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.NET: Neural Network, Supervised Deep Machine Learning Example in C#

Supervised learning is a machine learning paradigm for problems where the available data consists of labeled examples, meaning that each data point contains features (covariates) and an associated label. The goal of supervised learning algorithms is learning a function that maps feature vectors (inputs) to labels (output), based on example input-output pairs.

This neural network code is available in a C++ library.

See unsupervised learning version. Also, see convolutional neural network example.

Learn about feedforward neural networks. Learn more about the lambda parameter.

Visit the playground for related.

This network supports both Categorical Cross-Entropy (CCE) and Sparse Categorical Cross-Entropy (SCCE). To support CCE, supply a one-hot vector, for SCCE, supply the index into the output layer — that's it! Oh, plus make sure to use a linear output neuron activation for SCCE, and SoftMax for CCE.

Download these files including how to train the network and the MNIST image files of hand-written digits with their labels: NEURALNETWORK.zip. Experiment with the number of neurons and layers. This example usage code requires SixLabors.ImageSharp (NuGet package).

The neural network code:

// NOTE: This network only supports forms of Categorical Cross-Entropy loss a.k.a. SoftMax loss for multi-class classification.
// NOTE: This network supports "Sparse Categorical Cross-Entropy" (SCCE) which requires a Linear final layer activation.
// NOTE: SCCE requires much less memory for networks with a large number of outputs, like NLP networks, than having to
// create a one-hot array for each output. SCCE networks are more efficient with resources.

// CceNeuralNetwork.cs
// Compatible with .NET Core 3.1 and later
using System;
using System.IO;
using System.Collections.Generic;
using System.Threading;

namespace ML
{
	namespace FeedForward
	{
		public class NeuralNetwork
		{
			public List<Matrix> Weights { get; set; }

			public List<Matrix> Biases { get; set; }

			public int LayerCount { get; set; }

			public NeuralNetwork()
			{
				Biases = new List<Matrix>();
				Weights = new List<Matrix>();
			}

			public NeuralNetwork(uint[] neuronCounts, IActivationMethods activationMethods, bool biases = true)
			{
				Biases = new List<Matrix>();
				if (biases)
					for (int i = 1; i < neuronCounts.Length; i++)
					{
						if (neuronCounts[i] == 0)
							throw new ArgumentException("Neuron count cannot be 0.");
						Biases.Add(new Matrix(neuronCounts[i], 1));
					}

				Weights = new List<Matrix>();
				for (int i = 0; i < neuronCounts.Length - 1; i++)
				{
					if (neuronCounts[i + 1] == 0 || neuronCounts[i] == 0)
						throw new ArgumentException("Neuron count cannot be 0.");
					Weights.Add(new Matrix(neuronCounts[i + 1], neuronCounts[i]));
				}

				LayerCount = neuronCounts.Length;

				activationMethods.Randomize(Weights, Biases);
			}

			// lambda is for the L2 regularization term, and should be a very small fraction (between zero and one) to help prevent overfitting and exploding gradients
			// at zero, it provides no regularization and risks exploding gradients, use a clipThreshold, such as 5.0
			// training might be slowed with multiple threads because the batch of training data of each thread is smaller and therefore has less to learn from; consider decreasing the number of threads as the epoch count increases and experimenting with the learning rate
			// learningRate can decrease by using an algorithm such as: 0.1 ^ (epoch / (float)epochCount) * initialLearningRate
			public void Train(List<Matrix> givenInputs, List<Matrix> desiredOutputs, IActivationMethods activationObject, float learningRate, float lambda, int threadCount = 0, float clipThreshold = 0f)
			{
				if (givenInputs.Count != desiredOutputs.Count)
					throw new ArgumentException("\"givenInputs\" count must match \"desiredOutputs\" count.");

				if (threadCount < 1)
					threadCount = Environment.ProcessorCount;
				int mini_batch_size = givenInputs.Count / threadCount;
				if (mini_batch_size > 0)
				{
					var bps = new List<BatchParams>();
					for (int x = 0; x < threadCount; x++)
					{
						var bp = new BatchParams()
						{
							network = this,
							givenInputs = givenInputs.GetRange(x * mini_batch_size, mini_batch_size),
							desiredOutputs = desiredOutputs.GetRange(x * mini_batch_size, mini_batch_size),
							local_weights = new List<Matrix>(),
							local_biases = new List<Matrix>(),
							activationObject = activationObject,
							learningRate = learningRate,
							lambda = lambda,
							delta_gradient_w = new Matrix[Weights.Count],
							delta_gradient_b = new Matrix[Biases.Count],
							clipThreshold = clipThreshold,
							thread = new Thread(TrainMiniBatch) { IsBackground = true },
							threadCount = threadCount
						};

						foreach (var weight in Weights)
							bp.local_weights.Add(new Matrix(weight));

