tags: - colorclass/a thermodynamic theory of statistical learning ---# Thermodynamic Extension of Chinchilla Scaling
1. Chinchilla’s Base Framework
1.1 Original Scaling Law
L(N,D) ≈ A + B(N⁻α + D⁻β)
where: - N: number of parameters - D: dataset tokens - L: loss - A,B,α,β: empirical constants
1.2 Compute Budget Constraint
C = κND
where: - C: compute budget (FLOPs) - κ: FLOPs per token per parameter
2. Thermodynamic Reinterpretation
2.1 Energy-Based Formulation
Total energy budget:
E_total = P_avg * t_train = C * E_FLOP
where: - E_FLOP: energy per FLOP - P_avg: average power - t_train: training time
2.2 Information-Theoretic View
Information gained:
ΔI = D * I_token - S_noise
where: - I_token: information per token - S_noise: entropy from training noise
3. Thermodynamic Constraints
3.1 Landauer Bound
E_min = k_B T * ΔI * ln(2)
3.2 Fisher Information Bound
E_total ≥ k_B T * ∫tr(I_F(θ,t))dt
3.3 Power-Limited Learning Rate
dI/dt ≤ P_max/(k_B T ln(2))
4. Enhanced Scaling Analysis
4.1 Modified Loss Model
L(N,D,T) ≈ A + B(N⁻α + D⁻β) + γT⁻δ
where: - T: effective temperature - γ,δ: temperature scaling parameters
4.2 Energy-Aware Compute Budget
C_effective = κND * f(T)
where f(T) accounts for temperature-dependent efficiency
5. Optimality Conditions
5.1 Traditional Chinchilla
∂L/∂N = λ∂C/∂N
∂L/∂D = λ∂C/∂D
5.2 Thermodynamic Extension
∂L/∂N = λ₁∂C/∂N + λ₂∂E/∂N
∂L/∂D = λ₁∂C/∂D + λ₂∂E/∂D
∂L/∂T = λ₁∂C/∂T + λ₂∂E/∂T
6. Improved Training Strategies
6.1 Temperature Schedule
T(t) = T₀ * (tr(I_F(θ₀))/tr(I_F(θ(t))))^(1/2)
6.2 Power-Aware Batch Sizing
B(t) = min(
√(P_max/(ν₀ * tr(I_F(θ)))),
B_max
)
6.3 Information-Optimal Learning Rate
η(t) = min(
1/max_eigenvalue(I_F(θ)),
P_max/(ν₀ * tr(I_F(θ)))
)
7. Potential Improvements
7.1 Energy Redistribution
Optimize energy allocation:
dE/dt = P_max * √(tr(I_F(θ(t)))/tr(I_F(θ₀)))
7.2 Information-Aware Sampling
Token selection probability:
p(token) ∝ exp(β * ΔI_token)
7.3 Barrier-Aware Training
Adjust temperature near barriers:
T_local = T_base * exp(ΔL_barrier/T_base)
8. Expected Benefits
8.1 Improved Final Loss
ΔL_improvement ≈ k_B T * log(η_thermo/η_chinchilla)
8.2 Energy Efficiency Gain
η_thermo/η_chinchilla ≈ exp(-ΔS_irreversible/k_B)
8.3 Training Time Reduction
t_thermo/t_chinchilla ≈ √(tr(I_F_chinchilla)/tr(I_F_thermo))
9. Practical Implementation
9.1 Adaptive Algorithm
while not converged:
# Estimate local Fisher Information
I_F = estimate_fisher(model)
# Update temperature
T = update_temperature(I_F)
# Adjust batch size
B = compute_optimal_batch(I_F, P_max)
# Set learning rate
η = min(1/max_eigenval(I_F), P_max/(ν₀ * tr(I_F)))
# Update parameters
θ = θ - η * I_F⁻¹ * ∇L(θ)9.2 Monitoring Metrics
- Fisher trace: tr(I_F(θ)) - Energy efficiency: ΔI/E_spent - Irreversible entropy: ΔS_irreversible