Entropy Production in Machine Learning Under Fokker-Planck Probability Flow

TL;DR

Data drift in deployed ML models is reframed as nonequilibrium stochastic dynamics under the Fokker–Planck framework, linking feature evolution to probability flow via $dX_t = a(X_t,t)\,dt + B(X_t,t)\,dW_t$ and $\partial_t p = -\nabla\cdot J$ with $J = a\,p - \nabla\cdot(D p)$ and $D=\tfrac{1}{2}BB^\top$. The mismatch between the deployment distribution and a fixed reference is captured by $D(t)=D_{KL}(p(\cdot,t)\| q_{\mathrm{ref}})$ whose rate admits an entropy-balance decomposition $\frac{d}{dt}D(t) = -\dot{\Sigma}_{\mathrm{tot}}(t) + \dot{Q}_{\mathrm{hk}}(t)$ with $\dot{\Sigma}_{\mathrm{tot}}(t) \ge 0$, i.e., entropy production driven by probability currents. This motivates entropy-triggered retraining as a label-free intervention that resets the reference distribution to reduce accumulated mismatch, balancing predictive performance against retraining cost. In controlled nonstationary classification experiments, entropy-triggered retraining achieves near daily-baseline performance with substantially fewer retraining events, demonstrating a principled deployment strategy grounded in nonequilibrium thermodynamics and offering practical efficiency gains in drift scenarios.

Abstract

Machine learning models deployed in nonstationary environments experience performance degradation due to data drift. While many drift detection heuristics exist, most lack a principled dynamical interpretation and provide limited guidance on how retraining frequency should be balanced against operational cost. In this work, we propose an entropy--based retraining framework grounded in nonequilibrium stochastic dynamics. Modeling deployment--time data drift as probability flow governed by a Fokker--Planck equation, we quantify model--data mismatch using a time--evolving Kullback--Leibler divergence. We show that the time derivative of this mismatch admits an entropy--balance decomposition featuring a nonnegative entropy production term driven by probability currents. This interpretation motivates entropy--triggered retraining as a label--free intervention strategy that responds to accumulated mismatch rather than delayed performance collapse. In a controlled nonstationary classification experiment, entropy--triggered retraining achieves predictive performance comparable to high--frequency retraining while reducing retraining events by an order of magnitude relative to daily and label--based policies.

Figures (4)

• Figure 1: Evolution of mismatch entropy under nonstationary drift. The threshold defines entropy-triggered retraining events. Small negative empirical values arise from finite-sample estimation noise; the population Kullback--Leibler divergence is nonnegative.
• Figure 1: Predictive log loss over time under nonstationary drift for different retraining strategies. Entropy-triggered retraining tracks daily retraining while requiring far fewer interventions.
• Figure 2: Cumulative number of retraining events over time. Entropy-triggered retraining reduces retraining frequency by an order of magnitude relative to daily and performance-triggered baselines.
• Figure 3: Cost--performance tradeoff (Pareto view). Each point corresponds to a policy, with cost measured by the number of retraining events and performance measured by average log loss.

Theorems & Definitions (1)

• Remark 1: Lyapunov and nonequilibrium interpretation