Logistic Regression for Binary Classification With Core APIs

Content Overview Setup Load the data Preprocess the data Logistic regression Logistic regression fundamentals The log loss function The gradient descent update rule Train the model Performance evaluation Save the model Conclusion Setup Load the data Preprocess the data Logistic regression Logistic regression fundamentals The log loss function The gradient descent update rule Train the model Performance evaluation Save the model Conclusion This guide demonstrates how to use the TensorFlow Core low-level APIs to perform binary classification with logistic regression. It uses the Wisconsin Breast Cancer Dataset for tumor classification. Wisconsin Breast Cancer Dataset Logistic regression is one of the most popular algorithms for binary classification. Given a set of examples with features, the goal of logistic regression is to output values between 0 and 1, which can be interpreted as the probabilities of each example belonging to a particular class. Logistic regression Setup This tutorial uses pandas for reading a CSV file into a DataFrame, seaborn for plotting a pairwise relationship in a dataset, Scikit-learn for computing a confusion matrix, and matplotlib for creating visualizations. pandas DataFrame seaborn Scikit-learn matplotlib pip install -q seaborn pip install -q seaborn import tensorflow as tf import pandas as pd import matplotlib from matplotlib import pyplot as plt import seaborn as sns import sklearn.metrics as sk_metrics import tempfile import os # Preset matplotlib figure sizes. matplotlib.rcParams['figure.figsize'] = [9, 6] print(tf.__version__) # To make the results reproducible, set the random seed value. tf.random.set_seed(22) import tensorflow as tf import pandas as pd import matplotlib from matplotlib import pyplot as plt import seaborn as sns import sklearn.metrics as sk_metrics import tempfile import os # Preset matplotlib figure sizes. matplotlib.rcParams['figure.figsize'] = [9, 6] print(tf.__version__) # To make the results reproducible, set the random seed value. tf.random.set_seed(22) 2024-08-15 02:45:41.468739: E external/local_xla/xla/stream_executor/cuda/cuda_fft.cc:485] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered 2024-08-15 02:45:41.489749: E external/local_xla/xla/stream_executor/cuda/cuda_dnn.cc:8454] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered 2024-08-15 02:45:41.496228: E external/local_xla/xla/stream_executor/cuda/cuda_blas.cc:1452] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered 2.17.0 2024-08-15 02:45:41.468739: E external/local_xla/xla/stream_executor/cuda/cuda_fft.cc:485] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered 2024-08-15 02:45:41.489749: E external/local_xla/xla/stream_executor/cuda/cuda_dnn.cc:8454] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered 2024-08-15 02:45:41.496228: E external/local_xla/xla/stream_executor/cuda/cuda_blas.cc:1452] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered 2.17.0 Load the data Next, load the Wisconsin Breast Cancer Dataset from the UCI Machine Learning Repository. This dataset contains various features such as a tumor's radius, texture, and concavity. Wisconsin Breast Cancer Dataset UCI Machine Learning Repository url = 'https://archive.ics.uci.edu/ml/machine-learning-databases/breast-cancer-wisconsin/wdbc.data' features = ['radius', 'texture', 'perimeter', 'area', 'smoothness', 'compactness', 'concavity', 'concave_poinits', 'symmetry', 'fractal_dimension'] column_names = ['id', 'diagnosis'] for attr in ['mean', 'ste', 'largest']: for feature in features: column_names.append(feature + "_" + attr) url = 'https://archive.ics.uci.edu/ml/machine-learning-databases/breast-cancer-wisconsin/wdbc.data' features = ['radius', 'texture', 'perimeter', 'area', 'smoothness', 'compactness', 'concavity', 'concave_poinits', 'symmetry', 'fractal_dimension'] column_names = ['id', 'diagnosis'] for attr in ['mean', 'ste', 'largest']: for feature in features: column_names.append(feature + "_" + attr) Read the dataset into a pandas DataFrame using pandas.read_csv: DataFrame pandas.read_csv dataset = pd.read_csv(url, names=column_names) dataset = pd.read_csv(url, names=column_names) dataset.info() dataset.info() RangeIndex: 569 entries, 0 to 568 Data columns (total 32 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 id 569 non-null int64 1 diagnosis 569 non-null object 2 radius_mean 569 non-null float64 3 texture_mean 569 non-null float64 4 perimeter_mean 569 non-null float64 5 area_mean 569 non-null float64 6 smoothness_mean 569 non-null float64 7 compactness_mean 569 non-null float64 8 concavity_mean 569 non-null float64 9 concave_poinits_mean 569 non-null float64 10 symmetry_mean 569 