						foreach (var bias in Biases)
							bp.local_biases.Add(new Matrix(bias));

						bps.Add(bp);
						bp.thread.Start(bp);
					}
					bool loop = true;
					while (loop)
					{
						loop = false;
						for (int i = 0; i < bps.Count; i++)
						{
							if (bps[i].thread.IsAlive)
							{
								Thread.Sleep(100);
								loop = true;
								break;
							}
						}
					}

					for (int x = 0; x < Weights.Count; x++)
					{
						var weights = new List<Matrix>();
						for (int t = 0; t < bps.Count; t++)
							weights.Add(bps[t].local_weights[x]);
						ParameterAveraging(Weights[x], weights);
					}

					for (int x = 0; x < Biases.Count; x++)
					{
						var biases = new List<Matrix>();
						for (int t = 0; t < bps.Count; t++)
							biases.Add(bps[t].local_biases[x]);
						ParameterAveraging(Biases[x], biases);
					}
				}
				for (int x = threadCount * mini_batch_size; x < givenInputs.Count; x++)
					Train(givenInputs[x], desiredOutputs[x], activationObject, learningRate, lambda, clipThreshold);
			}

			// lambda is for the L2 regularization term, and should be a very small fraction (between zero and one) to help prevent overfitting and exploding gradients
			// at zero, it provides no regularization and risks exploding gradients, use a clipThreshold, such as 5.0
			// learningRate can decrease by using an algorithm such as: 0.1 ^ (epoch / (float)epochCount) * initialLearningRate
			public void Train(Matrix givenInputs, Matrix desiredOutputs, IActivationMethods activationObject, float learningRate, float lambda, float clipThreshold = 0f)
			{
				var delta_gradient_w = new Matrix[Weights.Count];
				var delta_gradient_b = new Matrix[Biases.Count];

				BackPropagate(this, Weights, Biases, givenInputs, desiredOutputs, activationObject, delta_gradient_w, delta_gradient_b, clipThreshold);

				var new_weights = new List<Matrix>();
				var new_biases = new List<Matrix>();

				for (int i = 0; i < delta_gradient_w.Length; i++)
				{
					for (uint j = 0; j < delta_gradient_w[i].rows; j++)
						for (uint k = 0; k < delta_gradient_w[i].columns; k++)
						{
							float w = Weights[i].GetValue(j, k);
							float nw = delta_gradient_w[i].GetValue(j, k);
							delta_gradient_w[i].SetValue(j, k, (1 - learningRate * lambda) * w - learningRate * nw);
						}
					new_weights.Add(delta_gradient_w[i]);
				}

				for (int i = 0; i < delta_gradient_b.Length; i++)
				{
					for (uint j = 0; j < delta_gradient_b[i].rows; j++)
					{
						float b = Biases[i].GetValue(j, 0);
						float nb = delta_gradient_b[i].GetValue(j, 0);
						delta_gradient_b[i].SetValue(j, 0, b - learningRate * nb);
					}
					new_biases.Add(delta_gradient_b[i]);
				}

				Weights = new_weights;
				Biases = new_biases;

			}
			public uint TrueIndex(Matrix desiredOutputs)
			{
				uint index;
				if (desiredOutputs.rows == 1) // SCCE (Linear, must apply SoftMax)
					index = (uint)desiredOutputs.GetValue(0, 0);
				else
					index = Functions.GetIndexMax(desiredOutputs);
				return index;
			}

			public uint PredictedIndex(Matrix givenInputs, IActivationMethods activationObject)
			{
				Matrix ff = FeedForward(givenInputs, activationObject);
				return Functions.GetIndexMax(ff);
			}

			public double CalculateLoss(Matrix givenInputs, Matrix desiredOutputs, IActivationMethods activationObject, out uint predictedIndex, out uint trueIndex)
			{
				uint iPredicted, iTrue;
				Matrix ff = FeedForward(givenInputs, activationObject);
				if (desiredOutputs.rows == 1 && givenInputs.rows != desiredOutputs.rows) // Calculate SCCE (Linear)
				{
					Functions.SoftMax(ff);
					iPredicted = Functions.GetIndexMax(ff);
					iTrue = (uint)desiredOutputs.GetValue(0, 0); // holds the index of the true element
				}
				else
				{
					iPredicted = Functions.GetIndexMax(ff);
					iTrue = Functions.GetIndexMax(desiredOutputs);
				}
				double loss, a = (iPredicted == iTrue) ? ff.GetValue(iPredicted, 0) : 0;
				if (a == 0)
					loss = 1.0f;
				else if (a == 1)
					loss = 0f;
				else
					loss = -(float)Math.Log(a);
				predictedIndex = iPredicted;
				trueIndex = iTrue;
				return loss;
			}

			public Matrix FeedForward(Matrix givenInputs, IActivationMethods activationObject)
			{
				for (int i = 0; i < LayerCount - 1; i++)
				{
					Matrix? temp;
					Matrix.Multiply(Weights[i], givenInputs, out temp);
					if (temp == null)
						throw new ArgumentException("Cannot multiply matrices.");

					if (Biases.Count > 0) // add bias
					{
						for (uint j = 0; j < temp.rows; j++)
							for (uint k = 0; k < temp.columns; k++)
								temp.SetValue(j, k, temp.GetValue(j, k) + Biases[i].GetValue(j, 0));
					}

					if (i < LayerCount - 2)
						activationObject.ActivationMethod(temp);
					else
						activationObject.OutputActivationMethod(temp);
					givenInputs = temp;
				}
				return givenInputs;
			}

