Add Das-Dennis weight initialization method

HISTORY.md

@@ -37,6 +37,8 @@
  * Introduce Policy Methods for MOEA/D-DE
    ([#293](https://github.com/mlpack/ensmallen/pull/293)).
 
+ * Add Das-Dennis weight initialization method
+   ([#295](https://github.com/mlpack/ensmallen/pull/295)).
 ### ensmallen 2.16.1: "Severely Dented Can Of Polyurethane"
 ###### 2021-03-02
  * Fix test compilation issue when `ENS_USE_OPENMP` is set

doc/optimizers.md

@@ -1581,6 +1581,7 @@ initialize the reference directions.
 
 The following types are available:
 
+ * **`Uniform`**
  * **`BayesianBootstrap`**
 
 The _`DecompPolicyType`_ template parameter refers to the strategy used to
@@ -1594,9 +1595,11 @@ The following types are available:
 
 For convenience the following types can be used:
 
- * **`DefaultMOEAD`** (equivalent to `MOEAD<BayesianBootstrap, Tchebycheff>`): utilizes BayesianBootstrap for weight init
+ * **`DefaultMOEAD`** (equivalent to `MOEAD<Uniform, Tchebycheff>`): utilizes Uniform method for weight initialization
  and Tchebycheff for weight decomposition.
 
+ * **`BBSMOEAD`** (equivalent to `MOEAD<BayesianBootstrap, Tchebycheff>`): utilizes Bayesian Bootstrap method for weight initialization and Tchebycheff for weight decomposition.
+
 #### Attributes
 
 | **type** | **name** | **description** | **default** |
@@ -1629,7 +1632,7 @@ Attributes of the optimizer may also be changed via the member methods
 SchafferFunctionN1<arma::mat> SCH;
 arma::vec lowerBound("-10 -10");
 arma::vec upperBound("10 10");
-DefaultMOEAD opt(150, 300, 1.0, 0.9, 20, 20, 0.5, 2, 1E-10, lowerBound, upperBound);
+DefaultMOEAD opt(300, 300, 1.0, 0.9, 20, 20, 0.5, 2, 1E-10, lowerBound, upperBound);
 typedef decltype(SCH.objectiveA) ObjectiveTypeA;
 typedef decltype(SCH.objectiveB) ObjectiveTypeB;
 arma::mat coords = SCH.GetInitialPoint();

include/ensmallen_bits/moead/moead.hpp

@@ -22,6 +22,8 @@
 
 //! Weight initialization policies.
 #include "weight_init_policies/bbs_init.hpp"
+#include "weight_init_policies/uniform_init.hpp"
+
 namespace ens {
 
 /**
@@ -45,7 +47,7 @@ namespace ens {
  *   year={2008},
  * @endcode
  */
-template<typename InitPolicyType = BayesianBootstrap,
+template<typename InitPolicyType = Uniform,
          typename DecompPolicyType = Tchebycheff>
 class MOEAD {
  public:
@@ -72,8 +74,8 @@ class MOEAD {
    * @param upperBound The upper bound on each variable of a member
    *    of the variable space.
    */
-  MOEAD(const size_t populationSize = 150,
-        const size_t maxGenerations = 300,
+  MOEAD(const size_t populationSize = 300,
+        const size_t maxGenerations = 500,
         const double crossoverProb = 1.0,
         const double neighborProb = 0.9,
         const size_t neighborSize = 20,
@@ -111,8 +113,8 @@ class MOEAD {
    * @param upperBound The upper bound on each variable of a member
    *    of the variable space.
    */
-    MOEAD(const size_t populationSize = 150,
-          const size_t maxGenerations = 300,
+    MOEAD(const size_t populationSize = 300,
+          const size_t maxGenerations = 500,
           const double crossoverProb = 1.0,
           const double neighborProb = 0.9,
           const size_t neighborSize = 20,
@@ -324,7 +326,8 @@ class MOEAD {
   DecompPolicyType decompPolicy;
 };
 
-using DefaultMOEAD = MOEAD<BayesianBootstrap, Tchebycheff>;
+using DefaultMOEAD = MOEAD<Uniform, Tchebycheff>;
+using BBSMOEAD = MOEAD<BayesianBootstrap, Tchebycheff>;
 } // namespace ens
 
 // Include implementation.

