深度学习中的初始化

深度学习中的初始化

layer {
  name: "conv1"
  type: "Convolution"
  bottom: "data"
  top: "conv1"
  param {
    lr_mult: 1
  }
  param {
    lr_mult: 2
  }
  convolution_param {
    num_output: 32
    pad: 2
    kernel_size: 5
    stride: 1
    weight_filler {
      type: "gaussian"
      std: 0.0001
    }
    bias_filler {
      type: "constant"
    }
  }
}

1.1 xavier初始化 （归一化初始化)

当激活函数是sigmoid时，使用标准初始化往往性能比较差，收敛较慢且容易陷入局部最优。

“xavier”初始化是一种有效的神经网络舒适化方法，来自于2010年 Xavier Glorot和Yoshua Bengio两人的一篇论文<< Understanding the difficulty of training deep feedforward neural networks >>，配合tanh等函数能够获得比较好的效果。

主要思想是：尽可能保证前向传播和反向传播时每一层的方差尽量相等

推导前提：激活函数是线性的 ReLU和PReLU不满足这一条件

定义参数所在层的输入维度为n,输出维度为m,则参数讲均匀分布在

W∼U[−6n+m‾‾‾‾‾‾√,6n+m‾‾‾‾‾‾√]

Caffe中的实现：

/**
 * @brief Fills a Blob with values @f$ x \sim U(-a, +a) @f$ where @f$ a @f$ is
 *        set inversely proportional to number of incoming nodes, outgoing
 *        nodes, or their average.
 *
 * A Filler based on the paper [Bengio and Glorot 2010]: Understanding
 * the difficulty of training deep feedforward neuralnetworks.
 *
 * It fills the incoming matrix by randomly sampling uniform data from [-scale,
 * scale] where scale = sqrt(3 / n) where n is the fan_in, fan_out, or their
 * average, depending on the variance_norm option. You should make sure the
 * input blob has shape (num, a, b, c) where a * b * c = fan_in and num * b * c
 * = fan_out. Note that this is currently not the case for inner product layers.
 *
 * TODO(dox): make notation in above comment consistent with rest & use LaTeX.
 */

template <typename Dtype>
class XavierFiller : public Filler {
 public:
  explicit XavierFiller(const FillerParameter& param)
      : Filler(param) {}
  virtual void Fill(Blob* blob) {
    CHECK(blob->count());
    int fan_in = blob->count() / blob->num();
    int fan_out = blob->count() / blob->channels();
    Dtype n = fan_in;  // default to fan_in
    if (this->filler_param_.variance_norm() ==
        FillerParameter_VarianceNorm_AVERAGE) {
      n = (fan_in + fan_out) / Dtype(2);
    } else if (this->filler_param_.variance_norm() ==
        FillerParameter_VarianceNorm_FAN_OUT) {
      n = fan_out;
    }
    Dtype scale = sqrt(Dtype(3) / n);
    caffe_rng_uniform(blob->count(), -scale, scale,
        blob->mutable_cpu_data());
    CHECK_EQ(this->filler_param_.sparse(), -1)
         << "Sparsity not supported by this Filler.";
  }
};

caffe中提供了3种方式：

（1）默认情况，只考虑输入

W∼U[−3n‾‾√,3n‾‾√]

（2）FillerParameter_VarianceNorm_FAN_OUT

W∼U[−3m‾‾‾√,3m‾‾‾√]

（3）FillerParameter_VarianceNorm_AVERAGE

W∼U[−6n+m‾‾‾‾‾‾√,6n+m‾‾‾‾‾‾√]

1.2 MSRA初始化

来自于MSRA研究员何恺明2015年论文

Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification

传统的固定方差的高斯分布初始化，在网络变深时使得模型很难收敛。
改善的方法有用预训练的模型去初始化网络的部分层；xavier也是不错的初始化方法，但是它需要满足激活函数线性的条件。在MSRA初始化中，考虑了ReLU和PReLU。

MSRA初始化的权重分布是一个均值为0，均值为 2n 的高斯分布，初始化满足下式

W∼G[0,2n‾‾√]

Caffe中的实现:

/**
 * @brief Fills a Blob with values @f$ x \sim N(0, \sigma^2) @f$ where
 *        @f$ \sigma^2 @f$ is set inversely proportional to number of incoming
 *        nodes, outgoing nodes, or their average.
 *
 * A Filler based on the paper [He, Zhang, Ren and Sun 2015]: Specifically
 * accounts for ReLU nonlinearities.
 *
 * Aside: for another perspective on the scaling factor, see the derivation of
 * [Saxe, McClelland, and Ganguli 2013 (v3)].
 *
 * It fills the incoming matrix by randomly sampling Gaussian data with std =
 * sqrt(2 / n) where n is the fan_in, fan_out, or their average, depending on
 * the variance_norm option. You should make sure the input blob has shape (num,
 * a, b, c) where a * b * c = fan_in and num * b * c = fan_out. Note that this
 * is currently not the case for inner product layers.
 */
template <typename Dtype>
class MSRAFiller : public Filler {
 public:
  explicit MSRAFiller(const FillerParameter& param)
      : Filler(param) {}
  virtual void Fill(Blob* blob) {
    CHECK(blob->count());
    int fan_in = blob->count() / blob->num();
    int fan_out = blob->count() / blob->channels();
    Dtype n = fan_in;  // default to fan_in
    if (this->filler_param_.variance_norm() ==
        FillerParameter_VarianceNorm_AVERAGE) {
      n = (fan_in + fan_out) / Dtype(2);
    } else if (this->filler_param_.variance_norm() ==
        FillerParameter_VarianceNorm_FAN_OUT) {
      n = fan_out;
    }
    Dtype std = sqrt(Dtype(2) / n);
    caffe_rng_gaussian(blob->count(), Dtype(0), std,
        blob->mutable_cpu_data());
    CHECK_EQ(this->filler_param_.sparse(), -1)
         << "Sparsity not supported by this Filler.";
  }
};

同样有3种方案：

（1）默认情况下，n是输入层的维度

（2）n取输出层的维度

（3）n取输入和输出层的均值

1.3 其他初始化

（1）“constant”: 常量初始化

（2）“gaussian”: 固定方差高斯分布初始化

（3）“positive_utiball”:每个值为在[0,1]之间，对于每个 i ：

∀i∑jxij=1

（4）“uniform”: 均匀分布初始化

（5）“bilinear”: 双线性初始化，通常用于反卷积核