深度學習：正則化-權重衰減-(2). AdamW實現

摘要：第一部分中解釋了權重衰減及其現有框架實現的不合理處，這部分從源碼出發，在不影響原始Adam優化器使用的情況下添加解耦權重衰減。使用Tensorflow框架。

目錄

1. Adam源碼修改
2. 結果對比

主要參考文獻

源碼的修改根據參考【1】簡化。

【1】“https://github.com/OverLordGoldDragon/keras-adamw”

【2】“Decoupled Weight Decay Regularization”

1. Adam源碼修改

直接給出修改後的源碼，增加行98-106，增加參數wd，wd_dir。

"""From built-in optimizer classes.
"""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import six
import copy
from six.moves import zip

from keras import backend as K
from keras.utils.generic_utils import serialize_keras_object
from keras.utils.generic_utils import deserialize_keras_object
from keras.legacy import interfaces

from keras.optimizers import Optimizer
    
class Adam(Optimizer):
    """Adam optimizer.

    Default parameters follow those provided in the original paper.

    # Arguments
        learning_rate: float >= 0. Learning rate.
        beta_1: float, 0 < beta < 1. Generally close to 1.
        beta_2: float, 0 < beta < 1. Generally close to 1.
        amsgrad: boolean. Whether to apply the AMSGrad variant of this
            algorithm from the paper "On the Convergence of Adam and
            Beyond".

    # References
        - [Adam - A Method for Stochastic Optimization](
           https://arxiv.org/abs/1412.6980v8)
        - [On the Convergence of Adam and Beyond](
           https://openreview.net/forum?id=ryQu7f-RZ)
    """

    def __init__(self, learning_rate=0.001, beta_1=0.9, beta_2=0.999, wd=0.0, wd_dir=None,
                 amsgrad=False, **kwargs):
        self.initial_decay = kwargs.pop('decay', 0.0)
        self.epsilon = kwargs.pop('epsilon', K.epsilon())
        learning_rate = kwargs.pop('lr', learning_rate)
        super(Adam, self).__init__(**kwargs)
        with K.name_scope(self.__class__.__name__):
            self.iterations = K.variable(0, dtype='int64', name='iterations')
            self.learning_rate = K.variable(learning_rate, name='learning_rate')
            self.beta_1 = K.variable(beta_1, name='beta_1')
            self.beta_2 = K.variable(beta_2, name='beta_2')
            self.decay = K.variable(self.initial_decay, name='decay')
        self.amsgrad = amsgrad
        self.wd = wd
        self.wd_dir = wd_dir

    @interfaces.legacy_get_updates_support
    @K.symbolic
    def get_updates(self, loss, params):
        grads = self.get_gradients(loss, params)
        self.updates = [K.update_add(self.iterations, 1)]

        lr = self.learning_rate
        if self.initial_decay > 0:
            lr = lr * (1. / (1. + self.decay * K.cast(self.iterations,
                                                      K.dtype(self.decay))))

        t = K.cast(self.iterations, K.floatx()) + 1
        lr_t = lr * (K.sqrt(1. - K.pow(self.beta_2, t)) /
                     (1. - K.pow(self.beta_1, t)))

        ms = [K.zeros(K.int_shape(p),
              dtype=K.dtype(p),
              name='m_' + str(i))
              for (i, p) in enumerate(params)]
        vs = [K.zeros(K.int_shape(p),
              dtype=K.dtype(p),
              name='v_' + str(i))
              for (i, p) in enumerate(params)]

        if self.amsgrad:
            vhats = [K.zeros(K.int_shape(p),
                     dtype=K.dtype(p),
                     name='vhat_' + str(i))
                     for (i, p) in enumerate(params)]
        else:
            vhats = [K.zeros(1, name='vhat_' + str(i))
                     for i in range(len(params))]
        self.weights = [self.iterations] + ms + vs + vhats

