Caffe源码-几种优化算法

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SGD简介

caffe中的SGDSolver类中实现了带动量的梯度下降法,其原理如下,\(lr\)为学习率,\(m\)为动量参数。

  1. 计算新的动量:history_data = local_rate * param_diff + momentum * history_data
    \(\nu_{t+1}=lr*\nabla_{\theta_{t}}+m*\nu_{t}\)
  2. 计算更新时使用的梯度:param_diff = history_data
    \(\Delta\theta_{t+1}=\nu_{t+1}\)
  3. 应用更新:param_data = param_data - param_diff
    \(\theta_{t+1}=\theta_{t}-\Delta\theta_{t+1}\)

步骤1和步骤2均在SGDSolver类的ComputeUpdateValue()函数中实现,步骤3对每个优化方法来说都是相同的,代码可参考之前的博客:Caffe源码-SGDSolver类

NAG(Nesterov Accelerated Gradient)简介

NAG算法在NesterovSolver类中实现,NAG与SGD相比唯一区别在于梯度的计算上。如上,SGD使用的梯度是参数\(\theta_{t}\)在当前位置的梯度\(\nabla_{\theta_{t}}\),而NAG中使用的是当前参数\(\theta_{t}\)在施加了动量之后的位置的梯度\(\nabla_{(\theta_{t}-m*\nu_{t})}\),其原理为:

  1. 应用临时更新:\(\tilde{\theta}_{t+1}=\theta_{t}-m*\nu_{t}\)
  2. 计算该位置的梯度:\(\nabla_{\tilde{\theta}_{t+1}}\)
  3. 计算新的动量:\(\nu_{t+1}=lr*\nabla_{\tilde{\theta}_{t+1}}+m*\nu_{t}\)
  4. 得到更新时使用的梯度:\(\Delta\theta_{t+1}=\nu_{t+1}\)
  5. 应用更新:\(\theta_{t+1}=\theta_{t}-\Delta\theta_{t+1}\)

网络上有一张常见的图用于表示SGD和NAG的过程。
Caffe源码-几种优化算法

对于SGD算法,蓝色向量\(p_{1}\)为当前参数\(\theta_{t}\)在该位置的梯度\(lr*\nabla_{\theta_{t}}\),蓝色向量\(p_{2}\)为动量\(m*\nu_{t}\),而\(p_{1}+p_{2}\)即为参数一次的更新量\(\Delta\theta_{t+1}\)
对于NAG算法,\(O_{1}\)为参数\(\theta_{t}\)的初始位置,棕色向量\(p_{3}=p_{2}\),先计算运用动量后的参数\(\tilde{\theta}_{t+1}\)的位置\(O_{2}\),然后计算该位置梯度\(\nabla_{\tilde{\theta}_{t+1}}\),即为图中的红色向量\(p_{4}\),而\(p_{5}=p_{3}+p_{4}\)即为参数一次的更新量\(\Delta\theta_{t+1}=\nu_{t+1}\)。之后仿照该步骤计算下一次迭代的动量\(m*\nu_{t+1}\)(棕色向量\(p_{6}\))和梯度\(\nabla_{\tilde{\theta}_{t+2}}\)(红色向量\(p_{7}\)),得到更新量\(p_{8}\)

NAG算法的原理还是很好理解的,但是实现起来却有一个非常难理解的地方,即如何计算参数临时更新位置的梯度\(\nabla_{\tilde{\theta}_{t+1}}\)?神经网络这种复杂的系统中想要根据当前位置的梯度\(\nabla_{\theta_{t}}\)来估算另一位置的梯度\(\nabla_{\tilde{\theta}_{t+1}}\)几乎是不可能的。网络上关于该算法的实现细节非常少,不过结合caffe代码和其他的开源代码等,可以判断出,NAG算法每次迭代时保存的参数是临时参数\(\tilde{\theta}_{t+1}\)(位置\(O_{2}\)),而非初始\(O_{1}\)位置处的参数\(\theta_{t}\),这样每次反向传播计算出的梯度实际上就是红色向量\(p_{4}\)。然后每次更新时,会根据动量\(p_{3}\)先将参数从位置\(O_{2}\)退回\(O_{1}\),然后计算得到一次迭代的更新量\(p_{5}\),使参数更新\(\theta_{t+1}\)(位置\(O_{3}\)),并保存下一次迭代时需要使用的临时参数\(\tilde{\theta}_{t+2}\)(位置\(O_{4}\))。

