7 Log loss.ipynb

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    "# Thinking in tensors, writing in PyTorch\n",
    "\n",
    "Hands-on training  by [Piotr Migdał](https://p.migdal.pl) (2019). Version for Uniwersytet Śląski.\n",
    "\n",
    "**WORK IN PROGRESS**\n",
    "\n",
    "\n",
    "## Log loss\n",
    "\n",
    "Multi-class logistic regression can be expressed as a shallow neural network consisting of one linear layer and a softmax activation function.\n",
    "\n",
    "For binary classification, we can use sigmoid (a.k.a. logistic function):\n",
    "\n",
    "$$ \\sigma(x) = \\frac{1}{1+\\exp(-x)} $$\n",
    "\n",
    "Softmax function transforms any vector into distribution vector (values in range (0., 1.) that sum up to 1.):\n",
    "$$\\text{softmax}(x_i) = \\frac{\\exp(x_i)}{\\sum_j \\exp(x_j)}$$\n",
    "\n",
    "We use a cross-entropy loss function:\n",
    "$$- \\sum_j p_{j, true} \\log(p_{j, pred})$$\n",
    "\n",
    "Note that we do not state explicitly the softmax function in the model class below. For details see [torch.nn.CrossEntropyLoss](https://pytorch.org/docs/stable/nn.html#torch.nn.CrossEntropyLoss).\n",
    "\n",
    "See also:\n",
    "\n",
    "* [Cross-entropy vs. mean-squared error loss](https://www.reddit.com/r/MachineLearning/comments/8im9eb/d_crossentropy_vs_meansquared_error_loss/)\n",
    "* [Understanding binary cross-entropy / log loss: a visual explanation](https://towardsdatascience.com/understanding-binary-cross-entropy-log-loss-a-visual-explanation-a3ac6025181a)\n",
    "* [Cross entropy](https://pandeykartikey.github.io/machine/learning/basics/2018/05/22/cross-entropy.html) - another explanation\n",
    "* [Softmax function](https://en.wikipedia.org/wiki/Softmax_function)\n",
    "* [Multiclass logistic regression](https://en.wikipedia.org/wiki/Multinomial_logistic_regression)"
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    "import numpy as np"
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