null @@ -1,183 +0,0 @@ -"""Flow matching, finite-map identities, Shortcut, MeanFlow, and latent maps. - -# %% 1. Learn local motion before learning an interval. -All times run noise -> data except MeanFlow, whose original backward clock -r <= t runs data at zero -> noise at one during training. -""" -import torch -from torch import nn -from torch.nn import functional as F - -from common import (MapNet, draw_batch, ema_copy, finite_map, interpolate, mlp, - optimize, ordered_times, time_like) - - -def diagonal_loss(model, data): - t = torch.rand(len(data), 1, device=data.device) - x, displacement = interpolate(data, t) - return F.mse_loss(model(x, t, t), displacement) - - -def lagrangian_residual(model, velocity, x, s, t): - # d_t F(s,t,x) = b(t,F(s,t,x)). JVP differentiates only arrival time. - y, dt = torch.func.jvp(lambda end: finite_map(model, x, s, end), - (t,), (torch.ones_like(t),)) - return dt - velocity(y.detach(), t).detach() - - -def eulerian_residual(model, velocity, x, s, t): - # (d_s + b_s dot grad_x)F = 0. No full Jacobian is materialized. - direction = velocity(x, s).detach() - _, residual = torch.func.jvp(lambda z, start: finite_map(model, z, start, t), - (x, s), (direction, torch.ones_like(s))) - return residual - - -def semigroup_loss(model, teacher, x, s, t, context=None): - u = (s + t) / 2 - with torch.no_grad(): - mid = finite_map(teacher, x, s, u, context) - target = finite_map(teacher, mid, u, t, context) - prediction = finite_map(model, x, s, t, context) - return ((prediction - target) / (t - s).clamp_min(.05)).square().mean() - - -def meanflow_loss(model, data): - # %% 2. Average backward velocity satisfies u = v - (t-r) D_t u. - noise = torch.randn_like(data) - r, t, _ = ordered_times(data, 1.) - # Include diagonal examples with nonzero probability, as in the paper. - r = torch.where(torch.rand_like(r) < .75, t, r) - z = (1 - t) * data + t * noise - v = noise - data - average, derivative = torch.func.jvp(model, (z, r, t), - (v, torch.zeros_like(r), torch.ones_like(t))) - target = (v - (t - r) * derivative).detach() - residual = (average - target).square().mean(-1) - # Detached adaptive weighting controls large self-distillation residuals. - weight = (residual.detach() + .01).pow(-.5) - return (weight * residual).mean(), {'mse': residual.mean()} - - -def shortcut_loss(model, teacher, data): - t = torch.rand(len(data), 1, device=data.device) - x, v = interpolate(data, t) - diag = F.mse_loss(model(x, t, t), v) - # d is the half-step. Learn the 2d shortcut from two d shortcuts. - powers = torch.randint(1, 6, (len(data), 1), device=data.device) - d = 2. ** (-powers) - start = torch.rand_like(d) * (1 - 2 * d) - x, _ = interpolate(data, start) - with torch.no_grad(): - first = teacher(x, start, start + d) - second = teacher(x + d * first, start + d, start + 2 * d) - target = .5 * (first + second) - finite = F.mse_loss(model(x, start, start + 2 * d), target) - return diag + finite, {'diagonal': diag, 'shortcut': finite} - - -def train_continuous(method, data, args): - model = MapNet(data.shape[1], args.width).to(data) - teacher_logs = [] - teacher = None - if method in {'fmm-lagrangian', 'fmm-eulerian', 'consistency'}: - teacher = MapNet(data.shape[1], args.width).to(data) - teacher_logs = optimize(teacher, lambda _: (diagonal_loss(teacher, draw_batch(data, args.batch_size)), {}), - args.teacher_steps, args.lr) - teacher.requires_grad_(False) - model.load_state_dict(teacher.state_dict()) - ema = ema_copy(model) - - def objective(step): - batch = draw_batch(data, args.batch_size) - if method == 'meanflow': - return meanflow_loss(model, batch) - if method == 'shortcut': - return shortcut_loss(model, ema, batch) - diag = diagonal_loss(model, batch) - if method == 'flow-matching': - return diag, {'diagonal': diag} - s, t, _ = ordered_times(batch) - x, _ = interpolate(batch, s) - if method == 'fmm-lagrangian': - finite = lagrangian_residual(model, lambda z, a: teacher(z, a, a), x, s, t).square().mean() - elif method == 'fmm-eulerian': - finite = eulerian_residual(model, lambda z, a: teacher(z, a, a), x, s, t).square().mean() - elif method == 'consistency': - # Endpoint consistency distillation on teacher-solved short intervals. - # At t=1 the residual endpoint parametrization is exactly identity. - t = (s + .1).clamp_max(1.) - with torch.no_grad(): - y = integrate_velocity(lambda z, a: teacher(z, a, a), x, s, t, 4) - target = finite_map(ema, y, t, torch.ones_like(t)) - finite = F.mse_loss(finite_map(model, x, s, torch.ones_like(s)), target) - else: - finite = semigroup_loss(model, ema, x, s, t) - weight = min(1., (step + 1) / max(1, args.train_steps // 5)) - return diag + weight * finite, {'diagonal': diag, 'finite': finite} - - logs = optimize(model, objective, args.train_steps, args.lr, ema) - # Use the trained student; EMA supplies fixed bootstrap targets during training. - state = {'model': model.state_dict(), 'dim': data.shape[1]} - if teacher is not None: - state['teacher'] = teacher.state_dict() - return model, state, logs, teacher_logs - - -@torch.no_grad() -def integrate_velocity(velocity, x, s, t, steps): - """Heun integration of a velocity, also supports batch-specific time bounds.""" - s, t = time_like(s, x), time_like(t, x) - h = (t - s) / steps - for i in range(steps): - a = s + i * h - k1 = velocity(x, a) - k2 = velocity(x + h * k1, a + h) - x = x + h * .5 * (k1 + k2) - return x - - -@torch.no_grad() -def sample_continuous(model, method, noise, steps): - if method == 'flow-matching': - return integrate_velocity(lambda z, a: model(z, a, a), noise, 0., 1., steps) - if method == 'consistency': - return finite_map(model, noise, 0., 1.) - x = noise - for i in range(steps): - if method == 'meanflow': - t, r = 1 - i / steps, 1 - (i + 1) / steps - x = x - (t - r) * model(x, r, t) - else: - x = finite_map(model, x, i / steps, (i + 1) / steps) - return x - - -class Autoencoder(nn.Module): - def __init__(self, width): - super().__init__() - self.encoder = mlp(3, 2, width) - self.decoder = mlp(2, 3, width) - - -def embed_surface(x): - return torch.cat([x, .3 * (x[:, :1].square() - x[:, 1:].square())], -1) - - -def train_latent(data, args): - # %% 3. First fit the representation, then freeze it while fitting its flow. - surface = embed_surface(data) - ae = Autoencoder(args.width).to(data) - def objective(_): - batch = draw_batch(surface, args.batch_size) - reconstruction = ae.decoder(ae.encoder(batch)) - return F.mse_loss(reconstruction, batch), {} - ae_logs = optimize(ae, objective, args.teacher_steps, args.lr) - ae.requires_grad_(False) - with torch.no_grad(): - z = ae.encoder(surface) - mean, std = z.mean(0), z.std(0).clamp_min(.05) - z = (z - mean) / std - model, state, logs, _ = train_continuous('self-distill', z, args) - state.update({'autoencoder': ae.state_dict(), 'latent_mean': mean, 'latent_std': std}) - return model, ae, state, logs, ae_logs diff --git a/lecture_6/data/README.md b/lecture_6/data/README.md deleted file mode 100644 index 5a228d477cf851b50054c6768dd229f1f74af43b..0000000000000000000000000000000000000000 --- a/lecture_6/data/README.md +++ /dev/null @@ -1,34 +0,0 @@ -# Data for Lecture 6 - -`phrases.txt` and `variable_text.txt` were generated deterministically with Python -`random.Random(6270)`. Each contains 640 examples. Their vocabulary describes -colored shapes and directions; no external dataset is needed. - -| Color | Shape | Direction | -| --- | --- | --- | -| red | circle | left | -| blue | square | right | -| green | triangle | up | -| gold | star | down | - -The fixed-length corpus has four tokens per line. The variable-length corpus -uses `COLOR SHAPE`, `COLOR SHAPE moves DIRECTION`, or the latter followed by -`slowly` or `twice`. The color, shape, and direction remain correlated. This -makes invalid independent combinations easy to inspect. - -Training uses a seeded shuffle and 80/20 example split. The grammar has a small -support, so examples repeat and both partitions contain the same grammar. -Training-support fraction measures membership in this small support; it is -not a test of open-ended language generalization. - -For a custom corpus, supply a UTF-8 text file with one whitespace-tokenized -sequence per line. Fixed-length methods require equal token counts. The -expanding method accepts variable lengths up to 32. Vocabulary and maximum -length are saved in the checkpoint, so sampling does not need the original file. - -The continuous dataset is sampled from four equally weighted Gaussians centered -at `(−1.5,−1.5)`, `(−1.5,1.5)`, `(1.5,−1.5)`, and `(1.5,1.5)`, with coordinate -standard deviation 0.22. The latent example embeds these points on -`(x, y, 0.3(x²−y²))` before learning its autoencoder. SSFM uses analytic -Ornstein-Uhlenbeck marginals with initial variance one, drift `−x`, and diffusion -coefficient 0.7. diff --git a/lecture_6/data/phrases.txt b/lecture_6/data/phrases.txt deleted file mode 100644 index 73a16f2eceba5a3178ce49355f20136f45c000c3..0000000000000000000000000000000000000000 --- a/lecture_6/data/phrases.txt +++ /dev/null @@ -1,640 +0,0 @@ -green triangle moves up -gold star moves down -green triangle moves up -gold star moves down -blue square moves right -gold star moves down -green triangle moves up -red circle moves left -blue square moves right -green triangle moves up -gold star moves down -gold star moves down -red circle moves left -blue square moves right -blue square moves right -red circle moves left -gold star moves down -blue square moves right -red circle moves left -gold star moves down -green triangle moves up -red circle moves left -red circle moves left -green triangle moves up -green triangle moves up -green triangle moves up -red circle moves left -red circle moves left -green triangle moves up -green triangle moves up -red circle moves left -gold star moves down -red circle moves left -red circle moves left -gold star moves down -gold star moves down -red circle moves left -red circle moves left -red circle moves left -blue square moves right -red circle moves left -blue square moves right -gold star moves down -blue square moves right -red circle moves left -gold star moves down -blue square moves right -blue square moves right -green triangle moves up -red circle moves left -green triangle moves up -blue square moves right -red circle moves left -green triangle moves up -red circle moves left -blue square moves right -green triangle moves up -green triangle moves up -green triangle moves up -red circle moves left -red circle moves left -red circle moves left -gold star moves down -gold star moves down -red circle moves left -gold star moves down -blue square moves right -blue square moves right -red circle moves left -red circle moves left -blue square moves right -green triangle moves up -blue square moves right -blue square moves right -gold star moves down -green triangle moves up -red circle moves left -green triangle moves up -gold star moves down -gold star moves down -gold star moves down -gold star moves down -blue square moves right -gold star moves down -red circle moves left -red circle moves left -green triangle moves up -gold star moves down -blue square moves right -gold star moves down -red circle moves left -red circle moves left -red circle moves left -green triangle moves up -blue square moves right -red circle moves left -blue square moves right -gold star moves down -gold star moves down -red circle moves left -green triangle moves up -green triangle moves up -gold star moves down -red circle moves left -green triangle moves up -green triangle moves up -blue square moves right -green triangle moves up -gold star moves down -green triangle moves up -green triangle moves up -blue square moves right -red circle moves left -blue square moves right -gold star moves down -blue square moves right -blue square moves right -blue square moves right -red circle moves left -blue square moves right -blue square moves right -blue square moves right -gold star moves down -red circle moves left -green triangle moves up -gold star moves down -red circle moves left -green triangle moves up -red circle moves left -green triangle moves up -gold star moves down -blue square moves right -red circle moves left -green triangle moves up -red circle moves left -blue square moves right -green triangle moves up -gold star moves down -gold star moves down -gold star moves down -green triangle moves up -gold star moves down -blue square moves right -blue square moves right -blue square moves right -red circle moves left -gold star moves down -green triangle moves up -green triangle moves up -red circle moves left -blue square moves right -gold star moves down -gold star moves down -gold star moves down -gold star moves down -gold star moves down -gold star moves down -red circle moves left -red circle moves left -blue square moves right -blue square moves right -blue square moves right -gold star moves down -blue square moves right -red circle moves left -blue square moves right -gold star moves down -green triangle moves up -blue square moves right -green triangle moves up -gold star moves down -blue square moves right -green triangle moves up -red circle moves left -blue square moves right -green triangle moves up -gold star moves down -gold star moves down -green triangle moves up -blue square moves right -red circle moves left -gold star moves down -gold star moves down -red circle moves left -red circle moves left -green triangle moves up -blue square moves right -gold star moves down -gold star moves down -green triangle moves up -red circle moves left -green triangle moves up -green triangle moves up -red circle moves left -red circle moves left -gold star moves down -red circle moves left -red circle moves left -green triangle moves up -blue square moves right -green triangle moves up -blue square moves right -red circle moves left -gold star moves down -blue square moves right -green triangle moves up -red circle moves left -red circle moves left -gold star moves down -green triangle moves up -blue square moves right -red circle moves left -gold star moves down -green triangle moves up -gold star moves down -red circle moves left -blue square moves right -red circle moves left -blue square moves right -green triangle moves up -blue square moves right -green triangle moves up -green triangle moves up -gold star moves down -blue square moves right -blue square moves right -blue square moves right -green triangle moves up -blue square moves right -blue square moves right -blue square moves right -green triangle moves up -blue square moves right -green triangle moves up -red circle moves left -blue square moves right -green triangle moves up -red circle moves left -blue square moves right -gold star moves down -gold star moves down -red circle moves left -red circle moves left -blue square moves right -red circle moves left -red circle moves left -gold star moves down -gold star moves down -green triangle moves up -red circle moves left -gold star moves down -red circle moves left -red circle moves left -gold