source: branches/2994-AutoDiffForIntervals/HeuristicLab.Problems.DataAnalysis.Regression.Symbolic.Extensions/NLOptEvaluator.cs @ 17200

#region License Information
/* HeuristicLab
 * Copyright (C) 2002-2019 Heuristic and Evolutionary Algorithms Laboratory (HEAL)
 *
 * This file is part of HeuristicLab.
 *
 * HeuristicLab is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * HeuristicLab is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with HeuristicLab. If not, see <http://www.gnu.org/licenses/>.
 */
#endregion

using System;
using System.Collections.Generic;
using System.Linq;
using HeuristicLab.Common;
using HeuristicLab.Core;
using HeuristicLab.Data;
using HeuristicLab.Encodings.SymbolicExpressionTreeEncoding;
using HeuristicLab.Optimization;
using HeuristicLab.Parameters;
using HEAL.Attic;
using System.Runtime.InteropServices;

namespace HeuristicLab.Problems.DataAnalysis.Symbolic.Regression {
  [Item("NLOpt Evaluator (with constraints)", "")]
  [StorableType("5FADAE55-3516-4539-8A36-BC9B0D00880D")]
  public class NLOptEvaluator : SymbolicRegressionSingleObjectiveEvaluator {
    private const string ConstantOptimizationIterationsParameterName = "ConstantOptimizationIterations";
    private const string ConstantOptimizationImprovementParameterName = "ConstantOptimizationImprovement";
    private const string ConstantOptimizationProbabilityParameterName = "ConstantOptimizationProbability";
    private const string ConstantOptimizationRowsPercentageParameterName = "ConstantOptimizationRowsPercentage";
    private const string UpdateConstantsInTreeParameterName = "UpdateConstantsInSymbolicExpressionTree";
    private const string UpdateVariableWeightsParameterName = "Update Variable Weights";

    private const string FunctionEvaluationsResultParameterName = "Constants Optimization Function Evaluations";
    private const string GradientEvaluationsResultParameterName = "Constants Optimization Gradient Evaluations";
    private const string CountEvaluationsParameterName = "Count Function and Gradient Evaluations";

    public IFixedValueParameter<IntValue> ConstantOptimizationIterationsParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[ConstantOptimizationIterationsParameterName]; }
    }
    public IFixedValueParameter<DoubleValue> ConstantOptimizationImprovementParameter {
      get { return (IFixedValueParameter<DoubleValue>)Parameters[ConstantOptimizationImprovementParameterName]; }
    }
    public IFixedValueParameter<PercentValue> ConstantOptimizationProbabilityParameter {
      get { return (IFixedValueParameter<PercentValue>)Parameters[ConstantOptimizationProbabilityParameterName]; }
    }
    public IFixedValueParameter<PercentValue> ConstantOptimizationRowsPercentageParameter {
      get { return (IFixedValueParameter<PercentValue>)Parameters[ConstantOptimizationRowsPercentageParameterName]; }
    }
    public IFixedValueParameter<BoolValue> UpdateConstantsInTreeParameter {
      get { return (IFixedValueParameter<BoolValue>)Parameters[UpdateConstantsInTreeParameterName]; }
    }
    public IFixedValueParameter<BoolValue> UpdateVariableWeightsParameter {
      get { return (IFixedValueParameter<BoolValue>)Parameters[UpdateVariableWeightsParameterName]; }
    }

    public IResultParameter<IntValue> FunctionEvaluationsResultParameter {
      get { return (IResultParameter<IntValue>)Parameters[FunctionEvaluationsResultParameterName]; }
    }
    public IResultParameter<IntValue> GradientEvaluationsResultParameter {
      get { return (IResultParameter<IntValue>)Parameters[GradientEvaluationsResultParameterName]; }
    }
    public IFixedValueParameter<BoolValue> CountEvaluationsParameter {
      get { return (IFixedValueParameter<BoolValue>)Parameters[CountEvaluationsParameterName]; }
    }
    public IConstrainedValueParameter<StringValue> SolverParameter {
      get { return (IConstrainedValueParameter<StringValue>)Parameters["Solver"]; }
    }


