# Optimizer

## Momentum

class `paddle.v2.optimizer.``Momentum`(momentum=None, sparse=False, **kwargs)

  SGD Optimizer.

  SGD is an optimization method, trying to find a neural network that minimize the “cost/error” of it by iteration. In paddle’s implementation SGD Optimizer is synchronized, which means all gradients will be wait to calculate and reduced into one gradient, then do optimize operation.

  The neural network consider the learning problem of minimizing an objective function, that has the form of a sum

  \[Q(w) = \sum_{i}^{n} Q_i(w)\]

  The value of function Q sometimes is the cost of neural network (Mean Square Error between prediction and label for example). The function Q is parametrised by w, the weight/bias of neural network. And weights is what to be learned. The i is the i-th observation in (trainning) data.

  So, the SGD method will optimize the weight by

  \[w = w - \eta \nabla Q(w) = w - \eta \sum_{i}^{n} \nabla Q_i(w)\]

  where \(\eta\) is learning rate. And \(n\) is batch size.

## Adam

class `paddle.v2.optimizer.``Adam`(beta1=0.9, beta2=0.999, epsilon=1e-08, **kwargs)

  Adam optimizer. The details of please refer Adam: A Method for Stochastic Optimization

  \[\begin{split}m(w, t) & = \beta_1 m(w, t-1) + (1 - \beta_1) \nabla Q_i(w) \\ v(w, t) & = \beta_2 v(w, t-1) + (1 - \beta_2)(\nabla Q_i(w)) ^2 \\ w & = w - \frac{\eta}{\sqrt{v(w,t) + \epsilon}}\end{split}\]

  参数:

  - beta1 (float) – the \(\beta_1\) in equation.
  - beta2 (float) – the \(\beta_2\) in equation.
  - epsilon (float) – the \(\epsilon\) in equation. It is used to prevent divided by zero.

## Adamax

class `paddle.v2.optimizer.``Adamax`(beta1=0.9, beta2=0.999, **kwargs)

  Adamax optimizer.

  The details of please refer this Adam: A Method for Stochastic Optimization

  \[\begin{split}m_t & = \beta_1 * m_{t-1} + (1-\beta_1)* \nabla Q_i(w) \\ u_t & = max(\beta_2*u_{t-1}, abs(\nabla Q_i(w))) \\ w_t & = w_{t-1} - (\eta/(1-\beta_1^t))*m_t/u_t\end{split}\]

  参数:

  - beta1 (float) – the \(\beta_1\) in the equation.
  - beta2 (float) – the \(\beta_2\) in the equation.

## AdaGrad

class `paddle.v2.optimizer.``AdaGrad`(**kwargs)

  Adagrad(for ADAptive GRAdient algorithm) optimizer.

  For details please refer this Adaptive Subgradient Methods for Online Learning and Stochastic Optimization.

  \[\begin{split}G &= \sum_{\tau=1}^{t} g_{\tau} g_{\tau}^T \\ w & = w - \eta diag(G)^{-\frac{1}{2}} \circ g\end{split}\]

## DecayedAdaGrad

class `paddle.v2.optimizer.``DecayedAdaGrad`(rho=0.95, epsilon=1e-06, **kwargs)

  AdaGrad method with decayed sum gradients. The equations of this method show as follow.

  \[\begin{split}E(g_t^2) &= \rho * E(g_{t-1}^2) + (1-\rho) * g^2 \\ learning\_rate &= 1/sqrt( ( E(g_t^2) + \epsilon )\end{split}\]

  参数:

  - rho (float) – The \(\rho\) parameter in that equation
  - epsilon (float) – The \(\epsilon\) parameter in that equation.

## AdaDelta

class `paddle.v2.optimizer.``AdaDelta`(rho=0.95, epsilon=1e-06, **kwargs)

  AdaDelta method. The details of adadelta please refer to this ADADELTA: AN ADAPTIVE LEARNING RATE METHOD.

  \[\begin{split}E(g_t^2) &= \rho * E(g_{t-1}^2) + (1-\rho) * g^2 \\ learning\_rate &= sqrt( ( E(dx_{t-1}^2) + \epsilon ) / ( \ E(g_t^2) + \epsilon ) ) \\ E(dx_t^2) &= \rho * E(dx_{t-1}^2) + (1-\rho) * (-g*learning\_rate)^2\end{split}\]

  参数:

  - rho (float) – \(\rho\) in equation
  - epsilon (float) – \(\rho\) in equation

## RMSProp

class `paddle.v2.optimizer.``RMSProp`(rho=0.95, epsilon=1e-06, **kwargs)

  RMSProp(for Root Mean Square Propagation) optimizer. For details please refer this slide.

  The equations of this method as follows:

  \[\begin{split}v(w, t) & = \rho v(w, t-1) + (1 - \rho)(\nabla Q_{i}(w))^2 \\ w & = w - \frac{\eta} {\sqrt{v(w,t) + \epsilon}} \nabla Q_{i}(w)\end{split}\]

  参数:

  - rho (float) – the \(\rho\) in the equation. The forgetting factor.
  - epsilon (float) – the \(\epsilon\) in the equation.