TACD: Distilling Efficient Text-to-Motion Models via Terminal Amplification Control

Segmented on-policy distillation

Given text $c$ and noise, the student generates a complete motion in $K$ evaluations, without real-motion training targets. Flow time runs from noise at $t=0$ to clean motion at $t=1$. The sampling grid $0=t_0<\cdots<t_K=1$ fixes the $K$ student evaluations and splits denoising into the segments $[t_k,t_{k+1}]$. Following the single-knot DX parameterization of $\pi$-Flow, one clean-motion prediction defines the student policy over each segment: at the segment start $x^{(k)}$, the student's velocity output $v_\theta$ gives

The prediction estimates the clean motion at $t=1$, not the state at the next boundary; holding it fixed within the segment yields the inference update

Supervision along the student rollout. Teacher supervision is obtained on states that the student itself visits. An EMA copy $\bar\theta$ of the student generates the rollout: starting from noise $\bar x^{(0)}$, it produces the segment starts $\bar x^{(k)}$. Within each segment we place $M$ supervision times

where the terminal margin $\delta>0$ keeps the last query below $t=1$, at which the policy is undefined. This supervision grid is used only during training and is distinct from the sampling grid, which always ends at $t_K=1$. The rollout states at the supervision times are

with $\operatorname{sg}[\cdot]$ the stop-gradient. The online student computes its prediction $\hat x_{1,\theta}^{(k)}$ once at each detached segment start and reuses it for the segment's $M$ queries; the frozen teacher velocity $v_{\mathrm{tea}}$ is queried at these states. Velocity matching penalizes, at each query, the residual

where $w_k$ are segment weights and $\|\cdot\|_m^2$ is the squared error averaged over valid frames and channels. Rollout states and teacher targets are constants for the update, so gradients pass only through the online predictions at the segment starts. This construction fixes where supervision is obtained; the matching space sets how strongly each query counts.

Terminal amplification in velocity matching

The teacher's instantaneous velocity implies a clean-motion estimate, $\hat x_1^{\mathrm{tea}}(x,t;c)=x+(1-t)\,v_{\mathrm{tea}}(x,t;c)$. Substituting it into the velocity residual at a supervision state $\bar x$ and time $s$ gives

with $\ell_x(s)=\|\hat x_{1,\theta}^{(k)}-\hat x_1^{\mathrm{tea}}(\bar x,s;c)\|_m^2$ the clean-motion loss: velocity matching is clean-motion matching with an inverse-square time-dependent weight. The objective is therefore

where the segment weights $w_k$ are a design choice but the coefficients $b_{k,j}$ are not: they are induced by matching in velocity space.

The last query of the supervision grid sits at $s_{K-1,M}=1-\delta$ and receives the terminal coefficient $b_{K-1,M}=\delta^{-2}$. On a uniform sampling grid every other coefficient is at most $(MK)^2$, so a small numerical margin makes one coefficient exceed all others by orders of magnitude: $\delta=10^{-3}$ gives $10^6$, while at $K=8$ and $M=2$ the other fifteen coefficients are at most $252$.

Because the supervision grid is reused at every update, this weight returns at the same supervision time in every update, instead of being met occasionally as under random time sampling. We call this persistent overweighting terminal amplification.

Terminal amplification control

TACD bounds every coefficient $b_{k,j}$ by a value tied to the student's step size. It retains the sampling grid, the inference update, and the training budget, changes only the supervision used during training, and adds no network module or training stage. For a flow teacher it ties the terminal margin to the step $\Delta t=1/K$ of the uniform sampling grid $t_k=k/K$:

The latest teacher query then sits at the midpoint of the final student step, so the teacher is never queried next to the singular time, while the sampling grid still ends at $t_K=1$. The terminal coefficient has the same order as the others on the grid, and the rule is closed-form in the step count, with no tuned threshold.

Two other enforcements meet the same bound at the original queries, each with a different loss. The three differ in how they meet the bound:

At $K=8$ the bound is $4K^2=256$, and TACD queries the final segment at $0.90625$ and $0.9375$.

Extension to diffusion teachers

For a diffusion teacher the student again predicts clean motion at each segment start and learns from frozen-teacher targets on detached rollout states, with two changes: the states follow deterministic DDIM sub-trajectories, and the loss is the objective above with $b_{k,j}\equiv1$. Let $\tau_0>\cdots>\tau_K$ be the teacher sampler's $K$-step timestep schedule with cumulative signal level $\bar\alpha_\tau$, and $\hat x_{0,\theta}^{(k)}$ the student's clean-motion prediction at the segment start ($x_0$ is the flow case's $x_1$ under reversed time). Holding the rollout prediction $\hat x_{0,\bar\theta}^{(k)}$ fixed over the segment also fixes the noise $\bar\epsilon^{(k)}$ it implies, and the DDIM states at $M$ supervision times $\tau_{k,j}$ within the segment are

The target is the frozen teacher's clean-motion estimate $\hat x_0^{\mathrm{tea}}$, computed with the teacher sampler's guidance. Written in clean-motion space from the start, this loss carries no $(1-s)^{-2}$ factor; matching noise predictions on the same states would instead weight the clean-motion error by the signal-to-noise ratio, which also grows without bound toward the clean end. Endpoint matching thus carries no time-dependent weight, and for diffusion teachers TACD denotes this endpoint-matching form.