include/ensmallen_bits/moead/weight_init_policies/uniform_init.hpp

@@ -0,0 +1,207 @@
+/**
+ * @file uniform_init.hpp
+ * @author Nanubala Gnana Sai
+ *
+ * The Uniform (Das Dennis) methodology of Weight Initialization.
+ *
+ * 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_MOEAD_UNIFORM_HPP
+#define ENSMALLEN_MOEAD_UNIFORM_HPP
+
+namespace ens {
+
+/**
+ * The Uniform (Das Dennis) method for initializing weights. This algorithm guarantees
+ * that the distance between adjacent points would be uniform.
+ *
+ * For more information, see the following:
+ *
+ * @code
+ * article{zhang2007moea,
+ *   title={MOEA/D: A multiobjective evolutionary algorithm based on decomposition},
+ *   author={Zhang, Qingfu and Li, Hui},
+ *   journal={IEEE Transactions on evolutionary computation},
+ *   pages={712--731},
+ *   year={2007}
+ * @endcode
+ */
+class Uniform
+{
+ public:
+  /**
+   * Constructor for Uniform Weight Initializatoin Policy.
+   */
+  Uniform()
+  {
+    /* Nothing to do. */
+  }
+
+  /**
+   * Generate the reference direction matrix.
+   *
+   * @tparam MatType The type of the matrix used for constructing weights.
+   * @param numObjectives The dimensionality of objective space.
+   * @param numPoints The number of reference directions requested.
+   * @param epsilon Handle numerical stability after weight initialization.
+   */
+  template<typename MatType>
+  MatType Generate(size_t numObjectives,
+                   size_t numPoints,
+                   double epsilon)
+  {
+    size_t numPartitions  = FindNumParitions(numObjectives, numPoints);
+    size_t validNumPoints = FindNumUniformPoints(numObjectives, numPartitions);
+
+    //! The requested number of points is not matching any partition number.
+    if (numPoints != validNumPoints)
+    {
+      size_t nextValidNumPoints = FindNumUniformPoints(numObjectives, numPartitions + 1);
+      std::ostringstream oss;
+      oss << "DasDennis::Generate(): " << "The requested numPoints " << numPoints
+          << " cannot be generated uniformly.\n " << "Either choose numPoints as "
+          << validNumPoints << "(numPartition = " << numPartitions << ") or "
+          << "numPoints as " << nextValidNumPoints << "(numPartition = "
+          << numPartitions + 1 << ").";
+      throw std::logic_error(oss.str());
+    }
+
+    return DasDennis<MatType>(numObjectives, numPoints,
+        numPartitions, epsilon);
+  }
+
+ private:
+  /**
+   * Finds the number of points which can be sampled uniformly from a 
+   * unit simplex given the number of partitions.
+   */
+  size_t FindNumUniformPoints(const size_t numObjectives,
+                              const size_t numPartitions)
+  {
+    //! O(N) algorithm to calculate binomial coefficient.
+    auto BinomialCoefficient =
+        [](size_t n, size_t k) -> size_t
+        {
+          size_t retval = 1;
+          // Since, C(n, k) = C(n, n - k).
+          if (k > n - k)
+            k = n - k;
+
+          // [n * (n - 1) * .... * (n - k + 1)] / [k * (k - 1) * .... * 1].
+          for (size_t i = 0; i < k; ++i)
+          {
+            retval *= (n - i);
+            retval /= (i + 1);
+          }
+
+          return retval;
+        };
+    return BinomialCoefficient(numObjectives + numPartitions - 1, numPartitions);
+  }
+
+  /**
+   *  Calculates the appropriate number of partitions such that, the binomial
+   *  coefficient value is closest to the number of points requested.
+   */
+  size_t FindNumParitions(size_t numObjectives, size_t numPoints)
+  {
+    if (numObjectives == 1) return 0;
+    // Iteratively increase numPartitions so that the binomial coefficient
+    // comes near to numPoints;
+    size_t numPartitions {1};
+    size_t sampledNumPoints = FindNumUniformPoints(numPartitions,
+        numObjectives);
+    while (sampledNumPoints <= numPoints)
+    {
+       ++numPartitions;
+       sampledNumPoints = FindNumUniformPoints(numObjectives,
+          numPartitions);
+    }
+
+    return numPartitions - 1;
+  }
+
+  /**
+   * A helper function for DasDennis
+   */
+  template<typename AuxInfoStackType,
+           typename MatType>
+  void DasDennisHelper(AuxInfoStackType& progressStack,
+                       MatType& weights,
+                       const size_t numObjectives,
+                       const size_t numPoints,
+                       const size_t numPartitions,
+                       const double epsilon)
+  {
+    typedef typename MatType::elem_type ElemType;
+    typedef typename arma::Row<ElemType> RowType;
+
+    size_t counter = 0;
+    const ElemType delta = 1.0 / (ElemType)numPartitions;
+
+    while ((counter < numPoints) && !progressStack.empty())
+    {
+      MatType point{};
+      size_t beta{};
+      std::tie(point, beta) = progressStack.back();
+      progressStack.pop_back();
+
+      if (point.size() + 1 == numObjectives)
+      {
+        point.insert_rows(point.n_rows, RowType(1).fill(
+            delta * static_cast<ElemType>(beta)));
+        weights.col(counter) = point + epsilon;
+        ++counter;
+      }
+
+      else
+      {
+        for (size_t i = 0; i <= beta; ++i)
+        {
+          MatType pointClone(point);
+          pointClone.insert_rows(pointClone.n_rows, RowType(1).fill(
+              delta * static_cast<ElemType>(i)));
+          progressStack.push_back({pointClone, beta - i});
+        }
+      }
+    }
+  }
+
+  /**
+   * Generates the weight matrix after verifying the
+   * validity of the parameters.
+   */
+  template <typename MatType>
+  MatType DasDennis(const size_t numObjectives,
+                    const size_t numPoints,
+                    const size_t numPartitions,
+                    const double epsilon)
+  {
+    //! Holds auxillary information required for the helper function. 
+    //! Holds the current point and beta value.
+    using AuxContainer = std::pair<MatType, size_t>;
+
+    std::vector<AuxContainer> progressStack{};
+    //! Init the progress stack.
+    progressStack.push_back({{}, numPartitions});
+    MatType weights(numObjectives, numPoints);
+    weights.fill(arma::datum::nan);
+    DasDennisHelper<decltype(progressStack), MatType>(
+        progressStack,
+        weights,
+        numObjectives,
+        numPoints,
+        numPartitions,
+        epsilon);
+
+    return weights;
+  }
+
+};
+
+} // namespace ens
+
+#endif