nesterov_solver.cpp源码

//根据当前迭代次数对应的学习率rate,计算网络中第param_id个可学习参数在更新时使用的梯度
template <typename Dtype>
void NesterovSolver<Dtype>::ComputeUpdateValue(int param_id, Dtype rate) {
  const vector<Blob<Dtype>*>& net_params = this->net_->learnable_params();  //网络中的所有可学习参数
  const vector<float>& net_params_lr = this->net_->params_lr();   //网络中每个参数对应的学习率系数
  Dtype momentum = this->param_.momentum();                       //求解器设置的动量
  Dtype local_rate = rate * net_params_lr[param_id];              //得到当前参数对应的学习率
  switch (Caffe::mode()) {
  case Caffe::CPU: {    //CPU模式
    // save history momentum for stepping back
    caffe_copy(net_params[param_id]->count(), this->history_[param_id]->cpu_data(),
        this->update_[param_id]->mutable_cpu_data());   //将历史数据history_拷贝至update_中,update_data = history_data

    // update history     //history_data = local_rate * net_params_diff + momentum * history_data
    caffe_cpu_axpby(net_params[param_id]->count(), local_rate, net_params[param_id]->cpu_diff(), momentum,
        this->history_[param_id]->mutable_cpu_data());

    // compute update: step back then over step   //update_data = (1 + momentum) * history_data + (-momentum) * update_data
    caffe_cpu_axpby(net_params[param_id]->count(), Dtype(1) + momentum,
        this->history_[param_id]->cpu_data(), -momentum,
        this->update_[param_id]->mutable_cpu_data());

    // copy   //net_params_diff = update_data
    caffe_copy(net_params[param_id]->count(), this->update_[param_id]->cpu_data(),
        net_params[param_id]->mutable_cpu_diff());
    break;
  }
  case Caffe::GPU: {
#ifndef CPU_ONLY
    // gpu的操作同理
    // h_temp = history_data
    // history_data = momentum * h_temp + local_rate * net_params_diff
    // net_params_diff = (1+momentum) * history_data - momentum * h_temp
    nesterov_update_gpu(net_params[param_id]->count(), net_params[param_id]->mutable_gpu_diff(),
        this->history_[param_id]->mutable_gpu_data(), momentum, local_rate);
#else
    NO_GPU;
#endif
    break;
  }
  default:
    LOG(FATAL) << "Unknown caffe mode: " << Caffe::mode();
  }
}

对应上述的说明,代码中的各步操作为:

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  1. 当前迭代的动量\(\nu_{t}\)update_data = history_data
  2. net_params_diff临时位置的参数的梯度\(\nabla_{\tilde{\theta}_{t+1}}\),计算新的动量:history_data = local_rate * net_params_diff + momentum * history_data
    \(\nu_{t+1}=lr*\nabla_{\tilde{\theta}_{t+1}}+m*\nu_{t}\)
  3. 计算下一次迭代的临时参数相对于当前临时参数的更新量\(\Delta\tilde{\theta}_{t+2}\)update_data = (1 + momentum) * history_data + (-momentum) * update_data
    \(\Delta\tilde{\theta}_{t+2}=(1+m)*\nu_{t+1}-m*\nu_{t}\)
    注意,当前临时参数在位置\(O_{2}\),需要减去向量\(p_{3}\)\(p_{3}=m*\nu_{t}\)),再加上向量\(p_{5}\)\(p_{6}\)\(p_{5}=\nu_{t+1},p_{6}=m*\nu_{t+1}\))才能得到新的临时位置\(O_{4}\)
  4. 保存参数更新量:net_params_diff = update_data
  5. 应用更新:\(\tilde{\theta}_{t+2}=\tilde{\theta}_{t+1}-\Delta\tilde{\theta}_{t+2}\)