star moves down -gold star moves down -gold star moves down -red circle moves left -green triangle moves up -gold star moves down -green triangle moves up -blue square moves right -gold star moves down -green triangle moves up -red circle moves left -green triangle moves up -red circle moves left -green triangle moves up -green triangle moves up -blue square moves right -red circle moves left -blue square moves right -gold star moves down -gold star moves down -blue square moves right -blue square moves right -red circle moves left -red circle moves left -blue square moves right -green triangle moves up -green triangle moves up -blue square moves right -red circle moves left -green triangle moves up -blue square moves right -red circle moves left -green triangle moves up -gold star moves down -red circle moves left -red circle moves left -blue square moves right -gold star moves down -red circle moves left -red circle moves left -blue square moves right -red circle moves left -gold star moves down -red circle moves left -blue square moves right -red circle moves left -gold star moves down -green triangle moves up -blue square moves right -gold star moves down -gold star moves down -gold star moves down -green triangle moves up -gold star moves down -red circle moves left -gold star moves down -blue square moves right -red circle moves left -green triangle moves up -green triangle moves up -gold star moves down -green triangle moves up -gold star moves down -green triangle moves up -red circle moves left -blue square moves right -red circle moves left -red circle moves left -blue square moves right -green triangle moves up -red circle moves left -red circle moves left -gold star moves down -green triangle moves up -gold star moves down -green triangle moves up -green triangle moves up -gold star moves down -red circle moves left -red circle moves left -green triangle moves up -red circle moves left -green triangle moves up -blue square moves right -blue square moves right -green triangle moves up -red circle moves left -green triangle moves up -green triangle moves up -blue square moves right -green triangle moves up -gold star moves down -blue square moves right -green triangle moves up -green triangle moves up -blue square moves right -red circle moves left -gold star moves down -red circle moves left -red circle moves left -gold star moves down -blue square moves right -green triangle moves up -red circle moves left -green triangle moves up -green triangle moves up -green triangle moves up -gold star moves down -green triangle moves up -gold star moves down -blue square moves right -blue square moves right -green triangle moves up -green triangle moves up -blue square moves right -red circle moves left -red circle moves left -blue square moves right -blue square moves right -red circle moves left -red circle moves left -red circle moves left -red circle moves left -blue square moves right -green triangle moves up -blue square moves right -gold star moves down -blue square moves right -blue square moves right -blue square moves right -green triangle moves up -red circle moves left -blue square moves right -gold star moves down -green triangle moves up -gold star moves down -gold star moves down -blue square moves right -gold star moves down -blue square moves right -gold star moves down -gold star moves down -gold star moves down -blue square moves right -gold star moves down -green triangle moves up -gold star moves down -blue square moves right -gold star moves down -blue square moves right -red circle moves left -gold star moves down -green triangle moves up -green triangle moves up -green triangle moves up -red circle moves left -green triangle moves up -blue square moves right -blue square moves right -green triangle moves up -gold star moves down -green triangle moves up -red circle moves left -gold star moves down -gold star moves down -blue square moves right -red circle moves left -red circle moves left -blue square moves right -blue square moves right -gold star moves down -gold star moves down -blue square moves right -blue square moves right -gold star moves down -blue square moves right -gold star moves down -gold star moves down -red circle moves left -red circle moves left -blue square moves right -blue square moves right -green triangle moves up -green triangle moves up -gold star moves down -green triangle moves up -green triangle moves up -green triangle moves up -blue square moves right -green triangle moves up -gold star moves down -red circle moves left -red circle moves left -gold star moves down -gold star moves down -gold star moves down -blue square moves right -blue square moves right -green triangle moves up -gold star moves down -blue square moves right -green triangle moves up -gold star moves down -gold star moves down -blue square moves right -blue square moves right -blue square moves right -gold star moves down -red circle moves left -blue square moves right -green triangle moves up -green triangle moves up -green triangle moves up -blue square moves right -gold star moves down -blue square moves right -gold star moves down -red circle moves left -blue square moves right -green triangle moves up -green triangle moves up -green triangle moves up -green triangle moves up -gold star moves down -blue square moves right -red circle moves left -green triangle moves up -blue square moves right -blue square moves right -green triangle moves up -green triangle moves up -red circle moves left -blue square moves right -green triangle moves up -red circle moves left -blue square moves right -gold star moves down -gold star moves down -blue square moves right -blue square moves right -red circle moves left -gold star moves down -blue square moves right -green triangle moves up -blue square moves right -green triangle moves up -gold star moves down -green triangle moves up -green triangle moves up -green triangle moves up -blue square moves right -gold star moves down -green triangle moves up -gold star moves down -red circle moves left -blue square moves right -gold star moves down -gold star moves down -red circle moves left -gold star moves down -green triangle moves up -gold star moves down -red circle moves left -red circle moves left -blue square moves right -gold star moves down -green triangle moves up -blue square moves right -red circle moves left -green triangle moves up -red circle moves left -green triangle moves up -blue square moves right -red circle moves left -green triangle moves up -gold star moves down -green triangle moves up -green triangle moves up -gold star moves down -green triangle moves up -blue square moves right -green triangle moves up -red circle moves left -green triangle moves up -green triangle moves up -gold star moves down -red circle moves left -green triangle moves up -green triangle moves up -gold star moves down -green triangle moves up -blue square moves right -green triangle moves up -green triangle moves up -green triangle moves up -blue square moves right -red circle moves left -blue square moves right -red circle moves left -green triangle moves up -green triangle moves up -red circle moves left -green triangle moves up -blue square moves right -red circle moves left -red circle moves left -green triangle moves up -gold star moves down -blue square moves right -blue square moves right -red circle moves left -green triangle moves up -red circle moves left -green triangle moves up -red circle moves left -red circle moves left -blue square moves right -red circle moves left -red circle moves left -red circle moves left -gold star moves down -red circle moves left -green triangle moves up -green triangle moves up -green triangle moves up -red circle moves left -gold star moves down -gold star moves down -gold star moves down -red circle moves left -gold star moves down -green triangle moves up -green triangle moves up -blue square moves right -green triangle moves up -gold star moves down -gold star moves down -green triangle moves up -gold star moves down -red circle moves left -gold star moves down -green triangle moves up -blue square moves right -red circle moves left -green triangle moves up -green triangle moves up -red circle moves left -green triangle moves up -green triangle moves up -red circle moves left -red circle moves left -gold star moves down -blue square moves right -red circle moves left -gold star moves down -gold star moves down -blue square moves right -red circle moves left -blue square moves right -blue square moves right -blue square moves right -red circle moves left -blue square moves right -blue square moves right -green triangle moves up -blue square moves right -green triangle moves up -blue square moves right -gold star moves down -gold star moves down -blue square moves right -red circle moves left -red circle moves left -blue square moves right -red circle moves left -green triangle moves up -red circle moves left -red circle moves left -red circle moves left -gold star moves down -green triangle moves up -gold star moves down -red circle moves left -blue square moves right -red circle moves left -green triangle moves up -green triangle moves up diff --git a/lecture_6/data/variable_text.txt b/lecture_6/data/variable_text.txt deleted file mode 100644 index 7eb78e7b02d9510af0ea0f701ddf2fabcc54edda..0000000000000000000000000000000000000000 --- a/lecture_6/data/variable_text.txt +++ /dev/null @@ -1,640 +0,0 @@ -green triangle moves up -gold star moves down twice -green triangle moves up -gold star -blue square moves right -gold star -green triangle -red circle moves left twice -blue square -green triangle moves up slowly -gold star moves down twice -gold star moves down slowly -red circle moves left -blue square moves right slowly -blue square moves right twice -red circle moves left -gold star moves down slowly -blue square moves right twice -red circle -gold star moves down -green triangle -red circle -red circle moves left twice -green triangle moves up twice -green triangle moves up -green triangle moves up slowly -red circle -red circle moves left slowly -green triangle -green triangle -red circle moves left twice -gold star moves down twice -red circle moves left twice -red circle -gold star moves down -gold star moves down -red circle -red circle moves left slowly -red circle moves left -blue square moves right twice -red circle moves left slowly -blue square -gold star -blue square moves right twice -red circle moves left -gold star moves down -blue square moves right -blue square moves right -green triangle -red circle -green triangle moves up slowly -blue square moves right -red circle -green triangle moves up slowly -red circle moves left -blue square moves right slowly -green triangle -green triangle moves up slowly -green triangle moves up slowly -red circle moves left slowly -red circle moves left slowly -red circle moves left slowly -gold star moves down -gold star -red circle moves left -gold star moves down -blue square moves right slowly -blue square moves right -red circle moves left -red circle -blue square moves right twice -green triangle moves up slowly -blue square moves right slowly -blue square -gold star moves down -green triangle moves up slowly -red circle moves left -green triangle -gold star -gold star moves down -gold star moves down slowly -gold star moves down twice -blue square moves right -gold star moves down -red circle moves left -red circle -green triangle -gold star moves down -blue square moves right slowly -gold star -red circle moves left -red circle moves left slowly -red circle -green triangle moves up slowly -blue square moves right -red circle moves left twice -blue square moves right twice -gold star moves down -gold star moves down twice -red circle moves left twice -green triangle moves up slowly -green triangle -gold star moves down twice -red circle moves left -green triangle moves up slowly -green triangle moves up -blue square -green triangle -gold star moves down -green triangle moves up twice -green triangle moves up slowly -blue square -red circle moves left slowly -blue square moves right twice -gold star -blue square moves right -blue square -blue square -red circle -blue square moves right -blue square moves right -blue square moves right slowly -gold star moves down -red circle -green triangle moves up slowly -gold star moves down -red circle moves left -green triangle moves up -red circle moves left -green triangle moves up -gold star -blue square moves right -red circle -green triangle moves up -red circle moves left twice -blue square -green triangle -gold star -gold star moves down twice -gold star -green triangle moves up -gold star -blue square moves right slowly -blue square -blue square -red circle -gold star moves down slowly -green triangle moves up -green triangle -red circle moves left -blue square moves right slowly -gold star moves down -gold star -gold star moves down twice -gold star moves down slowly -gold star moves down slowly -gold star moves down -red circle moves left -red circle -blue square moves right -blue square moves right slowly -blue square -gold star -blue square -red circle -blue square moves right -gold star moves down slowly -green triangle moves up -blue square moves right -green triangle moves up -gold star -blue square moves right slowly -green triangle moves up twice -red circle moves left slowly -blue square moves right slowly -green triangle moves up twice -gold star moves down -gold star -green triangle moves up slowly -blue square -red circle moves left slowly -gold star -gold star moves down -red circle moves left -red circle moves left -green triangle moves up -blue square -gold star moves down -gold star moves down -green triangle moves up twice -red circle moves left -green triangle -green triangle -red circle moves left slowly -red circle moves left -gold star moves down slowly -red circle moves left -red circle moves left twice -green triangle moves up slowly -blue square moves right -green triangle moves up slowly -blue square -red circle moves left twice -gold star -blue square -green triangle -red circle moves left slowly -red circle -gold star moves down slowly -green triangle -blue square moves right -red circle -gold star -green triangle moves up twice -gold star moves down twice -red circle -blue square moves right slowly -red circle moves left slowly -blue square -green triangle -blue square moves right -green triangle moves up twice -green triangle moves up slowly -gold star moves down -blue square -blue square moves right -blue square moves right -green triangle -blue square moves right slowly -blue square moves right slowly -blue square -green triangle moves up slowly -blue square moves right -green triangle moves up -red circle moves left slowly -blue square moves right slowly -green triangle -red circle -blue square moves right -gold star moves down twice -gold star moves down -red circle moves left -red circle moves left slowly -blue square moves right slowly -red circle -red circle moves left -gold star -gold star moves down slowly -green triangle moves up -red circle moves left -gold star moves down twice -red circle moves left -red circle moves left slowly -gold star moves down -gold star moves down slowly -gold star -red circle moves left twice -green triangle -gold star moves down slowly -green triangle -blue square moves right -gold star moves down slowly -green triangle moves up twice -red circle -green triangle -red circle moves left twice -green triangle -green triangle moves up slowly -blue square moves right -red circle -blue square -gold star moves down -gold star -blue square -blue square moves right slowly -red circle moves left -red circle moves left slowly -blue square moves right -green triangle moves up -green triangle moves up -blue square moves right -red circle moves left twice -green triangle moves up -blue square moves right twice -red circle -green triangle -gold star moves down twice -red circle -red circle moves left -blue square moves right -gold star moves down twice -red circle moves left slowly -red circle moves left slowly -blue square moves right -red circle moves left -gold star moves down -red circle moves left slowly -blue square -red circle moves left twice -gold star moves down -green triangle -blue square -gold star moves down -gold star moves down -gold star -green triangle moves up -gold star moves down -red circle moves left -gold star moves down -blue square -red circle -green triangle -green triangle moves up -gold star moves down -green triangle moves up slowly -gold star moves down -green triangle moves up twice -red circle -blue square -red circle moves left twice -red circle moves left slowly -blue square moves right -green triangle moves up -red circle moves left -red circle -gold star moves down -green triangle moves up slowly -gold star -green triangle moves up -green triangle moves up -gold star moves down slowly -red circle -red circle moves left -green triangle moves up twice -red circle -green triangle moves up -blue square moves right -blue square moves right slowly -green triangle moves up slowly -red circle moves left -green triangle moves up twice -green triangle moves up slowly -blue square moves right slowly -green triangle moves up slowly -gold star moves down twice -blue square -green triangle moves up -green triangle moves up slowly -blue square moves right twice -red circle moves left slowly -gold star moves down -red circle moves left twice -red circle moves left slowly -gold star moves down twice -blue square moves right -green triangle moves up -red circle moves left -green triangle moves up -green triangle -green triangle -gold star -green triangle moves up -gold star moves down -blue square moves right twice -blue square moves right slowly -green triangle -green triangle moves up slowly -blue square moves right twice -red circle -red circle moves left slowly -blue square moves right twice -blue square -red circle -red circle moves left -red circle moves left -red circle moves left -blue square moves right twice -green triangle moves up slowly -blue square moves right slowly -gold star moves down slowly -blue square moves right -blue square -blue square moves right slowly -green triangle moves up -red circle moves left -blue square moves right slowly -gold star -green triangle moves up -gold star moves down slowly -gold star moves down -blue square -gold star moves down -blue square moves right slowly -gold star moves down twice -gold star moves down twice -gold star moves down slowly -blue square -gold star moves down -green triangle -gold star moves down twice -blue square moves right slowly -gold star moves down -blue square moves right -red circle moves left twice -gold star moves down -green triangle -green triangle -green triangle -red circle -green triangle -blue square -blue square moves right -green triangle moves up slowly -gold star moves down -green triangle moves up -red circle -gold star -gold star moves down -blue square moves right twice -red circle moves left -red circle -blue square moves right -blue square moves right twice -gold star moves down -gold star moves down twice -blue square moves right -blue square moves right twice -gold star -blue square -gold star moves down slowly -gold star moves down slowly -red circle moves left -red circle moves left -blue square moves right -blue square moves right -green triangle moves up twice -green triangle moves up -gold star moves down -green triangle moves up -green triangle moves up -green triangle -blue square moves right slowly -green triangle moves up slowly -gold star moves down -red circle -red circle moves left twice -gold star moves down twice -gold star -gold star -blue square moves right twice -blue square moves right twice -green triangle moves up -gold star moves down -blue square moves right twice -green triangle moves up slowly -gold star moves down slowly -gold star moves down -blue square moves right twice -blue square moves right -blue square moves right slowly -gold star moves down twice -red circle moves left twice -blue square -green triangle moves up twice -green triangle -green triangle -blue square moves right -gold star moves down twice -blue square moves right -gold star -red circle moves left twice -blue square moves right -green triangle moves up -green triangle -green triangle moves up -green triangle moves up -gold star -blue square moves right -red circle -green triangle moves up -blue square moves right twice -blue square moves right slowly -green triangle moves up -green triangle -red circle -blue square -green triangle moves up -red circle moves left -blue square -gold star -gold star -blue square moves right -blue square moves right slowly -red circle -gold star -blue square -green triangle -blue square moves right -green triangle moves up -gold star moves down twice -green triangle moves up -green triangle moves up slowly -green triangle -blue square moves right -gold star moves down slowly -green triangle -gold star -red circle moves left -blue square moves right -gold star moves down -gold star -red circle moves left twice -gold star moves down twice -green triangle moves up slowly -gold star moves down -red circle moves left -red circle moves left slowly -blue square moves right twice -gold star moves down -green triangle moves up slowly -blue square -red circle moves left twice -green triangle moves up twice -red circle moves left -green triangle -blue square -red circle -green triangle -gold star moves down twice -green triangle moves up twice -green triangle -gold star moves down -green triangle moves up -blue square moves right twice -green triangle -red circle -green triangle moves up twice -green triangle moves up -gold star moves down -red circle moves left -green triangle moves up twice -green triangle -gold star -green triangle moves up twice -blue square moves right slowly -green triangle -green triangle moves up -green triangle moves up -blue square -red circle moves left -blue square moves right twice -red circle -green triangle -green triangle moves up -red circle -green triangle moves up twice -blue square moves right slowly -red circle moves left slowly -red circle moves left twice -green triangle moves up slowly -gold star moves down slowly -blue square moves right -blue square moves right twice -red circle -green triangle moves up slowly -red circle -green triangle -red circle moves left slowly -red circle -blue square moves right slowly -red circle moves left slowly -red circle -red circle moves left twice -gold star -red circle moves left twice -green triangle moves up twice -green triangle moves up -green triangle moves up -red circle -gold star moves down slowly -gold star -gold star -red circle -gold star moves down twice -green triangle moves up -green triangle -blue square -green triangle moves up slowly -gold star moves down -gold star -green triangle -gold star -red circle moves left -gold star moves down -green triangle -blue square moves right slowly -red circle moves left -green triangle moves up -green triangle moves up slowly -red circle moves left -green triangle moves up twice -green triangle moves up -red circle moves left twice -red circle moves left -gold star moves down twice -blue square moves right -red circle moves left twice -gold star -gold star -blue square -red circle -blue square -blue square -blue square moves right twice -red circle moves left -blue square moves right slowly -blue square moves right slowly -green triangle -blue square moves right slowly -green triangle moves up twice -blue square -gold star moves down -gold star moves down twice -blue square moves right -red circle -red circle moves left twice -blue square moves right twice -red circle moves left -green triangle moves up slowly -red circle -red circle moves left -red circle moves left twice -gold star moves down slowly -green triangle moves up twice -gold star moves down -red circle -blue square -red circle -green triangle -green triangle diff --git a/lecture_6/examples/categorical.py b/lecture_6/examples/categorical.py deleted file mode 100644 index 6d2d03b25da05cb26c6073fd26485e62b3013925..0000000000000000000000000000000000000000 --- a/lecture_6/examples/categorical.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete categorical training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'categorical', *sys.argv[1:]]) diff --git a/lecture_6/examples/consistency.py b/lecture_6/examples/consistency.py deleted file mode 100644 index bb0297d5886d96e2cbda850f77580a792aa898fa..0000000000000000000000000000000000000000 --- a/lecture_6/examples/consistency.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete consistency training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'consistency', *sys.argv[1:]]) diff --git a/lecture_6/examples/diamond.py b/lecture_6/examples/diamond.py deleted file mode 100644 index 47a9835aad9e90f405a463c6a4d6e8e3386f136d..0000000000000000000000000000000000000000 --- a/lecture_6/examples/diamond.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete diamond training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'diamond', *sys.argv[1:]]) diff --git a/lecture_6/examples/discrete_esd.py b/lecture_6/examples/discrete_esd.py deleted file mode 100644 index a4ed0e4c930b151f72c7f2c005915d4fbb0f116f..0000000000000000000000000000000000000000 --- a/lecture_6/examples/discrete_esd.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete discrete-esd training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'discrete-esd', *sys.argv[1:]]) diff --git a/lecture_6/examples/discrete_lsd.py b/lecture_6/examples/discrete_lsd.py deleted file mode 100644 index e5efe11236e63d6cfa58c34ca3dbc792b14a2ce9..0000000000000000000000000000000000000000 --- a/lecture_6/examples/discrete_lsd.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete discrete-lsd training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'discrete-lsd', *sys.argv[1:]]) diff --git a/lecture_6/examples/expanding.py b/lecture_6/examples/expanding.py deleted file mode 100644 index 45822d3c22715cb948be713240e06c64745b334b..0000000000000000000000000000000000000000 --- a/lecture_6/examples/expanding.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete expanding training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'expanding', *sys.argv[1:]]) diff --git a/lecture_6/examples/flow_matching.py b/lecture_6/examples/flow_matching.py deleted file mode 100644 index 37d43941670493963b672e97d92c54e688ec68ca..0000000000000000000000000000000000000000 --- a/lecture_6/examples/flow_matching.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete flow-matching training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'flow-matching', *sys.argv[1:]]) diff --git a/lecture_6/examples/fmlm.py b/lecture_6/examples/fmlm.py deleted file mode 100644 index f970b0e64f22eb58a34abd57fcb53064dad714ef..0000000000000000000000000000000000000000 --- a/lecture_6/examples/fmlm.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete fmlm training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'fmlm', *sys.argv[1:]]) diff --git a/lecture_6/examples/fmm_eulerian.py b/lecture_6/examples/fmm_eulerian.py deleted file mode 100644 index 22099957c4d58ea5db40eb524a2aed3108f56543..0000000000000000000000000000000000000000 --- a/lecture_6/examples/fmm_eulerian.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete fmm-eulerian training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'fmm-eulerian', *sys.argv[1:]]) diff --git a/lecture_6/examples/fmm_lagrangian.py b/lecture_6/examples/fmm_lagrangian.py deleted file mode 100644 index d42ae727efdfecf8577f583a4e42ee0eb8040f2a..0000000000000000000000000000000000000000 --- a/lecture_6/examples/fmm_lagrangian.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete fmm-lagrangian training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'fmm-lagrangian', *sys.argv[1:]]) diff --git a/lecture_6/examples/latent.py b/lecture_6/examples/latent.py deleted file mode 100644 index 5b57c05aab7fd082eb050a73a75deda2c715cde0..0000000000000000000000000000000000000000 --- a/lecture_6/examples/latent.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete latent training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'latent', *sys.argv[1:]]) diff --git a/lecture_6/examples/meanflow.py b/lecture_6/examples/meanflow.py deleted file mode 100644 index 39867b02e3af7b45a53bc92802d47b4aa2294b62..0000000000000000000000000000000000000000 --- a/lecture_6/examples/meanflow.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete meanflow training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'meanflow', *sys.argv[1:]]) diff --git a/lecture_6/examples/meta.py b/lecture_6/examples/meta.py deleted file mode 100644 index 7b49369834e9bc1607d55e70f27124d8848cdb77..0000000000000000000000000000000000000000 --- a/lecture_6/examples/meta.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete meta training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'meta', *sys.argv[1:]]) diff --git a/lecture_6/examples/self_distill.py b/lecture_6/examples/self_distill.py deleted file mode 100644 index 609d36bb1ad1b50c6df3aa3aead32f1ec7eb0b9b..0000000000000000000000000000000000000000 --- a/lecture_6/examples/self_distill.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete self-distill training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'self-distill', *sys.argv[1:]]) diff --git a/lecture_6/examples/shortcut.py b/lecture_6/examples/shortcut.py deleted file mode 100644 index 4eeb2bcfc2f5399499407c1f174d6417772182fa..0000000000000000000000000000000000000000 --- a/lecture_6/examples/shortcut.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete shortcut training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'shortcut', *sys.argv[1:]]) diff --git a/lecture_6/examples/ssfm.py b/lecture_6/examples/ssfm.py deleted file mode 100644 index 0fcb46dd6fd1dc064683a531b736f90f4fd4ba38..0000000000000000000000000000000000000000 --- a/lecture_6/examples/ssfm.py +++ /dev/null @@ -1,8 +0,0 @@ -#!