			#region Private_Decl
			private static void TrainMiniBatch(object? o)
			{
				if (o == null)
					return;

				BatchParams bp = (BatchParams)o;

				double invSqrtThreadCount = 1.0 / Math.Sqrt(bp.threadCount);

				for (int x = 0; x < bp.givenInputs.Count; x++)
				{
					BackPropagate(bp.network, bp.local_weights, bp.local_biases, bp.givenInputs[x], bp.desiredOutputs[x], bp.activationObject, bp.delta_gradient_w, bp.delta_gradient_b, bp.clipThreshold);


					for (int i = 0; i < bp.delta_gradient_w.Length; i++)
					{
						for (uint row = 0; row < bp.delta_gradient_w[i].rows; row++)
							for (uint column = 0; column < bp.delta_gradient_w[i].columns; column++)
							{
								float w = bp.local_weights[i].GetValue(row, column);
								float nw = bp.delta_gradient_w[i].GetValue(row, column);
								bp.local_weights[i].SetValue(row, column, (float)((1 - bp.learningRate * bp.lambda) * w - bp.learningRate * invSqrtThreadCount * nw));
							}
					}

					for (int i = 0; i < bp.delta_gradient_b.Length; i++)
					{
						for (uint row = 0; row < bp.delta_gradient_b[i].rows; row++)
						{
							float b = bp.local_biases[i].GetValue(row, 0);
							float nb = bp.delta_gradient_b[i].GetValue(row, 0);
							bp.local_biases[i].SetValue(row, 0, (float)(b - bp.learningRate * invSqrtThreadCount * nb));
						}
					}
				}
			}

			private static void ParameterAveraging(Matrix globalParameters, List<Matrix> localParametersOfThreads)
			{
				// Initialize a temporary matrix to hold the sum of local parameters
				double[,] sumOfLocalParams = new double[globalParameters.rows, globalParameters.columns];

				for (int threadId = 0; threadId < localParametersOfThreads.Count; threadId++)
					for (uint row = 0; row < localParametersOfThreads[threadId].rows; row++)
						for (uint column = 0; column < localParametersOfThreads[threadId].columns; column++)
							sumOfLocalParams[row, column] += localParametersOfThreads[threadId].GetValue(row, column);

				// Update the global parameter matrix using parameter averaging formula
				for (uint row = 0; row < globalParameters.rows; row++)
					for (uint column = 0; column < globalParameters.columns; column++)
						globalParameters.SetValue(row, column, (float)(sumOfLocalParams[row, column] / localParametersOfThreads.Count));
			}

			// uses Stochastic Gradient Descent
			private static void BackPropagate(NeuralNetwork network, List<Matrix> weights, List<Matrix> biases, Matrix givenInputs, Matrix desiredOutputs, IActivationMethods activationObject, Matrix[] delta_gradient_w, Matrix[] delta_gradient_b, float clipThreshold)
			{
				Matrix activation = givenInputs;

				List<Matrix> activations = new List<Matrix> { activation };

				List<Matrix> zs = new List<Matrix>();

				// feed forward
				for (int i = 0; i < network.LayerCount - 1; i++)
				{
					Matrix? z;
					Matrix.Multiply(weights[i], activation, out z);
					if (z == null)
						throw new ArgumentException("Cannot multiply matrices.");

					if (biases.Count > 0) // add bias
					{
						for (uint j = 0; j < z.rows; j++)
							for (uint k = 0; k < z.columns; k++)
								z.SetValue(j, k, z.GetValue(j, k) + biases[i].GetValue(j, 0));
					}

					zs.Add(new Matrix(z));

					if (i < network.LayerCount - 2)
						activationObject.ActivationMethod(z);
					else
						activationObject.OutputActivationMethod(z);
					activation = z;
					activations.Add(activation);
				}

				// Loss
				Matrix delta = new Matrix(activations[^1].rows, activations[^1].columns);
				Cost.Delta(activations[^1], desiredOutputs, delta);

				// backward pass
				if (delta_gradient_b.Length > 0)
					delta_gradient_b[^1] = new Matrix(delta);

				Matrix transposed = new Matrix(activations[^2]);
				transposed.Transpose();

