# Method All the architectures were built using Keras. The loss is categorical cross entropy, adam optimizer and learning rate of 0.001 was used ``` {.python language="Python" style="mystyle"} model.add(Dense(128, activation='relu', input_dim=input_shape)) model.add(Dropout(0.5)) model.add(Dense(256, activation='relu')) model.add(Dense(256, activation='relu')) model.add(Dropout(0.5)) model.add(Dense(128, activation='relu')) model.add(Dropout(0.5)) model.add(Dense(64, activation='relu')) model.add(Dropout(0.5)) model.add(Dense(output_shape, activation='softmax')) ``` ``` {.python language="Python" style="mystyle"} model = Sequential() model.add(Dense(25, kernel_initializer=keras.initializers.glorot_normal(seed=0), kernel_regularizer=keras.regularizers.l2(1e-4))) model.add(Activation('relu')) model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=0))), kernel_regularizer=keras.regularizers.l2(1e-4))) model.add(BatchNormalization()) model.add(Activation('relu')) model.add(Dropout(0.3)) model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='softmax')) ``` ``` {.python language="Python" style="mystyle"} model = Sequential() model.add(Dense(20, kernel_initializer=keras.initializers.glorot_normal(seed=5), activation='relu')) model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=5), activation='relu')) model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=5)))#, activation='softmax')) model.add(Activation('softmax')) ``` ``` {.python language="Python" style="mystyle"} model = Sequential() model.add(Dense(35, kernel_initializer=keras.initializers.glorot_normal(seed=5))) model.add(Activation('relu')) model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=5))) model.add(BatchNormalization()) model.add(Activation('relu')) model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=5)))#, activation='softmax')) model.add(Dropout(0.5)) model.add(Activation('softmax')) ``` ``` {.python language="Python" style="mystyle"} model = Sequential() model.add(Dense(15, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) model.add(Dropout(0.2)) model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='softmax')) ``` ``` {.python language="Python" style="mystyle"} model = Sequential() model.add(Dense(40, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) model.add(Dense(25, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) model.add(Dropout(0.2)) model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='softmax')) ``` ::: small ``` {caption="We present below the top 5 boolean rule based features used by Logistic regression (with L1 penalty) on the Sky Survey Dataset along with their feature importances."} GBFL rank 1 feature 1.51 >= dirf1 >= 0.0 & 0.66 >= dirf2 >= 0.0 & 0.26 >= dirf3 >= 0.0 & 629.05 >= fiberid >= 419.36 & 0.80 >= redshift >= 0.0 & 233.35 >= ra >= 137.26 & 57481 >= mjd >= 42354.42 & 14.42 >= dec >= 0.0 & 1770.52 >= plate >= 0.0 GBFL rank 2 feature 2.53 >= dirf1 >= 0.50 & 0.49 >= dirf2 >= 0.0 & 0.35 >= dirf3 >= 0.0 & 2213.15 >= plate >= 442.63 & 14.42 >= dec >= 0.0 & 0.80 redshift >= 0.0 & 262.10 >= fiberid >= 52.42 & 164.72 >= ra >= 109.81 & 57481 >= mjd >= 42354.42 GBFL rank 3 feature 1.51 >= dirf1 >= 0.0 & 0.49 >= dirf2 >= 0.0 & 0.35 >= dirf3 >= 0.0 & 260.81 >= ra >= 233.35 & 0.80 >= redshift >= 0.0 & 524.21 >= fiberid >= 157.26 & 14.42 >= dec >= 0.0 & 1770.52 >= plate >= 0.0 & 57481 >= mjd >= 42354.42 GBFL rank 4 feature 2.53 >= dirf1 >= 0.50 & 0.49 >= dirf2 >= 0.0 & 0.44 >= dirf3 >= 0.089 & 576.63 >= fiberid >= 366.94 & 260.81 >= ra >= 233.35 & 14.42 >= dec >= 0.0 & 0.80 >= redshift >= 0.0 & 1770.52 >= plate >= 0.0 & 57481.0 >= mjd >= 42354.42 GBFL rank 5 feature 2.53 >= dirf1 >= 0.50 & 0.49 >= dirf2 >= 0.0 & 0.35 >= dirf3 >= 0.0 & 10.82 >= dec >= 0.0 & 52.42 >= fiberid >= 0.0 & 178.44 >= ra >= 123.54 & 0.80 >= redshift >= 0.0 & 1770.52 >= plate >= 0.0 & 57481.0 >= mjd >= 42354.42 ``` ::: ::: small ``` {caption="We present below the top 5 boolean rule based features used by the decision tree on the WDBC dataset ranked by their importance."} GBFL rank 1 feature 3575.86>=n2_area>=863.33 & 36.04>=n2_radius>=17.29 & n1_fractald<=0.01 & n0_concavity<=0.28 & n1_area<=363.87 & n1_compactness<=0.09 & n1_concavepts<=0.03 & n0_symmetry>=0.13 & n2_concavity<=0.83 & n2_fractald<=0.15 & n0_area<=2108.08 & n0_smoothness<=0.14 & n0_fractald>=0.04 & n0_concavepts<=0.16 & n2_texture<=43.28 & n2_smoothness<=0.19 & n0_perimeter<=188.5 & n2_symmetry>=0.15 & n2_concavepts<=0.27 & n0_texture>=9.71 & n0_compactness<=0.29 & n1_radius>=0.11 & n1_texture>=0.36 & n1_perimeter>=0.75 & n1_smoothness>=0 & n1_concavity>=0 & n1_symmetry>=0 & n2_perimeter<=251.2 & n0_radius>=10.5 & n2_compactness<=0.71 GBFL rank 2 feature n0_concavity>=0.07 & n2_concavepts>=0.13 & 7.22<=n1_area<=274.71 & 0<=n1_fractald<=0.01 & 185.2<=n2_area<=2219.6 & n0_perimeter<=140.26 & 0<=n1_compactness<=0.09 & 0<=n1_concavepts<=0.03 & 7.93<=n2_radius<=26.66 & 0.01<=n0_compactness<=0.29 & 0<=n0_concavity<=0.35 & 12.02<=n2_texture<=43.28 & 0.02<=n2_compactness<=0.88 & 0.05<=n2_fractald<=0.18 & 0.07<=n0_smoothness<=0.16 & 0.12<=n2_smoothness<=0.22 & 0.2<=n2_concavity<=1.25 & 0.09<=n2_concavepts<=0.27 & 21.06>=n0_radius>=6.98 & n0_texture>=9.71 & n0_concavepts>=0.0 & 1322.25>=n0_area>=143.5 & n0_fractald>=0.04 & 1.49>=n1_radius>=0.11 & 2.62>=n1_texture>=0.36 & 11.36>=n1_perimeter>=0.75 & n1_smoothness>=0 & n1_concavity>=0 & 0.04>=n1_symmetry>=0 & n2_perimeter>=50.41 & n2_symmetry>=0.15 & n0_symmetry>=0.13 GBFL rank 3 feature n0_concavity>=0.07 & n0_texture>=19.56 & n1_area<=274.71 & n1_compactness<=0.06 & n1_fractald<=0.01 & n2_compactness<=0.54 & n2_fractald<=0.13 & n0_compactness<=0.23 & n1_concavepts<=0.03 & n2_area<=2897.73 & n2_concavity<=0.83 & n0_area<=2108.08 & n0_smoothness<=0.14 & n0_concavity<=0.35 & n0_concavepts<=0.16 & n1_smoothness<=0.01 & n2_radius<=31.35 & n0_texture<=39.28 & n0_perimeter<=188.5 & n2_texture<=49.54 & n2_smoothness<=0.2226 & n0_symmetry>=0.106 & n0_fractald>=0.04 & n1_radius>=0.11 & n1_texture>=0.36 & n1_perimeter>=0.75 & n1_concavity>=0 & n1_symmetry>=0 & n2_perimeter>=50.41 & n2_symmetry>=0.15 & n0_radius>=10.50 & n2_concavepts>=0.04 GBFL rank 4 feature nucleus2_area >= 863.33 & nucleus2_radius >= 17.29 & nucleus1_compactness <= 0.06 & nucleus1_fractal_dim <= 0.01 & nucleus2_fractal_dim <= 0.13 & nucleus1_area <= 363.87 & nucleus1_concave_pts <= 0.03 & nucleus2_compactness <= 0.71 & nucleus0_smoothness <= 0.14 & nucleus0_compactness <= 0.29 & nucleus2_texture <= 43.28 & nucleus2_concavity <= 1.04 & nucleus0_perimeter <= 188.5 & nucleus0_area <= 2501.0 & nucleus0_concavity <= 0.42 & nucleus0_concave_pts <= 0.20 & nucleus2_radius <= 36.04 & nucleus2_area <= 4254.0 & nucleus2_smoothness <= 0.22 & nucleus0_texture >= 9.71 & nucleus0_symmetry >= 0.10 & nucleus0_fractal_dim >= 0.04 & nucleus1_radius >= 0.11 & nucleus1_texture >= 0.36 & nucleus1_perimeter >= 0.75 & nucleus1_smoothness >= 0 & nucleus1_concavity >= 0 & nucleus1_symmetry >= 0 & nucleus2_symmetry >= 0.15 & nucleus0_radius >= 14.02 & nucleus2_perimeter >= 117.33 & nucleus2_concave_pts >= 0.09 GBFL rank 5 feature nucleus2_concave_pts >= 0.13 & 7.22 <= nucleus1_area <= 274.71 & 0 <= nucleus1_fractal_dim <= 0.01 & 185.2 <= nucleus2_area <= 2219.6 & 0.01 <= nucleus0_compactness <= 0.23 & 0 <= nucleus0_concavity <= 0.28 & 0 <= nucleus0_concave_pts <= 0.13 & 0 <= nucleus1_smoothness <= 0.01 & 0 <= nucleus1_compactness <= 0.09 & 0 <= nucleus1_concave_pts <= 0.03 & 7.93 <= nucleus2_radius <= 26.66 & 0.02 <= nucleus2_compactness <= 0.71 & 0 <= nucleus2_concavity <= 0.83 & 0.05 <= nucleus2_fractal_dim <= 0.15 & 43.79 <= nucleus0_perimeter <= 164.38 & 0.05 <= nucleus0_smoothness <= 0.14 & 0.07 <= nucleus2_smoothness <= 0.19 & 18.27 <= nucleus2_texture <= 49.54 & 0.04 <= nucleus2_concave_pts <= 0.27 & nucleus0_radius >= 6.98 & nucleus0_texture >= 9.71 & nucleus0_area >= 143.5 & nucleus0_symmetry >= 0.10 & nucleus0_fractal_dim >= 0.04 & 1.49 >= nucleus1_radius >= 0.11 & nucleus1_texture >= 0.36 & nucleus1_perimeter >= 0.75 & nucleus1_concavity >= 0.0 & 0.04 >= nucleus1_symmetry >= 0 & nucleus2_perimeter >= 50.41 & 0.42 >= nucleus2_symmetry >= 0.15 ``` ::: ::: small ``` {caption="Top 5 boolean rule based features used by the decision tree on the Magic dataset ranked by their importance."} GBFL rank 1 feature fSize >= 2.937951724137931 & fSize <= 3.629234482758621 & fAlpha <= 6.206896551724138 & fLength >= 0.0 & fWidth >= 0.0 & fM3Long >= 0.0 & fAlpha >= 0.0 GBFL rank 2 feature fM3Long <= 0.0 & fAlpha <= 9.310344827586206 & fLength >= 0.0 & fWidth >= 0.0 & fAlpha >= 0.0 & fSize >= 2.073848275862069 GBFL rank 3 feature fSize >= 2.7651310344827587 & fAlpha <= 34.13793103448276 & fSize <= 3.456413793103448 & fWidth <= 15.207889655172414 & fLength >= 0.0 & fWidth >= 0.0 & fM3Long >= 0.0 GBFL rank 4 feature fWidth >= 30.415779310344828 & fSize >= 2.937951724137931 & fWidth <= 60.831558620689655 & fSize <= 3.629234482758621 & fM3Long <= 0.0 & fLength >= 0.0 & fAlpha >= 12.413793103448276 GBFL rank 5 features fM3Long >= 15.961793103448276 & fM3Long <= 47.88537931034483 & fWidth <= 7.603944827586207 & fLength <= 34.57003448275862 & fAlpha <= 21.724137931034484 & fLength >= 0.0 & fWidth >= 0.0 & fAlpha >= 9.310344827586206 & fSize >= 2.073848275862069 ``` ::: **Remark:** Although, the method in [@macem] does not really perform the constrained optimization but uses regularization like 'Elasticnet' penalty to impose sparsity, we will assume that our PPs and PNs are the result of these optimizations just for simplicity of exposition. The only difference is that the sparsity $k$ cannot be pre-determined but is typically a constant for many training samples in practice.