ensmallen_bits/qhadam/qhadam_update.hpp

/**
 * @file qhadam_update.hpp
 * @author Niteya Shah
 *
 * Implments the QHAdam Optimizer. QHAdam is a variant of Adam which introduces
 * quasi hyperbolic moment terms to improve paramterisation and performance.
 *
 * 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_ADAM_QHADAM_UPDATE_HPP
#define ENSMALLEN_ADAM_QHADAM_UPDATE_HPP

namespace ens {

/**
 * QHAdam is a optimising strategy based on the Quasi-Hyperbolic step when
 * applied to the Adam Optimiser.QH updates can be considered to a weighted
 * average of the momentum.QHAdam,based on its paramterisation can recover
 * many algorithms such as NAdam and RMSProp.
 *
 * For more information, see the following.
 *
 * @code
 * @inproceedings{ma2019qh,
 *   title={Quasi-hyperbolic momentum and Adam for deep learning},
 *   author={Jerry Ma and Denis Yarats},
 *   booktitle={International Conference on Learning Representations},
 *   year={2019}
 * }
 * @endcode
 */
class QHAdamUpdate
{
 public:
  /**
   * Construct the QHAdam 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.
   * @param v1 The first quasi-hyperbolic term.
   * @param v1 The second quasi-hyperbolic term.
   */
  QHAdamUpdate(const double epsilon = 1e-8,
               const double beta1 = 0.9,
               const double beta2 = 0.999,
               const double v1 = 0.7,
               const double v2 = 1) :
    epsilon(epsilon),
    beta1(beta1),
    beta2(beta2),
    v1(v1),
    v2(v2),
    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; }

  //! Get the first quasi-hyperbolic term.
  double V1() const { return v1; }
  //! Modify the first quasi-hyperbolic term.
  double& V1() { return v1; }

  //! Get the second quasi-hyperbolic term.
  double V2() const { return v2; }
  //! Modify the second quasi-hyperbolic term.
  double& V2() { return v2; }

  /**
   * 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 AdamUpdate object.
     * @param rows Number of rows in the gradient matrix.
     * @param cols Number of columns in the gradient matrix.
     */
    Policy(QHAdamUpdate& parent, const size_t rows, const size_t cols) :
        parent(parent)
    {
      m.zeros(rows, cols);
      v.zeros(rows, cols);
    }

    /**
     * Update step for QHAdam.
     *
     * @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);

      GradType mDash = m / biasCorrection1;
      GradType vDash = v / biasCorrection2;

      // QHAdam recovers Adam when v2 = v1 = 1.
      iterate -= stepSize *
          ((((1 - parent.v1) * gradient) + parent.v1 * mDash) /
           (arma::sqrt(((1 - parent.v2) * (gradient % gradient)) +
            parent.v2 * vDash) + parent.epsilon));
    }

   private:
    //! Instantiated parent object.
    QHAdamUpdate& parent;

    //! The exponential moving average of gradient values.
    GradType m;

    // The exponential moving average of squared gradient values.
    GradType v;
  };

 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 first quasi-hyperbolic term.
  double v1;

  // The second quasi-hyperbolic term.
  double v2;

  // The number of iterations.
  size_t iteration;
};

} // namespace ens

#endif