				Matrix? temp;
				Matrix.Multiply(delta, transposed, out temp);
				if (temp == null)
					throw new ArgumentException("Cannot multiply matrices.");
				delta_gradient_w[^1] = temp;

				for (int i = 2; i < network.LayerCount; i++)
				{
					var t = network.LayerCount - i;
					transposed = new Matrix(weights[t]);
					transposed.Transpose();

					Matrix.Multiply(transposed, delta, out temp);
					if (temp == null)
						throw new ArgumentException("Cannot multiply matrices.");

					// multiply the derivative function on "temp"
					Matrix z = zs[^i];
					for (uint j = 0; j < temp.rows; j++)
						for (uint k = 0; k < temp.columns; k++)
							temp.SetValue(j, k, temp.GetValue(j, k) * activationObject.Derivative(z.GetValue(j, 0)));

					delta.Copy(temp);

					if (delta_gradient_b.Length > 0)
						delta_gradient_b[^i] = temp;

					t = network.LayerCount - i - 1;
					transposed = new Matrix(activations[t]);
					transposed.Transpose();
					Matrix.Multiply(delta, transposed, out temp);
					if (temp == null)
						throw new ArgumentException("Cannot multiply matrices.");

					delta_gradient_w[^i] = temp;
				}
				if (clipThreshold > 0f) // if greater than zero then will take care of exploding gradients
				{
					double gradients_norm, scale_factor;

					gradients_norm = 0;
					for (int i = 0; i < delta_gradient_b.Length; i++)
						for (uint j = 0; j < delta_gradient_b[i].rows; j++)
							for (uint k = 0; k < delta_gradient_b[i].columns; k++)
								gradients_norm += delta_gradient_b[i].GetValue(j, k) * delta_gradient_b[i].GetValue(j, k);

					gradients_norm = Math.Sqrt(gradients_norm);
					if (gradients_norm > clipThreshold)
					{
						scale_factor = clipThreshold / gradients_norm;
						for (uint i = 0; i < delta_gradient_b.Length; i++)
							for (uint j = 0; j < delta_gradient_b[i].rows; j++)
								for (uint k = 0; k < delta_gradient_b[i].columns; k++)
									delta_gradient_b[i].SetValue(j, k, (float)(delta_gradient_b[i].GetValue(j, k) * scale_factor));
					}

					// weights
					gradients_norm = 0;
					for (uint i = 0; i < delta_gradient_w.Length; i++)
						for (uint j = 0; j < delta_gradient_w[i].rows; j++)
							for (uint k = 0; k < delta_gradient_w[i].columns; k++)
								gradients_norm += delta_gradient_w[i].GetValue(j, k) * delta_gradient_w[i].GetValue(j, k);

					gradients_norm = Math.Sqrt(gradients_norm);
					if (gradients_norm > clipThreshold)
					{
						scale_factor = clipThreshold / gradients_norm;
						for (uint i = 0; i < delta_gradient_w.Length; i++)
							for (uint j = 0; j < delta_gradient_w[i].rows; j++)
								for (uint k = 0; k < delta_gradient_w[i].columns; k++)
									delta_gradient_w[i].SetValue(j, k, (float)(delta_gradient_w[i].GetValue(j, k) * scale_factor));
					}
				}
			}

			public string ToJson()
			{
				return System.Text.Json.JsonSerializer.Serialize(this);
			}

			public static NeuralNetwork? FromJson(Stream jsonStream)
			{
				return System.Text.Json.JsonSerializer.Deserialize<NeuralNetwork>(jsonStream);
			}

			public static NeuralNetwork? FromJson(string json)
			{
				return System.Text.Json.JsonSerializer.Deserialize<NeuralNetwork>(json);
			}
			#endregion

		}
		internal class BatchParams
		{
			public FeedForward.NeuralNetwork network;
			public List<Matrix> givenInputs, desiredOutputs, local_weights;
			public List<Matrix> local_biases;
			public IActivationMethods activationObject;
			public Matrix[] delta_gradient_w;
			public Matrix[] delta_gradient_b;
			public float learningRate;
			public float lambda;
			public float clipThreshold;
			public Thread? thread;
			public int threadCount;
		}
	}