    public IntValue ConstantOptimizationIterations {
      get { return ConstantOptimizationIterationsParameter.Value; }
    }
    public DoubleValue ConstantOptimizationImprovement {
      get { return ConstantOptimizationImprovementParameter.Value; }
    }
    public PercentValue ConstantOptimizationProbability {
      get { return ConstantOptimizationProbabilityParameter.Value; }
    }
    public PercentValue ConstantOptimizationRowsPercentage {
      get { return ConstantOptimizationRowsPercentageParameter.Value; }
    }
    public bool UpdateConstantsInTree {
      get { return UpdateConstantsInTreeParameter.Value.Value; }
      set { UpdateConstantsInTreeParameter.Value.Value = value; }
    }

    public bool UpdateVariableWeights {
      get { return UpdateVariableWeightsParameter.Value.Value; }
      set { UpdateVariableWeightsParameter.Value.Value = value; }
    }

    public bool CountEvaluations {
      get { return CountEvaluationsParameter.Value.Value; }
      set { CountEvaluationsParameter.Value.Value = value; }
    }

    public string Solver {
      get { return SolverParameter.Value.Value; }
    }
    public override bool Maximization {
      get { return false; }
    }

    [StorableConstructor]
    protected NLOptEvaluator(StorableConstructorFlag _) : base(_) { }
    protected NLOptEvaluator(NLOptEvaluator original, Cloner cloner)
      : base(original, cloner) {
    }
    public NLOptEvaluator()
      : base() {
      Parameters.Add(new FixedValueParameter<IntValue>(ConstantOptimizationIterationsParameterName, "Determines how many iterations should be calculated while optimizing the constant of a symbolic expression tree (0 indicates other or default stopping criterion).", new IntValue(10)));
      Parameters.Add(new FixedValueParameter<DoubleValue>(ConstantOptimizationImprovementParameterName, "Determines the relative improvement which must be achieved in the constant optimization to continue with it (0 indicates other or default stopping criterion).", new DoubleValue(0)) { Hidden = true });
      Parameters.Add(new FixedValueParameter<PercentValue>(ConstantOptimizationProbabilityParameterName, "Determines the probability that the constants are optimized", new PercentValue(1)));
      Parameters.Add(new FixedValueParameter<PercentValue>(ConstantOptimizationRowsPercentageParameterName, "Determines the percentage of the rows which should be used for constant optimization", new PercentValue(1)));
      Parameters.Add(new FixedValueParameter<BoolValue>(UpdateConstantsInTreeParameterName, "Determines if the constants in the tree should be overwritten by the optimized constants.", new BoolValue(true)) { Hidden = true });
      Parameters.Add(new FixedValueParameter<BoolValue>(UpdateVariableWeightsParameterName, "Determines if the variable weights in the tree should be  optimized.", new BoolValue(true)) { Hidden = true });

      Parameters.Add(new FixedValueParameter<BoolValue>(CountEvaluationsParameterName, "Determines if function and gradient evaluation should be counted.", new BoolValue(false)));

      var validSolvers = new ItemSet<StringValue>(new[] { "MMA", "COBYLA", "CCSAQ", "ISRES" }.Select(s => new StringValue(s).AsReadOnly()));
      Parameters.Add(new ConstrainedValueParameter<StringValue>("Solver", "The solver algorithm", validSolvers, validSolvers.First()));
      Parameters.Add(new ResultParameter<IntValue>(FunctionEvaluationsResultParameterName, "The number of function evaluations performed by the constants optimization evaluator", "Results", new IntValue()));
      Parameters.Add(new ResultParameter<IntValue>(GradientEvaluationsResultParameterName, "The number of gradient evaluations performed by the constants optimization evaluator", "Results", new IntValue()));
    }

    public override IDeepCloneable Clone(Cloner cloner) {
      return new NLOptEvaluator(this, cloner);
    }