AdaGrad简介

AdaGrad算法通过缩放每个参数反比于其所有梯度历史平方值总和的平方跟,可使得具有较大梯度的参数能够快速下降,使具有小偏导的参数能够缓慢下降。
其原理如下,初始累积变量\(r=0\)\(\delta\)为较小常数,防止除法除数过小而不稳定。

  1. 累加平方梯度(\(\odot\)为逐元素点乘):\(r_{t+1}=r_{t}+\nabla_{\theta_{t}}\odot\nabla_{\theta_{t}}\)
  2. 计算梯度的更新量:\(\Delta\theta_{t+1}=\frac{lr}{\delta+\sqrt{r_{t+1}}}\odot\nabla_{\theta_{t}}\)
  3. 应用更新:\(\theta_{t+1}=\theta_{t}-\Delta\theta_{t+1}\)

adagrad_solver.cpp源码

template <typename Dtype>
void AdaGradSolver<Dtype>::ComputeUpdateValue(int param_id, Dtype rate) {
  const vector<Blob<Dtype>*>& net_params = this->net_->learnable_params();
  const vector<float>& net_params_lr = this->net_->params_lr();
  Dtype delta = this->param_.delta();
  Dtype local_rate = rate * net_params_lr[param_id];
  switch (Caffe::mode()) {
  case Caffe::CPU: {
    // compute square of gradient in update
    caffe_powx(net_params[param_id]->count(),
        net_params[param_id]->cpu_diff(), Dtype(2),
        this->update_[param_id]->mutable_cpu_data());   //update_data = net_params ^ 2

    // update history
    caffe_add(net_params[param_id]->count(),
        this->update_[param_id]->cpu_data(),
        this->history_[param_id]->cpu_data(),
        this->history_[param_id]->mutable_cpu_data());  //history_data = update_data + history_data

    // prepare update
    caffe_powx(net_params[param_id]->count(), this->history_[param_id]->cpu_data(), Dtype(0.5),
        this->update_[param_id]->mutable_cpu_data());   //update_data = history_data ^ 0.5

    caffe_add_scalar(net_params[param_id]->count(),
        delta, this->update_[param_id]->mutable_cpu_data());  //update_data += delta

    caffe_div(net_params[param_id]->count(),
              net_params[param_id]->cpu_diff(),
              this->update_[param_id]->cpu_data(),
              this->update_[param_id]->mutable_cpu_data());   //update_data = net_params_diff / update_data

    // scale and copy
    caffe_cpu_axpby(net_params[param_id]->count(), local_rate,
        this->update_[param_id]->cpu_data(), Dtype(0),
        net_params[param_id]->mutable_cpu_diff());    //net_params_diff = local_rate * update_data + 0 * net_params_diff
    break;
  }
  case Caffe::GPU: {    //gpu操作同理
#ifndef CPU_ONLY
    // gi = net_params_diff;
    // hi = history_data = history_data + gi*gi;
    // net_params_diff = local_rate * gi / (sqrt(hi) + delta);
    adagrad_update_gpu(net_params[param_id]->count(),
        net_params[param_id]->mutable_gpu_diff(),
        this->history_[param_id]->mutable_gpu_data(), delta, local_rate);
#else
    NO_GPU;
#endif
    break;
  }
  default:
    LOG(FATAL) << "Unknown caffe mode: " << Caffe::mode();
  }
}

AdaGrad/RMSProp/AdaDelta/Adam算法的caffe代码很容易找到对应的公式,不再详细介绍。

RMSProp简介

RMSProp算法在AdaGrad基础上增加一个衰减系数\(\rho\),以便将很早之前的历史梯度数据丢弃。
其原理如下,初始累积变量\(r=0\)\(\delta\)同样为较小常数。