/usr/bin/env python3 -"""Run the complete ssfm training and sampling example.""" -import sys -from pathlib import Path -sys.path.insert(0, str(Path(__file__).resolve().parents[1])) -from run import main -if __name__ == '__main__': - main(['--method', 'ssfm', *sys.argv[1:]]) diff --git a/lecture_6/expanding.py b/lecture_6/expanding.py deleted file mode 100644 index cb25f4143318c74bf8fe0eceeb0aa0f181e4ac21..0000000000000000000000000000000000000000 --- a/lecture_6/expanding.py +++ /dev/null @@ -1,192 +0,0 @@ -"""Expanding Flow Maps on variable-length text with learned gap insertions. - -Based on paper Algorithms 2-4. The state carries birth times alongside token -coordinates. Finite transport uses local clocks on a shared destination canvas. -The insertion head learns both remaining-count and interval-count expectations. -""" -import torch -from torch import nn -from torch.nn import functional as F - -from common import ema_copy, mlp, optimize - - -class ExpandingNet(nn.Module): - def __init__(self, length, vocab, width): - super().__init__() - self.length, self.vocab = length, vocab - # Every token contributes state, two local clocks, birth, and active mask. - self.denoiser = mlp(length * (vocab+4), length*vocab, width) - self.insertion = mlp(length*(vocab+2)+2, length+1, width) - - def predict(self, x, local_s, local_t, births, mask): - features = torch.cat([x, local_s[..., None], local_t[..., None], - births[..., None], mask[..., None].to(x)], -1) - return self.denoiser(features.flatten(1)).reshape(-1, self.length, self.vocab) - - def counts(self, x, births, mask, s, t, diagonal=False): - features = torch.cat([x, births[..., None], mask[..., None].to(x)], -1).flatten(1) - features = torch.cat([features, s, t], -1) - remaining = F.softplus(self.insertion(features)) + 1e-5 - # Linear birth CDF: rho=(t-s)/(1-s). The diagonal predicts the - # remaining count; off-diagonal predictions include the interval factor. - return remaining if diagonal else (t-s)/(1-s)*remaining - - -def local_clock(t, births): - return ((t-births)/(1-births)).clamp(0, 1) - - -def local_map(model, x, a, b, births, mask): - p = model.predict(x, a, b, births, mask).softmax(-1) - eta = ((b-a)/(1-a).clamp_min(1e-6))[..., None] - y = (x + eta*(p-x)) * mask[..., None] - return y, p - - -def gap_counts(active_indices, born_indices, length): - """Assign each missing index to a gap while retaining original token order.""" - result = torch.zeros(len(active_indices)+1, device=active_indices.device) - if len(born_indices): - gap = torch.searchsorted(active_indices, born_indices) - result.scatter_add_(0, gap, torch.ones_like(gap, dtype=result.dtype)) - return result - - -def compact(full, births, selected, max_length): - x = full.new_zeros(max_length, full.shape[-1]) - b = births.new_zeros(max_length) - mask = torch.zeros(max_length, dtype=torch.bool, device=full.device) - n = len(selected) - x[:n], b[:n], mask[:n] = full[selected], births[selected], True - return x, b, mask - - -def count_divergence(target, mean): - # Poisson NLL up to target-only constants. Finite at target=0. - # Avoid xlogy(0,0)'s undefined intermediate derivative in autodiff. - return mean - target + target * (target.clamp_min(1e-12).log()-mean.log()) - - -def expansion_batch(ids, lengths, vocab, batch_size): - index = torch.randint(len(ids), (batch_size,), device=ids.device) - fields = {k: [] for k in ['xs','bs','ms','xe','be','me','ys','ye','missing','interval','gapmask']} - pairs = (.96*torch.rand(batch_size, 2, device=ids.device)).sort(-1).values - s, t = pairs[:, :1], pairs[:, 1:] - t = torch.maximum(t, s+1e-4) - L = ids.shape[1] - for j, row in enumerate(index): - n = int(lengths[row]) - clean = F.one_hot(ids[row, :n], vocab).float() - births = .999*torch.rand(n, device=ids.device) - noise = torch.randn_like(clean) - clock = local_clock(s[j], births) - full = (1-clock[:, None])*noise+clock[:, None]*clean - active = torch.where(births <= s[j])[0] - later = torch.where(births <= t[j])[0] - xs, bs, ms = compact(full, births, active, L) - xe, be, me = compact(full, births, later, L) - ys, _, _ = compact(clean, births, active, L) - ye, _, _ = compact(clean, births, later, L) - missing = full.new_zeros(L+1) - interval = full.new_zeros(L+1) - gm = torch.arange(L+1, device=ids.device) <= len(active) - missing[gm] = gap_counts(active, torch.where(births>s[j])[0], n) - interval[gm] = gap_counts(active, torch.where((births>s[j]) & (births<=t[j]))[0], n) - for k, value in zip(fields, [xs,bs,ms,xe,be,me,ys,ye,missing,interval,gm]): - fields[k].append(value) - return {k: torch.stack(v) for k,v in fields.items()}, s, t - - -def train_expanding(ids, lengths, vocab, args): - model = ExpandingNet(ids.shape[1], len(vocab), args.width).to(ids.device) - teacher = ema_copy(model) - def objective(step): - b, s, t = expansion_batch(ids, lengths, len(vocab), args.batch_size) - a = local_clock(s, b['bs']) - logits = model.predict(b['xs'], a, a, b['bs'], b['ms']) - ce = -(b['ys']*logits.log_softmax(-1)).sum(-1) - diagonal = (ce*b['ms']).sum()/b['ms'].sum().clamp_min(1) - # Lift both paths to the same destination canvas, with the same noise. - la, lb = local_clock(s,b['be']), local_clock(t,b['be']) - lu = local_clock((s+t)/2,b['be']) - with torch.no_grad(): - middle, p1 = local_map(teacher,b['xe'],la,lu,b['be'],b['me']) - _, p2 = local_map(teacher,middle,lu,lb,b['be'],b['me']) - gamma = ((1-lb)*(lu-la)/((1-lu)*(lb-la)).clamp_min(1e-6)).clamp(0,1) - target = gamma[...,None]*p1+(1-gamma[...,None])*p2 - logits = model.predict(b['xe'],la,lb,b['be'],b['me']) - kl = F.kl_div(logits.log_softmax(-1),target,reduction='none').sum(-1) - finite = (kl*b['me']).sum()/b['me'].sum().clamp_min(1) - m = model.counts(b['xs'],b['bs'],b['ms'],s,s,diagonal=True) - interval = model.counts(b['xs'],b['bs'],b['ms'],s,t) - count = count_divergence(b['missing'],m)+count_divergence(b['interval'],interval) - insertion = (count*b['gapmask']).sum()/b['gapmask'].sum() - ramp = min(1.,(step+1)/max(1,args.train_steps//4)) - return diagonal+ramp*finite+insertion, {'diagonal_ce':diagonal,'finite':finite,'insertion':insertion} - logs = optimize(model,objective,args.train_steps,args.lr,teacher) - return model, {'model':model.state_dict(),'length':ids.shape[1],'vocab':vocab}, logs - - -def bounded_counts(means, budget): - """Independent binomial proposals, then left-to-right joint budget capping. - - Before capping, each proposal has its predicted mean after [0,budget] - clipping. Conditional means alone do not determine the joint count law. - """ - if budget == 0: - return torch.zeros_like(means, dtype=torch.long), 0 - raw = torch.distributions.Binomial(budget, probs=means.clamp(0,budget)/budget).sample().long() - kept = raw.clone() - remaining = budget - for i in range(len(kept)): - kept[i] = min(int(kept[i]),remaining) - remaining -= int(kept[i]) - return kept, int((raw-kept).sum()) - - -def insert_tokens(state, births, counts, noise, birth_time): - pieces, times, offset = [], [], 0 - for gap, count in enumerate(counts.tolist()): - if count: - pieces.append(noise[offset:offset+count]) - times.append(births.new_full((count,),birth_time)) - offset += count - if gap < len(state): - pieces.append(state[gap:gap+1]); times.append(births[gap:gap+1]) - if offset != len(noise): - raise ValueError('Noise rows must match insertion counts.') - if not pieces: - return state, births - return torch.cat(pieces),torch.cat(times) - - -@torch.no_grad() -def sample_expanding(model,count,steps,device): - result = torch.zeros(count,model.length,dtype=torch.long,device=device) - lengths = torch.zeros(count,dtype=torch.long,device=device) - capped = 0 - traces = [] - for j in range(count): - x = torch.zeros(0,model.vocab,device=device) - births = torch.zeros(0,device=device) - trace = [0] - for k in range(steps): - s,t = k/steps,(k+1)/steps - padded,bt,mask = compact(x,births,torch.arange(len(x),device=device),model.length) - start,end = torch.tensor([[s]],device=device),torch.tensor([[t]],device=device) - means = model.counts(padded[None],bt[None],mask[None],start,end)[0,:len(x)+1] - counts,discarded = bounded_counts(means,model.length-len(x));capped+=discarded - noise = torch.randn(int(counts.sum()),model.vocab,device=device) - # Finite expand-then-transport convention: new coordinates enter at s. - x,births = insert_tokens(x,births,counts,noise,s) - if len(x): - padded,bt,mask=compact(x,births,torch.arange(len(x),device=device),model.length) - y,_=local_map(model,padded[None],local_clock(start,bt[None]), - local_clock(end,bt[None]),bt[None],mask[None]) - x=y[0,:len(x)] - trace.append(len(x)) - lengths[j]=len(x) - if len(x):result[j,:len(x)]=x.argmax(-1) - traces.append(trace) - return result,lengths,{'capped_proposed_insertions':capped,'length_trajectories':traces} diff --git a/lecture_6/lecture_core.py b/lecture_6/lecture_core.py deleted file mode 100644 index 195e1b55f42065d1870aa945ba224e2fcffb335b..0000000000000000000000000000000000000000 --- a/lecture_6/lecture_core.py +++ /dev/null @@ -1,14 +0,0 @@ -"""Stable imports connecting the lecture's code walkthroughs to full methods. - -Implementation lives in the named modules so training and sampling use the -same functions demonstrated on the slides. See SLIDE_CODE_MAP.md. -""" -from common import finite_map -from continuous import (diagonal_loss,lagrangian_residual,eulerian_residual, - semigroup_loss,meanflow_loss,shortcut_loss) -from categorical import (categorical_map,composition_target,corrected_logit_teacher, - probability_kl,DecodingClock) -from posterior import (glass_denoiser,posterior_samples,posterior_value, - fine_tune_surrogate,weighted_diamond_samples) -from expanding import (local_clock,local_map,count_divergence,bounded_counts,insert_tokens) -from stochastic import chen_two,sample_coefficients diff --git a/lecture_6/numerical_examples.py b/lecture_6/numerical_examples.py deleted file mode 100644 index d9be9ffcfb6229e59240fe7d33f79a462fb4648f..0000000000000000000000000000000000000000 --- a/lecture_6/numerical_examples.py +++ /dev/null @@ -1,187 +0,0 @@ -"""CIS 6270 Lecture 6. Small, reproducible flow-map teaching experiments. - -Run: python numerical_examples.py --output outputs/numerical -Dependencies: Python 3.11+, numpy, torch. -The toy experiments validate the lecture mathematics. They do not reproduce -the large-scale training or benchmark claims of the cited papers. -""" -import argparse, json, math, pathlib -import numpy as np -import torch -from torch import nn -from torch.nn import functional as F - -def flow(model, x, s, t): - """Residual map with an exactly identity diagonal.""" - return x + (t-s)*model(torch.cat([x,s,t],dim=-1)) - -def train_scalar(output, iterations=6000): - """Self-distill the flow of dx/dt=x, using diagonal and composition only.""" - torch.manual_seed(6270) - torch.set_num_threads(2) - model=nn.Sequential(nn.Linear(3,64),nn.SiLU(),nn.Linear(64,64),nn.SiLU(),nn.Linear(64,1)) - opt=torch.optim.Adam(model.parameters(),lr=1e-3) - history=[] - for step in range(iterations): - # Broad state coverage includes states reached by the split map. - x=6*torch.rand(256,1)-3 - times=torch.rand(256,2).sort(dim=-1).values - s,t=times[:,:1],times[:,1:] - u=s+(t-s)*torch.rand_like(s) - diagonal=model(torch.cat([x,s,s],dim=-1)) - diag_loss=(diagonal-x).square().mean() - with torch.no_grad(): - target=flow(model,flow(model,x,s,u),u,t) - prediction=flow(model,x,s,t) - # Normalize the interval residual to prevent tiny intervals dominating - # the count of nearly zero-error examples. Keep a finite floor. - cons_loss=((prediction-target)/(t-s).clamp_min(0.1)).square().mean() - loss=diag_loss + (0 if step<500 else 1)*cons_loss - opt.zero_grad();loss.backward();opt.step() - if step%500==0:history.append({'step':step,'diagonal':diag_loss.item(),'composition':cons_loss.item()}) - with torch.no_grad(): - x=torch.linspace(-1.5,1.5,501)[:,None] - s=torch.zeros_like(x);t=torch.ones_like(x);u=0.5*t - pred=flow(model,x,s,t);truth=x*math.e - split=flow(model,flow(model,x,s,u),u,t) - rmse=(pred-truth).square().mean().sqrt().item() - composition_rmse=(pred-split).square().mean().sqrt().item() - np.savez(output/'scalar_predictions.npz',x=x.numpy().ravel(),pred=pred.numpy().ravel(),truth=truth.numpy().ravel()) - torch.save(model.state_dict(),output/'scalar_map_weights.pt') - return {'endpoint_rmse':rmse,'composition_rmse':composition_rmse,'iterations':iterations,'history':history} - -def categorical_map(net,x,s,t): - psi=net(x,s,t).softmax(dim=-1) - h=(t-s)/(1-s) - return (1-h)*x+h*psi,psi - -def check_categorical(): - x=torch.tensor([-.2,.6,1.1],dtype=torch.float64) - psi=torch.tensor([.1,.7,.2],dtype=torch.float64) - y=(1/3)*x+(2/3)*psi - assert torch.allclose(y,torch.tensor([0,2/3,.5],dtype=torch.float64)) - target=(1/3)*torch.tensor([.8,.2])+(2/3)*torch.tensor([.2,.8]) - assert torch.allclose(target,torch.tensor([.4,.6])) - logits=torch.tensor([.2,-.1],requires_grad=True) - loss=F.kl_div(logits.log_softmax(-1),target,reduction='sum') - loss.backward() - assert torch.allclose(logits.grad,logits.softmax(-1)-target,atol=1e-7) - return {'mapped_state':y.tolist(),'state_sum':y.sum().item(),'target':target.tolist()} - -def check_meanflow_jvp(): - # Exact backward average for dz/dt=z. Stay off diagonal for this check. - def exact_average(z,r,t): - h=t-r - return z*(-torch.expm1(-h))/h - z=torch.tensor([[1.7]],dtype=torch.float64) - r=torch.tensor([[.2]],dtype=torch.float64) - t=torch.tensor([[.8]],dtype=torch.float64) - value,derivative=torch.func.jvp(exact_average,(z,r,t),(z,torch.zeros_like(r),torch.ones_like(t))) - target=z-(t-r)*derivative - assert torch.allclose(value,target,atol=1e-10) - return {'average':value.item(),'identity_residual':(value-target).abs().item()} - -def posterior_value_demo(): - # Prior Z~N(0,1), observation x=beta*Z+alpha*eps. - # Reward r(Z)=c*Z gives an analytic log moment-generating function. - torch.manual_seed(13) - alpha,beta,c=.7,.6,.4 - x=torch.tensor(.3,dtype=torch.float64,requires_grad=True) - gain=beta/(alpha**2+beta**2) - variance=alpha**2/(alpha**2+beta**2) - eps=torch.randn(100000,dtype=torch.float64) - z=gain*x+math.sqrt(variance)*eps - logw=c*z - estimate=torch.logsumexp(logw,0)-math.log(len(eps)) - gradient=torch.autograd.grad(estimate,x)[0] - exact=c*gain*x.detach()+.5*c*c*variance - exact_gradient=c*gain - assert abs(gradient.item()-exact_gradient)<1e-10 - assert abs(estimate.item()-exact.item())<.006 - return {'estimated_value':estimate.item(),'exact_value':exact.item(),'gradient':gradient.item(),'exact_gradient':exact_gradient} - -def sample_gap_counts(means,remaining_budget): - """Paper-style bounded proposals followed by left-to-right budget capping. - - These proposals match the per-gap means before joint truncation. They are - not asserted to identify the full conditional count law from means alone. - """ - counts=[];remaining=int(remaining_budget) - for mean in means: - if remaining_budget==0:count=0 - else: - prob=float(torch.as_tensor(mean).clamp(0,remaining_budget))/remaining_budget - count=int(torch.distributions.Binomial(remaining_budget,probs=prob).sample()) - count=min(count,remaining);counts.append(count);remaining-=count - return torch.tensor(counts,dtype=torch.long) - -def insert_by_gap(state,old_birth_times,counts,new_noise,birth_time): - """Insert ordered noise rows into the n+1 gaps and preserve clock alignment.""" - assert len(counts)==len(state)+1 - pieces=[];clocks=[];offset=0 - for gap,count in enumerate(counts.tolist()): - if count: - pieces.append(new_noise[offset:offset+count]);offset+=count - clocks.append(torch.full((count,),float(birth_time))) - if gap 1e-12, value, torch.zeros_like(value)) - - -def glass_velocity(inner, s, outer, t): - return (glass_denoiser(inner, s, outer, t) - inner) / (1-s) - - -def posterior_context(outer, t): - return torch.cat([outer, time_like(t, outer)], -1) - - -def train_posterior(method, data, args): - model = MapNet(2, args.width, context_dim=3).to(data) - teacher = ema_copy(model) - - def objective(step): - clean = draw_batch(data, args.batch_size) - outer_t = .94 * torch.rand(len(clean), 1, device=data.device) - outer, _ = interpolate(clean, outer_t) - context = posterior_context(outer, outer_t) - s, t, _ = ordered_times(clean, .97) - # Independent inner noise; reusing the outer noise changes the posterior. - inner, displacement = interpolate(clean, s) - if method == 'diamond': - diagonal_target = glass_velocity(inner, s, outer, outer_t).detach() - else: - diagonal_target = displacement - diagonal = F.mse_loss(model(inner, s, s, context), diagonal_target) - if method == 'diamond': - y, derivative = torch.func.jvp(lambda end: finite_map(model, inner, s, end, context), - (t,), (torch.ones_like(t),)) - target = glass_velocity(y.detach(), t, outer, outer_t).detach() - finite = F.mse_loss(derivative, target) - else: - finite = semigroup_loss(model, teacher, inner, s, t, context) - weight = min(1., (step+1) / max(1, args.train_steps//4)) - return diagonal + weight * finite, {'diagonal': diagonal, 'finite': finite} - - logs = optimize(model, objective, args.train_steps, args.lr, teacher) - return model, {'model': model.state_dict(), 'dim': 2}, logs - - -def posterior_samples(model, outer, t, count, steps=1, noise=None): - context = posterior_context(outer, t) - context = context[:, None].expand(-1, count, -1).reshape(-1, 3) - x = torch.randn(len(outer)*count, 2, device=outer.device) if noise is None else noise.reshape(-1, 2) - for i in range(steps): - x = finite_map(model, x, i/steps, (i+1)/steps, context) - return x.reshape(len(outer), count, 2) - - -def reward(x, strength=1.): - """Bounded differentiable preference for positive first coordinates.""" - return strength * torch.tanh(x[..., 0]) - - -def posterior_value(model, x, t, particles=32, strength=1., steps=1): - samples = posterior_samples(model, x, t, particles, steps) - log_weights = reward(samples, strength) - value = torch.logsumexp(log_weights, -1) - math.log(particles) - weighted_mean = (log_weights.softmax(-1)[..., None] * samples).sum(1) - return value, weighted_mean, samples - - -def fine_tune_surrogate(delta, weights, grad_weights, coefficient): - """Equation 43 of Meta Flow Maps. Both Monte Carlo terms are detached. - - d/d(delta) E[loss] = 2 E[w*delta - coefficient*grad(w)]. - Differentiating (w*delta - coefficient*grad(w))^2 instead adds a wrong w. - """ - residual = delta + (weights.detach()-1) * delta.detach() - coefficient * grad_weights.detach() - return residual.square().mean() - - -def train_reward_drift(posterior, data, args): - posterior.requires_grad_(False) - drift = MapNet(2, args.width).to(data) - def objective(_): - clean = draw_batch(data, args.batch_size) - t = .05 + .8 * torch.rand(len(clean), 1, device=data.device) - x, _ = interpolate(clean, t) - x.requires_grad_(True) - endpoint = posterior_samples(posterior, x, t, 1, args.posterior_steps)[:, 0] - w = reward(endpoint, args.reward_strength).exp().unsqueeze(-1) - grad_w = torch.autograd.grad(w.sum(), x)[0] - baseline = exact_velocity(x, t).detach() - delta = drift(x.detach(), t, t) - baseline - # Linear interpolant's compatible probability-flow correction g^2/2=(1-t)/t. - # Use t>=.05 to avoid singular coefficients in this demonstration. - coefficient = (1-t) / t - loss = fine_tune_surrogate(delta, w, grad_w, coefficient) - return loss, {'mean_weight': w.mean()} - logs = optimize(drift, objective, args.finetune_steps, args.lr) - return drift, logs - - -@torch.no_grad() -def guided_samples(model, count, steps, particles, strength, posterior_steps=1): - """Derivative-free reward-weighted posterior mean in the linear flow drift. - - v*(x,t)=(E_reward[X1|x,t]-x)/(1-t). It equals the compatible - probability-flow correction for an exact posterior. Finite learned maps - and self-normalized Monte Carlo introduce approximation and ratio bias. - """ - x = torch.randn(count, 2, device=next(model.parameters()).device) - h = 1/steps - for i in range(steps): - t = torch.full((count, 1), i/steps, device=x.device) - _, mean, _ = posterior_value(model, x, t, particles, strength, posterior_steps) - x = x + h * (mean-x) / (1-t) - return x - - -def weighted_diamond_samples(outer, t, particles=128): - """Exact tractable demonstration of weighted Diamond proposal correction. - - Proposal q=N(0,4I) has evaluable density and full support. Reweight with - p1(z)*p_t(outer|z)/q(z), then resample. This supplies an independent - check of posterior recovery without assuming an arbitrary learned map's - Jacobian density is known. Finite self-normalized importance sampling is biased. - """ - from common import CENTERS, DATA_STD - t = time_like(t, outer) - z = 2 * torch.randn(len(outer), particles, 2, device=outer.device) - delta = z[:, :, None] - CENTERS.to(z) - lp = torch.logsumexp(-delta.square().sum(-1)/(2*DATA_STD**2), -1) - lp -= math.log(4 * 2 * math.pi * DATA_STD**2) - noise = (outer[:, None]-t[:, None]*z)/(1-t[:, None]) - likelihood = -.5*noise.square().sum(-1) - 2*(1-t).log() - lq = -z.square().sum(-1)/8 - math.log(8*math.pi) - weights = (lp + likelihood - lq).softmax(-1) - mean = (weights[..., None]*z).sum(1) - indices = torch.multinomial(weights, 1) - selected = z.gather(1, indices[..., None].expand(-1, -1, 2))[:, 0] - return selected, mean, 1/weights.square().sum(-1) - - -@torch.no_grad() -def posterior_diagnostics(model, steps=1): - device = next(model.parameters()).device - x = torch.tensor([[0., 0.], [1., -.5], [-1., 1.]], device=device) - t = torch.tensor([[.15], [.5], [.8]], device=device) - p, means, var = mixture_posterior(x, t) - exact_mean = (p[..., None]*means).sum(1) - exact_variance = (p[..., None]*(var[:, None]+means.square())).sum(1)-exact_mean.square() - samples = posterior_samples(model, x, t, 512, steps) - mean, variance = samples.mean(1), samples.var(1) - return {'posterior_mean_rmse': float((mean-exact_mean).square().mean().sqrt()), - 'posterior_variance_rmse': float((variance-exact_variance).square().mean().sqrt()), - 'posterior_reference_mean': exact_mean.tolist(), 'posterior_estimated_mean': mean.tolist()} diff --git a/lecture_6/requirements.txt b/lecture_6/requirements.txt deleted file mode 100644 index 2ff91cf64bf30e5bad4befa43e72caecc5516759..0000000000000000000000000000000000000000 --- a/lecture_6/requirements.txt +++ /dev/null @@ -1,3 +0,0 @@ -# Match the course's PyTorch baseline; torch.func.jvp is required. -torch==2.9.1 -numpy>=1.24,<3 diff --git a/lecture_6/run.py b/lecture_6/run.py deleted file mode 100644 index e6fad4b2ff044eb63c11a066613e06d395fa8579..0000000000000000000000000000000000000000 --- a/lecture_6/run.py +++ /dev/null @@ -1,245 +0,0 @@ -#!/usr/bin/env python3 -"""Train, save, reload, and sample the Lecture 6 flow-map methods.""" -import argparse -import json -import platform -import time -from pathlib import Path - -import torch -from torch.nn import functional as F - -from common import (MapNet, SequenceNet, interpolate, load_text, mixture_data, - mixture_metrics, seed_all, text_metrics, write_json) -from continuous import (Autoencoder, embed_surface, integrate_velocity, sample_continuous, - train_continuous, train_latent) -from categorical import sample_categorical, train_categorical -from posterior import (guided_samples, posterior_diagnostics, posterior_samples, - posterior_value, reward, train_posterior, train_reward_drift, - weighted_diamond_samples) -from expanding import ExpandingNet, sample_expanding, train_expanding -from stochastic import StrongMap, evaluate_stochastic, train_stochastic - -ROOT = Path(__file__).resolve().parent -CONTINUOUS = ['flow-matching','fmm-lagrangian','fmm-eulerian','self-distill', - 'consistency','shortcut','meanflow','latent'] -CATEGORICAL = ['fmlm','categorical','discrete-lsd','discrete-esd'] -METHODS = CONTINUOUS+CATEGORICAL+['diamond','meta','expanding','ssfm'] - - -def parser(): - p=argparse.ArgumentParser(description=__doc__) - p.add_argument('--method',choices=METHODS) - p.add_argument('--mode',choices=['train-sample','train','sample'],default='train-sample') - p.add_argument('--train-steps',type=int,default=1000) - p.add_argument('--teacher-steps',type=int,default=1000) - p.add_argument('--finetune-steps',type=int,default=0,help='Optional Meta reward-drift fine-tuning') - p.add_argument('--sample-steps',type=int,default=8) - p.add_argument('--posterior-steps',type=int,default=4) - p.add_argument('--particles',type=int,default=32) - p.add_argument('--reward-strength',type=float,default=1.) - p.add_argument('--ssfm-target',choices=['official-code','paper'],default='official-code') - p.add_argument('--batch-size',type=int,default=64) - p.add_argument('--samples',type=int,default=128) - p.add_argument('--width',type=int,default=64) - p.add_argument('--lr',type=float,default=1e-3) - p.add_argument('--seed',type=int,default=6270) - p.add_argument('--threads',type=int,default=1) - p.add_argument('--device',choices=['cpu','cuda'],default='cpu') - p.add_argument('--data',help='Whitespace-separated text, one sequence per line; or continuous CSV with two numeric columns') - p.add_argument('--out',help='Run directory; defaults to lecture_6/outputs/METHOD') - return p - - -def restore_model(checkpoint,device): - method=checkpoint['method'];width=checkpoint['width'] - if method in CATEGORICAL: - model=SequenceNet(checkpoint['length'],len(checkpoint['vocab']),width) - elif method=='expanding': - model=ExpandingNet(checkpoint['length'],len(checkpoint['vocab']),width) - elif method=='ssfm': - model=StrongMap(width,checkpoint['sigma']) - else: - model=MapNet(checkpoint['dim'],width,3 if method in ['diamond','meta'] else 0) - model.load_state_dict(checkpoint['model']) - model=model.to(device).eval() - return model - - -def generate(model,state,args): - """All metrics use fresh, seeded samples; no training examples stand in for output.""" - seed_all(args.seed+100,args.threads) - method=state['method'];device=args.device - report={'method':method,'sample_steps':args.sample_steps,'samples':args.samples, - 'sampling_seed':args.seed+100} - if method in CATEGORICAL: - ids=sample_categorical(model,args.samples,args.sample_steps,device).cpu() - texts,metrics=text_metrics(ids,state['vocab'],reference=set(state['reference_text'])) - report.update(metrics) - return texts,report - if method=='expanding': - ids,lengths,extra=sample_expanding(model,args.samples,args.sample_steps,device) - texts,metrics=text_metrics(ids.cpu(),state['vocab'],lengths.cpu(),set(state['reference_text'])) - report.update(metrics);report.update(extra) - return texts,report - if method=='ssfm': - x,extra=evaluate_stochastic(model,args.samples,args.sample_steps,device) - report.update(extra) - elif method in ['diamond','meta']: - with torch.no_grad(): - outer=torch.randn(args.samples,2,device=device) - t=torch.zeros(args.samples,1,device=device) - x=posterior_samples(model,outer,t,1,args.posterior_steps)[:,0] - guided=guided_samples(model,args.samples,args.sample_steps,args.particles, - args.reward_strength,args.posterior_steps) - report.update(posterior_diagnostics(model,args.posterior_steps)) - report['unconditional_mean_reward']=float(reward(x,args.reward_strength).mean()) - report['guided_mean_reward']=float(reward(guided,args.reward_strength).mean()) - report['guided_distribution']=mixture_metrics(guided) - report['posterior_steps']=args.posterior_steps - report['particles']=args.particles - observation=torch.tensor([[.5,-.3]],device=device) - _,estimate,ess=weighted_diamond_samples(observation,.5,4096) - report['importance_posterior_mean']=estimate.tolist() - report['importance_effective_sample_size']=ess.tolist() - # Verify differentiability through the learned posterior's context. - observation=observation.detach().requires_grad_(True) - value,_,_=posterior_value(model,observation,.5,args.particles,args.reward_strength,args.posterior_steps) - gradient=torch.autograd.grad(value.sum(),observation)[0] - report['posterior_value_context_gradient']=gradient.tolist() - report['finite_value_gradient']=bool(torch.isfinite(gradient).all()) - report['guided_samples']=guided.detach().cpu().tolist() - if 'reward_drift' in state: - drift=MapNet(2,state['width']).to(device) - drift.load_state_dict(state['reward_drift']) - # A complete sampler for the fitted drift; boundary-time queries - # extrapolate beyond the [.05,.85] fine-tuning interval. - aligned=integrate_velocity(lambda z,a:drift(z,a,a),outer,0.,1.,args.sample_steps) - report['finetuned_mean_reward']=float(reward(aligned,args.reward_strength).mean()) - report['finetuned_finite_samples']=bool(torch.isfinite(aligned).all()) - report['finetuned_samples']=aligned.detach().cpu().tolist() - report['finetuned_sampling_note']='Full-interval Heun integration; boundary times extrapolate beyond fine-tuning support.' - else: - noise=torch.randn(args.samples,state['dim'],device=device) - x=sample_continuous(model,method,noise,args.sample_steps) - if method=='latent': - ae=Autoencoder(state['width']).to(device) - ae.load_state_dict(state['autoencoder']) - with torch.no_grad(): - x=ae.decoder(x*state['latent_std'].to(device)+state['latent_mean'].to(device)) - heldout=state['heldout_data'].to(device) - reconstruction=ae.decoder(ae.encoder(embed_surface(heldout))) - report['heldout_reconstruction_mse']=float((reconstruction-embed_surface(heldout)).square().mean()) - report['surface_residual_rmse']=float((x[:,2:]-.3*(x[:,:1].square()-x[:,1:2].square())).square().mean().sqrt()) - if method not in ['consistency','meanflow','flow-matching']: - from common import finite_map - with torch.no_grad(): - direct=finite_map(model,noise,0.,1.) - split=finite_map(model,finite_map(model,noise,0.,.5),.5,1.) - report['composition_rmse']=float((direct-split).square().mean().sqrt()) - report['finite_samples']=bool(torch.isfinite(x).all()) - if not report['finite_samples']:raise FloatingPointError('Sampling produced nonfinite coordinates.') - if x.shape[1]>=2 and state['data_kind']=='four-gaussian-mixture': - report.update(mixture_metrics(x[:,:2])) - rows=[' '.join(f'{value:.7f}' for value in row) for row in x.detach().cpu().tolist()] - return rows,report - - -def main(argv=None): - p=parser();args=p.parse_args(argv) - if min(args.train_steps,args.teacher_steps,args.sample_steps,args.posterior_steps, - args.particles,args.batch_size,args.samples,args.width,args.threads)<1 or args.finetune_steps<0: - p.error('Step counts, sizes, and width must be positive; finetune steps must be nonnegative.') - if args.lr<=0:p.error('Learning rate must be positive.') - if args.device=='cuda' and not torch.cuda.is_available():p.error('CUDA is unavailable; select cpu.') - if args.mode=='sample' and args.out is None:p.error('Sample mode requires --out for its checkpoint.') - if args.mode!