    [StorableHook(HookType.AfterDeserialization)]
    private void AfterDeserialization() { }

    private static readonly object locker = new object();

    public override IOperation InstrumentedApply() {
      var solution = SymbolicExpressionTreeParameter.ActualValue;
      double quality;
      if (RandomParameter.ActualValue.NextDouble() < ConstantOptimizationProbability.Value) {
        IEnumerable<int> constantOptimizationRows = GenerateRowsToEvaluate(ConstantOptimizationRowsPercentage.Value);
        var counter = new EvaluationsCounter();
        quality = OptimizeConstants(SymbolicDataAnalysisTreeInterpreterParameter.ActualValue, solution, ProblemDataParameter.ActualValue,
           constantOptimizationRows, ApplyLinearScalingParameter.ActualValue.Value, Solver, ConstantOptimizationIterations.Value, updateVariableWeights: UpdateVariableWeights, lowerEstimationLimit: EstimationLimitsParameter.ActualValue.Lower, upperEstimationLimit: EstimationLimitsParameter.ActualValue.Upper, updateConstantsInTree: UpdateConstantsInTree, counter: counter);

        if (ConstantOptimizationRowsPercentage.Value != RelativeNumberOfEvaluatedSamplesParameter.ActualValue.Value) {
          throw new NotSupportedException();
        }

        if (CountEvaluations) {
          lock (locker) {
            FunctionEvaluationsResultParameter.ActualValue.Value += counter.FunctionEvaluations;
            GradientEvaluationsResultParameter.ActualValue.Value += counter.GradientEvaluations;
          }
        }

      } else {
        throw new NotSupportedException();
      }
      QualityParameter.ActualValue = new DoubleValue(quality);

      return base.InstrumentedApply();
    }

    public override double Evaluate(IExecutionContext context, ISymbolicExpressionTree tree, IRegressionProblemData problemData, IEnumerable<int> rows) {
      SymbolicDataAnalysisTreeInterpreterParameter.ExecutionContext = context;
      EstimationLimitsParameter.ExecutionContext = context;
      ApplyLinearScalingParameter.ExecutionContext = context;
      FunctionEvaluationsResultParameter.ExecutionContext = context;
      GradientEvaluationsResultParameter.ExecutionContext = context;

      // MSE evaluator is used on purpose instead of the const-opt evaluator, 
      // because Evaluate() is used to get the quality of evolved models on 
      // different partitions of the dataset (e.g., best validation model)
      double mse = SymbolicRegressionSingleObjectiveMeanSquaredErrorEvaluator.Calculate(SymbolicDataAnalysisTreeInterpreterParameter.ActualValue, tree, double.MinValue, double.MaxValue, problemData, rows, applyLinearScaling: false);

      SymbolicDataAnalysisTreeInterpreterParameter.ExecutionContext = null;
      EstimationLimitsParameter.ExecutionContext = null;
      ApplyLinearScalingParameter.ExecutionContext = null;
      FunctionEvaluationsResultParameter.ExecutionContext = null;
      GradientEvaluationsResultParameter.ExecutionContext = null;

      return mse;
    }

    public class EvaluationsCounter {
      public int FunctionEvaluations = 0;
      public int GradientEvaluations = 0;
    }

    private static void GetParameterNodes(ISymbolicExpressionTree tree, out List<ISymbolicExpressionTreeNode> thetaNodes, out List<double> thetaValues) {
      thetaNodes = new List<ISymbolicExpressionTreeNode>();
      thetaValues = new List<double>();

      var nodes = tree.IterateNodesPrefix().ToArray();
      for (int i = 0; i < nodes.Length; ++i) {
        var node = nodes[i];
        if (node is VariableTreeNode variableTreeNode) {
          thetaValues.Add(variableTreeNode.Weight);
          thetaNodes.Add(node);
        } else if (node is ConstantTreeNode constantTreeNode) {
          thetaNodes.Add(node);
          thetaValues.Add(constantTreeNode.Value);
        }
      }
    }

