tags: - colorclass/a thermodynamic theory of statistical learning ---# Fundamental Trade-offs in Neural Network Training
1. Speed vs. Efficiency Trade-off
1.1 Basic Relations
Power consumption scales with clock rate:
P ∝ ν₀²
Energy per FLOP:
E_FLOP = k_B T ln(2) + α ν₀²
where: - k_B T ln(2) is Landauer limit - α ν₀² is dynamic power cost
1.2 Efficiency Metric
FLOPS per joule:
η = ν₀/E_FLOP = 1/(k_B T ln(2) + α ν₀²)
1.3 Optimal Points
Maximum efficiency (η_max) occurs at:
ν₀_efficient = √(k_B T ln(2)/α)
Maximum speed (ν₀_max) from thermal limits:
ν₀_max = √(T_max/α)
2. Batch Size vs. Hardware Trade-off
2.1 Total Compute Rate
R_compute = ν₀ * B * N_parallel
where N_parallel is hardware parallelism
2.2 Memory Bandwidth Limit
B_max = min(M_bandwidth/(ν₀ * d * bits_per_param), M_total/d)
2.3 Statistical Efficiency
Learning rate per step:
ΔL ∝ 1/√B
2.4 Optimal Batch Size
For fixed compute budget C:
B_opt = min(√(C * ν₀ * exp(ΔL/T)), B_max)
3. Time vs. Quality Trade-off
3.1 Training Time
Time to reach loss L*:
t_train = (L₀ - L*)/(d * ν₀ * B * exp(-ΔL/T) * ΔL_step)
3.2 Final Loss Bound
L* - L_min ≥ (T/2) * log(ν₀ * t_max)
3.3 Optimal Temperature
T_opt = ΔL/log(ν₀ * t_available)
4. Memory vs. Computation Trade-off
4.1 Memory Access Cost
Energy per parameter access:
E_mem = β * d_mem
where d_mem is memory distance
4.2 Computation vs. Memory Trade-off
Total energy per update:
E_total = N_compute * E_FLOP + N_mem * E_mem
4.3 Optimal Recompute Ratio
Recompute when:
N_compute * E_FLOP < E_mem
5. Parallelism vs. Communication Trade-off
5.1 Parallel Speedup
S(N) = N/(1 + γ * N * log(N))
where γ is communication overhead
5.2 Critical Batch Size
B_crit = 1/(γ * log(N_parallel))
5.3 Communication-Computation Ratio
CCR = (β * d_mem * N_comm)/(α * ν₀² * N_compute)
6. Combined Optimization
6.1 Total Cost Function
C_total = w_time * t_train + w_energy * E_total + w_quality * (L* - L_min)
6.2 Optimal Parameter Set
{ν₀*, B*, T*} = argmin(C_total)
subject to:
ν₀ ≤ ν₀_max
B ≤ B_max
t_train ≤ t_max
6.3 Practical Operating Points
1. Speed-Optimized:
ν₀ = ν₀_max
B = B_max
T = ΔL/log(ν₀_max * t_min)
2. Efficiency-Optimized:
ν₀ = ν₀_efficient
B = B_crit
T = ΔL/log(ν₀_efficient * t_available)
3. Quality-Optimized:
ν₀ = ν₀_efficient
B = B_opt
T = T_opt/2