tags: - colorclass/statistical mechanics ---see also: - Thermodynamics - Physics
Work and Power in Neural Network Training
1. Thermodynamic Work
1.1 Work Done During Training
Total work has multiple components:
W_total = W_useful + W_dissipative
Where: - W_useful: Information-theoretic minimum work to encode learning - W_dissipative: Heat dissipated during computation
1.2 Useful Work (Landauer Bound)
Minimum work to encode one bit of information:
W_useful_per_bit = k_B T ln(2)
For parameter updates:
W_useful = k_B T ΔS_information
where ΔS_information is information gain in parameters
1.3 Dissipative Work
From gradient computations:
W_dissipative = ∫ F·dθ = ∫ η|∇L(θ)|²dt
2. Power Analysis
2.1 Instantaneous Power
P(t) = dW/dt = P_compute + P_memory + P_communication
2.2 Computational Power
P_compute = ν₀ * N_params * (E_FLOP + T * ΔS_per_op)
where: - E_FLOP is energy per operation - ΔS_per_op is entropy generation per operation
2.3 Memory Power
P_memory = f_memory * N_params * E_mem_access
where: - f_memory is memory access frequency - E_mem_access is energy per memory access
3. Efficiency Metrics
3.1 Thermodynamic Efficiency
η_thermo = W_useful/W_total
= ΔS_information/(ΔS_information + ΔS_dissipative)
3.2 Power Efficiency
η_power = Information_gained/Energy_spent
= ΔI/(P_total * Δt)
4. Work-Loss Trade-offs
4.1 Work-Loss Relationship
L(θ) - L* ≥ (T/2)log(W_min/W_actual)
4.2 Optimal Work Path
Minimum work path follows:
dW_opt = T * |∇S_information|* |dθ|
5. Power-Speed Trade-offs
5.1 Speed Limit from Power
For fixed power budget P_max:
ν₀_max = P_max/(N_params * E_FLOP)
5.2 Training Time vs Power
t_train ≥ ΔS_information * k_B T ln(2)/P_max
6. Optimization Under Constraints
6.1 Fixed Energy Budget
For total energy budget E_budget:
∫P(t)dt ≤ E_budget
Optimal batch size:
B_opt = √(E_budget * η_power/(ν₀ * t_train))
6.2 Fixed Power Budget
Under P_max constraint:
ν₀ * B ≤ P_max/(N_params * E_FLOP)
6.3 Fixed Time Budget
Under t_max constraint:
P_min = ΔS_information * k_B T ln(2)/t_max
7. Energy Landscapes
7.1 Work Done Against Gradient
W_gradient = ∫|∇L(θ)|* |dθ|
7.2 Heat Dissipation
Q = W_total - ΔF
where ΔF is free energy change
8. Practical Implications
8.1 Energy-Aware Training
Optimal power schedule:
P(t) = P_max * (1 - t/t_train)^(1/2)
8.2 Work-Optimal Learning Rate
η_opt = min(1/L, P_max/(ν₀ * N_params * |∇L|²))
8.3 Energy-Efficient Batch Size
B_efficient = √(P_max * η_power/(ν₀ * E_FLOP))
9. Work-Information Balance
9.1 Information-Work Principle
I_gained ≤ W_total/(k_B T ln(2))
9.2 Optimal Work Distribution
dW/dt ∝ |∇I(θ,t)|
where I(θ,t) is Fisher information
9.3 Power-Limited Learning Rate
dI/dt ≤ P_max/(k_B T ln(2))