├── PE-LTR.py
└── README.md


/PE-LTR.py:
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 1 | import tensorflow as tf
 2 | import numpy as np
 3 | from scipy.optimize import minimize
 4 | from scipy.optimize import nnls
 5 | 
 6 | seed = 3456
 7 | tf.set_random_seed(seed)
 8 | np.random.seed(seed)
 9 | 
10 | x_data = np.float32(np.random.rand(100, 4))
11 | y_data = np.dot(x_data, [[0.100], [0.200], [0.3], [0.4]]) + 0.300
12 | 
13 | weight_a = tf.placeholder(tf.float32)
14 | weight_b = tf.placeholder(tf.float32)
15 | 
16 | b = tf.Variable(tf.zeros([1]))
17 | W = tf.Variable(tf.random_uniform([4, 1], -1.0, 1.0))
18 | y = tf.matmul(x_data, W) + b
19 | 
20 | loss_a = tf.reduce_mean(tf.square(y - y_data))
21 | loss_b = tf.reduce_mean(tf.square(W) + tf.square(b))
22 | loss = weight_a * loss_a + weight_b * loss_b
23 | 
24 | optimizer = tf.train.GradientDescentOptimizer(0.1)
25 | 
26 | a_gradients = tf.gradients(loss_a, W)
27 | b_gradients = tf.gradients(loss_b, W)
28 | 
29 | train = optimizer.minimize(loss)
30 | 
31 | init = tf.initialize_all_variables()
32 | 
33 | sess = tf.Session()
34 | sess.run(init)
35 | 
36 | 
37 | def pareto_step(w, c, G):
38 |     """
39 |     ref:http://ofey.me/papers/Pareto.pdf
40 |     K : the number of task
41 |     M : the dim of NN's params
42 |     :param W: # (K,1)
43 |     :param C: # (K,1)
44 |     :param G: # (K,M)
45 |     :return:
46 |     """
47 |     GGT = np.matmul(G, np.transpose(G))  # (K, K)
48 |     e = np.mat(np.ones(np.shape(w)))  # (K, 1)
49 |     m_up = np.hstack((GGT, e))  # (K, K+1)
50 |     m_down = np.hstack((np.transpose(e), np.mat(np.zeros((1, 1)))))  # (1, K+1)
51 |     M = np.vstack((m_up, m_down))  # (K+1, K+1)
52 |     z = np.vstack((-np.matmul(GGT, c), 1 - np.sum(c)))  # (K+1, 1)
53 |     hat_w = np.matmul(np.matmul(np.linalg.inv(np.matmul(np.transpose(M), M)), M), z)  # (K+1, 1)
54 |     hat_w = hat_w[:-1]  # (K, 1)
55 |     hat_w = np.reshape(np.array(hat_w), (hat_w.shape[0],))  # (K,)
56 |     c = np.reshape(np.array(c), (c.shape[0],))  # (K,)
57 |     new_w = ASM(hat_w, c)
58 |     return new_w
59 | 
60 | 
61 | def ASM(hat_w, c):
62 |     """
63 |     ref:
64 |     http://ofey.me/papers/Pareto.pdf,
65 |     https://stackoverflow.com/questions/33385898/how-to-include-constraint-to-scipy-nnls-function-solution-so-that-it-sums-to-1
66 | 
67 |     :param hat_w: # (K,)
68 |     :param c: # (K,)
69 |     :return:
70 |     """
71 |     A = np.array([[0 if i != j else 1 for i in range(len(c))] for j in range(len(c))])
72 |     b = hat_w
73 |     x0, _ = nnls(A, b)
74 | 
75 |     def _fn(x, A, b):
76 |         return np.linalg.norm(A.dot(x) - b)
77 | 
78 |     cons = {'type': 'eq', 'fun': lambda x: np.sum(x) + np.sum(c) - 1}
79 |     bounds = [[0., None] for _ in range(len(hat_w))]
80 |     min_out = minimize(_fn, x0, args=(A, b), method='SLSQP', bounds=bounds, constraints=cons)
81 |     new_w = min_out.x + c
82 |     return new_w
83 | 
84 | 
85 | w_a, w_b = 0.5, 0.5
86 | c_a, c_b = 0.2, 0.2
87 | for step in range(0, 10):
88 |     res = sess.run([a_gradients, b_gradients, train], feed_dict={weight_a: w_a, weight_b: w_b})
89 |     weights = np.mat([[w_a], [w_b]])
90 |     paras = np.hstack((res[0][0], res[1][0]))
91 |     paras = np.transpose(paras)
92 |     w_a, w_b = pareto_step(weights, np.mat([[c_a], [c_b]]), paras)
93 |     la = sess.run(loss_a)
94 |     lb = sess.run(loss_b)
95 |     print("{:0>2d} {:4f} {:4f} {:4f} {:4f} {:4f}".format(step, w_a, w_b, la, lb, la / lb))
96 |     # print(np.reshape(sess.run(W), (4,)), sess.run(b))
97 | 


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/README.md:
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1 | # PE-LTR
2 | Demo implementation for PE-LTR.
3 | 
4 | paper: "A Pareto-Efficient Algorithm for Multiple Objective Optimization in E-Commerce Recommendation". RecSys, 2019, Alibaba
5 | 
6 | link:  http://www.yongfeng.me/attach/lin-recsys2019.pdf
7 | 
8 | ref: https://github.com/jackielinxiao/PE-LTR
9 | 


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