Smooth and Convex Proof:-x=Ix [:-mVf()]-x*2 (GD) llxt-nt Vf(xt)-x*(Pythagoras Theorem) =xt-x*2-2meVf(x),x-x*》+n呢I7f(x)I2 ≤x-xP+(-2) Vf(x)I2 exploiting coercivity of smoothness and unconstrained first-order optimality (fx,x4-x*)=(fx)-Vf(x),x-x)≥元IVf6x)-Vf6x*)川2=Vf(x)2 →x+1-x*2≤x-x*2+(2-22)I川Vf(x)2 ≤x-x*2-2IVf(x)2 (by picking=刀=是to minimize ther.h.s) ≤x4-x*2≤.≤x1-x*2 which already implies the convergence Advanced Optimization(Fall 2023) Lecture 4.Gradient Descent Method II 9 Advanced Optimization (Fall 2023) Lecture 4. Gradient Descent Method II 9 Smooth and Convex Proof: (Pythagoras Theorem) (GD) exploiting coercivity of smoothness and unconstrained first-order optimality which already implies the convergence