    // for data exchange to/from optimizer in native code
    [StructLayout(LayoutKind.Sequential)]
    private struct ConstraintData {
      public ISymbolicExpressionTree Tree;
      public ISymbolicExpressionTreeNode[] ParameterNodes;
    }

    public static double OptimizeConstants(ISymbolicDataAnalysisExpressionTreeInterpreter interpreter,
      ISymbolicExpressionTree tree, IRegressionProblemData problemData, IEnumerable<int> rows, bool applyLinearScaling,
      string solver,
      int maxIterations, bool updateVariableWeights = true,
      double lowerEstimationLimit = double.MinValue, double upperEstimationLimit = double.MaxValue,
      bool updateConstantsInTree = true, Action<double[], double, object> iterationCallback = null, EvaluationsCounter counter = null) {

      if (!updateVariableWeights) throw new NotSupportedException("not updating variable weights is not supported");
      if (!updateConstantsInTree) throw new NotSupportedException("not updating tree parameters is not supported");
      if (!applyLinearScaling) throw new NotSupportedException("application without linear scaling is not supported");

      // we always update constants, so we don't need to calculate initial quality
      // double originalQuality = SymbolicRegressionSingleObjectiveMeanSquaredErrorEvaluator.Calculate(interpreter, tree, lowerEstimationLimit, upperEstimationLimit, problemData, rows, applyLinearScaling: false);

      if (counter == null) counter = new EvaluationsCounter();
      var rowEvaluationsCounter = new EvaluationsCounter();

      var intervalConstraints = problemData.IntervalConstraints;
      var dataIntervals = problemData.VariableRanges.GetIntervals();
      var trainingRows = problemData.TrainingIndices.ToArray();
      // buffers 
      var target = problemData.TargetVariableTrainingValues.ToArray();
      var targetStDev = target.StandardDeviationPop();
      var targetVariance = targetStDev * targetStDev;
      var targetMean = target.Average();
      var pred = interpreter.GetSymbolicExpressionTreeValues(tree, problemData.Dataset, trainingRows).ToArray();
      if (pred.Any(pi => double.IsInfinity(pi) || double.IsNaN(pi))) return targetVariance;

      #region linear scaling
      var predStDev = pred.StandardDeviationPop();
      if (predStDev == 0) return targetVariance; // constant expression
      var predMean = pred.Average();

      var scalingFactor = targetStDev / predStDev;
      var offset = targetMean - predMean * scalingFactor;

      ISymbolicExpressionTree scaledTree = null;
      if (applyLinearScaling) scaledTree = CopyAndScaleTree(tree, scalingFactor, offset);
      #endregion

      // convert constants to variables named theta...
      var treeForDerivation = ReplaceConstWithVar(scaledTree, out List<string> thetaNames, out List<double> thetaValues); // copies the tree

      // create trees for relevant derivatives
      Dictionary<string, ISymbolicExpressionTree> derivatives = new Dictionary<string, ISymbolicExpressionTree>();
      var allThetaNodes = thetaNames.Select(_ => new List<ConstantTreeNode>()).ToArray();
      var constraintTrees = new List<ISymbolicExpressionTree>();
      foreach (var constraint in intervalConstraints.Constraints) {
        if (constraint.IsDerivation) {
          if (!problemData.AllowedInputVariables.Contains(constraint.Variable))
            throw new ArgumentException($"Invalid constraint: the variable {constraint.Variable} does not exist in the dataset.");
          var df = DerivativeCalculator.Derive(treeForDerivation, constraint.Variable);