='sample' and args.method is None:args.method='flow-matching' - out=Path(args.out) if args.out else ROOT/'outputs'/args.method - out.mkdir(parents=True,exist_ok=True) - seed_all(args.seed,args.threads) - start=time.perf_counter() - if args.mode=='sample': - state=torch.load(out/'checkpoint.pt',map_location=args.device,weights_only=True) - if args.method is not None and args.method!=state['method']: - p.error('Requested method differs from the saved checkpoint.') - args.method=state['method'] - if args.method=='ssfm' and args.sample_steps & (args.sample_steps-1): - p.error('SSFM sample steps must be a power of two.') - model=restore_model(state,args.device) - rows,report=generate(model,state,args) - (out/'resampled.txt').write_text('\n'.join(rows)+'\n') - write_json(out/'sample_report.json',report) - print(json.dumps({'method':args.method,'mode':'sample','out':str(out),'seconds':time.perf_counter()-start})) - return report - if args.method=='ssfm' and args.sample_steps & (args.sample_steps-1): - p.error('SSFM sample steps must be a power of two.') - if args.finetune_steps and args.method!='meta':p.error('--finetune-steps applies to meta.') - if args.data and args.method in ['diamond','meta','ssfm']: - p.error('Analytic posterior/OU examples use their specified reference distributions.') - stage_logs=[] - if args.method in CATEGORICAL+['expanding']: - default='variable_text.txt' if args.method=='expanding' else 'phrases.txt' - data_path=Path(args.data) if args.data else ROOT/'data'/default - ids,lengths,vocab=load_text(data_path,args.method=='expanding') - perm=torch.randperm(len(ids));ids,lengths=ids[perm],lengths[perm] - split=max(1,int(.8*len(ids))) - training=ids[:split].to(args.device) - if args.method=='expanding': - model,state,logs=train_expanding(training,lengths[:split].to(args.device),vocab,args) - else: - model,state,logs=train_categorical(args.method,training,vocab,args) - reference,_=text_metrics(ids[:split],vocab,lengths[:split]) - state.update({'reference_text':reference,'data_kind':'synthetic-text' if args.data is None else 'custom-text'}) - with torch.no_grad(): - if args.method in CATEGORICAL: - heldout=ids[split:].to(args.device) - clean=F.one_hot(heldout,len(vocab)).float();t=torch.full((len(clean),1),.5,device=args.device) - x,_=interpolate(clean,t) - state['validation_ce']=float(F.cross_entropy(model(x,t,t).flatten(0,1),heldout.flatten())) - elif args.method=='ssfm': - model,state,logs=train_stochastic(args,args.device) - state['data_kind']='ornstein-uhlenbeck' - else: - if args.data: - import numpy as np - data=torch.tensor(np.loadtxt(args.data,delimiter=','),dtype=torch.float32) - if data.ndim!=2 or data.shape[1]!=2 or len(data)<8 or not torch.isfinite(data).all(): - p.error('Continuous CSV needs at least eight finite rows and exactly two columns, without a header.') - data=data[torch.randperm(len(data))];kind='custom-continuous' - else: - data=mixture_data(4096);kind='four-gaussian-mixture' - split=int(.8*len(data));train=data[:split].to(args.device) - if args.method=='latent': - model,_,state,logs,stage_logs=train_latent(train,args) - elif args.method in ['diamond','meta']: - model,state,logs=train_posterior(args.method,train,args) - if args.finetune_steps: - drift,finetune_logs=train_reward_drift(model,train,args) - state['reward_drift']=drift.state_dict() - write_json(out/'finetune_losses.json',finetune_logs) - else: - model,state,logs,stage_logs=train_continuous(args.method,train,args) - state.update({'data_kind':kind,'heldout_data':data[split:]}) - state.update({'format_version':1,'method':args.method,'width':args.width,'seed':args.seed}) - torch.save(state,out/'checkpoint.pt') - write_json(out/'config.json',vars(args)) - write_json(out/'losses.json',logs) - if stage_logs:write_json(out/'teacher_losses.json',stage_logs) - report={'method':args.method,'data_kind':state['data_kind'], - 'train_loss_first_20_mean':sum(x['loss'] for x in logs[:20])/len(logs[:20]), - 'train_loss_last_20_mean':sum(x['loss'] for x in logs[-20:])/len(logs[-20:]), - 'training_steps':args.train_steps,'python':platform.python_version(),'torch':str(torch.__version__)} - if 'validation_ce' in state:report['heldout_diagonal_ce_at_half_time']=state['validation_ce'] - if args.mode=='train-sample': - model.eval() - rows,metrics=generate(model,state,args) - (out/'samples.txt').write_text('\n'.join(rows)+'\n') - report.update(metrics) - report['elapsed_seconds']=time.perf_counter()-start - write_json(out/'report.json',report) - print(json.dumps({'method':args.method,'out':str(out),'loss':report['train_loss_last_20_mean'], - 'seconds':report['elapsed_seconds']}),flush=True) - return report - - -if __name__=='__main__':main() diff --git a/lecture_6/run_all.py b/lecture_6/run_all.py deleted file mode 100644 index 49bcedb22110467bf784a9b8d9bd06d02a31585c..0000000000000000000000000000000000000000 --- a/lecture_6/run_all.py +++ /dev/null @@ -1,32 +0,0 @@ -#!/usr/bin/env python3 -"""Run every method, then verify generation from each saved checkpoint.""" -import argparse -from pathlib import Path - -from run import METHODS, ROOT, main - - -def run_all(): - parser=argparse.ArgumentParser(description=__doc__) - parser.add_argument('--quick',action='store_true',help='20-step execution check; not a quality benchmark') - parser.add_argument('--out',default=str(ROOT/'outputs')) - parser.add_argument('--train-steps',type=int,default=1000) - args=parser.parse_args() - for method in METHODS: - out=Path(args.out)/method - command=['--method',method,'--out',str(out)] - if args.quick: - command+=['--train-steps','20','--teacher-steps','20','--batch-size','8', - '--samples','8','--width','32','--sample-steps','4','--posterior-steps','2','--particles','4'] - else:command+=['--train-steps',str(args.train_steps),'--teacher-steps',str(args.train_steps)] - main(command) - sample_args=['--mode','sample','--out',str(out)] - if args.quick: - sample_args+=['--samples','8','--sample-steps','4','--posterior-steps','2','--particles','4'] - main(sample_args) - if (out/'samples.txt').read_bytes()!=(out/'resampled.txt').read_bytes(): - raise AssertionError(f'{method}: seeded generation changed after checkpoint reload') - print(f'All {len(METHODS)} methods trained, sampled, and reproduced samples after checkpoint reload.') - - -if __name__=='__main__':run_all() diff --git a/lecture_6/source_manifest.json b/lecture_6/source_manifest.json deleted file mode 100644 index 38259ab28f36ba766f254a439aceae0be39b43ba..0000000000000000000000000000000000000000 --- a/lecture_6/source_manifest.json +++ /dev/null @@ -1,132 +0,0 @@ -[ - { - "paper_title": "Flow Map Matching", - "paper_url": "https://arxiv.org/abs/2406.07507v2", - "methods": [ - "fmm-lagrangian", - "fmm-eulerian" - ], - "scope": "Frozen learned velocity teachers; residual-map JVPs; small MLP and bounded time range.", - "repository": "nmboffi/flow-maps", - "commit": "2f115a07fa9073553193e4b265dfc303827af2b0", - "inspected_file": "py/common/losses.py", - "inspected_file_sha256": "55d5a6cd2b7602afc93f848bda687aa4a287bfb7d1a4095d5071c7df59334d59" - }, - { - "paper_title": "How to build a consistency model: Learning flow maps via self-distillation", - "paper_url": "https://arxiv.org/abs/2505.18825v2", - "methods": [ - "self-distill" - ], - "scope": "Diagonal regression plus an EMA progressive target, normalized by a floored interval length. Learned uncertainty weights are omitted.", - "repository": "nmboffi/flow-maps", - "commit": "2f115a07fa9073553193e4b265dfc303827af2b0", - "inspected_file": "py/common/losses.py", - "inspected_file_sha256": "55d5a6cd2b7602afc93f848bda687aa4a287bfb7d1a4095d5071c7df59334d59" - }, - { - "paper_title": "One Step Diffusion via Shortcut Models", - "paper_url": "https://arxiv.org/abs/2410.12557v3", - "methods": [ - "shortcut" - ], - "scope": "Two half-step average target and dyadic intervals retained. Continuous start-time sampling and small MLP replace image-specific schedules and clipping.", - "repository": "kvfrans/shortcut-models", - "commit": "601004348667094e1b71f30942199759412d4432", - "inspected_file": "targets_shortcut.py", - "inspected_file_sha256": "adb2ac83febc1de012a7cdb713b9117d53616473ccd38f1104fd2475d8b1abc3" - }, - { - "paper_title": "Mean Flows for One-step Generative Modeling", - "paper_url": "https://arxiv.org/abs/2505.13447", - "methods": [ - "meanflow" - ], - "scope": "Backward clock, conditional-velocity JVP, detached target, adaptive loss, and 75% diagonal proportion retained. Uniform time sampling; no class guidance.", - "repository": "Gsunshine/meanflow", - "commit": "d70cb55d298ee03c53bf6da67bec281082e4e2d9", - "inspected_file": "meanflow.py", - "inspected_file_sha256": "b8dcee1033e6ba01de0019d14fad846a399d356d67e549ee311a5e19d2a7c3a4" - }, - { - "paper_title": "Flow Map Language Models: One-step Language Modeling via Continuous Denoising", - "paper_url": "https://arxiv.org/abs/2602.16813v3", - "methods": [ - "fmlm" - ], - "scope": "PSD denoiser target and Gaussian decoding clock retained. Independent clean-data diagonal supervision; EMA targets; quadrature lookup and MLP replace full language architecture.", - "repository": "david3684/flm", - "commit": "a1918d5164e5038e37d0b7a4fb2010ce75b863b3", - "inspected_file": "algo.py", - "inspected_file_sha256": "65206adc72013bb9b09b68fd847239b3193dd65fd59669bbdee42e061583d76a" - }, - { - "paper_title": "Categorical Flow Maps", - "paper_url": "https://arxiv.org/abs/2602.12233v1", - "methods": [ - "categorical" - ], - "scope": "Released ECLD endpoint CE plus sum-of-squares time energy. Detached-target KL has identical student gradient. EMA target replaces the current network target.", - "repository": "olsdavis/semicat", - "commit": "558602a0fa722514e4a6012f5c46a8ae178b3068", - "inspected_file": "semicat/models/semicat.py", - "inspected_file_sha256": "82f4bf2cab2fcfa78ccfc59d9094e6c3132f5c2933c1cd7e61ff0b644ac71425" - }, - { - "paper_title": "Discrete Flow Maps", - "paper_url": "https://arxiv.org/abs/2604.09784v1", - "methods": [ - "discrete-lsd", - "discrete-esd" - ], - "scope": "Derived from the paper. The linked project page has a placeholder Code link, so no author implementation was verified. Positive log denominators are explicitly clamped and counted; no gradient surgery." - }, - { - "paper_title": "Diamond Maps: Stochastic Flow Maps", - "paper_url": "https://arxiv.org/abs/2602.05993", - "methods": [ - "diamond" - ], - "scope": "GLASS conditional velocity and Lagrangian posterior distillation. Exact Gaussian-mixture denoiser replaces a pretrained image teacher. Weighted posterior recovery uses an explicit Gaussian proposal with a known density.", - "repository": "PeterHolderrieth/diamond_maps", - "commit": "d30f65c75a169a2ed624f146b2770aeef882543a", - "inspected_file": "posterior_diamond_maps/py/common/losses.py", - "inspected_file_sha256": "acb26f1e717e43a648661c1641cadd4a573ed2672b53210ad1ddb207c6a6ac1e" - }, - { - "paper_title": "Meta Flow Maps enable scalable reward alignment", - "paper_url": "https://arxiv.org/abs/2601.14430v2", - "methods": [ - "meta" - ], - "scope": "Data training with independent inner/outer noises and a shared endpoint; conditional semigroup objective. Optional Equation 43 fine-tuning also checked against src/mfm/losses/finetune.py.", - "repository": "adh1s/mfm", - "commit": "53c0f60db695cad88bbace8fb26469614e9e7d7d", - "inspected_file": "src/mfm/losses/losses.py", - "inspected_file_sha256": "52b99238199f5d6ac691c15566463d52725a6f7ebcdef1e252b2d28da5eb1df2" - }, - { - "paper_title": "Expanding Flow Maps", - "paper_url": "https://arxiv.org/abs/2607.21585v1", - "methods": [ - "expanding" - ], - "scope": "Paper Algorithms 2-4 and local-clock equations. The inspected repository contains README and artwork only. Classroom implementation includes a shared lifted canvas, remaining/interval count heads, explicit local clocks, and budget-capped binomial insertions.", - "repository": "sophtang/ExpandingFlowMaps", - "commit": "4defc1ff168d12526d85b551bea78150b9aec605", - "inspected_file": "README.md", - "inspected_file_sha256": "d2a55e221a909f7282609cc661e49c07a241570dcd1dab7b55c6df207f821f4a" - }, - { - "paper_title": "Strong Stochastic Flow Maps", - "paper_url": "https://arxiv.org/abs/2606.01086v1", - "methods": [ - "ssfm" - ], - "scope": "First two Legendre integrals, Chen composition, known diffusion, small-step matching. The released loss detaches an EMA split target; paper Algorithm 1 detaches the direct target. Both orientations are selectable. OU replaces image/molecular experiments; no learned uncertainty weights.", - "repository": "sammccallum/ssfm", - "commit": "24b563620a8ee61c683d791fe802b56668f98f9b", - "inspected_file": "ssfm/losses.py", - "inspected_file_sha256": "3c4414ffb29c906272831e3642827131582c4741fb2457cb71d122564da29233" - } -] diff --git a/lecture_6/stochastic.py b/lecture_6/stochastic.py deleted file mode 100644 index 8d310f22b68d6a58c8acbd8248be0906210d866e..0000000000000000000000000000000000000000 --- a/lecture_6/stochastic.py +++ /dev/null @@ -1,110 +0,0 @@ -"""Strong stochastic flow maps for dX=-X dt + sigma dW. - -The first two shifted-Legendre Brownian integrals have variances h and h/3. -Chen composition makes coarse and fine evaluations use the same path. -""" -import math - -import torch -from torch import nn -from torch.nn import functional as F - -from common import ema_copy, mlp, optimize, time_like - - -def sample_coefficients(h, count=None): - h = torch.as_tensor(h) - if h.ndim == 0: - if count is None: raise ValueError('count is required for scalar h') - h = h.expand(count,1) - scale = torch.cat([h.sqrt(),(h/3).sqrt()],-1) - return torch.randn_like(scale)*scale - - -def chen_two(left,right,h_left,h_right): - h=h_left+h_right - return torch.stack([left[...,0]+right[...,0], - (h_left*left[...,1]+h_right*right[...,1]-h_right*left[...,0]+h_left*right[...,0])/h],-1) - - -class StrongMap(nn.Module): - def __init__(self,width=64,sigma=.7): - super().