  1. 累加平方梯度:\(r_{t+1}=\rho*r_{t}+(1-\rho)*\nabla_{\theta_{t}}\odot\nabla_{\theta_{t}}\)
  2. 计算梯度的更新量:\(\Delta\theta_{t+1}=\frac{lr}{\delta+\sqrt{r_{t+1}}}\odot\nabla_{\theta_{t}}\)
  3. 应用更新:\(\theta_{t+1}=\theta_{t}-\Delta\theta_{t+1}\)

rmsprop_solver.cpp源码

template <typename Dtype>
void RMSPropSolver<Dtype>::ComputeUpdateValue(int param_id, Dtype rate) {
  const vector<Blob<Dtype>*>& net_params = this->net_->learnable_params();  //所有可学习参数
  const vector<float>& net_params_lr = this->net_->params_lr();     //参数对应的学习率系数
  // get the learning rate
  Dtype delta = this->param_.delta();                   //常数delta
  Dtype rms_decay = this->param_.rms_decay();           //衰减速率
  Dtype local_rate = rate * net_params_lr[param_id];    //参数对应的学习率

  switch (Caffe::mode()) {
  case Caffe::CPU:
    // compute square of gradient in update
    caffe_powx(net_params[param_id]->count(), net_params[param_id]->cpu_diff(), Dtype(2),
        this->update_[param_id]->mutable_cpu_data());   //update_data = net_params_diff ^ 2

    // update history       //history_data = (1-rms_decay) * update_data + rms_decay * history_data
    caffe_cpu_axpby(net_params[param_id] -> count(), Dtype(1-rms_decay), this->update_[param_id]->cpu_data(),
        rms_decay, this->history_[param_id]-> mutable_cpu_data());

    // prepare update
    caffe_powx(net_params[param_id]->count(), this->history_[param_id]->cpu_data(), Dtype(0.5),
        this->update_[param_id]->mutable_cpu_data());   //update_data = history_data ^ 0.5

    caffe_add_scalar(net_params[param_id]->count(),
        delta, this->update_[param_id]->mutable_cpu_data());    //update_data += delta

    //update_data = net_params_diff / update_data
    caffe_div(net_params[param_id]->count(), net_params[param_id]->cpu_diff(),
        this->update_[param_id]->cpu_data(), this->update_[param_id]->mutable_cpu_data());

    // scale and copy
    caffe_cpu_axpby(net_params[param_id]->count(), local_rate,
        this->update_[param_id]->cpu_data(), Dtype(0),
        net_params[param_id]->mutable_cpu_diff());  //net_params_diff = local_rate * update_data + 0 * net_params_diff
    break;
  case Caffe::GPU:
#ifndef CPU_ONLY
    // g = net_params_diff
    // h = history_data
    // gi = g[i];
    // hi = h[i] = rms_decay*h[i] + (1-rms_decay)*gi*gi;
    // g[i] = local_rate * g[i] / (sqrt(hi) + delta);
    rmsprop_update_gpu(net_params[param_id]->count(),
        net_params[param_id]->mutable_gpu_diff(),
        this->history_[param_id]->mutable_gpu_data(),
        rms_decay, delta, local_rate);
#else
    NO_GPU;
#endif
    break;
  default:
    LOG(FATAL) << "Unknown caffe mode: " << Caffe::mode();
  }
}

AdaDelta简介

AdaDelta也像RMSProp算法一样在AdaGrad基础上增加一个衰减系数\(\rho\),并且还额外维护一个状态量\(x\)
其原理如下,初始累积变量\(x=0, r=0\)\(\delta\)同样为较小常数。

  1. 累加平方梯度:\(r_{t+1}=\rho*r_{t}+(1-\rho)*\nabla_{\theta_{t}}\odot\nabla_{\theta_{t}}\)
  2. 计算不带学习率的梯度的更新量:\(\Delta\tilde{\theta}_{t+1}=\sqrt{\frac{x_{t}+\delta}{r_{t+1}+\delta}}\odot\nabla_{\theta_{t}}\)
  3. 更新状态量:\(x_{t+1}=\rho*x_{t}+(1-\rho)*\Delta\tilde{\theta}_{t+1}\odot\Delta\tilde{\theta}_{t+1}\)
  4. 计算带学习率的梯度的更新量:\(\Delta\theta_{t+1}=lr*\Delta\tilde{\theta}_{t+1}\)
    参考 4中的说明不同,caffe代码中仍然有使用学习率\(lr\)
  5. 应用更新:\(\theta_{t+1}=\theta_{t}-\Delta\theta_{t+1}\)

adadelta_solver.cpp源码

template <typename Dtype>
void AdaDeltaSolver<Dtype>::AdaDeltaPreSolve() {        //AdaDeltaSolver类在构造时会调用该函数
  // Add the extra history entries for AdaDelta after those from SGDSolver::PreSolve
  const vector<Blob<Dtype>*>& net_params = this->net_->learnable_params();  //当前网络中的所有可学习参数
  for (int i = 0; i < net_params.size(); ++i) {
    const vector<int>& shape = net_params[i]->shape();  //第i个可学习参数的形状
    //在SGDSolver<Dtype>::PreSolve中history_已经存入一个与参数blob相同形状的空blob,此处再存入一个
    this->history_.push_back(shared_ptr<Blob<Dtype> >(new Blob<Dtype>(shape)));
  }
}

