text: /**
   * @file amsgrad_update.hpp
   * @author Haritha Nair
   *
   * Implementation of AMSGrad optimizer. AMSGrad is an exponential moving average 
   * optimizer that dynamically adapts over time with guaranteed convergence.
   *
   * ensmallen is free software; you may redistribute it and/or modify it under
   * the terms of the 3-clause BSD license.  You should have received a copy of
   * the 3-clause BSD license along with ensmallen.  If not, see
   * http://www.opensource.org/licenses/BSD-3-Clause for more information.
   */
  #ifndef ENSMALLEN_AMS_GRAD_AMS_GRAD_UPDATE_HPP
  #define ENSMALLEN_AMS_GRAD_AMS_GRAD_UPDATE_HPP
  
  namespace ens {
  
  /**
   * AMSGrad is an exponential moving average variant which along with having
   * benefits of optimizers like Adam and RMSProp, also guarantees convergence.
   * Unlike Adam, it uses maximum of past squared gradients rather than their
   * exponential average for updation.
   *
   * For more information, see the following.
   *
   * @code
   * @article{
   *   title   = {On the convergence of Adam and beyond},
   *   url     = {https://openreview.net/pdf?id=ryQu7f-RZ}
   *   year    = {2018}
   * }
   * @endcode
   */
  class AMSGradUpdate
  {
   public:
    /**
     * Construct the AMSGrad update policy with the given parameters.
     *
     * @param epsilon The epsilon value used to initialise the squared gradient
     *        parameter.
     * @param beta1 The smoothing parameter.
     * @param beta2 The second moment coefficient.
     */
    AMSGradUpdate(const double epsilon = 1e-8,
                  const double beta1 = 0.9,
                  const double beta2 = 0.999) :
      epsilon(epsilon),
      beta1(beta1),
      beta2(beta2),
      iteration(0)
    {
      // Nothing to do.
    }
  
    //! Get the value used to initialise the squared gradient parameter.
    double Epsilon() const { return epsilon; }
    //! Modify the value used to initialise the squared gradient parameter.
    double& Epsilon() { return epsilon; }
  
    //! Get the smoothing parameter.
    double Beta1() const { return beta1; }
    //! Modify the smoothing parameter.
    double& Beta1() { return beta1; }
  
    //! Get the second moment coefficient.
    double Beta2() const { return beta2; }
    //! Modify the second moment coefficient.
    double& Beta2() { return beta2; }
  
    //! Get the current iteration number.
    size_t Iteration() const { return iteration; }
    //! Modify the current iteration number.
    size_t& Iteration() { return iteration; }
  
    /**
     * The UpdatePolicyType policy classes must contain an internal 'Policy'
     * template class with two template arguments: MatType and GradType.  This is
     * instantiated at the start of the optimization, and holds parameters
     * specific to an individual optimization.
     */
    template<typename MatType, typename GradType>
    class Policy
    {
     public:
      /**
       * This constructor is called by the SGD Optimize() method before the start
       * of the iteration update process.
       *
       * @param parent Instantiated AMSGradUpdate parent object.
       * @param rows Number of rows in the gradient matrix.
       * @param cols Number of columns in the gradient matrix.
       */
      Policy(AMSGradUpdate& parent, const size_t rows, const size_t cols) :
          parent(parent)
      {
        m.zeros(rows, cols);
        v.zeros(rows, cols);
        vImproved.zeros(rows, cols);
      }
  
      /**
       * Update step for AMSGrad.
       *
       * @param iterate Parameters that minimize the function.
       * @param stepSize Step size to be used for the given iteration.
       * @param gradient The gradient matrix.
       */
      void Update(MatType& iterate,
                  const double stepSize,
                  const GradType& gradient)
      {
        // Increment the iteration counter variable.
        ++parent.iteration;
  
        // And update the iterate.
        m *= parent.beta1;
        m += (1 - parent.beta1) * gradient;
  
        v *= parent.beta2;
        v += (1 - parent.beta2) * (gradient % gradient);
  
        const double biasCorrection1 = 1.0 - std::pow(parent.beta1,
            parent.iteration);
        const double biasCorrection2 = 1.0 - std::pow(parent.beta2,
            parent.iteration);
  
        // Element wise maximum of past and present squared gradients.
        vImproved = arma::max(vImproved, v);
  
        iterate -= (stepSize * std::sqrt(biasCorrection2) / biasCorrection1) *
                    m / (arma::sqrt(vImproved) + parent.epsilon);
      }
  
     private:
      // Instantiated parent AMSGradUpdate object.
      AMSGradUpdate& parent;
  
      // The exponential moving average of gradient values.
      GradType m;
  
      // The exponential moving average of squared gradient values.
      GradType v;
  
      // The optimal squared gradient value.
      GradType vImproved;
    };
  
   private:
    // The epsilon value used to initialise the squared gradient parameter.
    double epsilon;
  
    // The smoothing parameter.
    double beta1;
  
    // The second moment coefficient.
    double beta2;
  
    // The number of iterations.
    size_t iteration;
  };
  
  } // namespace ens
  
  #endif
  