__init__() - self.net=mlp(5,1,width) - self.sigma=sigma - - def average_drift(self,x,s,t,coefficients): - return self.net(torch.cat([x,time_like(s,x),time_like(t,x),coefficients],-1)) - - def forward(self,x,s,t,coefficients): - s,t=time_like(s,x),time_like(t,x) - return x+(t-s)*self.average_drift(x,s,t,coefficients)+self.sigma*coefficients[:,:1] - - -def train_stochastic(args,device): - model=StrongMap(args.width).to(device) - teacher=ema_copy(model) - def objective(step): - B=args.batch_size - s=.9*torch.rand(B,1,device=device) - # Exact one-time OU marginals with X0~N(0,1), not simulated trajectories. - variance=torch.exp(-2*s)+model.sigma**2/2*(1-torch.exp(-2*s)) - x=variance.sqrt()*torch.randn(B,1,device=device) - h=.001+.019*torch.rand_like(s) - c=sample_coefficients(h) - target=x-h*x+model.sigma*c[:,:1] - matching=((model(x,s,s+h,c)-target).square()/h).mean() - diagonal=F.mse_loss(model.average_drift(x,s,s,torch.zeros_like(c)),-x) - t=s+.025+(1-s-.025)*torch.rand_like(s) - u=(s+t)/2 - L,R=sample_coefficients(u-s),sample_coefficients(t-u) - C=chen_two(L,R,(u-s).squeeze(-1),(t-u).squeeze(-1)) - if args.ssfm_target=='paper': - # Paper Algorithm 1: EMA direct target; differentiate both splits. - with torch.no_grad():direct=teacher(x,s,t,C) - split=model(model(x,s,u,L),u,t,R) - else: - # Released code: EMA split target; differentiate the direct map. - with torch.no_grad():split=teacher(teacher(x,s,u,L),u,t,R) - direct=model(x,s,t,C) - consistency=((split-direct).square()/(t-s)).mean() - ramp=min(1.,(step+1)/max(1,args.train_steps//4)) - return diagonal+matching+ramp*consistency,{'drift':diagonal,'small_step':matching,'strong_consistency':consistency} - logs=optimize(model,objective,args.train_steps,args.lr,teacher) - return model,{'model':model.state_dict(),'sigma':model.sigma},logs - - -def aggregate_tree(coefficients): - """Coefficients are [B,leaves,2], with uniform leaf lengths.""" - c=coefficients - h=1/c.shape[1] - while c.shape[1]>1: - c=chen_two(c[:,::2],c[:,1::2],h,h) - h*=2 - return c[:,0] - - -@torch.no_grad() -def evaluate_stochastic(model,count,steps,device): - if steps & (steps-1):raise ValueError('SSFM sample steps must be a power of two.') - leaves=max(512,steps) - c=sample_coefficients(torch.tensor(1/leaves,device=device),count*leaves).reshape(count,leaves,2) - x0=torch.randn(count,1,device=device) - reference=x0.clone() - for i in range(leaves):reference=reference-reference/leaves+model.sigma*c[:,i,:1] - all_coeff=aggregate_tree(c) - direct=model(x0,0.,1.,all_coeff) - x=x0.clone() - block=leaves//steps - for i in range(steps): - # aggregate_tree assumes total duration one; equal-half Chen weights are - # scale invariant, so its output is valid for these smaller blocks too. - ci=aggregate_tree(c[:,i*block:(i+1)*block]) - x=model(x,i/steps,(i+1)/steps,ci) - exact_variance=math.exp(-2)+model.sigma**2/2*(1-math.exp(-2)) - return x,{'same_noise_reference_rmse':float((x-reference).square().mean().sqrt()), - 'direct_vs_split_rmse':float((direct-x).square().mean().sqrt()), - 'one_step_reference_rmse':float((direct-reference).square().mean().sqrt()), - 'reference_em_steps':leaves,'sample_variance':float(x.var()), - 'exact_terminal_variance':exact_variance, - 'reference_note':'Same Brownian increments; EM reference retains finite discretization error.'} diff --git a/lecture_6/tests/test_mathematics.py b/lecture_6/tests/test_mathematics.py deleted file mode 100644 index 42722f7a3abc1362cc58bff35f9756af03d233e0..0000000000000000000000000000000000000000 --- a/lecture_6/tests/test_mathematics.py +++ /dev/null @@ -1,189 +0,0 @@ -"""Independent mathematical checks, including failure cases from the lecture.""" -import math -import sys -import unittest -from pathlib import Path - -import torch -from torch import nn -from torch.nn import functional as F - -sys.path.insert(0,str(Path(__file__).resolve().parents[1])) -from common import (MapNet, SequenceNet, finite_map, exact_denoiser, exact_velocity, - mixture_posterior, seed_all, DATA_STD, CENTERS) -from continuous import lagrangian_residual, eulerian_residual, semigroup_loss -from categorical import (DecodingClock, categorical_map, composition_target, - corrected_logit_teacher, probability_kl) -from posterior import glass_denoiser, fine_tune_surrogate, weighted_diamond_samples -from expanding import (ExpandingNet, local_clock, local_map, gap_counts, - bounded_counts, insert_tokens, count_divergence) -from stochastic import sample_coefficients, chen_two, aggregate_tree - - -class ExactExponential(nn.Module): - def forward(self,x,s,t,context=None): - h=t-s - # Stable at h=0 while keeping the exact off-diagonal expression. - safe=h.clamp_min(1e-12) - ratio=torch.where(h.abs()<1e-8,1+h/2+h.square()/6,torch.expm1(h)/safe) - return x*ratio - - -class ConstantLogits(nn.Module): - def forward(self,x,s,t): - return torch.zeros_like(x)+torch.tensor([.2,-.1],dtype=x.dtype,device=x.device) - - -class MathematicsTests(unittest.TestCase): - def setUp(self):seed_all(6270) - - def test_exact_identity_and_semigroup(self): - model=ExactExponential();x=torch.tensor([[1.2],[-.4]],dtype=torch.float64) - s=torch.zeros_like(x);u=s+.3;t=s+.9 - torch.testing.assert_close(finite_map(model,x,s,s),x) - expected=x*math.exp(.9) - torch.testing.assert_close(finite_map(model,x,s,t),expected) - torch.testing.assert_close(finite_map(model,finite_map(model,x,s,u),u,t),expected) - - def test_lagrangian_and_eulerian_jvp(self): - x=torch.tensor([[1.7]],dtype=torch.float64);s=x*0+.2;t=x*0+.8 - for residual in [lagrangian_residual,eulerian_residual]: - value=residual(ExactExponential(),lambda z,a:z,x,s,t) - self.assertLess(float(value.abs().max()),1e-10) - - def test_jvp_losses_reach_parameters(self): - for residual in [lagrangian_residual,eulerian_residual]: - m=MapNet(2,16);x=torch.randn(8,2);s=torch.rand(8,1)*.3;t=s+.5 - loss=residual(m,lambda z,a:z,x,s,t).square().mean();loss.backward() - self.assertTrue(all(p.grad is not None and torch.isfinite(p.grad).all() for p in m.parameters())) - self.assertGreater(sum(float(p.grad.abs().sum()) for p in m.parameters()),0) - - def test_composition_alone_does_not_identify_motion(self): - m=MapNet(2,8) - for p in m.parameters():p.data.zero_() - x=torch.randn(8,2);s=torch.zeros(8,1);t=s+1 - self.assertEqual(float(semigroup_loss(m,m,x,s,t)),0.) - self.assertGreater(float((m(x,s,s)-x).square().mean()),0.) - - def test_meanflow_backward_identity(self): - def average(z,r,t):return z*(-torch.expm1(-(t-r)))/(t-r) - z=torch.tensor([[1.7]],dtype=torch.float64);r=z*0+.2;t=z*0+.8 - u,du=torch.func.jvp(average,(z,r,t),(z,torch.zeros_like(r),torch.ones_like(t))) - torch.testing.assert_close(u,z-(t-r)*du,atol=1e-11,rtol=1e-11) - - def test_analytic_mixture_velocity(self): - x=torch.randn(20,2,dtype=torch.float64);t=torch.rand(20,1,dtype=torch.float64)*.9 - torch.testing.assert_close(exact_velocity(x,t),(exact_denoiser(x,t)-x)/(1-t),atol=1e-10,rtol=1e-10) - - def test_categorical_numerical_step(self): - class Net(nn.Module): - def forward(self,x,s,t):return torch.tensor([.1,.7,.2]).log().expand_as(x) - x=torch.tensor([[[-.2,.6,1.1]]]);y,p=categorical_map(Net(),x,.25,.75) - torch.testing.assert_close(y,torch.tensor([[[0.,2/3,.5]]]),atol=1e-7,rtol=1e-6) - self.assertGreater(float(y.sum()),1.1) - torch.testing.assert_close(p.sum(-1),torch.ones(1,1)) - - def test_weighted_probability_composition_matches_maps(self): - net=SequenceNet(3,4,12);x=torch.randn(6,3,4) - s=torch.zeros(6,1);u=s+.5;t=s+.75 - q=composition_target(net,x,s,u,t) - mid,_=categorical_map(net,x,s,u);split,_=categorical_map(net,mid,u,t) - direct=(1-t[...,None])*x+t[...,None]*q - torch.testing.assert_close(direct,split) - torch.testing.assert_close(q.sum(-1),torch.ones(6,3)) - - def test_detached_kl_gradient(self): - logits=torch.tensor([.2,-.1],requires_grad=True);p=torch.tensor([.4,.6]) - probability_kl(logits,p).backward() - torch.testing.assert_close(logits.grad,logits.softmax(-1)-p) - - def test_discrete_teachers_with_zero_derivative(self): - x=torch.randn(5,3,2);s=torch.zeros(5,1)+.2;t=s+.5;net=ConstantLogits() - for kind in ['discrete-lsd','discrete-esd']: - logits,target,clipped=corrected_logit_teacher(net,net,x,s,t,kind) - torch.testing.assert_close(target,logits.softmax(-1)) - self.assertEqual(float(clipped),0.) - - def test_endpoint_match_does_not_remove_temporal_residual(self): - dt=torch.tensor([.2,-.2]);residual=.5*.5*dt - self.assertAlmostEqual(float(residual.square().sum()),.005,places=7) - - def test_decoding_clock_inverse(self): - clock=DecodingClock(13);tau=torch.linspace(0,1,101) - time=clock.inverse(tau) - self.assertTrue((time[1:]>=time[:-1]).all()) - self.assertEqual(float(time[0]),0.);self.assertEqual(float(time[-1]),1.) - - def test_glass_matches_two_observation_bayes_rule(self): - inner=torch.tensor([[.3,-.2]],dtype=torch.float64);outer=inner+.4 - s=torch.tensor([[.35]],dtype=torch.float64);t=s+.2 - # Independent derivation by conditioning each Gaussian component twice. - variance=1/(1/DATA_STD**2+(s/(1-s))**2+(t/(1-t))**2) - means=variance[:,None]*(CENTERS.double()/DATA_STD**2+s[:,None]*inner[:,None]/(1-s[:,None])**2+t[:,None]*outer[:,None]/(1-t[:,None])**2) - covariance=torch.tensor([[float((1-s)**2+s**2*DATA_STD**2),float(s*t*DATA_STD**2)], - [float(s*t*DATA_STD**2),float((1-t)**2+t**2*DATA_STD**2)]],dtype=torch.float64) - residual=torch.stack([inner[:,None]-s[:,None]*CENTERS,outer[:,None]-t[:,None]*CENTERS],-1) - logits=-.5*torch.einsum('bkdi,ij,bkdj->bk',residual,torch.linalg.inv(covariance),residual) - expected=(logits.softmax(-1)[...,None]*means).sum(1) - torch.testing.assert_close(glass_denoiser(inner,s,outer,t),expected,atol=1e-9,rtol=1e-9) - - def test_glass_no_inner_observation(self): - outer=torch.randn(5,2);t=torch.full((5,1),.4) - torch.testing.assert_close(glass_denoiser(torch.randn(5,2),torch.zeros_like(t),outer,t),exact_denoiser(outer,t)) - - def test_meta_surrogate_gradient(self): - delta=torch.tensor([[.3]],requires_grad=True) - w=torch.tensor([[1.],[4.]]);gw=torch.tensor([[.2],[.8]]) - loss=fine_tune_surrogate(delta,w,gw,.5);loss.backward() - torch.testing.assert_close(delta.grad,2*(w*delta.detach()-.5*gw).mean().reshape(1,1)) - - def test_importance_weights_recover_posterior_mean(self): - x=torch.tensor([[.5,-.3]]) - _,mean,ess=weighted_diamond_samples(x,.5,50000) - self.assertLess(float((mean-exact_denoiser(x,.5)).abs().max()),.2) - self.assertGreater(float(ess),100) - - def test_insertion_preserves_order_and_clocks(self): - x=torch.tensor([[1.,0.],[0.,1.]]);b=torch.tensor([0.,.2]);counts=torch.tensor([1,0,1]) - noise=torch.tensor([[-.2,.4],[.3,-.1]]) - y,bt=insert_tokens(x,b,counts,noise,.5) - torch.testing.assert_close(y[1:3],x) - torch.testing.assert_close(bt,torch.tensor([.5,0.,.2,.5])) - torch.testing.assert_close(local_clock(.75,bt),torch.tensor([.5,.75,.6875,.5])) - - def test_gap_labels_and_global_budget(self): - torch.testing.assert_close(gap_counts(torch.tensor([1,4]),torch.tensor([0,2,3,5]),6),torch.tensor([1.,2.,1.])) - counts,_=bounded_counts(torch.tensor([100.,100.,100.]),4) - self.assertLessEqual(int(counts.sum()),4) - self.assertTrue((counts>=0).all()) - - def test_count_divergence_minimizes_conditional_mean(self): - prediction=torch.tensor(2.,requires_grad=True) - count_divergence(torch.tensor([0.,1.,5.]),prediction).mean().backward() - self.assertAlmostEqual(float(prediction.grad),0.,places=6) - - def test_insertion_interval_zero_and_terminal(self): - net=ExpandingNet(4,3,8);x=torch.zeros(1,4,3);b=torch.zeros(1,4);m=b.bool();s=torch.tensor([[.3]]) - self.assertEqual(float(net.counts(x,b,m,s,s).sum()),0.) - t=torch.ones_like(s) - self.assertTrue(torch.isfinite(net.counts(x,b,m,s,t)).all()) - - def test_chen_arithmetic_and_covariance(self): - left=torch.tensor([.2,.04]);right=torch.tensor([-.1,-.02]) - torch.testing.assert_close(chen_two(left,right,.5,.5),torch.tensor([.1,-.14])) - L=sample_coefficients(torch.tensor(.3),100000);R=sample_coefficients(torch.tensor(.7),100000) - C=chen_two(L,R,.3,.7) - torch.testing.assert_close(torch.cov(C.T),torch.diag(torch.tensor([1.,1/3])),atol=.015,rtol=0) - - def test_same_noise_constant_sde_and_wrong_noise_counterexample(self): - direct=1+.5+.8*(.2-.1) - split=(1+.25+.8*.2)+.25+.8*(-.1) - self.assertAlmostEqual(direct,split);self.assertAlmostEqual(direct,1.58) - self.assertNotAlmostEqual(direct,1+.5+.8*(-.3)) - - def test_chen_tree_matches_increment_sum(self): - c=torch.randn(6,16,2) - torch.testing.assert_close(aggregate_tree(c)[:,0],c[:,:,0].sum(1)) - - -if __name__=='__main__':unittest.main() diff --git a/lecture_6/verified_examples/README.md b/lecture_6/verified_examples/README.md deleted file mode 100644 index 11bc73cf3944c0cecbe3801380b047e21132b77a..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/README.md +++ /dev/null @@ -1,76 +0,0 @@ -# Verified Lecture 6 examples - -These are actual outputs from seeded CPU runs on 2026-09-13. All 16 default -methods completed 1,000 training steps, sampled, and reproduced identical -seeded samples after checkpoint reload. Teacher/autoencoder stages also used -1,000 steps where applicable. The complete suite passed on PyTorch 2.9.1+cpu -and PyTorch 2.14.0+cpu with Python 3.12.14. These checked-in outputs use 2.9.1. -All 23 independent mathematical tests passed in both environments. - -Reproduce the default runs from `lecture_6/`: - -```bash -python run_all.py --out outputs/verified-torch291 --train-steps 1000 -python -m unittest discover -s tests -v -python numerical_examples.py --output outputs/numerical -``` - -The baseline reports use 128 samples, eight sampling steps, width 64, batch size -64, and seed 6270. Posterior methods use four inner-map steps and 32 reward -particles. Training losses are not comparable across different objectives. - -| Method | First 20 loss mean | Last 20 loss mean | Selected diagnostic | -| --- | ---: | ---: | --- | -| [categorical](categorical/report.json) | 2.47260 | 0.53130 | grammar support 1.000; mean length 4.00 | -| [consistency](consistency/report.json) | 1.97463 | 1.97996 | mean center distance 0.753; mode entropy 1.386 | -| [diamond](diamond/report.json) | 2.30034 | 1.39093 | posterior mean RMSE 0.285; variance RMSE 0.314 | -| [discrete-esd](discrete-esd/report.json) | 2.47265 | 0.55223 | grammar support 1.000; mean length 4.00 | -| [discrete-lsd](discrete-lsd/report.json) | 2.47260 | 0.52785 | grammar support 0.992; mean length 4.00 | -| [expanding](expanding/report.json) | 4.25715 | 2.91940 | grammar support 0.617; mean length 4.09 | -| [flow-matching](flow-matching/report.json) | 3.24047 | 2.02429 | mean center distance 0.575; mode entropy 1.383 | -| [fmlm](fmlm/report.json) | 2.47259 | 0.52103 | grammar support 1.000; mean length 4.00 | -| [fmm-eulerian](fmm-eulerian/report.json) | 1.99308 | 2.03712 | mean center distance 0.474; mode entropy 1.386 | -| [fmm-lagrangian](fmm-lagrangian/report.json) | 1.98769 | 2.05232 | mean center distance 0.461; mode entropy 1.384 | -| [latent](latent/report.json) | 1.95472 | 1.47142 | mean center distance 1.024; mode entropy 1.383 | -| [meanflow](meanflow/report.json) | 1.67703 | 1.57479 | mean center distance 0.480; mode entropy 1.382 | -| [meta](meta/report.json) | 3.04584 | 1.50320 | posterior mean RMSE 0.277; variance RMSE 0.487 | -| [self-distill](self-distill/report.json) | 3.19830 | 2.18310 | mean center distance 0.729; mode entropy 1.381 | -| [shortcut](shortcut/report.json) | 3.22197 | 2.18371 | mean center distance 0.738; mode entropy 1.385 | -| [ssfm](ssfm/report.json) | 0.40952 | 0.00012 | same-noise RMSE 0.00483 | - -The fixed-length text examples recover almost all phrases in the four-family -grammar at eight steps. The expanding example still produces invalid -combinations; its support fraction is retained. Posterior means and variances -both have measurable error. Continuous samples cover the modes but retain -finite distributional error. These observations describe the actual small runs, -and are not evidence of reproducing paper-scale quality. - -## Additional verified recipes - -```bash -python examples/meta.py --finetune-steps 300 --out outputs/meta-finetune-verified -python run.py --mode sample --out outputs/meta-finetune-verified -python examples/ssfm.py --ssfm-target paper --lr 0.0001 --train-steps 2000 --out outputs/ssfm-paper-stable -python run.py --mode sample --out outputs/ssfm-paper-stable -``` - -`meta-finetune/` records the complete optional reward-drift stage and its -sampling metrics. `ssfm-paper/` records the paper's gradient orientation with -its separate lower-learning-rate recipe. Both reproduced seeded samples after -reloading. The paper-orientation recipe is optimization-sensitive: an initial -1,000-step run at learning rate 0.001 diverged in accuracy despite finite -numbers. Its same-noise RMSE was 51.94. This is why the default implementation -uses the released code's EMA split target. The lower-learning-rate paper run -also remained inaccurate, with same-noise RMSE 3.85 and one-step RMSE 22.40. -It is retained as an optimization-sensitivity comparison, not a successful -strong-solution fit. These failures concern this small-model training recipe; -they do not establish a general failure of the paper's objective. - -`numerical_checks.json` retains the lecture's scalar-flow, posterior-gradient, -probability-update, insertion-clock, and Brownian-composition checks. The scalar -map's one-step endpoint RMSE is 0.011255 after 6,000 steps. - -The JSON files preserve each executed configuration and the full loss history. -`checkpoint.pt` files are regenerated by the commands and excluded from Git, -matching the prior lecture folders. `training_reload_log.txt` and -`mathematics_test_log.txt` record the completed verification. diff --git a/lecture_6/verified_examples/categorical/config.json b/lecture_6/verified_examples/categorical/config.json deleted file mode 100644 index ced2b9c0d2c98a3e7c1cc712e0c7c62dd4ffa5f1..