힌지 손실의 그라디언트

기본 그라디언트 하강을 구현하려고하는데 힌지 손실 기능으로 테스트하고 있습니다. . 그러나 힌지 손실의 그라디언트에 대해 혼란 스럽습니다. 나는 그것이$l_{\text{hinge}} = \max(0,1-y\ \boldsymbol{x}\cdot\boldsymbol{w})$

$$\frac{\partial }{\partial w}l_{\text{hinge}} = \begin{cases} -y\ \boldsymbol{x} &\text{if } y\ \boldsymbol{x}\cdot\boldsymbol{w} < 1 \\ 0&\text{if } y\ \boldsymbol{x}\cdot\boldsymbol{w} \geq 1 \end{cases}$$

그러나 이것은 \ boldsymbol {x} 와 같은 크기의 행렬을 반환하지 $\boldsymbol{x}$않습니까? 길이가 \ boldsymbol {w} 인 벡터를 반환하려고 생각했습니다 $\boldsymbol{w}$. 분명히, 어딘가에 혼란스러워하는 것이 있습니다. 누군가 올바른 방향으로 여기를 가리킬 수 있습니까?

작업에 대한 설명이 명확하지 않은 경우 기본 코드를 포함 시켰습니다.

#Run standard gradient descent
gradient_descent<-function(fw, dfw, n, lr=0.01)
{
    #Date to be used
    x<-t(matrix(c(1,3,6,1,4,2,1,5,4,1,6,1), nrow=3))
    y<-c(1,1,-1,-1)
    w<-matrix(0, nrow=ncol(x))

    print(sprintf("loss: %f,x.w: %s",sum(fw(w,x,y)),paste(x%*%w, collapse=',')))
    #update the weights 'n' times
    for (i in 1:n)
    {
      w<-w-lr*dfw(w,x,y)
      print(sprintf("loss: %f,x.w: %s",sum(fw(w,x,y)),paste(x%*%w,collapse=',')))
    }
}
#Hinge loss
hinge<-function(w,x,y) max(1-y%*%x%*%w, 0)
d_hinge<-function(w,x,y){ dw<-t(-y%*%x); dw[y%*%x%*%w>=1]<-0; dw}
gradient_descent(hinge, d_hinge, 100, lr=0.01)

업데이트 : 아래 답변이 문제를 이해하는 데 도움이되었지만이 알고리즘의 출력은 여전히 ​​주어진 데이터에 대해 올바르지 않습니다. 손실 함수는 매번 0.25 씩 감소하지만 너무 빨리 수렴되며 결과 가중치는 분류가 양호하지 않습니다. 현재 출력은 다음과 같습니다

#y=1,1,-1,-1
"loss: 1.000000, x.w: 0,0,0,0"
"loss: 0.750000, x.w: 0.06,-0.1,-0.08,-0.21"
"loss: 0.500000, x.w: 0.12,-0.2,-0.16,-0.42"
"loss: 0.250000, x.w: 0.18,-0.3,-0.24,-0.63"
"loss: 0.000000, x.w: 0.24,-0.4,-0.32,-0.84"
"loss: 0.000000, x.w: 0.24,-0.4,-0.32,-0.84"
"loss: 0.000000, x.w: 0.24,-0.4,-0.32,-0.84"
...  

그래디언트를 얻으려면 w의 번째 구성 요소 와 관련하여 손실을 구분합니다 .$i$$w$

측면에서 힌지 손실 재 작성 로서 F ( g ( w ) ) 여기서 F ( z는 ) = 최대 ( 0 , 1 - Y , Z ) 와 g ( w를 ) = X ⋅ $w$$f(g(w))$$f(z)=\max(0,1-y\ z)$$g(w)=\mathbf{x}\cdot \mathbf{w}$

체인 규칙을 사용하면

$$\frac{\partial}{\partial w_i} f(g(w))=\frac{\partial f}{\partial z} \frac{\partial g}{\partial w_i}$$

