Question

我正在尝试使用MNIST数据集通过前馈神经网络和反向传播来解决经典的手写数字识别问题。我正在使用Michael Nielsen's book学习基本知识，并使用3Blue1Brown's youtube video学习反向传播算法。

我前些时候写完了，并从那时开始进行调试，因为结果很糟糕。最好的情况下，网络可以在1个历元后识别出大约4000/10000个样本，并且该数目仅在随后的历元中下降，这使我相信反向传播算法存在一些问题。在最近的几天里，我一直沉迷于索引地狱，试图对其进行调试，但无法弄清楚问题出在哪里，我很感谢您指出问题的任何帮助。

背景知识：1）我没有使用任何矩阵乘法，也没有使用任何外部框架，而是使用for循环进行了所有操作，因为这就是我从视频中学到的方法。 2）与本书不同，我将权重和偏差都存储在同一数组中。每层的偏差在该层的权重矩阵的末尾是一列。

最后是代码，这是NeuralNetwork类的Backpropagate方法，在UpdateMiniBatch中调用，该方法本身在SGD中调用：

/// <summary>
/// Returns the partial derivative of the cost function on one sample with respect to every weight in the network.
/// </summary>
public List<double[,]> Backpropagate(ITrainingSample sample)
{
    // Forwards pass
    var (weightedInputs, activations) = GetWeightedInputsAndActivations(sample.Input);

    // The derivative with respect to the activation of the last layer is simple to compute: activation - expectedActivation
    var errors = activations.Last().Select((a, i) => a - sample.Output[i]).ToArray();

    // Backwards pass
    List<double[,]> delCostOverDelWeights = Weights.Select(x => new double[x.GetLength(0), x.GetLength(1)]).ToList();
    List<double[]> delCostOverDelActivations = Weights.Select(x => new double[x.GetLength(0)]).ToList();
    delCostOverDelActivations[delCostOverDelActivations.Count - 1] = errors;

    // Comment notation:
    // Cost function: C
    // Weight connecting the i-th neuron on the (l + 1)-th layer to the j-th neuron on the l-th layer: w[l][i, j]
    // Bias of the i-th neuron on the (l + 1)-th layer: b[l][i]
    // Activation of the i-th neuon on the l-th layer: a[l][i]
    // Weighted input of the i-th neuron on the l-th layer: z[l][i] // which doesn't make sense on layer 0, but is left for index convenience
    // Notice that weights, biases, delCostOverDelWeights and delCostOverDelActivation all start at layer 1 (the 0-th layer is irrelevant to their meanings) while activations and weightedInputs strat at the 0-th layer

    for (int l = Weights.Count - 1; l >= 0; l--)
    {
        //Calculate ∂C/∂w for the current layer:
        for (int i = 0; i < Weights[l].GetLength(0); i++)
            for (int j = 0; j < Weights[l].GetLength(1); j++)
                delCostOverDelWeights[l][i, j] = // ∂C/∂w[l][i, j]
                    delCostOverDelActivations[l][i] * // ∂C/∂a[l + 1][i]
                    SigmoidPrime(weightedInputs[l + 1][i]) * // ∂a[l + 1][i]/∂z[l + 1][i] = ∂(σ(z[l + 1][i]))/∂z[l + 1][i] = σ′(z[l + 1][i])
                    (j < Weights[l].GetLength(1) - 1 ? activations[l][j] : 1); // ∂z[l + 1][i]/∂w[l][i, j] = a[l][j] ||OR|| ∂z[l + 1][i]/∂b[l][i] = 1

        // Calculate ∂C/∂a for the previous layer(a[l]):
        if (l != 0)
            for (int i = 0; i < Weights[l - 1].GetLength(0); i++)
                for (int j = 0; j < Weights[l].GetLength(0); j++)
                    delCostOverDelActivations[l - 1][i] += // ∂C/∂a[l][i] = sum over j:
                        delCostOverDelActivations[l][j] * // ∂C/∂a[l + 1][j]
                        SigmoidPrime(weightedInputs[l + 1][j]) * // ∂a[l + 1][j]/∂z[l + 1][j] = ∂(σ(z[l + 1][j]))/∂z[l + 1][j] = σ′(z[l + 1][j])
                        Weights[l][j, i]; // ∂z[l + 1][j]/∂a[l][i] = w[l][j, i]
    }

    return delCostOverDelWeights;
}

GetWeightedInputsAndActivations：

public (List<double[]>, List<double[]>) GetWeightedInputsAndActivations(double[] input)
{
    List<double[]> activations = new List<double[]>() { input }.Concat(Weights.Select(x => new double[x.GetLength(0)])).ToList();
    List<double[]> weightedInputs = activations.Select(x => new double[x.Length]).ToList();

    for (int l = 0; l < Weights.Count; l++)
        for (int i = 0; i < Weights[l].GetLength(0); i++)
        {
            double value = 0;
            for (int j = 0; j < Weights[l].GetLength(1) - 1; j++)
                value += Weights[l][i, j] * activations[l][i];// weights
            weightedInputs[l + 1][i] = value + Weights[l][i, Weights[l].GetLength(1) - 1];// bias
            activations[l + 1][i] = Sigmoid(weightedInputs[l + 1][i]);
        }

    return (weightedInputs, activations);
}

出于完整性考虑，可以here找到整个NeuralNetwork以及其他所有内容。

Answer 1

已修复。问题是：我没有将像素输入除以255。其他所有内容似乎都可以正常运行，现在我在第一个时期获得+9000/10000。

Answer 2

有几种主要方法可通过使用归一化，正则化等方法来提高模型的准确性并减少损失。

反向传播算法产生不好的结果

2 个答案: