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ml:loss_functions

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ml:loss_functions [2023/10/27 21:07] – jmflanigml:loss_functions [2024/07/23 00:32] (current) – jmflanig
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     * Lots of different ways to write this loss function.  One way is minimize $L(\mathcal{D}) = -\sum_{i=1}^{N} log(p(y_i|x_i))$, where $p(y|x) = \frac{e^{score(x,y)}}{\sum_{y} e^{score(x,y)}}$, where $p(y|x) = \frac{e^{score(x,y)}}{\sum_{y} e^{score(x,y)}}$     * Lots of different ways to write this loss function.  One way is minimize $L(\mathcal{D}) = -\sum_{i=1}^{N} log(p(y_i|x_i))$, where $p(y|x) = \frac{e^{score(x,y)}}{\sum_{y} e^{score(x,y)}}$, where $p(y|x) = \frac{e^{score(x,y)}}{\sum_{y} e^{score(x,y)}}$
     * The cross-entropy version writes it as $L(\mathcal{D}) = -\sum_{i=1}^{N}\sum_{y} p(y|x_i) log(p_\theta(y|x_i))$, but usually we put in the empirical distribution $p(y|x_i) = I[y=y_i]$ which gives us the log-loss above.     * The cross-entropy version writes it as $L(\mathcal{D}) = -\sum_{i=1}^{N}\sum_{y} p(y|x_i) log(p_\theta(y|x_i))$, but usually we put in the empirical distribution $p(y|x_i) = I[y=y_i]$ which gives us the log-loss above.
 +    * Cross-entropy loss can be written as
 +\[
 +L(\theta,\mathcal{D}) = \sum_{(x_i,y_i)\in\mathcal{D}} \Big( -score_\theta(x_i,y_i) + log( \sum_{y \in \mathcal{Y}(x_i)} e^{score_\theta(x_i,y)} ) \Big)
 +\]
 +    * This is often call the Conditional Random Field (CRF) loss
     * The minimum of cross-entropy loss does not always exist, and does not exist if the data training data can be completely separated.  See for example, section 1.1 of [[https://arxiv.org/pdf/1804.09753.pdf|this paper]].     * The minimum of cross-entropy loss does not always exist, and does not exist if the data training data can be completely separated.  See for example, section 1.1 of [[https://arxiv.org/pdf/1804.09753.pdf|this paper]].
   * Perceptron loss \[   * Perceptron loss \[
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     * [[https://www.aclweb.org/anthology/N10-1112.pdf|Gimple & Smith 2010 - Softmax-Margin CRFs: Training Log-Linear Models with Cost Functions]]     * [[https://www.aclweb.org/anthology/N10-1112.pdf|Gimple & Smith 2010 - Softmax-Margin CRFs: Training Log-Linear Models with Cost Functions]]
     * [[https://arxiv.org/pdf/1612.02295.pdf|Large-Margin Softmax Loss for Convolutional Neural Networks]] L-Softmax.  Doesn't cite [[https://www.aclweb.org/anthology/N10-1112.pdf|Gimple & Smith]].  I suspect it may be different, but need to check.     * [[https://arxiv.org/pdf/1612.02295.pdf|Large-Margin Softmax Loss for Convolutional Neural Networks]] L-Softmax.  Doesn't cite [[https://www.aclweb.org/anthology/N10-1112.pdf|Gimple & Smith]].  I suspect it may be different, but need to check.
 +    * The softmax margin loss is obtained by replacing the max in the SVM loss with a softmax: \[
 +L(\theta,\mathcal{D}) = \sum_{(x_i,y_i)\in\mathcal{D}} \Big( -score_\theta(x_i,y_i) + log( \sum_{y \in \mathcal{Y}(x_i)} e^{score_\theta(x_i,y) + cost(y_i,y)} ) \Big)
 +\]
   * Risk\[   * Risk\[
 L(\theta,\mathcal{D}) = \sum_{(x_i,y_i)\in\mathcal{D}} \frac{\sum_{y\in\mathcal{Y}(x_i)} cost(y_i,y) e^{score_\theta(x_i,y_i)}}{\sum_{y\in\mathcal{Y}(x_i)} e^{score_\theta(x_i,y_i)}} L(\theta,\mathcal{D}) = \sum_{(x_i,y_i)\in\mathcal{D}} \frac{\sum_{y\in\mathcal{Y}(x_i)} cost(y_i,y) e^{score_\theta(x_i,y_i)}}{\sum_{y\in\mathcal{Y}(x_i)} e^{score_\theta(x_i,y_i)}}
ml/loss_functions.1698440878.txt.gz · Last modified: 2023/10/27 21:07 by jmflanig
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