          // alglib requires constraint expressions of the form c(x) <= 0
          // -> we make two expressions, one for the lower bound and one for the upper bound

          if (constraint.Interval.UpperBound < double.PositiveInfinity) {
            var df_smaller_upper = Subtract((ISymbolicExpressionTree)df.Clone(), CreateConstant(constraint.Interval.UpperBound));
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(df_smaller_upper, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
          if (constraint.Interval.LowerBound > double.NegativeInfinity) {
            var df_larger_lower = Subtract(CreateConstant(constraint.Interval.LowerBound), (ISymbolicExpressionTree)df.Clone());
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(df_larger_lower, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
        } else {
          if (constraint.Interval.UpperBound < double.PositiveInfinity) {
            var f_smaller_upper = Subtract((ISymbolicExpressionTree)treeForDerivation.Clone(), CreateConstant(constraint.Interval.UpperBound));
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(f_smaller_upper, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
          if (constraint.Interval.LowerBound > double.NegativeInfinity) {
            var f_larger_lower = Subtract(CreateConstant(constraint.Interval.LowerBound), (ISymbolicExpressionTree)treeForDerivation.Clone());
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(f_larger_lower, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
        }
      }

      var preparedTree = ReplaceVarWithConst(treeForDerivation, thetaNames, thetaValues, allThetaNodes);
      var preparedTreeParameterNodes = GetParameterNodes(preparedTree, allThetaNodes);

      // local function
      void UpdateThetaValues(double[] theta) {
        for (int i = 0; i < theta.Length; ++i) {
          foreach (var constNode in allThetaNodes[i]) constNode.Value = theta[i];
        }
      }

      var fi_eval = new double[target.Length];
      var jac_eval = new double[target.Length, thetaValues.Count];

      double calculate_obj(uint dim, double[] curX, double[] grad, IntPtr data) {
        UpdateThetaValues(curX);

        if (grad != null) {
          var autoDiffEval = new VectorAutoDiffEvaluator();
          autoDiffEval.Evaluate(preparedTree, problemData.Dataset, trainingRows,
            preparedTreeParameterNodes, fi_eval, jac_eval);

          // calc sum of squared errors and gradient
          var sse = 0.0;
          for (int j = 0; j < grad.Length; j++) grad[j] = 0;
          for (int i = 0; i < target.Length; i++) {
            var r = target[i] - fi_eval[i];
            sse += 0.5 * r * r;
            for (int j = 0; j < grad.Length; j++) {
              grad[j] -= r * jac_eval[i, j];
            }
          }
          if (double.IsNaN(sse)) return double.MaxValue;
          // average
          for (int j = 0; j < grad.Length; j++) { grad[j] /= target.Length; }
          return sse / target.Length;
        } else {
          var eval = new VectorEvaluator();
          var prediction = eval.Evaluate(preparedTree, problemData.Dataset, trainingRows);

          // calc sum of squared errors and gradient
          var sse = 0.0;
          for (int i = 0; i < target.Length; i++) {
            var r = target[i] - prediction[i];
            sse += 0.5 * r * r;
          }
          if (double.IsNaN(sse)) return double.MaxValue;
          // average
          return sse / target.Length;
        }

      }

      double calculate_constraint(uint dim, double[] curX, double[] grad, IntPtr data) {
        UpdateThetaValues(curX);
        var intervalEvaluator = new IntervalEvaluator();
        var constraintData = Marshal.PtrToStructure<ConstraintData>(data);

        if (grad != null) for (int j = 0; j < grad.Length; j++) grad[j] = 0; // clear grad

        var interval = intervalEvaluator.Evaluate(constraintData.Tree, dataIntervals, constraintData.ParameterNodes,
          out double[] lowerGradient, out double[] upperGradient);

        // we transformed this to a constraint c(x) <= 0, so only the upper bound is relevant for us
        if (grad != null) for (int j = 0; j < grad.Length; j++) { grad[j] = upperGradient[j]; }
        if (double.IsNaN(interval.UpperBound)) return double.MaxValue;
        else return interval.UpperBound;
      }

      var minVal = Math.Min(-1000.0, thetaValues.Min());
      var maxVal = Math.Max(1000.0, thetaValues.Max());
      var lb = Enumerable.Repeat(minVal, thetaValues.Count).ToArray();
      var up = Enumerable.Repeat(maxVal, thetaValues.Count).ToArray();
      IntPtr nlopt_opt = NLOpt.nlopt_create(GetAlgFromIdentifier(solver), (uint)thetaValues.Count); /* algorithm and dimensionality */

