**     **之前在https://blog.csdn.net/fengbingchun/article/details/124766283 介绍过深度学习中的优化算法AdaGrad，这里介绍下深度学习的另一种优化算法Adadelta。论文名字为：《ADADELTA: AN ADAPTIVE LEARNING RATE METHOD》，论文地址：https://arxiv.org/pdf/1212.5701.pdf

**      Adadelta一种自适应学习率方法，是AdaGrad的扩展，建立在AdaGrad****的基础上，旨在减少其过激的、单调递减的学习率**。Adadelta不是积累所有过去的平方梯度，而是将积累的过去梯度的窗口限制为某个固定大小。如下图所示，截图来自：https://arxiv.org/pdf/1609.04747.pdf

**     

**

     使用Adadelta，我们甚至不需要设置默认学习率这一超参数，因为它已从更新规则中消除，它使用参数本身的变化率来调整学习率。

**      Adadelta可被认为是梯度下降的进一步扩展，它建立在AdaGrad和RMSProp的基础上，并改变了自定义步长的计算(changes the calculation of the custom step size)****，进而不再需要初始学习率超参数**。

**     **Adadelta旨在加速优化过程，例如减少达到最优值所需的迭代次数，或提高优化算法的能力，例如获得更好的最终结果。

**     **最好将Adadelta理解为AdaGrad和RMSProp算法的扩展。Adadelta是RMSProp的进一步扩展，旨在提高算法的收敛性并消除对手动指定初始学习率的需要。

**     **与RMSProp一样，Adadelta为每个参数计算平方偏导数的衰减移动平均值。关键区别在于使用delta的衰减平均值或参数变化来计算参数的步长。(The decaying moving average of the squared partial derivative is calculated for each parameter, as with RMSProp. The key difference is in the calculation of the step size for a parameter that uses the decaying average of the delta or change in parameter.)

**     **以上内容主要参考：https://machinelearningmastery.com

**     **以下是与AdaGrad不同的代码片段：

**     **1.在原有枚举类Optimizaiton的基础上新增Adadelta：

enum class Optimization {
	BGD, // Batch Gradient Descent
	SGD, // Stochastic Gradient Descent
	MBGD, // Mini-batch Gradient Descent
	SGD_Momentum, // SGD with Momentum
	AdaGrad, // Adaptive Gradient
	RMSProp, // Root Mean Square Propagation
	Adadelta // an adaptive learning rate method
};

