FTRL代码实现

FTRL（Follow The Regularized Leader）是一种优化方法，就如同SGD（Stochastic Gradient Descent）一样。这里直接给出用FTRL优化LR（Logistic Regression）的步骤：

其中$p_t=\sigma(X_t\cdot w)$是LR的预测函数，求出$p_t$的唯一目的是为了求出目标函数（在LR中采用交叉熵损失函数作为目标函数）对参数$w$的一阶导数$g$，$g_i=(p_t-y_t)x_i$。上面的步骤同样适用于FTRL优化其他目标函数，唯一的不同就是求次梯度$g$（次梯度是左导和右导之间的集合，函数可导--左导等于右导时，次梯度就等于一阶梯度）的方法不同。

下面的python代码把FTRL和LR进行了解耦：

# coding=utf-8
__author__ = "orisun"

import numpy as np


class LR(object):

    @staticmethod
    def fn(w, x):
        '''决策函数为sigmoid函数
        '''
        return 1.0 / (1.0 + np.exp(-w.dot(x)))

    @staticmethod
    def loss(y, y_hat):
        '''交叉熵损失函数
        '''
        return np.sum(np.nan_to_num(-y * np.log(y_hat) - (1 - y) * np.log(1 - y_hat)))

    @staticmethod
    def grad(y, y_hat, x):
        '''交叉熵损失函数对权重w的一阶导数
        '''
        return (y_hat - y) * x


class FTRL(object):

    def __init__(self, dim, l1, l2, alpha, beta, decisionFunc=LR):
        self.dim = dim
        self.decisionFunc = decisionFunc
        self.z = np.zeros(dim)
        self.n = np.zeros(dim)
        self.w = np.zeros(dim)
        self.l1 = l1
        self.l2 = l2
        self.alpha = alpha
        self.beta = beta

    def predict(self, x):
        return self.decisionFunc.fn(self.w, x)

    def update(self, x, y):
        self.w = np.array([0 if np.abs(self.z[i]) <= self.l1 else (np.sign(
            self.z[i]) * self.l1 - self.z[i]) / (self.l2 + (self.beta + np.sqrt(self.n[i])) / self.alpha) for i in xrange(self.dim)])
        y_hat = self.predict(x)
        g = self.decisionFunc.grad(y, y_hat, x)
        sigma = (np.sqrt(self.n + g * g) - np.sqrt(self.n)) / self.alpha
        self.z += g - sigma * self.w
        self.n += g * g
        return self.decisionFunc.loss(y, y_hat)

    def train(self, trainSet, verbos=False, max_itr=100000000, eta=0.01, epochs=100):
        itr = 0
        n = 0
        while True:
            for x, y in trainSet:
                loss = self.update(x, y)
                if verbos:
                    print "itr=" + str(n) + "\tloss=" + str(loss)
                if loss < eta:
                    itr += 1
                else:
                    itr = 0
                if itr >= epochs:  # 损失函数已连续epochs次迭代小于eta
                    print "loss have less than", eta, " continuously for ", itr, "iterations"
                    return
                n += 1
                if n >= max_itr:
                    print "reach max iteration", max_itr
                    return


class Corpus(object):

    def __init__(self, file, d):
        self.d = d
        self.file = file

    def __iter__(self):
        with open(self.file, 'r') as f_in:
            for line in f_in:
                arr = line.strip().split()
                if len(arr) >= (self.d + 1):
                    yield (np.array([float(x) for x in arr[0:self.d]]), float(arr[self.d]))

if __name__ == '__main__':
    d = 4
    corpus = Corpus("train.txt", d)
    ftrl = FTRL(dim=d, l1=1.0, l2=1.0, alpha=0.1, beta=1.0)
    ftrl.train(corpus, verbos=False, max_itr=100000, eta=0.01, epochs=100)
    w = ftrl.w
    print w

    correct = 0
    wrong = 0
    for x, y in corpus:
        y_hat = 1.0 if ftrl.predict(x) > 0.5 else 0.0
        if y == y_hat:
            correct += 1
        else:
            wrong += 1
    print "correct ratio", 1.0 * correct / (correct + wrong)

输出：

reach max iteration 100000
w= [  4.08813934   1.84596245  10.83446088   3.12315268]
correct ratio 0.9946

当把参数调为$\lambda_1=0,\lambda_2=0,\alpha=0.5,\beta=1$时，准确率能达到0.9976。

train.txt文件前4列是特征，第5列是标签。内容形如：

-0.567811945258 0.899305436215 0.501926599477 -0.222973905568 1.0
-0.993964260114 0.261988294216 -0.349167046026 -0.923759536056 0.0
0.300707261785 -0.90855090557 -0.248270600228 0.879134142054 0.0
-0.311566995194 -0.698903141283 0.369841040784 0.175901270771 1.0
0.0245841670644 0.782128080056 0.542680482068 0.44897929707 1.0
0.344387543846 0.297686731698 0.338210312887 0.175049733038 1.0