随机梯度下降

机器学习 / Tensorflow
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梯度下降

Gradient

  • 导数derivative:一维
  • 偏微分partial derivate:多元函数中按照某一方向的导数
  • 梯度gradient:是多轴组合而成的向量

gradient的方向就代表了函数值增大的方向，loss找到最小值即是向梯度相反的方向更新

AutoGrad

自动求梯度

  • with tf.GradientTape() as tape: 将计算过程包在tape计算环境当中
  • [w_grad] = tape.gradient(loss,[w]) 传入loss函数和需要求解的参数就会返回梯度
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In [2]: w = tf.constant(1.)
In [3]: x = tf.constant(2.)
In [4]: y = x*w

In [7]: with tf.GradientTape() as tape:
   ...:     tape.watch([w])
   ...:     y2 = x*w
   ...: grad1 = tape.gradient(y,[w])

In [8]: grad1
Out[8]: [None]

In [9]: with tf.GradientTape() as tape:
   ...:     tape.watch([w])
   ...:     y2 = x*w
   ...: grad2 = tape.gradient(y2,[w])

In [10]: grad2
Out[10]: [<tf.Tensor: shape=(), dtype=float32, numpy=2.0>]

Persistent GradientTape

  • tape.gradient()只能调用求解一次，求解后就会自动释放掉一大部分资源

  • 如果需要调用两次，则要使用Persistent GradientTape

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with tf.GradientTape(persistent=True) as tape:
  • 如果要不使用watch的话需要将变量改为tf.Variable

2nd-order 二阶梯度

也就是求二阶导数

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In [47]: with tf.GradientTape() as t1:
    ...:     t1.watch([w,b])
    ...:     with tf.GradientTape() as t2:
    ...:         t2.watch([w,b])
    ...:         y = x*w +b
    ...:     dy_dw,dy_db = t2.gradient(y,[w,b])
    ...:     print(dy_dw,dy_db)
    ...: d2y_dw2 = t1.gradient(dy_dw,w)
    ...: print(d2y_dw2)
tf.Tensor(2.0, shape=(), dtype=float32) tf.Tensor(1.0, shape=(), dtype=float32)
None  #x对w的导数

Activation Functions 激活函数

  • 神经元机制并不是一个简单的输入加权求和，而是有一个预值响应机制，只有大于某个预值时才会输出，而且输出是固定的。这个预值就是激活函数的产生来源，这种到预值输出的激活函数是一种不可导的激活函数

Sigmod/Logistic

解决激活函数不可导的问题

函数非常光滑，相当于一个压缩功能，将(-∞,+∞)的值压缩到有限的范围当中(0,1)

sigmod当x趋近于无穷时，导数趋近于0，梯度会长时间得不到更新，会造成梯度离散

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In [4]: a = tf.linspace(-10.,10.,10)
<tf.Tensor: shape=(10,), dtype=float32, numpy=
array([-10.       ,  -7.7777777,  -5.5555553,  -3.333333 ,  -1.1111107,
         1.1111116,   3.333334 ,   5.5555563,   7.7777786,  10.       ],
      dtype=float32)>
      
In [7]: with tf.GradientTape() as tape:
   ...:     tape.watch(a)
   ...:     y = tf.sigmoid(a)
   ...: grads = tape.gradient(y,[a])
#y
tf.Tensor(
[4.5397872e-05 4.1876672e-04 3.8510333e-03 3.4445208e-02 2.4766390e-01
 7.5233626e-01 9.6555483e-01 9.9614894e-01 9.9958128e-01 9.9995458e-01], shape=(10,), dtype=float32)
#grads
[<tf.Tensor: shape=(10,), dtype=float32, numpy=
 array([4.5395809e-05, 4.1859134e-04, 3.8362027e-03, 3.3258736e-02,
        1.8632649e-01, 1.8632641e-01, 3.3258699e-02, 3.8362255e-03,
        4.1854731e-04, 4.5416677e-05], dtype=float32)>]

可以看出在|x|>3时会出现梯度离散

Tanh

往往用在循环神经网络当中，它可以有sigmod表示 tanh(x) = 2sigmod(2x)-1

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In [10]: a = tf.linspace(-5.,5.,10)

In [11]: tf.tanh(a)
Out[11]:
<tf.Tensor: shape=(10,), dtype=float32, numpy=
array([-0.99990916, -0.9991625 , -0.99229795, -0.9311096 , -0.5046722 ,
        0.5046726 ,  0.93110967,  0.99229795,  0.9991625 ,  0.99990916],
      dtype=float32)>

Relu (Rectified Linear Unit整型线性单元)

