tags: - colorclass/a thermodynamic theory of statistical learning ---# Conservation Laws and Invariants in Learning Dynamics
1. Noether’s Theorem for Learning
1.1 Action Functional
For learning trajectory θ(t):
S[θ] = ∫dt [(1/2)θ̇ᵀI_F(θ)θ̇ - L(θ)]
where: - I_F(θ): Fisher Information metric - L(θ): Loss function
1.2 Continuous Symmetries
For transformation θ → θ + εξ(θ):
δS = 0 ⟨=⟩ ∃ conserved quantity Q
2. Energy Conservation
2.1 Time Translation Invariance
Theorem 1: If loss is explicitly time-independent, total energy is conserved.
Proof: 1. Energy functional:
E = (1/2)θ̇ᵀI_F(θ)θ̇ + L(θ)
2. Time derivative:
dE/dt = θ̇ᵀI_F(θ)θ̈ + (1/2)θ̇ᵀ(dI_F/dt)θ̇ + ∇L(θ)ᵀθ̇
3. Using equations of motion:
d/dt[(1/2)θ̇ᵀI_F(θ)θ̇ + L(θ)] = 0
2.2 Modified Conservation
With learning rate η:
dE/dt = -η|∇L(θ)|²
3. Information Conservation
3.1 KL-Divergence Conservation
Theorem 2: For reversible dynamics, relative entropy is conserved.
Proof: 1. Define relative entropy:
D(p||q) = ∫p(x)log(p(x)/q(x))dx
2. For Hamiltonian flow ϕt:
D(ϕt(p)||ϕt(q)) = D(p||q)
3.2 Fisher Memory
M(t) = ∫₀ᵗ tr(I_F(θ(s)))ds
is monotonically increasing.
4. Geometric Invariants
4.1 Symplectic Structure
Theorem 3: Natural gradient preserves symplectic form.
Proof: 1. Define symplectic form:
ω = dθ ∧ I_F(θ)dθ̇
2. Under natural gradient flow:
dω/dt = 0
4.2 Volume Preservation
Phase space volume element:
dV = √det(I_F(θ))dθ
is preserved under natural gradient.
5. Statistical Invariants
5.1 Fluctuation Theorem
Theorem 4: Entropy production satisfies:
⟨exp(-ΔS_tot)⟩ = 1
Proof: 1. Path probability ratio:
P[θ(t)]/P[θ(-t)] = exp(ΔS_tot)
2. Average:
∫Dθ P[θ(t)]exp(-ΔS_tot) = ∫Dθ P[θ(-t)] = 1
5.2 Detailed Balance
For equilibrium:
p(θ→θ')/p(θ'→θ) = exp(-β[L(θ') - L(θ)])
6. Dynamical Invariants
6.1 Lyapunov Functions
Theorem 5: Free energy is a Lyapunov function.
Proof: 1. Free energy:
F(θ) = L(θ) - TS(θ)
2. Time derivative:
dF/dt = -η|∇L(θ)|² ≤ 0
6.2 Action Invariants
Adiabatic invariant:
J = ∮p·dθ
where p = I_F(θ)θ̇
7. Symmetry-Based Conservation
7.1 Parameter Space Symmetries
For transformation θ → R(θ):
L(R(θ)) = L(θ) ⟹ conserved momentum
7.2 Batch Symmetries
For permutation π of batch B:
L(θ,π(B)) = L(θ,B) ⟹ conserved batch statistics
8. Practical Conservation Laws
8.1 Energy-Information Trade-off
E_total * S_information = constant
8.2 Learning Rate Balance
η(t) * tr(I_F(θ(t))) = constant
8.3 Batch Size Scaling
B(t) * T(t) = constant
9. Applications
9.1 Constrained Optimization
Using conservation laws for constraints:
minimize: L(θ)
subject to: E(θ) = E₀
J(θ) = J₀
9.2 Adaptive Algorithms
Conservation-aware updates:
θ(t+dt) = θ(t) - η∇L(θ) * (Q₀/Q(t))^(1/2)
where Q is conserved quantity.
9.3 Monitoring Invariants
Track conservation violation:
ε(t) = |Q(t) - Q₀|/|Q₀|
for debugging and adaptation.