#ifndef CPU_ONLY
template <typename Dtype>
void adadelta_update_gpu(int N, Dtype* g, Dtype* h, Dtype* h2, Dtype momentum,
    Dtype delta, Dtype local_rate);
#endif

template <typename Dtype>
void AdaDeltaSolver<Dtype>::ComputeUpdateValue(int param_id, Dtype rate) {
  const vector<Blob<Dtype>*>& net_params = this->net_->learnable_params();  //网络中的所有可学习参数
  const vector<float>& net_params_lr = this->net_->params_lr();     //每个参数对应的学习率系数
  Dtype delta = this->param_.delta();                   //AdaDelta方法中的一个参数
  Dtype momentum = this->param_.momentum();             //动量系数
  Dtype local_rate = rate * net_params_lr[param_id];    //得到当前参数对应的学习率
  size_t update_history_offset = net_params.size();     //网络的参数个数
  //history_在AdaDeltaPreSolve()中又存入了一次与所有参数形状相同的空blob,下面将
  //history_[param_id]表示成 history_former, history_[update_history_offset + param_id]表示成 history_latter
  switch (Caffe::mode()) {
  case Caffe::CPU: {
    // compute square of gradient in update
    caffe_powx(net_params[param_id]->count(), net_params[param_id]->cpu_diff(), Dtype(2),
        this->update_[param_id]->mutable_cpu_data());       //update_data = net_params_diff ^ 2

    // update history of gradients    //history_former_data = (1 - momentum) * update_data + momentum * history_former_data
    caffe_cpu_axpby(net_params[param_id]->count(), Dtype(1) - momentum, this->update_[param_id]->cpu_data(),
         momentum, this->history_[param_id]->mutable_cpu_data());

    // add delta to history to guard against dividing by zero later
    caffe_set(net_params[param_id]->count(), delta,
        this->temp_[param_id]->mutable_cpu_data());     //temp_中每个元素都置为delta, temp_data = delta

    caffe_add(net_params[param_id]->count(),
        this->temp_[param_id]->cpu_data(),
        this->history_[update_history_offset + param_id]->cpu_data(),
        this->update_[param_id]->mutable_cpu_data());   //update_data = temp_data + history_latter_data

    caffe_add(net_params[param_id]->count(),
        this->temp_[param_id]->cpu_data(),
        this->history_[param_id]->cpu_data(),
        this->temp_[param_id]->mutable_cpu_data()); //temp_data = temp_data + history_former_data

    // divide history of updates by history of gradients
    caffe_div(net_params[param_id]->count(),
        this->update_[param_id]->cpu_data(),
        this->temp_[param_id]->cpu_data(),
        this->update_[param_id]->mutable_cpu_data());   //update_data = update_data / temp_data

    // jointly compute the RMS of both for update and gradient history
    caffe_powx(net_params[param_id]->count(),
        this->update_[param_id]->cpu_data(), Dtype(0.5),
        this->update_[param_id]->mutable_cpu_data());   //update_data = update_data ^ 0.5

    // compute the update
    caffe_mul(net_params[param_id]->count(),
        net_params[param_id]->cpu_diff(),
        this->update_[param_id]->cpu_data(),
        net_params[param_id]->mutable_cpu_diff());  //net_params_diff = net_params_diff * update_data

    // compute square of update
    caffe_powx(net_params[param_id]->count(),
        net_params[param_id]->cpu_diff(), Dtype(2),
        this->update_[param_id]->mutable_cpu_data());   //update_data = net_params_diff ^ 2