Relu导数非常简单，而且保持梯度不变，很大程度减轻梯度离散和梯度爆炸

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In [12]: a = tf.linspace(-1.,1.,10)
<tf.Tensor: shape=(10,), dtype=float32, numpy=
array([-1.        , -0.7777778 , -0.5555556 , -0.3333333 , -0.1111111 ,
        0.11111116,  0.33333337,  0.5555556 ,  0.7777778 ,  1.        ],
      dtype=float32)>
      
In [13]: tf.nn.relu(a)
Out[13]:
<tf.Tensor: shape=(10,), dtype=float32, numpy=
array([0.        , 0.        , 0.        , 0.        , 0.        ,
       0.11111116, 0.33333337, 0.5555556 , 0.7777778 , 1.        ],
      dtype=float32)>
   
In [15]: tf.nn.leaky_relu(a)
Out[15]:
<tf.Tensor: shape=(10,), dtype=float32, numpy=
array([-0.2       , -0.15555556, -0.11111112, -0.06666666, -0.02222222,
        0.11111116,  0.33333337,  0.5555556 ,  0.7777778 ,  1.        ],
      dtype=float32)>

Typical Loss

Mean Squared Error 均方差

MSE

Derivative

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In [3]: x=tf.random.normal([2,4])
In [4]: w=tf.random.normal([4,3])
In [6]: b=tf.zeros([3])
In [7]: y = tf.constant([2,0])

In [10]: with tf.GradientTape() as tape:
    ...:     tape.watch([w,b])
    ...:     prob = tf.nn.softmax(x@w+b,axis=1)
    ...:     loss = tf.reduce_mean(tf.losses.MSE(tf.one_hot(y,depth=3),prob))
    ...: grads = tape.gradient(loss,[w,b])
    
In [11]: grads[0]
Out[11]:
<tf.Tensor: shape=(4, 3), dtype=float32, numpy=
array([[-0.11654878,  0.0167967 ,  0.09975209],
       [-0.03289018, -0.0634746 ,  0.09636479],
       [ 0.02359655, -0.01446564, -0.00913091],
       [ 0.01675941, -0.05286441,  0.03610501]], dtype=float32)>
In [12]: grads[1]
Out[12]: <tf.Tensor: shape=(3,), dtype=float32, numpy=array([-0.00793203,  0.06198916, -0.05405714], dtype=float32)>

axis为求softmax的维度。[b,3],如果是axis求0维无意义，这里softmax函数axis默认为-1

softmax

当n分类的n个输出通过softmax 函数后，便满足了概率分布的要求使得每一个元素的范围都在(0,1)之间，并且所有元素的和为1

CrossEntropy CE

配合softmax

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In [22]: x=tf.random.normal([2,4])
    ...: w=tf.random.normal([4,3])
    ...: b=tf.zeros([3])
    ...: y = tf.constant([2,0])

In [23]: with tf.GradientTape() as tape:
    ...:     tape.watch([w,b])
    ...:     logits = x@w+b
    ...:     loss = tf.reduce_mean(tf.losses.categorical_crossentropy(tf.one_hot(y,depth=3),logits,from_logits=True))
    ...:
    ...: grads = tape.gradient(loss,[w,b])
    ...: print(grads)
[<tf.Tensor: shape=(4, 3), dtype=float32, numpy=
array([[-0.66708755,  1.213526  , -0.5464386 ],
       [ 0.09902969,  0.15032136, -0.24935105],
       [-0.49425828,  1.421092  , -0.92683387],
       [-0.87035865,  1.703031  , -0.83267236]], dtype=float32)>, 
       <tf.Tensor: shape=(3,), dtype=float32, numpy=array([-0.49924132,  0.9968054 , -0.49756414], dtype=float32)>]

链式法则

通过使用链式法则可以将最后一层的误差，一层层输出到中间层的权值当中，从而得到中间层的梯度信息，更好的优化权值，达到最优化的效果

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In [3]: x = tf.constant(1.)
   ...: w1 = tf.constant(2.)
   ...: b1 = tf.constant(1.)
   ...: w2 = tf.constant(2.)
   ...: b2 = tf.constant(1.)
   ...:
   ...: with tf.GradientTape(persistent=True) as tape:
   ...:     tape.watch([w1,b1,w2,b2])
   ...:     y1 = x*w1 + b1
   ...:     y2 = y1*w2 + b2
   ...: dy2_dy1 = tape.gradient(y2,[y1])[0]
   ...: dy1_dw1 = tape.gradient(y1,[w1])[0]
   ...: dy2_dw1 = tape.gradient(y2,[w1])[0]#这里可以自动进行这样的链式运算
   ...: print(dy2_dy1*dy1_dw1,dy2_dw1)#链式法则结果相同
tf.Tensor(2.0, shape=(), dtype=float32) tf.Tensor(2.0, shape=(), dtype=float32)