	static class Cost
	{
		public static void Delta(Matrix outputs, Matrix desiredOutputs, Matrix deltaValue)
		{
			if (desiredOutputs.rows == 1 && deltaValue.rows != desiredOutputs.rows) // this is the same as "Sparse Categorical Cross-Entropy" (SCCE) and requires a Linear final activation; it requires that the calling program FeedForward(givenInputs, activationObject), SoftMax(feedForward) and then find the index of the maximum of the feedForward.
			{
				uint desiredIndex = (uint)desiredOutputs.GetValue(0, 0);
				for (uint i = 0; i < deltaValue.rows; i++)
					deltaValue.SetValue(i, 0, outputs.GetValue(i, 0) - (i == desiredIndex ? 1 : 0));
			}
			else // this is "Categorical Cross-Entropy" (CCE) and it requires a one-hot array for the desired outputs; it is not memory efficient.
			{
				for (uint i = 0; i < deltaValue.rows; i++)
					deltaValue.SetValue(i, 0, outputs.GetValue(i, 0) - desiredOutputs.GetValue(i, 0));
			}
		}
		public static uint Index(Matrix feedForwardOutputs)
		{
			// Sparse Categorical Cross-Entropy
			Functions.SoftMax(feedForwardOutputs);
			return Functions.GetIndexMax(feedForwardOutputs);
		}
		public static double Loss(Matrix feedForwardOutputs, Matrix desiredOutputs)
		{
			// Categorical Cross-Entropy, desiredOutputs should be a one-hot vector with the same number of rows (classes) as feedForwardOutputs
			uint iO = Functions.GetIndexMax(feedForwardOutputs), iL = Functions.GetIndexMax(desiredOutputs);
			double a = (iO == iL) ? feedForwardOutputs.GetValue(iO, 0) : 0;
			if (a == 0)
				return 1;
			if (a == 1)
				return 0;
			return -Math.Log(a);
		}
	}

}

The neural network common code:

// NeuralNetworkCommon.cs
// Compatible with .NET Core 3.1 and later
using System;
using System.IO;
using System.Collections.Generic;
using System.Threading;

