對於支撐向量機中Truncated Pinball損失的平滑化函數

Abstract

我們的研究目的是探討truncated pinball 損失函數P_(τ,s) (x)以及它的平滑化函數φ_(τ,s) (x,μ)。我們推導了P_(τ,s) (x)可以被寫為絕對值函數跟仿射函數的和: -τ/2 |x+s|+(1+τ)/2 |x|+(τs+x)/2。再者，我們使用了來自於多篇參考文獻中的絕對值函數|x|的平滑化函數φ_abs^k (x,μ) (k=1,2,...,10)來產出我們關於truncated pinball損失函數的平滑化函數φ_(τ,s)^k (x,μ) (k=1,2,...,10)性質的主要結果。因此，我們可以把原先的最佳化問題min_(w,b)⁡〖1/2 ‖w‖^2+C∑_(i=1)^l〖P_(τ,s) (1-y_i (w^T Φ(x_i )+b))〗〗中的P_(τ,s) (x)替換成φ_(τ,s)^k (x,μ)來得到可微分的最佳化問題。我們得出的結論是當μ趨近於0^+的時候我們的可微分的最佳化問題min_(w,b)⁡〖1/2 ‖w‖^2+C∑_(i=1)^l〖φ_(τ,s)^k (1-y_i (w^T Φ(x_i )+b),μ)〗〗就變回原問題。更進一步地說，尋找可微分的最佳化問題的解將引出原問題的解。

The objective of our research was to investigate the truncated pinball loss function P_(τ,s) (x) and its smoothing function φ_(τ,s) (x,μ). We derived P_(τ,s) (x) can be rewritten as the sum of absolute value functions and an affine function: -τ/2 |x+s|+(1+τ)/2 |x|+(τs+x)/2. Moreover, we used the results of smoothing functions φ_abs^k (x,μ) (k=1,2,...,10) of absolute value function |x| from many references to produce our main results about properties of smoothing functions φ_(τ,s)^k (x,μ) (k=1,2,...,10) of the truncated pinball loss function P_(τ,s) (x). Hence, we can replace P_(τ,s) (x) with φ_(τ,s)^k (x,μ) for the original minimization problem min_(w,b)⁡〖1/2 ‖w‖^2+C∑_(i=1)^l〖P_(τ,s) (1-y_i (w^T Φ(x_i )+b))〗〗 to obtain a differentiable minimization problem. We concluded that as μ approaches 0^+ our differentiable minimization problem, min_(w,b)⁡〖1/2 ‖w‖^2+C∑_(i=1)^l〖φ_(τ,s)^k (1-y_i (w^T Φ(x_i )+b),μ)〗〗, becomes the original one. Furthermore, finding solutions to the differentiable minimization problem will lead to solution to the original one.