non-null float64 11 fractal_dimension_mean 569 non-null float64 12 radius_ste 569 non-null float64 13 texture_ste 569 non-null float64 14 perimeter_ste 569 non-null float64 15 area_ste 569 non-null float64 16 smoothness_ste 569 non-null float64 17 compactness_ste 569 non-null float64 18 concavity_ste 569 non-null float64 19 concave_poinits_ste 569 non-null float64 20 symmetry_ste 569 non-null float64 21 fractal_dimension_ste 569 non-null float64 22 radius_largest 569 non-null float64 23 texture_largest 569 non-null float64 24 perimeter_largest 569 non-null float64 25 area_largest 569 non-null float64 26 smoothness_largest 569 non-null float64 27 compactness_largest 569 non-null float64 28 concavity_largest 569 non-null float64 29 concave_poinits_largest 569 non-null float64 30 symmetry_largest 569 non-null float64 31 fractal_dimension_largest 569 non-null float64 dtypes: float64(30), int64(1), object(1) memory usage: 142.4+ KB RangeIndex: 569 entries, 0 to 568 Data columns (total 32 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 id 569 non-null int64 1 diagnosis 569 non-null object 2 radius_mean 569 non-null float64 3 texture_mean 569 non-null float64 4 perimeter_mean 569 non-null float64 5 area_mean 569 non-null float64 6 smoothness_mean 569 non-null float64 7 compactness_mean 569 non-null float64 8 concavity_mean 569 non-null float64 9 concave_poinits_mean 569 non-null float64 10 symmetry_mean 569 non-null float64 11 fractal_dimension_mean 569 non-null float64 12 radius_ste 569 non-null float64 13 texture_ste 569 non-null float64 14 perimeter_ste 569 non-null float64 15 area_ste 569 non-null float64 16 smoothness_ste 569 non-null float64 17 compactness_ste 569 non-null float64 18 concavity_ste 569 non-null float64 19 concave_poinits_ste 569 non-null float64 20 symmetry_ste 569 non-null float64 21 fractal_dimension_ste 569 non-null float64 22 radius_largest 569 non-null float64 23 texture_largest 569 non-null float64 24 perimeter_largest 569 non-null float64 25 area_largest 569 non-null float64 26 smoothness_largest 569 non-null float64 27 compactness_largest 569 non-null float64 28 concavity_largest 569 non-null float64 29 concave_poinits_largest 569 non-null float64 30 symmetry_largest 569 non-null float64 31 fractal_dimension_largest 569 non-null float64 dtypes: float64(30), int64(1), object(1) memory usage: 142.4+ KB Display the first five rows: dataset.head() dataset.head() id diagnosis radius_mean texture_mean perimeter_mean area_mean smoothness_mean compactness_mean concavity_mean concave_poinits_mean ... radius_largest texture_largest perimeter_largest area_largest smoothness_largest compactness_largest concavity_largest concave_poinits_largest symmetry_largest fractal_dimension_largest id diagnosis radius_mean texture_mean perimeter_mean area_mean smoothness_mean compactness_mean concavity_mean concave_poinits_mean ... radius_largest texture_largest perimeter_largest area_largest smoothness_largest compactness_largest concavity_largest concave_poinits_largest symmetry_largest fractal_dimension_largest 0 842302 M 17.99 10.38 122.80 1001.0 0.11840 0.27760 0.3001 0.14710 ... 25.38 17.33 184.60 2019.0 0.1622 0.6656 0.7119 0.2654 0.4601 0.11890 1 842517 M 20.57 17.77 132.90 1326.0 0.08474 0.07864 0.0869 0.07017 ... 24.99 23.41 158.80 1956.0 0.1238 0.1866 0.2416 0.1860 0.2750 0.08902 2 84300903 M 19.69 21.25 130.00 1203.0 0.10960 0.15990 0.1974 0.12790 ... 23.57 25.53 152.50 1709.0 0.1444 0.4245 0.4504 0.2430 0.3613 0.08758 3 84348301 M 11.42 20.38 77.58 386.1 0.14250 0.28390 0.2414 0.10520 ... 14.91 26.50 98.87 567.7 0.2098 0.8663 0.6869 0.2575 0.6638 0.17300 4 84358402 M 20.29 14.34 135.10 1297.0 0.10030 0.13280 0.1980 0.10430 ... 22.54 16.67 152.20 1575.0 0.1374 0.2050 0.4000 0.1625 0.2364 0.07678 5 rows × 32 columns Split the dataset into training and test sets using pandas.DataFrame.sample, pandas.DataFrame.drop and pandas.DataFrame.iloc. Make sure to split the features from the target labels. The test set is used to evaluate your model's generalizability to unseen data. pandas.DataFrame.sample pandas.DataFrame.drop pandas.DataFrame.iloc train_dataset = dataset.sample(frac=0.75, random_state=1) train_dataset = dataset.sample(frac=0.75, random_state=1) len(train_dataset) len(train_dataset) 427 427 test_dataset = dataset.drop(train_dataset.index) test_dataset = dataset.drop(train_dataset.index) len(test_dataset) len(test_dataset) 142 142 # The `id` column can be dropped since each row is unique x_train, y_train = train_dataset.iloc[:, 2:], train_dataset.iloc[:, 1] x_test, y_test = test_dataset.iloc[:, 2:], test_dataset.iloc[:, 1] # The `id` column can be dropped since each row is unique x_train, y_train = train_dataset.iloc[:, 2:], train_dataset.iloc[:, 