namespace ML
{
	public interface IActivationMethods
	{
		public void ActivationMethod(Matrix outputs);
		public void OutputActivationMethod(Matrix outputs);
		public float Derivative(float input);
		//public void OutputDerivative(Matrix z, Matrix derivatives);
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases);
	}
	public class ActivationReLUSoftMax : IActivationMethods
	{
		public void ActivationMethod(Matrix outputs) // Rectified Linear Unit function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.ReLU(outputs.GetValue(i, j)));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			Functions.SoftMax(outputs);
		}
		public float Derivative(float input)
		{
			return Functions.ReLUPrime(input);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives) // this is the SoftMax derivative and this may not work with the Quadratic Cost function
		//{
		//	float d;
		//	for (uint i = 0; i < z.rows; i++)
		//	{
		//		d = z.GetValue(i, 0);
		//		derivatives.SetValue(i, 0, d * (1.0 - d));
		//	}
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeHeNormal(Weights, Biases);
		}
	}
	public class ActivationELUSoftMax : IActivationMethods
	{
		public readonly float alpha;
		public ActivationELUSoftMax(float alpha = 1.0f)
		{
			this.alpha = alpha;
		}
		public void ActivationMethod(Matrix outputs) // Exponential Linear Unit function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.ELU(outputs.GetValue(i, j), alpha));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			Functions.SoftMax(outputs);
		}
		public float Derivative(float input)
		{
			return Functions.ELUPrime(input, alpha);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives) // this is the SoftMax derivative and this may not work with the Quadratic Cost function
		//{
		//	float d;
		//	for (uint i = 0; i < z.rows; i++)
		//	{
		//		d = z.GetValue(i, 0);
		//		derivatives.SetValue(i, 0, d * (1.0 - d));
		//	}
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeHeNormal(Weights, Biases);
		}
	}
	public class ActivationLeakyReLUSoftMax : IActivationMethods
	{
		public readonly float alpha;
		public ActivationLeakyReLUSoftMax(float alpha = 0.1f)
		{
			this.alpha = alpha;
		}
		public void ActivationMethod(Matrix outputs) // Leaky Rectified Linear Unit function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.LeakyReLU(outputs.GetValue(i, j), alpha));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			Functions.SoftMax(outputs);
		}
		public float Derivative(float input)
		{
			return Functions.LeakyReLUPrime(input, alpha);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives) // this is the SoftMax derivative and this may not work with the Quadratic Cost function
		//{
		//	float d;
		//	for (uint i = 0; i < z.rows; i++)
		//	{
		//		d = z.GetValue(i, 0);
		//		derivatives.SetValue(i, 0, d * (1.0 - d));
		//	}
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeHeNormal(Weights, Biases);
		}
	}
	public class ActivationTanhSoftMax : IActivationMethods
	{
		public void ActivationMethod(Matrix outputs) // Tanh function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.Tanh(outputs.GetValue(i, j)));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			Functions.SoftMax(outputs);
		}
		public float Derivative(float input)
		{
			return Functions.TanhPrime(input);
		}
		public void OutputDerivative(Matrix z, Matrix derivatives) // this is the SoftMax derivative and this may not work with the Quadratic Cost function
		{
			float d;
			for (uint i = 0; i < z.rows; i++)
			{
				d = z.GetValue(i, 0);
				derivatives.SetValue(i, 0, d * (1.0f - d));
			}
		}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeGlorotXavier(Weights, Biases);
		}
	}
	public class ActivationSigmoidSoftMax : IActivationMethods
	{
		public void ActivationMethod(Matrix outputs) // Sigmoid function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.Sigmoid(outputs.GetValue(i, j)));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			Functions.SoftMax(outputs);
		}
		public float Derivative(float input)
		{
			return Functions.TanhPrime(input);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives) // this is the SoftMax derivative and this may not work with the Quadratic Cost function
		//{
		//	float d;
		//	for (uint i = 0; i < z.rows; i++)
		//	{
		//		d = z.GetValue(i, 0);
		//		derivatives.SetValue(i, 0, d * (1.0 - d));
		//	}
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeGlorotXavier(Weights, Biases);
		}
	}
	public class ActivationReLULinear : IActivationMethods
	{
		public void ActivationMethod(Matrix outputs) // Rectified Linear Unit function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.ReLU(outputs.GetValue(i, j)));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			// Linear
		}
		public float Derivative(float input)
		{
			return Functions.ReLUPrime(input);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives)
		//{
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeHeNormal(Weights, Biases);
		}
	}
	public class ActivationELULinear : IActivationMethods
	{
		public readonly float alpha;
		public ActivationELULinear(float alpha = 1.0f)
		{
			this.alpha = alpha;
		}
		public void ActivationMethod(Matrix outputs) // Exponential Linear Unit function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.ELU(outputs.GetValue(i, j), alpha));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			// Linear
		}
		public float Derivative(float input)
		{
			return Functions.ELUPrime(input, alpha);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives)
		//{
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeHeNormal(Weights, Biases);
		}
	}
	public class ActivationLeakyReLULinear : IActivationMethods
	{
		public readonly float alpha;
		public ActivationLeakyReLULinear(float alpha = 0.1f)
		{
			this.alpha = alpha;
		}
		public void ActivationMethod(Matrix outputs) // Leaky Rectified Linear Unit function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.LeakyReLU(outputs.GetValue(i, j), alpha));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			// Linear
		}
		public float Derivative(float input)
		{
			return Functions.LeakyReLUPrime(input, alpha);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives)
		//{
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeHeNormal(Weights, Biases);
		}
	}
	public class ActivationTanhLinear : IActivationMethods
	{
		public void ActivationMethod(Matrix outputs) // Tanh function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.Tanh(outputs.GetValue(i, j)));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			// Linear
		}
		public float Derivative(float input)
		{
			return Functions.TanhPrime(input);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives)
		//{
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeGlorotXavier(Weights, Biases);
		}
	}
	public class ActivationSigmoidLinear : IActivationMethods
	{
		public void ActivationMethod(Matrix outputs) // Sigmoid function applied to whole matrix
		{
			for (uint i = 0; i < outputs.rows; i++)
				for (uint j = 0; j < outputs.columns; j++)
					outputs.SetValue(i, j, Functions.Sigmoid(outputs.GetValue(i, j)));
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			// Linear
		}
		public float Derivative(float input)
		{
			return Functions.TanhPrime(input);
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives)
		//{
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeGlorotXavier(Weights, Biases);
		}
	}
	//public class ActivationSigmoid : IActivationMethods
	//{
	//	public void ActivationMethod(Matrix outputs) // Sigmoid function applied to whole matrix
	//	{
	//		for (uint i = 0; i < outputs.rows; i++)
	//			for (uint j = 0; j < outputs.columns; j++)
	//				outputs.SetValue(i, j, Functions.Sigmoid(outputs.GetValue(i, j)));
	//	}
	//	public void OutputActivationMethod(Matrix outputs)
	//	{
	//		ActivationMethod(outputs);
	//	}
	//	public float Derivative(float input)
	//	{
	//		return Functions.SigmoidPrime(input);
	//	}
	//	public void OutputDerivative(Matrix z, Matrix derivatives)
	//	{
	//		for (uint i = 0; i < z.rows; i++)
	//			derivatives.SetValue(i, 0, Functions.SigmoidPrime(z.GetValue(i, 0)));
	//	}
	//	public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
	//	{
	//		Randomization.RandomizeGlorotXavier(Weights, Biases);
	//	}
	//}
	public class ActivationLinear : IActivationMethods // this is used for the hidden layers of the Query Key Value Projections in a Transformer Neural Network
	{
		private float stdDev;
		public ActivationLinear(float stdDev = 0.02f)
		{
			this.stdDev = stdDev;
		}
		public void ActivationMethod(Matrix outputs)
		{
			// Linear
		}
		public void OutputActivationMethod(Matrix outputs)
		{
			// Linear
		}
		public float Derivative(float input)
		{
			return Functions.LinearPrime();
		}
		//public void OutputDerivative(Matrix z, Matrix derivatives)
		//{
		//}
		public void Randomize(List<Matrix> Weights, List<Matrix> Biases)
		{
			Randomization.RandomizeLinear(Weights, Biases, stdDev);
		}
	}

