tags: - colorclass/a thermodynamic theory of statistical learning ---# Relationship Between FLOP Rate and Training Dynamics
1. Fundamental Quantities
1.1 Basic Rates
- ν₀: FLOPs/token/parameter/s (fundamental attempt frequency) - d: number of parameters - l: sequence length - B: batch size
1.2 Derived Quantities
Total attempt rate per parameter:
ω = ν₀ * l * B (attempts/parameter/s)
2. Transition Rates
2.1 Single Parameter Dynamics
Probability of successful update per attempt:
p_success = exp(-ΔL/T)
Effective transition rate per parameter:
k_param = ν₀ * l * B * exp(-ΔL/T)
2.2 System-wide Evolution
Total transition rate across all parameters:
k_total = d * ν₀ * l * B * exp(-ΔL/T)
3. Training Progress
3.1 Loss Evolution
Rate of loss decrease:
dL/dt = -k_total * ⟨ΔL⟩
= -d * ν₀ * l * B * exp(-ΔL/T) * ⟨ΔL⟩
where ⟨ΔL⟩ is average improvement per successful update
3.2 Learning Time Scale
Characteristic learning time:
τ_learn = 1/(ν₀ * l * B * exp(-ΔL/T))
4. Scaling Relations
4.1 Compute-optimal Batch Size
For fixed compute budget C:
B_opt ∝ √(C * ν₀ * exp(ΔL/T))
4.2 Training Time
Time to reach loss threshold L*:
t_train = (L₀ - L*)/(d * ν₀ * l * B * exp(-ΔL/T) * ⟨ΔL⟩)
5. Hardware Utilization
5.1 Effective Rate
Actual progress rate with hardware efficiency η:
ν_eff = η * ν₀
5.2 Hardware Scaling
Required compute for fixed training time:
C ∝ 1/(η * ν₀)
6. Information Flow
6.1 Information Per Update
Bits extracted per successful update:
ΔI = ΔL/T (nats)
6.2 Information Processing Rate
dI/dt = ν₀ * l * B * exp(-ΔL/T) * ΔI
7. Multi-scale Dynamics
7.1 Fast Processes
Parameter updates on time scale:
τ_fast = 1/ν₀
7.2 Slow Processes
Large-scale reorganization on time scale:
τ_slow = τ_fast * exp(ΔL/T)
8. Optimal Training Regime
8.1 Temperature Selection
Optimal temperature for given time budget t:
T_opt = ΔL/ln(ν₀ * l * B * t)
8.2 Batch Size Selection
Optimal batch size for fixed compute:
B_opt = min(√(C/(ν₀ * t)), C_max)
9. Practical Implications
9.1 Hardware Requirements
Minimum FLOP rate needed for training time t:
ν₀_min = ln(L₀/L*)/(d * l * B * t * exp(-ΔL/T))
9.2 Efficiency Bounds
Maximum achievable efficiency:
η_max = 1/(ν₀ * τ_memory)
where τ_memory is memory access time