这个不等式是怎么化简的啊
时间: 2024-04-19 20:23:55 浏览: 9
对于证明逻辑回归损失函数的凸性,我们可以使用Jensen不等式进行化简。下面是详细的化简过程:
首先,我们有逻辑回归的损失函数:
L(p) = -[ylog(p) + (1-y)log(1-p)]
现在,我们要使用Jensen不等式,我们需要选择两个样本点x1和x2以及对应的预测概率p1和p2。我们将x1设置为ylog(p1)和x2设置为ylog(p2)。然后,应用Jensen不等式,我们有:
L(wp1 + (1-w)p2) <= wL(p1) + (1-w)L(p2)
将逻辑回归损失函数的定义代入上述不等式,可以得到:
- [ylog(wp1 + (1-w)p2) + (1-y)log(1-(wp1 + (1-w)p2))] <= w[-(ylog(p1) + (1-y)log(1-p1))] + (1-w)[- (ylog(p2) + (1-y)log(1-p2))]
现在,我们可以对上述不等式进行化简。首先,展开并合并项:
- [y(log(wp1) + log(1-p2) + log(1-(wp1 + (1-w)p2)))] - [(1-y)(log(1-(wp1 + (1-w)p2)) + log(p1) + log(1-p2))] <= - [wy(log(p1) + log(1-p1))] - [(1-w)(ylog(p2) + (1-y)log(1-p2))]
接下来,我们可以进一步化简。注意到log(1-x) = log(1 - (1-x)) = log(x),我们可以得到:
- [y(log(wp1) + log(1-p2) + log(wp2))] - [(1-y)(log(wp1) + log(1-p2) + log(wp2))] <= - [wy(log(p1) + log(1-p1))] - [(1-w)(ylog(p2) + (1-y)log(1-p2))]
再次合并项,我们有:
y(log(wp1) - log(1-p2)) <= w[y(log(p1) - log(1-p1))] + (1-w)[(ylog(p2) - (1-y)log(1-p2))]
最后,我们可以得出结论,即上述不等式对于所有的w属于[0, 1]始终成立。这意味着逻辑回归的损失函数是凸函数。
希望这个化简过程能够清晰地展示如何利用Jensen不等式证明逻辑回归损失函数的凸性。