1] x_test, y_test = test_dataset.iloc[:, 2:], test_dataset.iloc[:, 1] Preprocess the data This dataset contains the mean, standard error, and largest values for each of the 10 tumor measurements collected per example. The "diagnosis" target column is a categorical variable with 'M' indicating a malignant tumor and 'B' indicating a benign tumor diagnosis. This column needs to be converted into a numerical binary format for model training. "diagnosis" 'M' 'B' The pandas.Series.map function is useful for mapping binary values to the categories. pandas.Series.map The dataset should also be converted to a tensor with the tf.convert_to_tensor function after the preprocessing is complete. tf.convert_to_tensor y_train, y_test = y_train.map({'B': 0, 'M': 1}), y_test.map({'B': 0, 'M': 1}) x_train, y_train = tf.convert_to_tensor(x_train, dtype=tf.float32), tf.convert_to_tensor(y_train, dtype=tf.float32) x_test, y_test = tf.convert_to_tensor(x_test, dtype=tf.float32), tf.convert_to_tensor(y_test, dtype=tf.float32) y_train, y_test = y_train.map({'B': 0, 'M': 1}), y_test.map({'B': 0, 'M': 1}) x_train, y_train = tf.convert_to_tensor(x_train, dtype=tf.float32), tf.convert_to_tensor(y_train, dtype=tf.float32) x_test, y_test = tf.convert_to_tensor(x_test, dtype=tf.float32), tf.convert_to_tensor(y_test, dtype=tf.float32) WARNING: All log messages before absl::InitializeLog() is called are written to STDERR I0000 00:00:1723689945.265757 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.269593 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.273290 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.276976 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.288712 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.292180 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.295550 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.299093 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.302584 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.306098 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.309484 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.312921 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.538105 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.540233 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.542239 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.544278 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.546323 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.548257 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.550168 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.552143 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.554591 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.556540 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.558447 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.560412 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.599852 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.601910 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.604061 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.606104 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.608094 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.610074 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.611985 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.613947 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.615903 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.618356 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.620668 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.623031 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 WARNING: All log messages before absl::InitializeLog() is called are written to STDERR I0000 00:00:1723689945.265757 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.269593 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.273290 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.276976 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.288712 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.292180 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.295550 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.299093 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.302584 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.306098 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.309484 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689945.312921 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.538105 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.540233 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.542239 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.544278 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.546323 