      NLOpt.nlopt_set_lower_bounds(nlopt_opt, lb);
      NLOpt.nlopt_set_upper_bounds(nlopt_opt, up);
      var calculateObjectiveDelegate = new NLOpt.nlopt_func(calculate_obj); // keep a reference to the delegate (see below)
      NLOpt.nlopt_set_min_objective(nlopt_opt, calculateObjectiveDelegate, IntPtr.Zero);


      var constraintDataPtr = new IntPtr[constraintTrees.Count];
      var calculateConstraintDelegates = new NLOpt.nlopt_func[constraintTrees.Count]; // make sure we keep a reference to the delegates (otherwise GC will free delegate objects see https://stackoverflow.com/questions/7302045/callback-delegates-being-collected#7302258)
      for (int i = 0; i < constraintTrees.Count; i++) {
        var constraintData = new ConstraintData() { Tree = constraintTrees[i], ParameterNodes = GetParameterNodes(constraintTrees[i], allThetaNodes) };
        constraintDataPtr[i] = Marshal.AllocHGlobal(Marshal.SizeOf<ConstraintData>());
        Marshal.StructureToPtr(constraintData, constraintDataPtr[i], fDeleteOld: false);
        calculateConstraintDelegates[i] = new NLOpt.nlopt_func(calculate_constraint);
        NLOpt.nlopt_add_inequality_constraint(nlopt_opt, calculateConstraintDelegates[i], constraintDataPtr[i], 1e-8);
      }

      NLOpt.nlopt_set_xtol_rel(nlopt_opt, 1e-4);
      NLOpt.nlopt_set_maxtime(nlopt_opt, 10.0); // 10 secs
      NLOpt.nlopt_set_maxeval(nlopt_opt, maxIterations);

      var x = thetaValues.ToArray();  /* initial guess */
      double minf = double.MaxValue; /* minimum objective value upon return */
      var res = NLOpt.nlopt_optimize(nlopt_opt, x, ref minf);
      if (res < 0) {
        throw new InvalidOperationException($"NLOpt failed {res} {NLOpt.nlopt_get_errmsg(nlopt_opt)}");
      } else {
        // update parameters in tree
        var pIdx = 0;
        // here we lose the two last parameters (for linear scaling)
        foreach (var node in tree.IterateNodesPostfix()) {
          if (node is ConstantTreeNode constTreeNode) {
            constTreeNode.Value = x[pIdx++];
          } else if (node is VariableTreeNode varTreeNode) {
            varTreeNode.Weight = x[pIdx++];
          }
        }
        // note: we keep the optimized constants even when the tree is worse.
        // assert that we lose the last two parameters 
        if (pIdx != x.Length - 2) throw new InvalidProgramException();
      }

      NLOpt.nlopt_destroy(nlopt_opt);
      for (int i = 0; i < constraintDataPtr.Length; i++)
        Marshal.FreeHGlobal(constraintDataPtr[i]);

      counter.FunctionEvaluations += NLOpt.nlopt_get_numevals(nlopt_opt);
      // counter.GradientEvaluations += NLOpt.nlopt_get; // TODO



      return Math.Min(minf, targetVariance);
    }

    private static NLOpt.nlopt_algorithm GetAlgFromIdentifier(string solver) {
      if (solver.Contains("MMA")) return NLOpt.nlopt_algorithm.NLOPT_LD_MMA;
      if (solver.Contains("COBYLA")) return NLOpt.nlopt_algorithm.NLOPT_LN_COBYLA;
      if (solver.Contains("CCSAQ")) return NLOpt.nlopt_algorithm.NLOPT_LD_CCSAQ;
      if (solver.Contains("ISRES")) return NLOpt.nlopt_algorithm.NLOPT_GN_ISRES;
      throw new ArgumentException($"Unknown solver {solver}");
    }

    private static ISymbolicExpressionTree CopyAndScaleTree(ISymbolicExpressionTree tree, double scalingFactor, double offset) {
      var m = (ISymbolicExpressionTree)tree.Clone();

      var add = MakeNode<Addition>(MakeNode<Multiplication>(m.Root.GetSubtree(0).GetSubtree(0), CreateConstant(scalingFactor)), CreateConstant(offset));
      m.Root.GetSubtree(0).RemoveSubtree(0);
      m.Root.GetSubtree(0).AddSubtree(add);
      return m;
    }