1 차 도함수 항은로 평가 되고 - Y 때 X ⋅ w < 1 , 0 , X ⋅ w > 1 . 2 차 미분 항은 x i가 됩니다. 결국 당신은 ∂ f ( g ( w ) ) 를 얻습니다. $g(w)=x\cdot w$$-y$$\mathbf{x}\cdot w<1$$\mathbf{x}\cdot w>1$$x_i$

$$\frac{\partial f(g(w))}{\partial w_i} = \begin{cases} -y\ x_i &\text{if } y\ \mathbf{x}\cdot \mathbf{w} < 1 \\ 0&\text{if } y\ \mathbf{x}\cdot \mathbf{w} > 1 \end{cases}$$

가 x 의 성분보다 범위가 넓으 므로 위의 벡터 수량으로보고 ∂를 쓸 수 있습니다$i$$x$ 속기($\frac{\partial}{\partial w}$$(\frac{\partial}{\partial w_1},\frac{\partial}{\partial w_2},\ldots)$

이것은 3 년 늦었지만 여전히 누군가와 관련이있을 수 있습니다 ...

$S$$x_i \in R^d$$y_i \in \{-1,1\}$$w$

$$\begin{equation} w^* = \underset{w}{\text{argmin }} L^{hinge}_S(w) = \underset{w}{\text{argmin }} \sum_i{l_{hinge}(w,x_i,y_i)}= \underset{w}{\text{argmin }} \sum_i{\max{\{0,1-y_iw\cdot x}\}} \end{equation}$$ $w^*$ $$\frac{\partial{l_{hinge}}}{\partial w}= \begin{cases} 0 & y_iw\cdot x \geq 1 \\ -y_ix & y_iw\cdot x < 1 \end{cases}$$

The gradient of the sum is a sum of gradients.

$$\frac{\partial{L_S^{hinge}}}{\partial{w}}=\sum_i{\frac{\partial{l_{hinge}}}{\partial w}}$$ Python example, which uses GD to find hinge-loss optimal separatinig hyperplane follows (its probably not the most efficient code, but it works)

import numpy as np
import matplotlib.pyplot as plt

def hinge_loss(w,x,y):
    """ evaluates hinge loss and its gradient at w

    rows of x are data points
    y is a vector of labels
    """
    loss,grad = 0,0
    for (x_,y_) in zip(x,y):
        v = y_*np.dot(w,x_)
        loss += max(0,1-v)
        grad += 0 if v > 1 else -y_*x_
    return (loss,grad)

def grad_descent(x,y,w,step,thresh=0.001):
    grad = np.inf
    ws = np.zeros((2,0))
    ws = np.hstack((ws,w.reshape(2,1)))
    step_num = 1
    delta = np.inf
    loss0 = np.inf
    while np.abs(delta)>thresh:
        loss,grad = hinge_loss(w,x,y)
        delta = loss0-loss
        loss0 = loss
        grad_dir = grad/np.linalg.norm(grad)
        w = w-step*grad_dir/step_num
        ws = np.hstack((ws,w.reshape((2,1))))
        step_num += 1
    return np.sum(ws,1)/np.size(ws,1)

def test1():
    # sample data points
    x1 = np.array((0,1,3,4,1))
    x2 = np.array((1,2,0,1,1))
    x  = np.vstack((x1,x2)).T
    # sample labels
    y = np.array((1,1,-1,-1,-1))
    w = grad_descent(x,y,np.array((0,0)),0.1)
    loss, grad = hinge_loss(w,x,y)
    plot_test(x,y,w)

