tags: - colorclass/a thermodynamic theory of statistical learning ---# Power-Time Trade-off Analysis in Neural Training
1. Trade-off Curve Derivation
1.1 Basic Relations
Training time to reach loss gap ΔL:
t(B,η,T) = ΔL/(η * |∇L|² * Q(B,T))
where Q(B,T) is update quality factor.
Power efficiency:
η_power(B) = min(1, B_crit/B)
1.2 Quality Factor
Q(B,T) = √(B_crit/B) * exp(-ΔL_barrier/(k_B T))
accounting for: - Batch size inefficiency - Temperature effects on barrier crossing
2. Pareto Frontier
2.1 Time-Efficiency Trade-off
t(α) * η_power(α) = t_opt * exp(f(α))
where: - α = B/B_crit (overcriticality parameter) - f(α) = (α-1) - ln(α) - t_opt is minimum time at optimal efficiency
2.2 Critical Points
Three key regimes:
α = 1: Maximum efficiency point
α = e: Equal time-efficiency trade-off
α → ∞: Minimum time point
3. Stability Analysis
3.1 Linear Stability Matrix
M(α) = [
∂_θ∂_θL ∂_θ∂_v L
-η*α/B_crit -γ(α)
]
where γ(α) is effective damping
3.2 Stability Conditions
System is stable when:
tr(M) < 0 and det(M) > 0
giving conditions:
γ(α) > 0
η*α/B_crit < λ_max(∂_θ∂_θL)
4. Operating Regimes
4.1 Maximum Efficiency (α = 1)
t = t_opt
η_power = 1
stability = maximum
4.2 Maximum Speed (α >> 1)
t = t_opt/√α
η_power = 1/α
stability ∝ 1/√α
4.3 Balanced Regime (α = e)
t = t_opt/√e
η_power = 1/e
stability = moderate
5. Instability Modes
5.1 Gradient Explosion
For α > α_crit:
P(gradient explosion) ∝ exp(α - α_crit)
5.2 Loss Oscillation
Frequency of oscillations:
ω(α) = √(η*α*λ_max(∂_θ∂_θL)/B_crit)
5.3 Mode Coupling
Critical coupling strength:
g_c(α) = √(γ(α)ω(α))
6. Stabilization Strategies
6.1 Temperature Scaling
Required temperature:
T(α) = T_opt * √α
6.2 Learning Rate Adjustment
η(α) = η_opt/√α
6.3 Momentum Adaptation
Optimal momentum:
β(α) = 1 - 1/√α
7. Practical Trade-off Curve
7.1 Time-to-Target
t(α) = t_opt * max(1/√α, exp(α/α_crit - 1))
7.2 Efficiency
η_power(α) = min(1/α, exp(1 - α/α_crit))
7.3 Combined Metric
Figure of merit:
FOM(α) = 1/(t(α) * P(α))
8. Operating Point Selection
8.1 Time-Limited Regime
Choose:
α = min(t_max/t_opt, α_crit)
8.2 Power-Limited Regime
Choose:
α = min(P_max/P_opt, α_crit)
8.3 Stability-Limited Regime
Choose:
α = min(γ_max/γ_opt, α_crit)