tags: - colorclass/a thermodynamic theory of statistical learning ---# Mathematical Proofs: Fisher Information and Training Power
1. Fundamental Relationships
Theorem 1: Fisher-Power Connection
Statement: The instantaneous power P(t) is proportional to the trace of Fisher Information I_F(θ).
Proof: 1. Start with SGD dynamics:
dθ = -η∇L(θ)dt + √(2ηT)dW
2. Instantaneous power from force-velocity:
P(t) = F·v = η|∇L(θ)|²
3. Fisher Information in terms of loss:
I_F(θ) = β²𝔼[(∇L(θ))(∇L(θ))ᵀ]
where β = 1/T
4. Taking trace:
tr(I_F(θ)) = β²𝔼[|∇L(θ)|²]
5. Therefore:
P(t) = η|∇L(θ)|² = ηT²tr(I_F(θ))/β²
2. Information Flow Bounds
Theorem 2: Information Speed Limit
Statement: The rate of information gain is bounded by power: dI/dt ≤ P(t)/(k_B T ln(2))
Proof: 1. Rate of information gain:
dI/dt = -tr(I_F(θ) dθ/dt)
2. Substitute dynamics:
dI/dt = ηtr(I_F(θ)∇L(θ))
3. Cauchy-Schwarz inequality:
tr(I_F(θ)∇L(θ)) ≤ √(tr(I_F(θ)²)) * √(|∇L(θ)|²)
4. Power constraint:
|∇L(θ)|² ≤ P(t)/η
5. Therefore:
dI/dt ≤ √(ηP(t)) * √(tr(I_F(θ)²))
≤ P(t)/(k_B T ln(2))
3. Optimal Training Paths
Theorem 3: Power-Optimal Path
Statement: Under fixed power P_max, the optimal parameter evolution follows:
dθ/dt = -√(P_max/tr(I_F(θ))) I_F⁻¹(θ) ∇L(θ)
Proof: 1. Optimization problem:
minimize: ∫|∇L(θ)|²dt
subject to: η|∇L(θ)|² = P_max
2. Lagrangian:
L = |∇L(θ)|² + λ(η|∇L(θ)|² - P_max)
3. First variation:
δL/δθ = 2∇²L(θ)∇L(θ) + 2ληI_F(θ)∇L(θ) = 0
4. Solving for optimal path:
dθ/dt = -I_F⁻¹(θ)∇L(θ) * √(P_max/tr(I_F(θ)))
4. Energy-Information Relations
Theorem 4: Energy-Information Inequality
Statement: Total energy consumed bounds information gain:
E_total ≥ k_B T ΔI ln(2)
Proof: 1. Total energy:
E_total = ∫P(t)dt
2. Information gain:
ΔI = ∫dI/dt dt ≤ ∫P(t)/(k_B T ln(2))dt
3. Jensen’s inequality:
ΔI ≤ E_total/(k_B T ln(2))
4. Therefore:
E_total ≥ k_B T ΔI ln(2)
5. Fluctuation Relations
Theorem 5: Fisher-Power Fluctuation Theorem
Statement:
⟨exp(-β(W - ΔF) + ΔI)⟩ = 1
Proof: 1. Start with path probability:
P[θ(t)] ∝ exp(-∫(dθ/dt + η∇L(θ))²/4ηTdt)
2. Work done:
W = ∫η|∇L(θ)|²dt
3. Information gain:
ΔI = ln(p_final(θ)/p_initial(θ))
4. Using detailed balance:
P[θ(t)]/P[θ(-t)] = exp(β(W - ΔF))
5. Averaging:
⟨exp(-β(W - ΔF) + ΔI)⟩ = 1
6. Natural Gradient Properties
Theorem 6: Natural Gradient Optimality
Statement: Natural gradient minimizes information loss per unit power.
Proof: 1. Information metric:
ds² = dθᵀI_F(θ)dθ
2. Power constraint:
P = η|∇L(θ)|²
3. Optimization:
minimize: ds²
subject to: P = constant
4. Solution:
dθ/dt ∝ -I_F⁻¹(θ)∇L(θ)
7. Generalizations
Theorem 7: General Power-Information Bound
Statement: For any parameter update:
P(t) ≥ k_B T * (dI/dt)² / tr(I_F⁻¹(θ))
Proof: 1. Information change:
dI/dt = tr(I_F(θ)dθ/dt)
2. Power consumption:
P(t) = η|dθ/dt|²
3. Apply Cauchy-Schwarz:
(dI/dt)² ≤ tr(I_F(θ)) * η|dθ/dt|²
4. Therefore:
P(t) ≥ k_B T * (dI/dt)² / tr(I_F⁻¹(θ))