FASHION_MNIST数据集 实践

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import  os
os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2'

import tensorflow as tf
from    tensorflow import keras
from    tensorflow.keras import datasets, layers, optimizers, Sequential, metrics

assert tf.__version__.startswith('2.')

def preprocess(x, y):

    x = tf.cast(x, dtype=tf.float32) / 255.
    y = tf.cast(y, dtype=tf.int32)
    return x,y


(x, y), (x_test, y_test) = datasets.fashion_mnist.load_data()
print(x.shape, y.shape)

batchsz = 128

db = tf.data.Dataset.from_tensor_slices((x,y))
db = db.map(preprocess).shuffle(10000).batch(batchsz)

db_test = tf.data.Dataset.from_tensor_slices((x_test,y_test))
db_test = db_test.map(preprocess).batch(batchsz)

db_iter = iter(db)
sample = next(db_iter)
print('batch:', sample[0].shape, sample[1].shape)

model = Sequential([
    layers.Dense(256, activation=tf.nn.relu), # [b, 784] => [b, 256]
    layers.Dense(128, activation=tf.nn.relu), # [b, 256] => [b, 128]
    layers.Dense(64, activation=tf.nn.relu), # [b, 128] => [b, 64]
    layers.Dense(32, activation=tf.nn.relu), # [b, 64] => [b, 32]
    layers.Dense(10) # [b, 32] => [b, 10], 330 = 32*10 + 10
])
model.build(input_shape=[None, 28*28])
model.summary()
# w = w - lr*grad
optimizer = optimizers.Adam(lr=1e-3)

def main():

    for epoch in range(30):

        for step, (x,y) in enumerate(db):

            # x: [b, 28, 28] => [b, 784]
            # y: [b]
            x = tf.reshape(x, [-1, 28*28])

            with tf.GradientTape() as tape:
                # [b, 784] => [b, 10]
                logits = model(x)#前向传播就在这一句话即可完成
                y_onehot = tf.one_hot(y, depth=10)
                # [b]
                loss_mse = tf.reduce_mean(tf.losses.MSE(y_onehot, logits))
                loss_ce = tf.losses.categorical_crossentropy(y_onehot, logits, from_logits=True)
                loss_ce = tf.reduce_mean(loss_ce)

            grads = tape.gradient(loss_ce, model.trainable_variables)

            #根据w = w-lr*grad对数据进行原地更新
            optimizer.apply_gradients(zip(grads, model.trainable_variables))

            if step % 100 == 0:
                print(epoch, step, 'loss:', float(loss_ce), float(loss_mse))


        # test
        total_correct = 0
        total_num = 0
        for x,y in db_test:

            # x: [b, 28, 28] => [b, 784]
            # y: [b]
            x = tf.reshape(x, [-1, 28*28])
            # [b, 10]
            logits = model(x)
            # logits => prob, [b, 10]
            prob = tf.nn.softmax(logits, axis=1)
            # [b, 10] => [b], int64
            pred = tf.argmax(prob, axis=1)
            pred = tf.cast(pred, dtype=tf.int32)
            # pred:[b]
            # y: [b]
            # correct: [b], True: equal, False: not equal
            correct = tf.equal(pred, y)
            correct = tf.reduce_sum(tf.cast(correct, dtype=tf.int32))

            total_correct += int(correct)
            total_num += x.shape[0]

        acc = total_correct / total_num
        print(epoch, 'test acc:', acc)

if __name__ == '__main__':
    main()
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(60000, 28, 28) (60000,)
batch: (128, 28, 28) (128,)
Model: "sequential"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
dense (Dense)                multiple                  200960    
_________________________________________________________________
dense_1 (Dense)              multiple                  32896     
_________________________________________________________________
dense_2 (Dense)              multiple                  8256      
_________________________________________________________________
dense_3 (Dense)              multiple                  2080      
_________________________________________________________________
dense_4 (Dense)              multiple                  330       
=================================================================
Total params: 244,522
Trainable params: 244,522
Non-trainable params: 0
_________________________________________________________________
0 0 loss: 2.3202524185180664 0.14212539792060852
0 100 loss: 0.7960939407348633 13.527359008789062
0 200 loss: 0.49242451786994934 15.604676246643066
0 300 loss: 0.3477419316768646 15.686136245727539
0 400 loss: 0.2666846513748169 15.745983123779297
0 test acc: 0.8465
1 0 loss: 0.45068711042404175 17.90788459777832
1 100 loss: 0.3727279603481293 15.884428977966309
1 200 loss: 0.4774554371833801 20.82514190673828
1 300 loss: 0.33690470457077026 19.180904388427734
1 400 loss: 0.39581966400146484 20.918468475341797
1 test acc: 0.8679 
...
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