	public static class Functions
	{
		public static float Linear(float x) // Linear function
		{
			return x;
		}
		public static float LinearPrime() // derivative of Linear function (the line's slope)
		{
			return 1;
		}
		// alpha default might be 0.01, but this can be modified, bigger or smaller; TensorFlow uses 0.2 while Keras uses 0.3
		public static float LeakyReLU(float x, float alpha) // Rectified Linear Unit function (Leaky variant)
		{
			return x >= 0 ? x : (alpha * x);
		}
		public static float LeakyReLUPrime(float x, float alpha) // derivative of Leaky ReLU function
		{
			return x >= 0 ? 1 : alpha;
		}
		public static float ReLU(float x) // Rectified Linear Unit function
		{
			return x > 0 ? x : 0;
		}
		public static float ReLUPrime(float x) // derivative of ReLU function
		{
			return x > 0 ? 1 : 0;
		}
		public static float ELU(float x, float alpha) // Exponential Linear Unit function
		{
			return (float)(x >= 0 ? x : (alpha * (Math.Exp(x) - 1)));
		}
		public static float ELUPrime(float x, float alpha) // derivative of ELU function
		{
			return (float)(x >= 0 ? 1 : (alpha * Math.Exp(x)));
		}
		public static float Tanh(float x)
		{
			return (float)((Math.Exp(x) - Math.Exp(-x)) / (Math.Exp(x) + Math.Exp(-x)));
		}
		public static float TanhPrime(float x)
		{
			return (float)(1 - ((Math.Exp(x) - Math.Exp(-x)) / (Math.Exp(x) + Math.Exp(-x))) * ((Math.Exp(x) - Math.Exp(-x)) / (Math.Exp(x) + Math.Exp(-x)))); // this is simply: 1 - (tanh(x) * tanh(x))
		}
		public static float Sigmoid(float x)
		{
			return (float)(1.0 / (1 + Math.Exp(-x)));
		}
		public static float SigmoidPrime(float x) // derivative of Sigmoid function
		{
			return (float)((1.0 / (1 + Math.Exp(-x))) * (1.0 - (1.0 / (1 + Math.Exp(-x))))); // this is simply: Sigmoid(x) * (1.0 - Sigmoid(x))
		}
		public static void SoftMax(Matrix input)
		{
			float max = input.GetValue(0, 0);

			for (uint i = 1; i < input.rows; i++)
				max = Math.Max(max, input.GetValue(i, 0));

			float val, sum = 0;
			for (uint i = 0; i < input.rows; i++)
			{
				val = (float)Math.Exp(input.GetValue(i, 0) - max);
				input.SetValue(i, 0, val);
				sum += val;
			}

			for (uint i = 0; i < input.rows; i++)
			{
				val = input.GetValue(i, 0) / sum;
				input.SetValue(i, 0, val);
			}
		}
		public static uint GetIndexMax(Matrix m) // only pass 1 dimension matrices
		{
			uint index = 0;
			float a, max = m.GetValue(0, 0);
			for (uint i = 1; i < m.rows; i++)
			{
				a = m.GetValue(i, 0);
				if (a > max)
				{
					max = a;
					index = i;
				}
			}
			return index;
		}
	}

	internal static class Randomization // play with learning rate when switching between these randomizations
	{
		static Random rand = new Random();
		private static float GetDouble()
		{
			return (float)rand.NextDouble();
		}
		public static void RandomizeHeNormal(List<Matrix> Weights, List<Matrix> Biases)
		{
			for (int a = 0; a < Weights.Count; a++)
			{
				float init = (float)Math.Sqrt(2.0 / Weights[a].columns); // HeNormal: good for ReLU activation

				for (uint i = 0; i < Weights[a].rows; i++)
					for (uint j = 0; j < Weights[a].columns; j++)
						Weights[a].SetValue(i, j, GetDouble() * init - init * 0.5f);
			}
			for (int a = 0; a < Biases.Count; a++)
			{
				for (uint i = 0; i < Biases[a].rows; i++)
					Biases[a].SetValue(i, 0, GetDouble() * 0.5f - 0.25f);
			}
		}
		public static void RandomizeGlorotXavier(List<Matrix> Weights, List<Matrix> Biases)
		{
			for (int a = 0; a < Weights.Count; a++)
			{
				float init = (float)Math.Sqrt(6.0 / (Weights[a].columns + Weights[a].rows)); // GlorotXavier: good for Tanh/Sigmoid activation

				for (uint i = 0; i < Weights[a].rows; i++)
					for (uint j = 0; j < Weights[a].columns; j++)
						Weights[a].SetValue(i, j, GetDouble() * init - init * 0.5f);
			}
			for (int a = 0; a < Biases.Count; a++)
			{
				for (uint i = 0; i < Biases[a].rows; i++)
					Biases[a].SetValue(i, 0, GetDouble() * 0.5f - 0.25f);
			}
		}
		public static void RandomizeLinear(List<Matrix> Weights, List<Matrix> Biases, float stdDev = 0.02f)
		{
			for (int a = 0; a < Weights.Count; a++)
			{
				for (uint i = 0; i < Weights[a].rows; i++)
					for (uint j = 0; j < Weights[a].columns; j++)
						Weights[a].SetValue(i, j, GetDouble() * stdDev * (GetDouble() * 2 < 1.0f ? -1 : 1));
			}
			for (int a = 0; a < Biases.Count; a++)
			{
				for (uint i = 0; i < Biases[a].rows; i++)
					Biases[a].SetValue(i, 0, 0f);
			}
		}
	}
}

