反向传播算法产生不好的结果

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的书来学习关于反向传播算法的要领和3Blue1Brown的youtube视频。

我写了一段时间才写完，自那时以来一直在调试，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。在过去的几天里，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，我很乐于指出问题。

背景知识：1）我没有使用任何矩阵乘法，也没有使用外部框架，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。每层的偏差是该层权重矩阵末尾的一列。

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] = ?(?(z[l + 1][i]))/?z[l + 1][i] = ??(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] = ?(?(z[l + 1][j]))/?z[l + 1][j] = ??(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][j];// 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);
}

整个NeuralNetwork以及其他所有内容都可以在此处找到。

编辑：对存储库进行许多重大更改后，以上链接可能不再起作用，但希望与答案无关。为了完整起见，这是指向已更改的存储库的功能链接。

Answer 1

固定的。问题是：我没有将像素输入除以 255。其他一切似乎都工作正常，我现在在第一个纪元上得到 +9000/10000。