Thermodynamic Concentration Inequalities: Controlling Uncertainty in Finite-Time and Small-Sample Thermodynamic Inference

What the math gives to ML

The paper provides nonasymptotic concentration inequalities for time-averaged currents of geometrically ergodic, possibly irreversible diffusions. The transferable mechanism is that finite-time uncertainty is controlled by a relaxation scale, local entropy-production rate, and local observable fluctuations, rather than by sample count alone. In neural-network training, stochastic gradient updates can be treated as a nonequilibrium diffusion, allowing a confidence-controlled stopping rule or adaptive batch/learning-rate controller based on an estimated concentration rate. A useful falsifiable test is whether the predicted exponential tail and effective relaxation time agree with observed gradient-current fluctuations across batch sizes and training phases.