# Dissipation-Budgeted Nonreversible Sampling

- ID: 3131
- Canonical URL: https://synthcore.org/idea/3131/dissipation-budgeted-nonreversible-sampling
- API JSON: https://synthcore.org/api/idea/3131.json
- API Markdown: https://synthcore.org/api/idea/3131.md
- Verification status: unverified
- Source: [arXiv:2609.03179](https://arxiv.org/abs/2609.03179)
- Category: sampling
- Solves: speedup, stability, accuracy
- ML areas: diffusion, diffusion-sampling, training-dynamics
- Math tags: statistical-mechanics, stochastic-processes, probability, dynamical-systems, control-theory
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

Add a controlled nonreversible drift to a Langevin or score-based diffusion sampler so trajectories reach a target high-probability region faster, while constraining pathwise entropy production or excess heat. The paper predicts that hazard-rate improvement has a thermodynamic ceiling: general time-dependent survival acceleration is at most linear in perturbation strength and prior entropy production, while rare-event acceleration is bounded exponentially by excess heat.

## Mathematical statement

Let $X_t$ be a sampler state, $\mathcal D$ a target set, $\mathcal T=\inf\{t:X_t\in\mathcal D\}$ the hitting time, $S(t)=\mathbb P(\mathcal T>t)$ the survival probability, and $r(t)=-d\log S(t)/dt$ the hazard. Use baseline dynamics $dX_t=b_0(X_t,t)dt+\sqrt{2D(X_t,t)}dW_t$ and driven dynamics $dX_t=[b_0(X_t,t)+u_\theta(X_t,t)]dt+\sqrt{2D(X_t,t)}dW_t$. The drift perturbation $u_\theta$ has pathwise quadratic dissipation $Q_T=\int_0^T u_\theta^\top(2D)^{-1}u_\theta\,dt$, where $D$ is the positive diffusion matrix. For approximately constant rare-event hazards, $S_i(t)=e^{-r_i t}$ and the speedup is $r_1/r_0$. The thermodynamic mechanism predicts a linear bound on general hazard improvement in perturbation strength and pre-perturbation entropy production, and an exponential bound of the form $r_1/r_0\leq\exp(CQ_{\rm ex})$ for rare events, where $Q_{\rm ex}$ is dimensionless excess heat and $C$ is the theorem-dependent coefficient. If the exact coefficient is unavailable from the implementation, estimate it on calibration trajectories and test the functional form independently on held-out trajectories.

## Key formulas

- $$S(t)=\mathbb P(\mathcal T>t)=\exp\left[-\int_0^t r(t')\,dt'\right],\qquad r(t)=-\frac{d}{dt}\log S(t).$$
- $$dX_t=b_0(X_t,t)dt+\sqrt{2D(X_t,t)}dW_t,\qquad dX_t^{(1)}=[b_0(X_t,t)+u_\theta(X_t,t)]dt+\sqrt{2D(X_t,t)}dW_t.$$
- $$Q_T=\int_0^T u_\theta(X_t,t)^\top(2D(X_t,t))^{-1}u_\theta(X_t,t)\,dt,$$
- $$\log\frac{r_1}{r_0}\lesssim C Q_{\rm ex},\qquad\text{equivalently}\qquad \frac{r_1}{r_0}\lesssim e^{C Q_{\rm ex}}.$$

## Implementation notes

(1) Integration point: modify the reverse-time drift of a small score-based diffusion sampler or the drift of an overdamped Langevin sampler. Keep the learned baseline drift $b_0$ fixed and add $u_\theta(x,t)$, represented by a small MLP. Define a target set as a ball around a desired mode, or as a high-density region under a known toy target. (2) Pseudocode: initialize paired baseline and driven states with the same $x_0$ and Brownian increments; at each step compute $x\leftarrow x+[b_0(x,t)+u_\theta(x,t)]\Delta t+\sqrt{2D\Delta t}\epsilon$; accumulate $q\leftarrow q+u_\theta^\top(2D)^{-1}u_\theta\Delta t$; stop a trajectory at its first entry into the target set; estimate $S(t)$ and $r(t)=-\Delta\log S(t)/\Delta t$. Train or tune $u_\theta$ using mean hitting time minus $\lambda q$, or enforce $q\leq Q_{\max}$ with a Lagrange multiplier or per-step projection. (3) Compute survival curves, first-passage times, terminal sample quality, and quadratic dissipation directly. Fit the unknown conversion coefficient $C$ only on calibration trajectories, then evaluate the bound on held-out trajectories. (4) First cheap experiment: use a two-dimensional double-well potential and compare baseline Langevin, an unconstrained driven sampler, and the dissipation-budgeted sampler over a sweep of drift scales. The falsifiable prediction is that $\log(r_1/r_0)$ versus $Q_{\rm ex}$ is approximately linear for rare events and does not systematically exceed the fitted exponential ceiling; for nonconstant hazards, integrated hazard gain should initially grow linearly with perturbation strength and then saturate. Also verify that faster hitting does not cause an unacceptable increase in terminal KL divergence.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.