    // update history of updates    //history_latter_data = (1 - momentum) * update_data + momentum * history_latter_data
    caffe_cpu_axpby(net_params[param_id]->count(), Dtype(1) - momentum,
        this->update_[param_id]->cpu_data(), momentum,
        this->history_[update_history_offset + param_id]->mutable_cpu_data());

    // apply learning rate
    caffe_cpu_scale(net_params[param_id]->count(), local_rate,
        net_params[param_id]->cpu_diff(),
        net_params[param_id]->mutable_cpu_diff());  //net_params_diff = local_rate * net_params_diff
    break;
  }
  case Caffe::GPU: {
#ifndef CPU_ONLY
    // g = net_params_diff;
    // h = history_former_data;
    // h2 = history_latter_data;
    // gi = g[i];
    // hi = h[i] = momentum * h[i] + (1-momentum) * gi * gi;
    // gi = gi * sqrt((h2[i] + delta) / (hi + delta));
    // h2[i] = momentum * h2[i] + (1-momentum) * gi * gi;
    // g[i] = local_rate * gi;
    adadelta_update_gpu(net_params[param_id]->count(),
        net_params[param_id]->mutable_gpu_diff(),
        this->history_[param_id]->mutable_gpu_data(),
        this->history_[update_history_offset + param_id]->mutable_gpu_data(),
        momentum, delta, local_rate);
#else
    NO_GPU;
#endif
    break;
  }
  default:
    LOG(FATAL) << "Unknown caffe mode: " << Caffe::mode();
  }
}

Adam简介

Adam算法包含两个衰减参数\(\rho_{1}\)\(\rho_{2}\),一般\(\rho_{1}=0.9, \rho_{2}=0.999\)。还包含一阶矩和二阶矩变量\(s, r\),时间步\(t\)
初始时\(s=0, r=0, t=0\)\(\delta\)同样为较小常数。

  1. 更新一阶矩:\(s_{t+1}=\rho_{1}*s_{t}+(1-\rho_{1})*\nabla_{\theta_{t}}\)
  2. 更新二阶矩:\(r_{t+1}=\rho_{2}*r_{t}+(1-\rho_{2})*\nabla_{\theta_{t}}\odot\nabla_{\theta_{t}}\)
  3. 修正一阶矩的偏差:\(\tilde{s}_{t+1}=\frac{s_{t+1}}{1-\rho_{1}^{t+1}}\)
  4. 修正二阶矩的偏差:\(\tilde{r}_{t+1}=\frac{r_{t+1}}{1-\rho_{2}^{t+1}}\)
  5. 计算梯度的更新量:\(\Delta\theta_{t+1}=lr*\frac{\tilde{s}_{t+1}}{\sqrt{\tilde{r}_{t+1}}+\delta}\)

  6. 应用更新:\(\theta_{t+1}=\theta_{t}-\Delta\theta_{t+1}\)

adam_solver.cpp源码

template <typename Dtype>
void AdamSolver<Dtype>::AdamPreSolve() {
  // Add the extra history entries for Adam after those from SGDSolver::PreSolve
  const vector<Blob<Dtype>*>& net_params = this->net_->learnable_params();  //所有可学习参数
  for (int i = 0; i < net_params.size(); ++i) {
    const vector<int>& shape = net_params[i]->shape();  //第i个可学习参数对应的形状
    this->history_.push_back(shared_ptr<Blob<Dtype> >(new Blob<Dtype>(shape))); //history_再存入一个与参数大小相同的空blob
  }
}

#ifndef CPU_ONLY
template <typename Dtype>
void adam_update_gpu(int N, Dtype* g, Dtype* m, Dtype* v, Dtype beta1,
    Dtype beta2, Dtype eps_hat, Dtype corrected_local_rate);
#endif

template <typename Dtype>
void AdamSolver<Dtype>::ComputeUpdateValue(int param_id, Dtype rate) {
  const vector<Blob<Dtype>*>& net_params = this->net_->learnable_params();  //所有可学习参数
  const vector<float>& net_params_lr = this->net_->params_lr(); //参数的学习率系数
  Dtype local_rate = rate * net_params_lr[param_id];            //当前参数的学习率
  const Dtype beta1 = this->param_.momentum();    //两个动量系数
  const Dtype beta2 = this->param_.momentum2();