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/categorical/config.json +++ /dev/null @@ -1,21 +0,0 @@ -{ - "method": "categorical", - "mode": "train-sample", - "train_steps": 1000, - "teacher_steps": 1000, - "finetune_steps": 0, - "sample_steps": 8, - "posterior_steps": 4, - "particles": 32, - "reward_strength": 1.0, - "ssfm_target": "official-code", - "batch_size": 64, - "samples": 128, - "width": 64, - "lr": 0.001, - "seed": 6270, - "threads": 1, - "device": "cpu", - "data": null, - "out": "outputs/verified-torch291/categorical" -} diff --git a/lecture_6/verified_examples/categorical/losses.json b/lecture_6/verified_examples/categorical/losses.json deleted file mode 100644 index 02b40bf2f05d6282a4700daad37e66f6b21244ea..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/categorical/losses.json +++ /dev/null @@ -1,8002 +0,0 @@ -[ - { - "step": 0, - "loss": 2.5544073581695557, - "diagonal_ce": 2.5544071197509766, - "finite": 7.632940832991153e-05, - "endpoint_kl": 7.518479833379388e-05, - "temporal_derivative": 1.1446123835412436e-06 - }, - { - "step": 1, - "loss": 2.542590379714966, - "diagonal_ce": 2.5425896644592285, - "finite": 8.30490025691688e-05, - "endpoint_kl": 8.175172843039036e-05, - "temporal_derivative": 1.2972749345863122e-06 - }, - { - "step": 2, - "loss": 2.534853219985962, - "diagonal_ce": 2.5348520278930664, - "finite": 0.00010586393909761682, - 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b/lecture_6/verified_examples/discrete-esd/samples.txt deleted file mode 100644 index 2074e254174c5f009332e8409fe783ee1969e485..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/discrete-esd/samples.txt +++ /dev/null @@ -1,128 +0,0 @@ -gold star moves down -blue square moves right -gold star moves down -green triangle moves up -green triangle moves up -gold star moves down -blue square moves right -green triangle moves up -gold star moves down -blue square moves right -blue square moves right -red circle moves left -gold star moves down -gold star moves down -blue square moves right -gold star moves down -red circle moves left -green triangle moves up -blue square moves right -gold star moves down -red circle moves left -green triangle moves up -green triangle moves up -green triangle moves up -red circle moves left -blue square moves right -green triangle moves up -green triangle moves up -gold star moves down -gold star moves down -blue square moves right 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"train_loss_last_20_mean": 0.5210289716720581, - "training_steps": 1000, - "python": "3.12.14", - "torch": "2.9.1+cpu", - "heldout_diagonal_ce_at_half_time": 0.4668397903442383, - "sample_steps": 8, - "samples": 128, - "sampling_seed": 6370, - "unique_fraction": 0.03125, - "token_entropy": 2.41172456741333, - "mean_length": 4.0, - "empty_fraction": 0.0, - "training_support_fraction": 1.0, - "elapsed_seconds": 3.229403880999598 -} diff --git a/lecture_6/verified_examples/fmlm/samples.txt b/lecture_6/verified_examples/fmlm/samples.txt deleted file mode 100644 index d9483d35f1340d583b1cf02eac645349983b2372..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/fmlm/samples.txt +++ /dev/null @@ -1,128 +0,0 @@ -gold star moves down -blue square moves right -gold star moves down -green triangle moves up -green triangle moves up -gold star moves down -blue square moves right -green triangle moves up -gold star moves down -blue square moves right -blue square moves right 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+0,0 @@ -test_analytic_mixture_velocity (test_mathematics.MathematicsTests.test_analytic_mixture_velocity) ... ok -test_categorical_numerical_step (test_mathematics.MathematicsTests.test_categorical_numerical_step) ... ok -test_chen_arithmetic_and_covariance (test_mathematics.MathematicsTests.test_chen_arithmetic_and_covariance) ... ok -test_chen_tree_matches_increment_sum (test_mathematics.MathematicsTests.test_chen_tree_matches_increment_sum) ... ok -test_composition_alone_does_not_identify_motion (test_mathematics.MathematicsTests.test_composition_alone_does_not_identify_motion) ... /workspace/scratch/e1464e1e4960/cis6270-code/lecture_6/tests/test_mathematics.py:65: UserWarning: Converting a tensor with requires_grad=True to a scalar may lead to unexpected behavior. -Consider using tensor.detach() first. (Triggered internally at /pytorch/torch/csrc/autograd/generated/python_variable_methods.cpp:836.) - self.assertEqual(float(semigroup_loss(m,m,x,s,t)),0.) -ok -test_count_divergence_minimizes_conditional_mean (test_mathematics.MathematicsTests.test_count_divergence_minimizes_conditional_mean) ... ok -test_decoding_clock_inverse (test_mathematics.MathematicsTests.test_decoding_clock_inverse) ... ok -test_detached_kl_gradient (test_mathematics.MathematicsTests.test_detached_kl_gradient) ... ok -test_discrete_teachers_with_zero_derivative (test_mathematics.MathematicsTests.test_discrete_teachers_with_zero_derivative) ... ok -test_endpoint_match_does_not_remove_temporal_residual (test_mathematics.MathematicsTests.test_endpoint_match_does_not_remove_temporal_residual) ... ok -test_exact_identity_and_semigroup (test_mathematics.MathematicsTests.test_exact_identity_and_semigroup) ... ok -test_gap_labels_and_global_budget (test_mathematics.MathematicsTests.test_gap_labels_and_global_budget) ... ok -test_glass_matches_two_observation_bayes_rule (test_mathematics.MathematicsTests.test_glass_matches_two_observation_bayes_rule) ... ok -test_glass_no_inner_observation (test_mathematics.MathematicsTests.test_glass_no_inner_observation) ... ok -test_importance_weights_recover_posterior_mean (test_mathematics.MathematicsTests.test_importance_weights_recover_posterior_mean) ... ok -test_insertion_interval_zero_and_terminal (test_mathematics.MathematicsTests.test_insertion_interval_zero_and_terminal) ... ok -test_insertion_preserves_order_and_clocks (test_mathematics.MathematicsTests.test_insertion_preserves_order_and_clocks) ... ok -test_jvp_losses_reach_parameters (test_mathematics.MathematicsTests.test_jvp_losses_reach_parameters) ... ok -test_lagrangian_and_eulerian_jvp (test_mathematics.MathematicsTests.test_lagrangian_and_eulerian_jvp) ... ok -test_meanflow_backward_identity (test_mathematics.MathematicsTests.test_meanflow_backward_identity) ... ok -test_meta_surrogate_gradient (test_mathematics.MathematicsTests.test_meta_surrogate_gradient) ... ok -test_same_noise_constant_sde_and_wrong_noise_counterexample (test_mathematics.MathematicsTests.test_same_noise_constant_sde_and_wrong_noise_counterexample) ... ok -test_weighted_probability_composition_matches_maps (test_mathematics.MathematicsTests.test_weighted_probability_composition_matches_maps) ... ok - ----------------------------------------------------------------------- -Ran 23 tests in 0.844s - -OK diff --git a/lecture_6/verified_examples/meanflow/config.json b/lecture_6/verified_examples/meanflow/config.json deleted file mode 100644 index 1d2df6f244e8f63ef02b947b37b9633e8bd71299..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/meanflow/config.json +++ /dev/null @@ -1,21 +0,0 @@ -{ - "method": "meanflow", - "mode": "train-sample", - "train_steps": 1000, - "teacher_steps": 1000, - "finetune_steps": 0, - "sample_steps": 8, - "posterior_steps": 4, - "particles": 32, - 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--git a/lecture_6/verified_examples/ssfm/config.json b/lecture_6/verified_examples/ssfm/config.json deleted file mode 100644 index 13fca913c27ef08e1fbc05536ab2042fcc13a809..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/ssfm/config.json +++ /dev/null @@ -1,21 +0,0 @@ -{ - "method": "ssfm", - "mode": "train-sample", - "train_steps": 1000, - "teacher_steps": 1000, - "finetune_steps": 0, - "sample_steps": 8, - "posterior_steps": 4, - "particles": 32, - "reward_strength": 1.0, - "ssfm_target": "official-code", - "batch_size": 64, - "samples": 128, - "width": 64, - "lr": 0.001, - "seed": 6270, - "threads": 1, - "device": "cpu", - "data": null, - "out": "outputs/verified-torch291/ssfm" -} diff --git a/lecture_6/verified_examples/ssfm/losses.json b/lecture_6/verified_examples/ssfm/losses.json deleted file mode 100644 index b11d60384857182dfb8046ff9f296fd823fea5bc..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/ssfm/losses.json +++ 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--0.8764173 --0.0864979 --0.2105043 -0.2688649 --0.3274689 --0.6257399 -0.2322355 -0.2492823 diff --git a/lecture_6/verified_examples/training_reload_log.txt b/lecture_6/verified_examples/training_reload_log.txt deleted file mode 100644 index b2c192cf5547879b2ff435965023048548545857..0000000000000000000000000000000000000000 --- a/lecture_6/verified_examples/training_reload_log.txt +++ /dev/null @@ -1,33 +0,0 @@ -{"method": "flow-matching", "out": "outputs/verified-torch291/flow-matching", "loss": 2.0242882668972015, "seconds": 2.210745528995176} -{"method": "flow-matching", "mode": "sample", "out": "outputs/verified-torch291/flow-matching", "seconds": 0.005338316004781518} -{"method": "fmm-lagrangian", "out": "outputs/verified-torch291/fmm-lagrangian", "loss": 2.0523160219192507, "seconds": 5.863147705000301} -{"method": "fmm-lagrangian", "mode": "sample", "out": "outputs/verified-torch291/fmm-lagrangian", "seconds": 0.005786456000350881} -{"method": "fmm-eulerian", "out": "outputs/verified-torch291/fmm-eulerian", "loss": 2.0371171474456786, "seconds": 5.261868300003698} -{"method": "fmm-eulerian", "mode": "sample", "out": "outputs/verified-torch291/fmm-eulerian", "seconds": 0.004115334995731246} -{"method": "self-distill", "out": "outputs/verified-torch291/self-distill", "loss": 2.1831001818180082, "seconds": 1.9765548720024526} -{"method": "self-distill", "mode": "sample", "out": "outputs/verified-torch291/self-distill", "seconds": 0.0036691740024252795} -{"method": "consistency", "out": "outputs/verified-torch291/consistency", "loss": 1.979964154958725, "seconds": 3.6416576160045224} -{"method": "consistency", "mode": "sample", "out": "outputs/verified-torch291/consistency", "seconds": 0.0030551349991583265} -{"method": "shortcut", "out": "outputs/verified-torch291/shortcut", "loss": 2.1837141811847687, "seconds": 2.735806157004845} -{"method": "shortcut", "mode": "sample", "out": "outputs/verified-torch291/shortcut", "seconds": 0.006531202998303343} -{"method": "meanflow", "out": "outputs/verified-torch291/meanflow", "loss": 1.5747917950153352, "seconds": 2.9319772759990883} -{"method": "meanflow", "mode": "sample", "out": "outputs/verified-torch291/meanflow", "seconds": 0.003650777005532291} -{"method": "latent", "out": "outputs/verified-torch291/latent", "loss": 1.4714181840419769, "seconds": 3.618038576998515} -{"method": "latent", "mode": "sample", "out": "outputs/verified-torch291/latent", "seconds": 0.0070175579967326485} -{"method": "fmlm", "out": "outputs/verified-torch291/fmlm", "loss": 0.5210289716720581, "seconds": 3.229403880999598} -{"method": "fmlm", "mode": "sample", "out": "outputs/verified-torch291/fmlm", "seconds": 0.009706972996355034} -{"method": "categorical", "out": "outputs/verified-torch291/categorical", "loss": 0.5313047975301742, "seconds": 5.423348005999287} -{"method": "categorical", "mode": "sample", "out": "outputs/verified-torch291/categorical", "seconds": 0.009315825001976918} -{"method": "discrete-lsd", "out": "outputs/verified-torch291/discrete-lsd", "loss": 0.5278535842895508, "seconds": 4.936540884998976} -{"method": "discrete-lsd", "mode": "sample", "out": "outputs/verified-torch291/discrete-lsd", "seconds": 0.009318361997429747} -{"method": "discrete-esd", "out": "outputs/verified-torch291/discrete-esd", "loss": 0.5522287711501122, "seconds": 4.619678268994903} -{"method": "discrete-esd", "mode": "sample", "out": "outputs/verified-torch291/discrete-esd", "seconds": 0.00982432599994354} -{"method": "diamond", "out": "outputs/verified-torch291/diamond", "loss": 1.3909348011016847, "seconds": 4.959251613996457} -{"method": "diamond", "mode": "sample", "out": "outputs/verified-torch291/diamond", "seconds": 0.07566004899854306} -{"method": "meta", "out": "outputs/verified-torch291/meta", "loss": 1.5032016336917877, "seconds": 2.076603032001003} -{"method": "meta", "mode": "sample", "out": "outputs/verified-torch291/meta", "seconds": 0.0813792570043006} -{"method": "expanding", "out": "outputs/verified-torch291/expanding", "loss": 2.9194031715393067, "seconds": 21.383618507999927} -{"method": "expanding", "mode": "sample", "out": "outputs/verified-torch291/expanding", "seconds": 0.5061805700024706} -{"method": "ssfm", "out": "outputs/verified-torch291/ssfm", "loss": 0.00012182250538899098, "seconds": 2.4875807970020105} -{"method": "ssfm", "mode": "sample", "out": "outputs/verified-torch291/ssfm", "seconds": 0.015510053002799395} -All 16 methods trained, sampled, and reproduced samples after checkpoint reload.