tags: - colorclass/a thermodynamic theory of statistical learning ---# Seifert’s Framework Applied to Neural Networks
1. Core Parallel: Single Trajectory Analysis
1.1 Seifert’s Original Framework
s(τ) = -ln p(x(τ),τ) # System entropy
sm(τ) = q(τ)/T # Medium entropy
stot(τ) = s(τ) + sm(τ) # Total entropy
1.2 Neural Network Mapping
s(θ,t) = -ln p(θ(t),t) # Parameter distribution entropy
sm(θ,t) = W(t)/T # Computational work entropy
stot(θ,t) = s(θ,t) + sm(θ,t) # Total training entropy
2. Fluctuation Theorems
2.1 Seifert’s Theorem
⟨exp(-Δstot)⟩ = 1
Valid for: - Arbitrary initial conditions - Time-dependent driving - Finite time intervals
2.2 Neural Network Version
⟨exp(-ΔL/T + ΔI)⟩ = 1
Where: - ΔL: Loss change - T: Training temperature - ΔI: Information gain
3. Medium Entropy Production
3.1 Seifert’s Original
s˙m(τ) = F(x,λ)ẋ/T # Force × velocity
3.2 Neural Network Analogue
s˙m(θ,t) = η|∇L(θ)|²/T # Gradient × learning rate
4. Jarzynski-like Relations
4.1 Seifert’s Form
⟨exp(-wd/T)⟩ = 1
where wd is dissipated work
4.2 Training Version
⟨exp(-C/T_eff)⟩ = 1
where: - C is compute cost - T_eff = T * B/B_crit
5. Key Extensions
5.1 Critical Batch Size
Analogous to critical damping in Seifert:
B_crit = P_max/(ν₀ * tr(I_F(θ))) # Our result
γ_crit = √(mK) # Seifert's damping
5.2 Power Limits
Our power analysis:
P(t) = η|∇L(θ)|² + D tr(I_F(θ))
Maps to Seifert’s heat dissipation:
q˙ = γẋ² + 2γT
6. Practical Implications
6.1 Optimal Schedules
Seifert: Optimal protocol minimizes dissipation Our version: Power-optimal training schedule:
η(t) = η₀(1 + ln(τ/t))/ln(τω₀)
6.2 Barrier Crossing
Seifert: Kramers escape rate Our version: Loss barrier crossing:
k = (ω₀/2π)(ωᵦ/γ)exp(-ΔL/T)
7. Novel Insights
7.1 Information Processing
Extension of Seifert through:
dI/dt ≤ P_max/(k_B T ln(2))
7.2 Batch Dynamics
New regime not in Seifert:
B > B_crit: P_effective = P_max * (B_crit/B)
8. Unified Picture
8.1 Common Principles
1. Path-dependent entropy production 2. Fluctuation theorems 3. Minimum work principles 4. Optimal protocols
8.2 Key Differences
1. Information processing limits 2. Parallel dynamics (batch) 3. Computational vs physical work 4. Discrete vs continuous updates