  // we create aliases for convenience
  size_t update_history_offset = net_params.size();     //history_的大小为2 * update_history_offset
  Blob<Dtype>* val_m = this->history_[param_id].get();
  Blob<Dtype>* val_v = this->history_[param_id + update_history_offset].get();
  Blob<Dtype>* val_t = this->temp_[param_id].get();

  const int t = this->iter_ + 1;                //步数
  const Dtype correction = std::sqrt(Dtype(1) - pow(beta2, t)) /
      (Dtype(1.) - pow(beta1, t));              //correction = sqrt(1 - beta2 ^ t) / (1 - beta1 ^ t)
  const int N = net_params[param_id]->count();  //参数的元素个数
  const Dtype eps_hat = this->param_.delta();   //微小值

  switch (Caffe::mode()) {
    case Caffe::CPU: {
    // update m <- \beta_1 m_{t-1} + (1-\beta_1)g_t
    caffe_cpu_axpby(N, Dtype(1)-beta1, net_params[param_id]->cpu_diff(), beta1,
        val_m->mutable_cpu_data()); //val_m = (1 - beta1) * net_params_diff + beta1 * val_m

    // update v <- \beta_2 m_{t-1} + (1-\beta_2)g_t^2
    caffe_mul(N, net_params[param_id]->cpu_diff(), net_params[param_id]->cpu_diff(),
        val_t->mutable_cpu_data()); //val_t = net_params_diff * net_params_diff
    caffe_cpu_axpby(N, Dtype(1)-beta2, val_t->cpu_data(), beta2,
        val_v->mutable_cpu_data()); //val_v = (1 - beta2) * val_t + beta2 * val_v

    // set update
    caffe_powx(N, val_v->cpu_data(), Dtype(0.5),
        val_t->mutable_cpu_data()); //val_t = val_v ^ 0.5
    caffe_add_scalar(N, eps_hat, val_t->mutable_cpu_data());  //val_t += eps_hat
    caffe_div(N, val_m->cpu_data(), val_t->cpu_data(),
        val_t->mutable_cpu_data()); //val_t = val_m / val_t

    caffe_cpu_scale(N, local_rate*correction, val_t->cpu_data(),
        net_params[param_id]->mutable_cpu_diff());  //net_params_diff = local_rate*correction * val_t
    break;
  }
  case Caffe::GPU: {
#ifndef CPU_ONLY
    // g = net_params_diff
    // m = val_m
    // v = val_v
    // gi = g[i];
    // mi = m[i] = m[i]*beta1 + gi*(1-beta1);
    // vi = v[i] = v[i]*beta2 + gi*gi*(1-beta2);
    // g[i] = local_rate * correction * mi / (sqrt(vi) + eps_hat);
    adam_update_gpu(N, net_params[param_id]->mutable_gpu_diff(),
        val_m->mutable_gpu_data(), val_v->mutable_gpu_data(), beta1, beta2,
        eps_hat, local_rate*correction);
#else
    NO_GPU;
#endif
    break;
  }
  default:
    LOG(FATAL) << "Unknown caffe mode: " << Caffe::mode();
  }
}

小结

  1. 很多地方的动量的符号与本文不用,是用\(\nu_{t+1}=-lr*\nabla_{\theta_{t}}+m*\nu_{t}\),然后\(\theta_{t+1}=\theta_{t}+\nu_{t+1}\),其实原理是一致的,本文只是为了保持与caffe的代码一致。

参考

  1. https://stats.stackexchange.com/questions/179915/whats-the-difference-between-momentum-based-gradient-descent-and-nesterovs-acc
  2. https://jlmelville.github.io/mize/nesterov.html
  3. https://zhuanlan.zhihu.com/p/22810533
  4. https://zh.d2l.ai/chapter_optimization/adadelta.html
  5. 《Deep Learning》– Ian Goodfellow and Yoshua Bengio and Aaron Courville

Caffe的源码笔者是第一次阅读,一边阅读一边记录,对代码的理解和分析可能会存在错误或遗漏,希望各位读者批评指正,谢谢支持!

《设计模式之禅》之代理模式

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原文地址:《Caffe源码-几种优化算法》 发布于2019-12-29

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