The matrix class code:

// NeuralNetworkMatrix.cs
// Compatible with .NET Core 3.1 and later
using System;
using System.IO;
using System.Collections.Generic;
using System.Threading;

namespace ML
{
	public class Matrix
	{
		public float[] data { get; set; }
		public uint rows { get; set; }
		public uint columns { get; set; }
		public Matrix()
		{
			data = new float[0];
		}
		public Matrix(uint rows, uint columns)
		{
			this.rows = rows;
			this.columns = columns;
			data = new float[rows * columns];
		}
		public Matrix(uint rows, uint columns, float[] values)
		{
			this.rows = rows;
			this.columns = columns;
			data = new float[rows * columns];
			Array.Copy(values, data, rows * columns);
		}
		public Matrix(Matrix m)
		{
			rows = m.rows;
			columns = m.columns;
			data = new float[rows * columns];
			Array.Copy(m.data, data, rows * columns);
		}
		public void Transpose()
		{
			var result = new Matrix(columns, rows);
			for (uint c = 0; c < columns; c++)
				for (uint r = 0; r < rows; r++)
					result.SetValue(c, r, GetValue(r, c));
			Copy(result);
		}
		public static void Add(Matrix M, float V)
		{
			for (uint i = 0; i < M.rows; i++)
				for (uint j = 0; j < M.columns; j++)
					M.SetValue(i, j, M.GetValue(i, j) + V);
		}
		public static void Multiply(Matrix A, Matrix B, out Matrix? C) // this is part of the dot product
		{
			C = null;
			if (A.columns == B.rows)
			{
				uint n = A.columns;
				C = new Matrix(A.rows, B.columns);
				for (uint i = 0; i < C.rows; i++)
					for (uint j = 0; j < C.columns; j++)
					{
						for (uint k = 0; k < n; k++)
							C.SetValue(i, j, C.GetValue(i, j) + A.GetValue(i, k) * B.GetValue(k, j));
					}
			}
		}
		public void Dropout(float dropoutRate) // apply a dropout to the matrix
		{
			var rand = new Random();
			for (uint i = 0; i < rows; i++)
				for (uint j = 0; j < columns; j++)
					if (rand.NextDouble() < dropoutRate)
						SetValue(i, j, 0f);
		}
		public float GetValue(uint row, uint column)
		{
			return data[row * columns + column];
		}
		public float SetValue(uint row, uint column, float value)
		{
			return data[row * columns + column] = value;
		}
		public void Copy(Matrix m)
		{
			if (rows * columns != m.rows * m.columns)
				data = new float[m.rows * m.columns];
			rows = m.rows;
			columns = m.columns;
			Array.Copy(m.data, data, rows * columns);
		}
	}

}

The confusion matrix code:

using System;
using System.Collections.Generic;

namespace ML
{
	public class Confusion
	{
        public List<Dictionary<string, string>> Samples { get; set; }
        public IEnumerable<string> Categories { get; set; }
        private Confusion()
		{
			Samples = new List<Dictionary<string, string>>();
			Categories = Array.Empty<string>();
		}
		public Confusion(IEnumerable<string> categories)
		{
			Samples = new List<Dictionary<string, string>>();
			Categories = categories;
		}
		public string ToJson()
		{
			return System.Text.Json.JsonSerializer.Serialize(this);
		}
        public void AddSample(string truth, string? label = null)
		{
			Samples.Add(new Dictionary<string, string>
			{
				{ "t", truth }, // t for true value
				{ "l", label ?? "?" } // l for label
			});
		}
		public void Reset()
		{
			Samples.Clear();
		}
		public string GetHtmlPage()
		{
			return $"<!DOCTYPE html><html lang='en'><head><meta charset='UTF-8' /><meta name='viewport' content='width=device-width, initial-scale=1.0' /><title>Confusion Matrix Chart</title><style>body{{margin:0;padding:0;}}</style></head><body><div id='confusion-container'></div><script 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'></script><script type='text/javascript'>var obj=JSON.parse('{ToJson()}');new ConfusionMatrixChart(document.getElementById('confusion-container'),obj.Samples,obj.Categories);</script></body></html>";
		}
	}
}

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