        for p, g, m, v, vhat in zip(params, grads, ms, vs, vhats):
            m_t = (self.beta_1 * m) + (1. - self.beta_1) * g
            v_t = (self.beta_2 * v) + (1. - self.beta_2) * K.square(g)
            if self.amsgrad:
                vhat_t = K.maximum(vhat, v_t)
                p_t = p - lr_t * m_t / (K.sqrt(vhat_t) + self.epsilon)
                self.updates.append(K.update(vhat, vhat_t))
            else:
                p_t = p - lr_t * m_t / (K.sqrt(v_t) + self.epsilon)

            eta_t = 1.0
            if self.wd != 0:
                '''Normalized weight decay according to the AdamW paper
                '''
                if p.name in self.wd_dir.keys():
                    print(self.wd_dir.keys())
#                     w_d = self.wd*K.sqrt(self.batch_size/(self.samples_per_epoch*self.epochs))
                    w_d = self.wd
                    p_t = p_t - eta_t*(w_d*p) 
            
            self.updates.append(K.update(m, m_t))
            self.updates.append(K.update(v, v_t))
            new_p = p_t

            # Apply constraints.
            if getattr(p, 'constraint', None) is not None:
                new_p = p.constraint(new_p)

            self.updates.append(K.update(p, new_p))
        return self.updates

    def get_config(self):
        config = {'learning_rate': float(K.get_value(self.learning_rate)),
                  'beta_1': float(K.get_value(self.beta_1)),
                  'beta_2': float(K.get_value(self.beta_2)),
                  'decay': float(K.get_value(self.decay)),
                  'epsilon': self.epsilon,
                  'amsgrad': self.amsgrad, 
                  'wd': self.wd, 
                  'wd_dir': self.wd_dir}
        base_config = super(Adam, self).get_config()
        return dict(list(base_config.items()) + list(config.items()))

2. 結果對比

對不加權重衰減，使用$L_2$範數正則化以及使用解耦權重衰減的三種情況進行對比，使用與第一部分內容相同的手工數據。

按照準備數據——選擇模型——計算代價函數、梯度，進行訓練的順序進行。

首先準備數據集。

import keras
from keras import layers
import tensorflow as tf
# 準備數據集
num_train, num_test = 20, 100
num_features = 200

true_w, true_b = tf.ones((num_features, 1)) * 0.01, 0.05

features = tf.random.normal((num_train + num_test, num_features))
noises = tf.random.normal((num_train + num_test, 1)) * 0.01
labels = tf.matmul(features, true_w) + tf.convert_to_tensor(true_b) + noises

train_data, test_data = features[:num_train, :], features[num_train:, :]
train_labels, test_labels = labels[:num_train], labels[num_train:]

其次，選擇過擬合的模型，模型中設置$L_2$正則化爲0。

# 選擇模型
model = keras.models.Sequential([
    layers.Dense(units=128, activation='relu', input_dim=200), 
    layers.Dense(128, activation='relu', kernel_regularizer=keras.regularizers.l2(0.00)),
    layers.Dense(1)
])
model.summary()

計算代價函數、梯度，進行訓練。

Adam的使用方式和原來稍有不同，需要查找網絡中包含$L_2$正則化的層，然後對這些層的參數進行權重衰減。

首先是無$L_2$正則化，無權重衰減的情況。

weight_decays_dir = get_weight_decays(model)
adam = Adam(learning_rate=0.001, wd=0.00, wd_dir=weight_decays_dir)
model.compile(optimizer=adam,
              loss='mse',
              metrics=['mse'])

hist1 = model.fit(train_data, train_labels, steps_per_epoch=2, epochs=100, validation_steps=1,validation_data=[test_data, test_labels])

修改網絡中的$L_2$正則化，得到僅包含$L_2$正則化的結果。

修改wd的值，得到僅包含權重衰減的結果。

結果如下：

可以看到，解耦權重衰減和$L_2$正則化在Adam中有不同的結果。這裏雖然兩者最終取得了一樣的結果，但根據【2】作者的結果，解耦權重衰減能使權重衰減超參數的選擇獨立於學習率，簡化優化難度，且泛化能力更強。