Minimal-Dissipation Learning for Energy-Based Models

TL;DR

This work connects the bias of approximate MLE training for persistent chain EBMs to the thermodynamic excess work, establishing a fundamental energy-efficiency bound for finite-time learning. By analyzing a harmonic-trap EBM, it shows that minimal-dissipation learning is achievable with carefully designed time-dependent learning-rate protocols, including continuous and discontinuous schemes, and that discontinuities enable learning of unknown targets under equilibrium initialization. The authors generalize these ideas to general potentials, deriving a learning-rate matrix that induces a natural gradient flow on the MLE objective, and demonstrating a deep link between stochastic thermodynamics, information geometry, and second-order optimization. The results offer principled guidance for energy-efficient training and illuminate how thermodynamic insights can inform learning-rate design and potential hardware implementations for thermodynamic computing.

Abstract

We show that the bias of the approximate maximum-likelihood estimation (MLE) objective of a persistent chain energy-based model (EBM) is precisely equal to the thermodynamic excess work of an overdamped Langevin dynamical system. We then answer the question of whether such a model can be trained with minimal excess work, that is, energy dissipation, in a finite amount of time. We find that a Gaussian energy function with constant variance can be trained with minimal excess work by controlling only the learning rate. This proves that it is possible to train a persistent chain EBM in a finite amount of time with minimal dissipation and also provides a lower bound on the energy required for the computation. We refer to such a learning process that minimizes the excess work as minimal-dissipation learning. We then provide a generalization of the optimal learning rate schedule to general potentials and find that it induces a natural gradient flow on the MLE objective, a well-known second-order optimization method.

Figures (6)

• Figure 1: The harmonic trap EBM is trained with a constant learning rate using the approximate MLE objective. The sample distribution, $p_\text{s}$, shown in blue using a ridge plot, evolves according to Langevin dynamics given by Eq. (\ref{['eqn_langevin']}), whereas the model distribution $p_\text{m}$ having centre of mass $\theta$, shown by the red curve, is updated according to the MLE gradient flow described by Eq. (\ref{['eqn_harmonic_trap_MLE']}) with a constant learning rate $\eta=10$. The data distribution $p_\text{d}$ is stationary, centred at $\theta^\ast$, as indicated by the vertical dotted line. The parameters used are $\theta_0=10$, $\theta^\ast=0$, $u_0=-10$, $\mu=10$, $\eta=10$, $\beta=1$, and $\tau=1$.
• Figure 2: The harmonic trap EBM is trained using the approximate MLE objective with a quasi-static protocol, i.e., with one with a constant learning rate $\eta \ll \mu$ so that $p_\text{s} \approx p_\text{m}$ at all times. The parameters chosen are $\theta_0=-10$, $\theta^\ast=0$, $u_0=-10$, $\mu=100$, $\eta=10$, $\beta=1$, and $\tau=1$.
• Figure 3: The harmonic trap EBM is trained with minimal entropy production using a continuous protocol, i.e., $\theta(t)$ (represented by a red curve) is continuous. However, this method does not strictly learn the value of $\theta^\ast$ (dotted line), as the value of $\theta^\ast$ must be known a priori to prepare $p_\text{s}$ and $p_\text{m}$ with the specific initial conditions required to minimize excess work. The parameters chosen are $\theta_0=-5$, $\theta^\ast=0$, $u_0=-10$, $\mu=1$, $\beta=1$, and $\tau=1$.
• Figure 4: The harmonic trap EBM is trained with minimal excess work, using a discontinuous protocol, i.e., the value of $\theta(t)$ (represented by the red curve) has jumps at the endpoints of the schedule. By initializing the system in equilibrium, i.e., $u_0 = \theta_0$, the value of the parameter $\theta^\ast$ (dotted line) is not needed to be known a priori, and so it is genuinely learned. The parameters chosen are $\theta_0=-10$, $\theta^\ast=0$, $u_0=-10$, $\mu=1$, $\beta=1$, and $\tau=1$.
• Figure 5: The harmonic trap EBM is trained using the slow driving protocol Eq. (\ref{['eqn_eta_general']}), which follows geodesics with respect to the thermodynamic metric. The parameters chosen are $\theta_0=-10$, $\theta^\ast=0$, $u_0=-10$, $\mu=5$, $\beta=1$, and $\tau=1$.

Theorems & Definitions (2)

• Theorem 1
• proof