**     **2.calculate_gradient_descent函数：

void LogisticRegression2::calculate_gradient_descent(int start, int end)
{
	switch (optim_) {
		case Optimization::Adadelta: {
			int len = end - start;
			std::vector<float> g(feature_length_, 0.), p(feature_length_, 0.);
			std::vector<float> z(len, 0.), dz(len, 0.);
			for (int i = start, x = 0; i < end; ++i, ++x) {
				z[x] = calculate_z(data_->samples[random_shuffle_[i]]);
				dz[x] = calculate_loss_function_derivative(calculate_activation_function(z[x]), data_->labels[random_shuffle_[i]]);
				for (int j = 0; j < feature_length_; ++j) {
					float dw = data_->samples[random_shuffle_[i]][j] * dz[x];
					g[j] = mu_ * g[j] + (1. - mu_) * (dw * dw); // formula 10
					float alpha = (eps_ + std::sqrt(p[j])) / (eps_ + std::sqrt(g[j]));
					float change = alpha * dw;
					p[j] = mu_ * p[j] +  (1. - mu_) * (change * change); // formula 15
					w_[j] = w_[j] - change;
				}
				b_ -= (eps_ * dz[x]);
			}
		}
			break;
		case Optimization::RMSProp: {
			int len = end - start;
			std::vector<float> g(feature_length_, 0.);
			std::vector<float> z(len, 0), dz(len, 0);
			for (int i = start, x = 0; i < end; ++i, ++x) {
				z[x] = calculate_z(data_->samples[random_shuffle_[i]]);
				dz[x] = calculate_loss_function_derivative(calculate_activation_function(z[x]), data_->labels[random_shuffle_[i]]);
				for (int j = 0; j < feature_length_; ++j) {
					float dw = data_->samples[random_shuffle_[i]][j] * dz[x];
					g[j] = mu_ * g[j] + (1. - mu_) * (dw * dw); // formula 18
					w_[j] = w_[j] - alpha_ * dw / (std::sqrt(g[j]) + eps_);
				}
				b_ -= (alpha_ * dz[x]);
			}
		}
			break;
		case Optimization::AdaGrad: {
			int len = end - start;
			std::vector<float> g(feature_length_, 0.);
			std::vector<float> z(len, 0), dz(len, 0);
			for (int i = start, x = 0; i < end; ++i, ++x) {
				z[x] = calculate_z(data_->samples[random_shuffle_[i]]);
				dz[x] = calculate_loss_function_derivative(calculate_activation_function(z[x]), data_->labels[random_shuffle_[i]]);
				for (int j = 0; j < feature_length_; ++j) {
					float dw = data_->samples[random_shuffle_[i]][j] * dz[x];
					g[j] += dw * dw;
					w_[j] = w_[j] - alpha_ * dw / (std::sqrt(g[j]) + eps_);
				}
				b_ -= (alpha_ * dz[x]);
			}
		}
			break;
		case Optimization::SGD_Momentum: {
			int len = end - start;
			std::vector<float> change(feature_length_, 0.);
			std::vector<float> z(len, 0), dz(len, 0);
			for (int i = start, x = 0; i < end; ++i, ++x) {
				z[x] = calculate_z(data_->samples[random_shuffle_[i]]);
				dz[x] = calculate_loss_function_derivative(calculate_activation_function(z[x]), data_->labels[random_shuffle_[i]]);
				for (int j = 0; j < feature_length_; ++j) {
					float new_change = mu_ * change[j] - alpha_ * (data_->samples[random_shuffle_[i]][j] * dz[x]);
					w_[j] += new_change;
					change[j] = new_change;
				}
				b_ -= (alpha_ * dz[x]);
			}
		}
			break;
		case Optimization::SGD:
		case Optimization::MBGD: {
			int len = end - start;
			std::vector<float> z(len, 0), dz(len, 0);
			for (int i = start, x = 0; i < end; ++i, ++x) {
				z[x] = calculate_z(data_->samples[random_shuffle_[i]]);
				dz[x] = calculate_loss_function_derivative(calculate_activation_function(z[x]), data_->labels[random_shuffle_[i]]);
				for (int j = 0; j < feature_length_; ++j) {
					w_[j] = w_[j] - alpha_ * (data_->samples[random_shuffle_[i]][j] * dz[x]);
				}
				b_ -= (alpha_ * dz[x]);
			}
		}
			break;
		case Optimization::BGD:
		default: // BGD
			std::vector<float> z(m_, 0), dz(m_, 0);
			float db = 0.;
			std::vector<float> dw(feature_length_, 0.);
			for (int i = 0; i < m_; ++i) {
				z[i] = calculate_z(data_->samples[i]);
				o_[i] = calculate_activation_function(z[i]);
				dz[i] = calculate_loss_function_derivative(o_[i], data_->labels[i]);
				for (int j = 0; j < feature_length_; ++j) {
					dw[j] += data_->samples[i][j] * dz[i]; // dw(i)+=x(i)(j)*dz(i)
				}
				db += dz[i]; // db+=dz(i)
			}
			for (int j = 0; j < feature_length_; ++j) {
				dw[j] /= m_;
				w_[j] -= alpha_ * dw[j];
			}
			b_ -= alpha_*(db/m_);
	}
}

**     **执行结果如下图所示：测试函数为test_logistic_regression2_gradient_descent，多次执行每种配置，最终结果都相同。图像集使用MNIST，其中训练图像总共10000张，0和1各5000张，均来自于训练集；预测图像总共1800张，0和1各900张，均来自于测试集。在AdaGrad中设置学习率为0.01，eps为1e-8及其它配置参数相同的情况下，AdaGrad耗时为17秒；在Adadelta中设置eps为1e-3时，Adadelta耗时为26秒；它们的识别率均为100%。

    GitHub： https://github.com/fengbingchun/NN_Test