Reference:
原住民健康問題不只有醫療 原健法立法衛福部盼多考量 /中央廣播電台
(https://www.rti.org.tw/news/view/id/2083929)
- 收入來源：二、菸品健康福利捐及公益彩券盈餘之百分之一。
關鍵評論- 為何《原住民族健康法》草案難以解決原住民族人的「文化安全」問題？ / 朱原慶
(https://phlib.org.tw/最新消息/2020110501/)
law of cosines
∣∣xt−x∣∣2=∣∣xt−xt+1∣∣2+∣∣xt+1−x∣∣2+2⟨xt−xt+1,xt+1−x⟩||x_t-x||^2=||x_t-x_{t+1}||^2+||x_{t+1}-x||^2\\ +2\langle x_t-x_{t+1},x_{t+1}-x\rangle∣∣xt−x∣∣2=∣∣xt−xt+1∣∣2+∣∣xt+1−x∣∣2+2⟨xt−xt+1,xt+1−x⟩
optimality condition
⟨∇f(xt)+L(xt+1−xt),x−xt+1⟩≥0\langle \nabla f(x_t) + L(x_{t+1}-x_t),x-x_{t+1} \rangle \geq 0⟨∇f(xt)+L(xt+1−xt),x−xt+1⟩≥0
proximal inequality 2L⟨∇f(xt),xt+1−x⟩≤∣∣xt−x∣∣2−∣∣xt−xt+1∣∣2−∣∣xt+1−x∣∣2\frac{2}{L}\langle \nabla f(x_t),x_{t+1}-x \rangle \leq \\ ||x_t-x||^2-||x_t-x_{t+1}||^2-||x_{t+1}-x||^2L2⟨∇f(xt),xt+1−x⟩≤∣∣xt−x∣∣2−∣∣xt−xt+1∣∣2−∣∣xt+1−x∣∣2
smoothness
f(xt+1)−f(x)≤L2(∣∣xt−x∣∣2−∣∣xt+1−x∣∣2)f(x_{t+1})-f(x) \leq \frac{L}{2} (||x_t-x||^2-||x_{t+1}-x||^2)f(xt+1)−f(x)≤2L(∣∣xt−x∣∣2−∣∣xt+1−x∣∣2)
f(xt+1)−f(x)≤L∣∣x1−x∣∣22t f(x_{t+1}) - f(x) \leq \frac{L||x_1-x||^2}{2t}f(xt+1)−f(x)≤2tL∣∣x1−x∣∣2
"telescopic sum"
smoothness strong convexity
f(xt+1)−f(x)≤L−μ2∣∣xt−x∣∣2−L2∣∣xt+1−x∣∣2f(x_{t+1})-f(x) \leq \frac{L-\mu}{2} ||x_t-x||^2-\frac{L}{2}||x_{t+1}-x||^2f(xt+1)−f(x)≤2L−μ∣∣xt−x∣∣2−2L∣∣xt+1−x∣∣2
f(xt+1)−f(x)≤μ∣∣x1−x∣∣22((1+μL−μ)t−1)f(x_{t+1}) - f(x) \leq \frac{\mu||x_1-x||^2}{2((1+\frac{\mu}{L-\mu})^t-1)}f(xt+1)−f(x)≤2((1+L−μμ)t−1)μ∣∣x1−x∣∣2
three point equality
Dh(x,xt)=Dh(x,xt+1)+Dh(xt+1,xt)+⟨∇h(xt+1)−∇h(xt),x−xt+1⟩D_h(x,x_t) = D_h(x,x_{t+1}) + D_h(x_{t+1},x_t) + \langle \nabla h(x_{t+1}) - \nabla h(x_t),x-x_{t+1}\rangleDh(x,xt)=Dh(x,xt+1)+Dh(xt+1,xt)+⟨∇h(xt+1)−∇h(xt),x−xt+1⟩
⟨∇f(xt)+L(∇h(xt+1)−∇h(xt)),x−xt+1⟩≥0\langle \nabla f(x_t) + L(\nabla h(x_{t+1})-\nabla h(x_t)),x-x_{t+1} \rangle \geq 0⟨∇f(xt)+L(∇h(xt+1)−∇h(xt)),x−xt+1⟩≥0
proximal inequality 1L⟨∇f(xt),xt+1−x⟩≤Dh(x,xt)−Dh(x,xt+1)−Dh(xt+1,xt)\frac{1}{L}\langle \nabla f(x_t),x_{t+1}-x \rangle \leq \\ D_h(x,x_t)-D_h(x,x_{t+1})-D_h(x_{t+1},x_t)L1⟨∇f(xt),xt+1−x⟩≤Dh(x,xt)−Dh(x,xt+1)−Dh(xt+1,xt)
relative smoothness
f(xt+1)−f(x)≤L(Dh(x,xt)−Dh(x,xt+1))f(x_{t+1})-f(x) \leq L(D_h(x,x_t)-D_h(x,x_{t+1}))f(xt+1)−f(x)≤L(Dh(x,xt)−Dh(x,xt+1))
f(xt+1)−f(x)≤LDh(x,x1)22t f(x_{t+1}) - f(x) \leq \frac{LD_h(x,x_1)^2}{2t}f(xt+1)−f(x)≤2tLDh(x,x1)2
EEE
E∗E^*E∗
xtx_txt
Dh(xt)Dh(x_t)Dh(xt)
Dh(xt)−ηDf(xt)Dh(x_t)-\eta Df(x_t)Dh(xt)−ηDf(xt)
Dh−1(Dh(xt)−ηDf(xt))Dh^{-1}(Dh(x_t)-\eta Df(x_t))Dh−1(Dh(xt)−ηDf(xt))
DhDhDh
Dh−1Dh^{-1}Dh−1
−ηDf(xt)-\eta Df(x_t)−ηDf(xt)
OH
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