Maximum Power Training Limits

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tags: - colorclass/a thermodynamic theory of statistical learning ---# Maximum Power Training Limits

1. Fundamental Bounds

1.1 Landauer-Shannon Limit

Maximum information processing rate:

(dI/dt)_max = P_max/(k_B T ln(2))

1.2 Power-Loss Trade-off

P(t) = η|∇L(θ)|² + D tr(I_F(θ))

where: - First term: directed power for optimization - Second term: diffusive power from exploration

2. Maximum Power Theorem

2.1 Power Transfer Analysis

Theorem 1: Maximum power transfer occurs when:

η|∇L(θ)|² = D tr(I_F(θ))

Proof: 1. Total power:

P_total = P_useful - P_dissipated

2. Useful power scaling:

P_useful ∝ η|∇L(θ)|²

3. Dissipation scaling:

P_dissipated ∝ (η|∇L(θ)|²)²/D tr(I_F(θ))

4. Maximum at equal terms.

2.2 Optimal Learning Rate

η_opt = D tr(I_F(θ))/|∇L(θ)|²

3. Speed Limits

3.1 Thermodynamic Speed Limit

Theorem 2: Minimum time to reduce loss by ΔL:

t_min = √(2ΔL * tr(I_F(θ))/P_max)

Proof: 1. Cramér-Rao bound:

var(Δθ) ≥ tr(I_F⁻¹(θ))

2. Power constraint:

|dθ/dt|² ≤ P_max/tr(I_F(θ))

3. Combine for minimum time.

3.2 Maximum Learning Rate

η_max = P_max/(|∇L(θ)|² * tr(I_F(θ)))

4. Batch Size Limits

4.1 Critical Batch Size

Theorem 3: Maximum useful batch size:

B_crit = P_max/(ν₀ * tr(I_F(θ)))

Proof: 1. Power per sample:

P_sample = ν₀ * tr(I_F(θ))

2. Total power constraint:

B * P_sample ≤ P_max

4.2 Optimal Batch Scaling

B_opt(t) = min(
    B_crit,
    √(P_max * t/E_sample)
)

5. Temperature Dynamics

5.1 Maximum Power Temperature

Theorem 4: Optimal temperature for power:

T_opt = ΔL/log(P_max/(ν₀ * tr(I_F(θ))))

5.2 Temperature Schedule

T(t) = T_opt * √(tr(I_F(θ₀))/tr(I_F(θ(t))))

6. Combined Strategy

6.1 Maximum Power Protocol

def max_power_update(model, P_max):
    # Estimate Fisher Information
    I_F = estimate_fisher(model)
    
    # Compute gradients
    grad = compute_gradients(model)
    
    # Set optimal parameters
    η = min(
        η_max,
        D * tr(I_F)/|grad|²
    )
    
    B = min(
        B_crit,
        √(P_max/(ν₀ * tr(I_F)))
    )
    
    T = ΔL/log(P_max/(ν₀ * tr(I_F)))
    
    return η, B, T

6.2 Efficiency Bound

Maximum efficiency:

η_max = (1 - T_c/T_h) * (1 - log(T_h/T_c))

7. Fluctuation Constraints

7.1 Power Fluctuations

var(P) ≤ (P_max)²/4

7.2 Stability Condition

T ≥ √(P_max * dt/k_B)

8. Practical Implementation

8.1 Hardware Limits

- Memory bandwidth: B_max - Compute capacity: ν₀_max - Power budget: P_max

8.2 Software Limits

- Gradient noise scale - Fisher trace estimation - Temperature control

8.3 Combined Limits

Effective power limit:

P_eff = min(
    P_max,
    B_max * ν₀_max * tr(I_F),
    Memory_bandwidth * E_access
)