Logloss va MSE uchun yo'qotish funktsiyasining hosilasini qanday aniqlash mumkin?

Mashina o'rganishdagi yo'qotish funksiyalari uchun, odatda, modelni optimallashtirish uchun hosilani (gradentni) hisoblash talab qilinadi.

1. MSE (O'rtacha kvadrat xato) — o'rtacha kvadrat xato:

Yo'qotish funktsiyasi:

[ L = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2 ]

Hosila prognozga ( \hat{y}_i ) nisbatan:

[ \frac{\partial L}{\partial \hat{y}_i} = -\frac{2}{n} (y_i - \hat{y}_i) ]

2. LogLoss (logarifmik yo'qotish funktsiyasi) — tasniflash vazifalarida ishlatiladi (masalan, logistika regressiyasi):

Ikki sinf uchun:

[ L = -\frac{1}{n} \sum_{i=1}^n \left[ y_i \log(\hat{y}_i) + (1 - y_i) \log(1 - \hat{y}_i) \right] ]

bu yerda ( y_i \in {0,1} ), ( \hat{y}_i ) 1 sinf uchun bashorat qilingan ehtimoldir.

Hosila ( \hat{y}_i ) ga:

[ \frac{\partial L}{\partial \hat{y}_i} = -\frac{1}{n} \left( \frac{y_i}{\hat{y}_i} - \frac{1 - y_i}{1 - \hat{y}_i} \right) ]

Logistik regressiya kontekstida, agar ( \hat{y}_i = \sigma(z_i) ) (sigmoid), unda modelga kirish ( z_i ) ga hosila:

[ \frac{\partial L}{\partial z_i} = \hat{y}_i - y_i ]

Bu, o'qitish davomida hisoblashlarni soddalashtiradi.

Shuning uchun, modelni optimallashtirish uchun, bu hosilalarni ishlatish va parametrlarni gradient usullari bilan yangilash zarur.