tags: - colorclass/a thermodynamic theory of statistical learning ---# Analysis of Training Beyond Critical Batch Size
1. Critical Point Analysis
1.1 Critical Point Definition
B_crit = P_max/(ν₀ * tr(I_F(θ)))
represents point where: - Power budget saturates - Information processing capacity maximizes - Gradient noise reaches optimal level
1.2 Physical Analogy
Like phase transitions: - B < B_crit: “liquid” phase (efficient mixing) - B = B_crit: “critical point” (optimal fluctuations) - B > B_crit: “gas” phase (inefficient mixing)
2. Supercritical Regime (B > B_crit)
2.1 Power Utilization
For B > B_crit:
P_effective = P_max * (B_crit/B)
Implications: - Power per sample decreases - Computational efficiency drops - Diminishing returns on parallelism
2.2 Information Processing
Information rate becomes sublinear:
dI/dt = (P_max/T) * √(B_crit/B)
Due to: - Redundant gradient computations - Reduced exploration effectiveness - Correlated sample statistics
3. Pathological Behaviors
3.1 Gradient Collapse
When B >> B_crit:
var(∇L_B) ∝ 1/B
std(∇L_B) ∝ 1/√B
Leading to: - Over-deterministic updates - Poor exploration - Premature convergence
3.2 Temperature Effects
Effective temperature drops:
T_eff = T * √(B_crit/B)
Resulting in: - Reduced barrier crossing - Trapped states - Loss plateaus
4. Energy Inefficiencies
4.1 Energy Waste
For B > B_crit:
E_waste = E_sample * (B - B_crit)
where E_sample is energy per sample
4.2 Efficiency Scaling
η(B) = η_max * (B_crit/B)
showing linear efficiency drop
5. Compensatory Mechanisms
5.1 Learning Rate Scaling
Required learning rate adjustment:
η(B) = η_opt * √(B_crit/B)
to maintain update magnitude
5.2 Temperature Compensation
Required temperature increase:
T(B) = T_opt * √(B/B_crit)
to maintain exploration
6. Optimization Dynamics
6.1 Update Quality
Quality factor Q:
Q(B) = min(1, √(B_crit/B))
measuring update effectiveness
6.2 Convergence Time
Time to convergence:
t_conv(B) = t_opt * max(1, B/B_crit)
7. Practical Implications
7.1 Resource Waste
For B > B_crit: - Compute waste: (B - B_crit)/B - Memory waste: (B - B_crit)/B - Time waste: 1 - Q(B)
7.2 Performance Impact
Loss degradation:
ΔL_excess ≈ k_B T * log(B/B_crit)
when B >> B_crit
7.3 Mitigation Strategies
1. Gradient accumulation 2. Layer-wise batch sizing 3. Dynamic batch adjustment
8. Detection Methods
8.1 Efficiency Metrics
Monitor:
η_actual/η_theoretical = min(1, B_crit/B)
8.2 Power Utilization
Track:
P_used/P_max = min(1, B_crit/B)
8.3 Information Rate
Measure:
(dI/dt)/(dI/dt)_max = √(B_crit/B)
9. Recommendations
9.1 Batch Size Selection
def optimal_batch(model, P_max, tolerance=0.9):
I_F = estimate_fisher(model)
B_crit = P_max/(ν₀ * tr(I_F))
return min(
B_crit/tolerance,
available_memory/sizeof(sample)
)9.2 Monitoring
def check_efficiency(B_actual, B_crit):
efficiency = min(1, B_crit/B_actual)
if efficiency < 0.8:
warn(f"Operating at {efficiency:.2%} efficiency")