Merge pull request #59 from deel-ai/joseba-theory-overview

Joseba theory overview

.pylintrc

@@ -16,6 +16,7 @@ disable=
     R0915, # allow too many statements
     R0917, # allow too many positional arguments
 
+
     W0105, # allow no effect string statement
     W0102, # allow dangerous default value []
     W0212, # allow access to protected member 

docs/source/theory_overview.rst

@@ -355,9 +355,30 @@ Regularized Adaptive Prediction Sets (RAPS)
 *******************************************
 .. _theory raps:
 
-Source: [Angelopoulos2021]_
+The RAPS algorithm introduced in [Angelopoulos2021]_ is a modification of the APS algorithm 
+that uses a regularization term in order to produce smaller and more stable prediction sets.
+Employing the same notations as for the APS algorithm above,
+the RAPS algorithm works in two stages:
+
+**Calibration**
+    #. For each example :math:`X_i` in the calibration data set, we compute the error :math:`R_i` as the probability mass needed for reaching the true label :math:`Y_i`, i.e. 
+
+        .. math::
+        
+            R_i=\widehat{\pi}_{(1)}+\cdots+\widehat{\pi}_{(k)} + \lambda(k-k_{reg}+1), 
+    
+        where :math:`(k)=Y_i`. The regularization term :math:`\lambda(k-k_{reg}+1)` is added to the APS error, where :math:`\lambda` and :math:`k_{reg}` are hyperparameters.
+
+    #. Stock all errors in a vector :math:`\mathcal{R}`.
+
+**Inference**
+    #. Compute the error margin :math:`\delta_{\alpha}` as the :math:`(1-\alpha)(1 + 1/n_{calib})`-th empirical quantile of :math:`\mathcal{R}`.
+    #. The prediction set for a test point :math:`X_{new}` is defined as :math:`\widehat{C}_{\alpha}(X_{new})=\big\lbrace (1),\dots,(k)\big\rbrace`, where
+
+        .. math::
+            k = \max\big\lbrace i : \widehat{\pi}_{(1)}+\cdots+\widehat{\pi}_{(i)} + \lambda(i-k_{reg}+1) \leq \delta_\alpha\big\rbrace + 1.
+
 
-TBC
 
 Conformal Anomaly Detection
 ---------------------------