def plot_test(x,y,w):
    plt.figure()
    x1, x2 = x[:,0], x[:,1]
    x1_min, x1_max = np.min(x1)*.7, np.max(x1)*1.3
    x2_min, x2_max = np.min(x2)*.7, np.max(x2)*1.3
    gridpoints = 2000
    x1s = np.linspace(x1_min, x1_max, gridpoints)
    x2s = np.linspace(x2_min, x2_max, gridpoints)
    gridx1, gridx2 = np.meshgrid(x1s,x2s)
    grid_pts = np.c_[gridx1.ravel(), gridx2.ravel()]
    predictions = np.array([np.sign(np.dot(w,x_)) for x_ in grid_pts]).reshape((gridpoints,gridpoints))
    plt.contourf(gridx1, gridx2, predictions, cmap=plt.cm.Paired)
    plt.scatter(x[:, 0], x[:, 1], c=y, cmap=plt.cm.Paired)
    plt.title('total hinge loss: %g' % hinge_loss(w,x,y)[0])
    plt.show()

if __name__ == '__main__':
    np.set_printoptions(precision=3)
    test1()

I fixed your code. The main problem is your definition of hinge and d_hinge functions. These should be applied one sample at a time. Instead your definition aggregates all the samples before taking the maximum.

#Run standard gradient descent
gradient_descent<-function(fw, dfw, n, lr=0.01)
{
    #Date to be used
    x<-t(matrix(c(1,3,6,1,4,2,1,5,4,1,6,1), nrow=3))
    y<-t(t(c(1,1,-1,-1)))
    w<-matrix(0, nrow=ncol(x))


    print(sprintf("loss: %f,x.w: %s",sum(mapply(function(xr,yr) fw(w,xr,yr), split(x,row(x)),split(y,row(y)))),paste(x%*%w, collapse=',')))
    #update the weights 'n' times
    for (i in 1:n)
    {
      w<-w-lr*dfw(w,x,y)
      print(sprintf("loss: %f,x.w: %s",sum(mapply(function(xr,yr) fw(w,xr,yr), split(x,row(x)),split(y,row(y)))),paste(x%*%w,collapse=',')))
    }
}

#Hinge loss
hinge<-function(w,xr,yr) max(1-yr*xr%*%w, 0)
d_hinge<-function(w,x,y){ dw<- apply(mapply(function(xr,yr) -yr * xr * (yr * xr %*% w < 1),split(x,row(x)),split(y,row(y))),1,sum); dw}
gradient_descent(hinge, d_hinge, 100, lr=0.01)

I need n=10000 to converge.

[1] "손실 : 0.090000, xw : 1.08999999999995,0.909999999999905, -1.19000000000008, -1.69000000000011"[1] "손실 : 0.100000, xw : 1.33999999999995,1.1199999999999, -0.900000000000075, -1.42000000000011"[1] "손실 : 0.230000, xw : 0.939999999999948,0.829999999999905, -1.32000000000007, -1.77000000000011 "[1]"손실 : 0.370000, xw : 1.64999999999995,1.2899999999999, -0.630000000000075, -1.25000000000011 "[1]"손실 : 0.000000, xw : 1.24999999999995,0.999999999999905, -1.0500000000000000011 " [1] "손실 : 0.240000, xw : 1.49999999999995,1.2099999999999, -0.760000000000075, -1.33000000000011"[1] "손실 : 0.080000, xw : 1.09999999999995,0.919999999999905, -1.18000000000007, -1.68000000000011"[1] "손실 : 0.110000, xw : 1.34999999999995,1.1299999999999, -0.890000000000075, -1.41000000000011 "[1] "손실 : 0.210000, xw : 0.949999999999948,0.839999999999905, -1.31000000000007, -1.76000000000011"[1] "손실 : 0.380000, xw : 1.65999999999995,1.2999999999999, -0.620000000000074, -1.24000000000011"[1] "손실 : 0.000000, xw : 1.25999999999995,1.0099999999999, -1.04000000000008, -1.59000000000011 "[1]"손실 : 0.000000, xw : 1.25999999999995,1.0099999999999, -1.04000000000008, -1.59000000000011 "