    #region helper
    private static ISymbolicExpressionTreeNode[] GetParameterNodes(ISymbolicExpressionTree tree, List<ConstantTreeNode>[] allNodes) {
      // TODO better solution necessary
      var treeConstNodes = tree.IterateNodesPostfix().OfType<ConstantTreeNode>().ToArray();
      var paramNodes = new ISymbolicExpressionTreeNode[allNodes.Length];
      for (int i = 0; i < paramNodes.Length; i++) {
        paramNodes[i] = allNodes[i].SingleOrDefault(n => treeConstNodes.Contains(n));
      }
      return paramNodes;
    }

    private static ISymbolicExpressionTree ReplaceVarWithConst(ISymbolicExpressionTree tree, List<string> thetaNames, List<double> thetaValues, List<ConstantTreeNode>[] thetaNodes) {
      var copy = (ISymbolicExpressionTree)tree.Clone();
      var nodes = copy.IterateNodesPostfix().ToList();
      for (int i = 0; i < nodes.Count; i++) {
        var n = nodes[i] as VariableTreeNode;
        if (n != null) {
          var thetaIdx = thetaNames.IndexOf(n.VariableName);
          if (thetaIdx >= 0) {
            var parent = n.Parent;
            if (thetaNodes[thetaIdx].Any()) {
              // HACK: REUSE CONSTANT TREE NODE IN SEVERAL TREES
              // we use this trick to allow autodiff over thetas when thetas occurr multiple times in the tree (e.g. in derived trees)
              var constNode = thetaNodes[thetaIdx].First();
              var childIdx = parent.IndexOfSubtree(n);
              parent.RemoveSubtree(childIdx);
              parent.InsertSubtree(childIdx, constNode);
            } else {
              var constNode = (ConstantTreeNode)CreateConstant(thetaValues[thetaIdx]);
              var childIdx = parent.IndexOfSubtree(n);
              parent.RemoveSubtree(childIdx);
              parent.InsertSubtree(childIdx, constNode);
              thetaNodes[thetaIdx].Add(constNode);
            }
          }
        }
      }
      return copy;
    }

    private static ISymbolicExpressionTree ReplaceConstWithVar(ISymbolicExpressionTree tree, out List<string> thetaNames, out List<double> thetaValues) {
      thetaNames = new List<string>();
      thetaValues = new List<double>();
      var copy = (ISymbolicExpressionTree)tree.Clone();
      var nodes = copy.IterateNodesPostfix().ToList();

      int n = 1;
      for (int i = 0; i < nodes.Count; ++i) {
        var node = nodes[i];
        if (node is ConstantTreeNode constantTreeNode) {
          var thetaVar = (VariableTreeNode)new Problems.DataAnalysis.Symbolic.Variable().CreateTreeNode();
          thetaVar.Weight = 1;
          thetaVar.VariableName = $"θ{n++}";

          thetaNames.Add(thetaVar.VariableName);
          thetaValues.Add(constantTreeNode.Value);

          var parent = constantTreeNode.Parent;
          if (parent != null) {
            var index = constantTreeNode.Parent.IndexOfSubtree(constantTreeNode);
            parent.RemoveSubtree(index);
            parent.InsertSubtree(index, thetaVar);
          }
        }
        if (node is VariableTreeNode varTreeNode) {
          var thetaVar = (VariableTreeNode)new Problems.DataAnalysis.Symbolic.Variable().CreateTreeNode();
          thetaVar.Weight = 1;
          thetaVar.VariableName = $"θ{n++}";

