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--- ml:loss_functions [2023/10/25 21:36] – [Loss Functions] jmflanig
+++ ml:loss_functions [2024/07/23 00:32] (current) – jmflanig
@@ Line 1: / Line 1: @@
 ====== Loss Functions ======
+A function that is minimized during training (using gradient descent or Adam, for example) is called a loss function.
+==== Code Examples =====
+  * Hugging Face
+    * Custom loss in Hugging Face trainer: [[https://huggingface.co/docs/transformers/main_classes/trainer|Trainer]]
+==== List of Loss Functions ====
   * Cross-entropy (aka log loss, conditional log-likelihood, CRF loss)
     * 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.
+    * 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]].
   * Perceptron loss \[
@@ Line 14: / Line 26: @@
     * [[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.
-  * Ramp loss\[
+    * 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\[
 L(\theta,\mathcal{D}) = \sum_{(x_i,y_i)\in\mathcal{D}} \frac{\sum_{y\in\mathcal{Y}(x_i)} cost(y_i,y) e^{score_\theta(x_i,y_i)}}{\sum_{y\in\mathcal{Y}(x_i)} e^{score_\theta(x_i,y_i)}}
 \]
+\[
+L(\theta,\mathcal{D}) = \sum_{(x_i,y_i)\in\mathcal{D}} \sum_{y\in\mathcal{Y}(x_i)} cost(y_i,y) p_\theta(y|x_i)
+\]
+  * Ramp loss
   * Soft ramp loss
   * Infinite ramp loss