tests/moead_test.cpp

@@ -46,7 +46,7 @@ TEST_CASE("MOEADSchafferN1DoubleTest", "[MOEADTest]")
   const double expectedUpperBound = 2.0;
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300, // Population size.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -108,7 +108,7 @@ TEST_CASE("MOEADSchafferN1TestVectorDoubleBounds", "[MOEADTest]")
   const double expectedUpperBound = 2.0;
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300, // Population size.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -168,7 +168,7 @@ TEST_CASE("MOEADFonsecaFlemingDoubleTest", "[MOEADTest]")
   const double expectedUpperBound = 1.0 / sqrt(3);
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300,  // Max generations.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -223,7 +223,7 @@ TEST_CASE("MOEADFonsecaFlemingTestVectorDoubleBounds", "[MOEADTest]")
   const double expectedUpperBound = 1.0 / sqrt(3);
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300,  // Max generations.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -278,7 +278,7 @@ TEST_CASE("MOEADSchafferN1FloatTest", "[MOEADTest]")
   const double expectedUpperBound = 2.0;
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300, // Population size.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -340,7 +340,7 @@ TEST_CASE("MOEADSchafferN1TestVectorFloatBounds", "[MOEADTest]")
   const double expectedUpperBound = 2.0;
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300, // Population size.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -400,7 +400,7 @@ TEST_CASE("MOEADFonsecaFlemingFloatTest", "[MOEADTest]")
   const float expectedUpperBound = 1.0 / sqrt(3);
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300,  // Max generations.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -455,7 +455,7 @@ TEST_CASE("MOEADFonsecaFlemingTestVectorFloatBounds", "[MOEADTest]")
   const float expectedUpperBound = 1.0 / sqrt(3);
 
   DefaultMOEAD opt(
-        150, // Population size.
+        300,  // Max generations.
         300,  // Max generations.
         1.0,  // Crossover probability.
         0.9, // Probability of sampling from neighbor.
@@ -499,8 +499,8 @@ TEST_CASE("MOEADFonsecaFlemingTestVectorFloatBounds", "[MOEADTest]")
 
 /**
  * Test against the first problem of ZDT Test Suite.  ZDT-1 is a 30 
- * variable-2 objective problem with a convex Pareto Front.
- * 
+ * variable-2 objective problem with a convex Pareto Front. 
+ *
  * NOTE: For the sake of runtime, only ZDT-1 is tested against the
  * algorithm. Others have been tested separately.
  */
@@ -512,8 +512,8 @@ TEST_CASE("MOEADZDTONETest", "[MOEADTest]")
   const double upperBound = 1;
 
   DefaultMOEAD opt(
-      150, // Population size.
-      300,  // Max generations.
+      300, // Population size.
+      150,  // Max generations.
       1.0,  // Crossover probability.
       0.9, // Probability of sampling from neighbor.
       20, // Neighborhood size.
@@ -538,6 +538,5 @@ TEST_CASE("MOEADZDTONETest", "[MOEADTest]")
   size_t numVariables = coords.size();
   double sum = arma::accu(coords(arma::span(1, numVariables - 1), 0));
   double g = 1. + 9. * sum / (static_cast<double>(numVariables - 1));
-
   REQUIRE(g == Approx(1.0).margin(0.99));
 }