B_power vs B_info

---

tags: - colorclass/a thermodynamic theory of statistical learning ---# Power vs Information Processing Dominance Analysis

1. Basic Relations

1.1 Power Limit

B_power = P_max/(ν₀ * tr(I_F(θ)))

1.2 Information Limit

B_info = √(P_max * T₀)/(k_B * ν₀ * tr(I_F(θ)))

2. Crossover Analysis

2.1 Dominance Condition

B_power > B_info when:

P_max/(ν₀ * tr(I_F(θ))) > √(P_max * T₀)/(k_B * ν₀ * tr(I_F(θ)))

2.2 Simplification

Rearranging:

P_max > T₀/(k_B)²

Define critical power:

P_crit = T₀/(k_B)²

3. Regime Analysis

3.1 Power-Limited Regime (P_max < P_crit)

When:

P_max < T₀/(k_B)²

Then:

B_crit = B_power = P_max/(ν₀ * tr(I_F(θ)))

3.2 Information-Limited Regime (P_max > P_crit)

When:

P_max > T₀/(k_B)²

Then:

B_crit = B_info = √(P_max * T₀)/(k_B * ν₀ * tr(I_F(θ)))

4. Physical Interpretation

4.1 Small Power Regime

When P_max < P_crit: - Raw compute is limiting - Information processing capacity exceeds available power - Linear scaling with power

4.2 Large Power Regime

When P_max > P_crit: - Information processing is limiting - Excess compute power available - Square root scaling with power

5. Practical Implications

5.1 Small Models/Hardware

If P_max << T₀/(k_B)²:
    B_crit ∝ P_max
    Focus on hardware efficiency

5.2 Large Models/Hardware

If P_max >> T₀/(k_B)²:
    B_crit ∝ √P_max
    Focus on algorithmic efficiency

6. Observable Differences

6.1 Power-Limited Signs

- Linear scaling with compute - High hardware utilization - Temperature-independent behavior

6.2 Information-Limited Signs

- Square root scaling with compute - Hardware underutilization - Strong temperature dependence

7. Critical Points

7.1 Power-Information Transition

Occurs at:

P_max = T₀/(k_B)²
B_transition = T₀/(k_B²ν₀ * tr(I_F(θ)))

7.2 Scaling Transitions

Efficiency ∝ {
    P_max            if P_max < P_crit
    √P_max           if P_max > P_crit
}