Gradient Descent
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Gradient descent is an algorithm to minimize a cost function by iteratively moving in the direction of steepest descent as defined by the negative of the gradient of the cost function. The rate of change or step size is represented by the learning rate parameter alpha. Potential limitations include getting stuck in local minima rather than finding the global minimum, slow convergence if features have very different scales, and instability if alpha is too large. Feature scaling and reducing alpha can help address these issues. Gradient descent is considered converged when the decrease in the cost function between iterations falls below a small threshold such as 0.001.
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![With respect to
With respect to
θ0
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Gradient Descent
- 1. Gradient Descent Glossary Cost Function Define An algorithm to find the parameters to minimize the Cost Function , a constant representing the rate of step Example 假設 Limit 只能得到區域最佳解，無法得到全域或絕對最佳解 變異太⼤，導致收斂緩慢 改善⽅式：Feature Scaling 太⼤，可能無限的震盪，⽽⼀直錯過最佳解 運⾏完 Gradient Descent 後 反⽽容易有上升的情況 太⼩，很緩慢地找到最佳解 如何決定何時收斂完成 當 在運⾏過⼀次 Gradient Descent 後所下降的成本⼩於千分之⼀就代表其已收斂 了。 建議從 0.001, 0.003, 0.01 ...，持續乘以 3 來試 Proof (參考1) 假設 = = J( , ) =θ0 θ1 1 2m ∑m i=1 ( ( ) − )hθ x(i) y(i) 2 := − α J(θ)θj θj ∂ ∂θj α (x) = + xhθ θ0 θ1 := − α J( , )θj θj ∂ ∂θj θ0 θ1 := − α ( ( ) − )θ0 θ0 1 m ∑ m i=1 hθ x(i) y(i) := − α ( ( ) − ) ⋅θ1 θ1 1 m ∑ m i=1 hθ x(i) y(i) x(i) x α J(θ) J(θ) (x) = + xhθ θ0 θ1 J( , ) =θ0 θ1 1 2m ∑m i=1 ( ( ) − )hθ x(i) y(i) 2 = 1 2m ∑m i=1 ( + − )θ0 θ1 x(i) y(i) 2 := − α J( , ) = − α g( , )θj θj ∂ ∂θj θ0 θ1 θj ∂ ∂θj θ0 θ1 g( , ) =θ0 θ1 1 2m ∑m i=1 (f( , )θ0 θ1 )(i) 2 f( , = + −θ0 θ1 )(i) θ0 θ1 x(i) y(i) g( , )∂ ∂θj θ0 θ1 g(f( , )∂ ∂θj θ0 θ1 )(i) g( , ) f( ,∂ ∂θj θ0 θ1 ∂ ∂θj θ0 θ1 )(i)
- 2. With respect to With respect to θ0 g( , )∂ ∂θ0 θ0 θ1 = ∂ ∂θ0 1 2m ∑m i=1 (f( , )θ0 θ1 )(i) 2 = 2 × 1 2m ∑m i=1 (f( , )θ0 θ1 )(i) 2−1 = f( ,1 m ∑m i=1 θ0 θ1 )(i) f( ,∂ ∂θ0 θ0 θ1 )(i) = ( + [a number][a number − [a number )∂ ∂θ0 θ0 ](i) ](i) = ∂ ∂θ0 θ0 = 1 g( , ) f( ,∂ ∂θ0 θ0 θ1 ∂ ∂θ0 θ0 θ1 )(i) = f( , f( ,1 m ∑m i=1 θ0 θ1 )(i) ∂ ∂θ0 θ0 θ1 )(i) = ( + − ) × 11 m ∑ m i=1 θ0 θ1 x(i) y(i) = ( + − ) 1 m ∑m i=1 θ0 θ1 x(i) y(i) θ1 g( , )∂ ∂θ1 θ0 θ1 = ∂ ∂θ1 1 2m ∑m i=1 (f( , )θ0 θ1 )(i) 2 = 2 × 1 2m ∑m i=1 (f( , )θ0 θ1 )(i) 2−1 = f( ,1 m ∑m i=1 θ0 θ1 )(i) f( ,∂ ∂θ1 θ0 θ1 )(i) = ([a number] + [a number, ] − [a number )∂ ∂θ1 θ1 x(i) ](i) = 0 + ( − 0d dθ1 θ1 )1 x(i) = 1 × θ(1−1=0) 1 x(i) = 1 × 1 × x(i) = x(i) g( , ) f( ,∂ ∂θ1 θ0 θ1 ∂ ∂θ1 θ0 θ1 )(i) = f( , f( ,1 m ∑ m i=1 θ0 θ1 )(i) ∂ ∂θ1 θ0 θ1 )(i) = ( + − ) ⋅1 m ∑m i=1 θ0 θ1 x(i) y(i) x(i) = ( + − ) 1 m ∑ m i=1 θ0 θ1 x(i) y(i) x(i)