At critical point: vTHv Hessian 1 L(8)≈L(8)t(8-8)H(8-8) For all v vTHv>0→ Around 0':L(O)>L(θ')→Local minima H is positive definite All eigen values are positive. For all v vTHv<0 Around0':L(8)<L(0)◆ Local maxima H is negative definite =All eigen values are negative. Sometimes vTHv>0,sometimes vHv<0 Saddle point Some eigen values are positive,and some are negative.Hessian At critical point: 𝐻 is positive definite 𝒗 𝑇𝐻𝒗 > 0 Around 𝜽 Local minima ′ : 𝐿 𝜽 > 𝐿 𝜽 ′ All eigen values are positive. 𝒗 𝑇𝐻𝒗 < 0 Local maxima Sometimes 𝒗 𝑇𝐻𝒗 > 0, sometimes 𝒗 𝑇𝐻𝒗 < 0 Saddle point 𝒗 𝑇𝐻𝒗 𝐿 𝜽 ≈ 𝐿 𝜽 ′ + 1 2 𝜽 − 𝜽 ′ 𝑇𝐻 𝜽 − 𝜽 ′ Around 𝜽 ′ : 𝐿 𝜽 < 𝐿 𝜽 ′ = = = 𝐻 is negative definite = All eigen values are negative. Some eigen values are positive, and some are negative. For all 𝒗 For all 𝒗