ADMM 方法介绍
Application of sGS-ADMM to SVM
SVM model:
min12∥w∥2+C⟨e,ξ⟩ s.t. ZTw+βy+ξ≥eξ≥0,y∈R,w∈Rd \min \quad \frac{1}{2}\|w\|^{2}+C\langle e, \xi\rangle\\ \text { s.t. } \quad Z^{T} w+\beta y+\xi \geq e\\ \xi \geq 0, y \in \mathbf{R}, w \in \mathbf{R}^{d} min21∥w∥2+C⟨e,ξ⟩ s.t. ZTw+βy+ξ≥eξ≥0,y∈R,w∈Rd
Derivation of the dual.
L(w,β,ξ;α,η)=12∥w∥2+C⟨e,ξ⟩−⟨α,ZTw+βy+ξ−e⟩−⟨η,ξ⟩=12∥w∥2+⟨ξ,Ce−α−η⟩−β⟨y,α⟩−⟨w,Zα⟩+⟨α,e⟩ where α,η≥0,α,η∈Rn. Now ∇ξL=−Ce−α−η=0⇒α=Ce−η≤Ce∇βL=−⟨y,α⟩=0∇βL=w−Zα=0 \begin{aligned} L(w, \beta, \xi ; \alpha, \eta) &=\frac{1}{2}\|w\|^{2}+C\langle e, \xi\rangle-\left\langle\alpha, Z^{T} w+\beta y+\xi-e\right\rangle-\langle\eta, \xi\rangle \\ &=\frac{1}{2}\|w\|^{2}+\langle\xi, C e-\alpha-\eta\rangle-\beta\langle y, \alpha\rangle-\langle w, Z \alpha\rangle+\langle\alpha, e\rangle \\ \text { where } \alpha, \eta \geq 0, \alpha, \eta & \in \mathbf{R}^{n} . \text { Now } \\ \nabla_{\xi} L &=-C e-\alpha-\eta=0 \Rightarrow \alpha=C e-\eta \leq C e \\ \nabla_{\beta} L &=-\langle y, \alpha\rangle= 0 \\ \nabla_{\beta} L &=w-Z \alpha=0 \end{aligned} L(w,β,
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