          thetaNames.Add(thetaVar.VariableName);
          thetaValues.Add(varTreeNode.Weight);

          var parent = varTreeNode.Parent;
          if (parent != null) {
            var index = varTreeNode.Parent.IndexOfSubtree(varTreeNode);
            parent.RemoveSubtree(index);
            var prodNode = MakeNode<Multiplication>();
            varTreeNode.Weight = 1.0;
            prodNode.AddSubtree(varTreeNode);
            prodNode.AddSubtree(thetaVar);
            parent.InsertSubtree(index, prodNode);
          }
        }
      }
      return copy;
    }

    private static ISymbolicExpressionTreeNode CreateConstant(double value) {
      var constantNode = (ConstantTreeNode)new Constant().CreateTreeNode();
      constantNode.Value = value;
      return constantNode;
    }

    private static ISymbolicExpressionTree Subtract(ISymbolicExpressionTree t, ISymbolicExpressionTreeNode b) {
      var sub = MakeNode<Subtraction>(t.Root.GetSubtree(0).GetSubtree(0), b);
      t.Root.GetSubtree(0).RemoveSubtree(0);
      t.Root.GetSubtree(0).InsertSubtree(0, sub);
      return t;
    }
    private static ISymbolicExpressionTree Subtract(ISymbolicExpressionTreeNode b, ISymbolicExpressionTree t) {
      var sub = MakeNode<Subtraction>(b, t.Root.GetSubtree(0).GetSubtree(0));
      t.Root.GetSubtree(0).RemoveSubtree(0);
      t.Root.GetSubtree(0).InsertSubtree(0, sub);
      return t;
    }

    private static ISymbolicExpressionTreeNode MakeNode<T>(params ISymbolicExpressionTreeNode[] fs) where T : ISymbol, new() {
      var node = new T().CreateTreeNode();
      foreach (var f in fs) node.AddSubtree(f);
      return node;
    }
    #endregion

    private static void UpdateConstants(ISymbolicExpressionTreeNode[] nodes, double[] constants) {
      if (nodes.Length != constants.Length) throw new InvalidOperationException();
      for (int i = 0; i < nodes.Length; i++) {
        if (nodes[i] is VariableTreeNode varNode) varNode.Weight = constants[i];
        else if (nodes[i] is ConstantTreeNode constNode) constNode.Value = constants[i];
      }
    }

    private static alglib.ndimensional_fvec CreateFunc(ISymbolicExpressionTree tree, VectorEvaluator eval, ISymbolicExpressionTreeNode[] parameterNodes, IDataset ds, string targetVar, int[] rows) {
      var y = ds.GetDoubleValues(targetVar, rows).ToArray();
      return (double[] c, double[] fi, object o) => {
        UpdateConstants(parameterNodes, c);
        var pred = eval.Evaluate(tree, ds, rows);
        for (int i = 0; i < fi.Length; i++)
          fi[i] = pred[i] - y[i];

        var counter = (EvaluationsCounter)o;
        counter.FunctionEvaluations++;
      };
    }

    private static alglib.ndimensional_jac CreateJac(ISymbolicExpressionTree tree, VectorAutoDiffEvaluator eval, ISymbolicExpressionTreeNode[] parameterNodes, IDataset ds, string targetVar, int[] rows) {
      var y = ds.GetDoubleValues(targetVar, rows).ToArray();
      return (double[] c, double[] fi, double[,] jac, object o) => {
        UpdateConstants(parameterNodes, c);
        eval.Evaluate(tree, ds, rows, parameterNodes, fi, jac);

        for (int i = 0; i < fi.Length; i++)
          fi[i] -= y[i];

        var counter = (EvaluationsCounter)o;
        counter.GradientEvaluations++;
      };
    }
    public static bool CanOptimizeConstants(ISymbolicExpressionTree tree) {
      return TreeToAutoDiffTermConverter.IsCompatible(tree);
    }
  }
}