tags: - colorclass/a thermodynamic theory of statistical learning ---# Optimal Training Through the Lens of Non-equilibrium Thermodynamics
1. Initial Setup
Let’s define our key quantities: - η(t): Time-dependent learning rate (mobility) - T(t): Effective temperature (from batch noise) - β(t) = 1/T(t): Inverse temperature - L(θ): Loss landscape - p(θ,t): Parameter distribution - s(θ,t) = -ln p(θ,t): Local entropy - j(θ,t): Probability current in parameter space
2. Dynamic Equations
The Fokker-Planck equation for parameter evolution:
∂ₜp(θ,t) = ∇·[η(t)p(θ,t)∇L(θ)] + D(t)∇²p(θ,t)
Where D(t) relates to temperature as:
D(t) = η(t)/β(t)
3. Quenching vs Annealing Analysis
3.1 Quenching Regime
In rapid quenching (aggressive early training):
η(t)|∇L(θ)| >> √(2D(t))
This leads to approximate dynamics:
∂ₜp(θ,t) ≈ ∇·[η(t)p(θ,t)∇L(θ)]
The entropy production rate becomes:
s˙tot(t) ≈ η(t)β(t)∫ p(θ,t)|∇L(θ)|² dθ
3.2 Annealing Regime
In proper annealing, we maintain:
η(t)|∇L(θ)| ∼ O(√(2D(t)))
The full entropy production has both terms:
s˙tot(t) = η(t)β(t)∫ p(θ,t)|∇L(θ)|² dθ - D(t)∫ |∇ln p(θ,t)|² p(θ,t) dθ
4. Optimal Cooling Conditions
4.1 Adiabatic Condition
For quasi-static optimization, need:
|∂ₜln p(θ,t)| << |∇L(θ)·∇ln p(θ,t)|
This ensures the distribution can relax locally before major changes.
4.2 Minimal Work Principle
The optimal schedule minimizes:
W = ∫₀ᵗ dt η(t)∫ p(θ,t)|∇L(θ)|² dθ
subject to maintaining sufficient exploration:
D(t) ≥ D_min(t) = c|∇²L(θ)|⁻¹
where c is a constant and |∇²L(θ)| is a measure of local curvature.
5. Optimal Schedule Derivation
5.1 Schedule Constraints
1. Initial condition: High temperature for exploration
T(0) = T₀ >> max(|∇²L(θ)|)
2. Final condition: Low temperature for exploitation
T(t_final) = T_f ∼ O(min(|∇²L(θ)|))
5.2 Optimal Form
The optimal schedule follows:
T(t) = T₀ exp(-t/τ)
where τ is the characteristic cooling time:
τ = t_final / ln(T₀/T_f)
5.3 Learning Rate Schedule
This implies a learning rate schedule:
η(t) = η₀ exp(-t/2τ)
To maintain the proper balance between drift and diffusion.
6. Practical Implementation
6.1 Estimating Parameters
1. Initial temperature T₀:
T₀ ≈ var(∇L(θ))/mean(|∇L(θ)|)
2. Final temperature T_f:
T_f ≈ min(eigenvalues(∇²L(θ)))
3. Cooling time τ:
τ ≈ d/min(eigenvalues(∇²L(θ)))
where d is the parameter dimension.
6.2 Schedule Adaptation
The schedule should be adjusted when:
|∂ₜs(θ,t)| > ε|s˙tot(t)|
This indicates too-rapid cooling causing trapped states.
7. Convergence Guarantees
Under this schedule: 1. The system maintains detailed balance approximately 2. Entropy production rate decreases monotonically 3. Final state approaches true minimum with probability:
P(|L(θ) - L_min| < ε) ≥ 1 - exp(-βε)
This provides exponentially better guarantees than quenching approaches.