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.548257 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.550168 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.552143 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.554591 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.556540 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.558447 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.560412 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.599852 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.601910 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.604061 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.606104 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.608094 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.610074 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.611985 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.613947 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.615903 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.618356 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.620668 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689946.623031 132290 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 Use seaborn.pairplot to review the joint distribution of a few pairs of mean-based features from the training set and observe how they relate to the target: seaborn.pairplot sns.pairplot(train_dataset.iloc[:, 1:6], hue = 'diagnosis', diag_kind='kde'); sns.pairplot(train_dataset.iloc[:, 1:6], hue = 'diagnosis', diag_kind='kde'); This pairplot demonstrates that certain features such as radius, perimeter and area are highly correlated. This is expected since the tumor radius is directly involved in the computation of both perimeter and area. Additionally, note that malignant diagnoses seem to be more right-skewed for many of the features. Make sure to also check the overall statistics. Note how each feature covers a vastly different range of values. train_dataset.describe().transpose()[:10] train_dataset.describe().transpose()[:10] count mean std min 25% 50% 75% max id 427.0 2.756014e+07 1.162735e+08 8670.00000 865427.500000 905539.00000 8.810829e+06 9.113205e+08 radius_mean 427.0 1.414331e+01 3.528717e+00 6.98100 11.695000 13.43000 1.594000e+01 2.811000e+01 texture_mean 427.0 1.924468e+01 4.113131e+00 10.38000 16.330000 18.84000 2.168000e+01 3.381000e+01 perimeter_mean 427.0 9.206759e+01 2.431431e+01 43.79000 75.235000 86.87000 1.060000e+02 1.885000e+02 area_mean 427.0 6.563190e+02 3.489106e+02 143.50000 420.050000 553.50000 7.908500e+02 2.499000e+03 smoothness_mean 427.0 9.633618e-02 1.436820e-02 0.05263 0.085850 0.09566 1.050000e-01 1.634000e-01 compactness_mean 427.0 1.036597e-01 5.351893e-02 0.02344 0.063515 0.09182 1.296500e-01 3.454000e-01 concavity_mean 427.0 8.833008e-02 7.965884e-02 0.00000 0.029570 0.05999 1.297500e-01 4.268000e-01 concave_poinits_mean 427.0 4.872688e-02 3.853594e-02 0.00000 0.019650 0.03390 7.409500e-02 2.012000e-01 symmetry_mean 427.0 1.804597e-01 2.637837e-02 0.12030 0.161700 0.17840 1.947000e-01 2.906000e-01 count mean std min 25% 50% 75% max id 427.0 2.756014e+07 1.162735e+08 8670.00000 865427.500000 905539.00000 8.810829e+06 9.113205e+08 radius_mean 427.0 1.414331e+01 3.528717e+00 6.98100 11.695000 13.43000 1.594000e+01 2.811000e+01 texture_mean 427.0 1.924468e+01 4.113131e+00 10.38000 16.330000 18.84000 2.168000e+01 3.381000e+01 perimeter_mean 427.0 9.206759e+01 2.431431e+01 43.79000 75.235000 86.87000 1.060000e+02 1.885000e+02 area_mean 427.0 6.563190e+02 3.489106e+02 143.50000 420.050000 553.50000 7.908500e+02 2.499000e+03 smoothness_mean 427.0 9.633618e-02 1.436820e-02 0.05263 0.085850 0.09566 1.050000e-01 1.634000e-01 compactness_mean 427.0 1.036597e-01 5.351893e-02 0.02344 0.063515 0.09182 1.296500e-01 3.454000e-01 concavity_mean 427.0 8.833008e-02 7.965884e-02 0.00000 0.029570 0.05999 1.297500e-01 4.268000e-01 concave_poinits_mean 427.0 4.872688e-02 3.853594e-02 0.00000 0.019650 0.03390 7.409500e-02 2.012000e-01 symmetry_mean 427.0 1.804597e-01 2.637837e-02 0.12030 0.161700 0.17840 1.947000e-01 2.906000e-01 count mean std min 25% 50% 75% max count count mean mean std std min min 25% 25% 50% 50% 75% 75% max max id 427.0 2.756014e+07 1.162735e+08 8670.00000 865427.500000 905539.00000 8.810829e+06 9.113205e+08 id id 427.0 427.0 2.756014e+07 2.756014e+07 1.162735e+08 1.162735e+08 8670.00000 8670.00000 865427.500000 865427.500000 905539.00000 905539.00000 8.810829e+06 8.810829e+06 9.113205e+08 9.113205e+08 radius_mean 427.0 1.414331e+01 3.528717e+00 6.98100 11.695000 13.43000 1.594000e+01 2.811000e+01 radius_mean radius_mean 427.0 427.0 1.414331e+01 1.414331e+01 3.528717e+00 3.528717e+00 6.98100 6.98100 11.695000 11.695000 13.43000 13.43000 1.594000e+01 1.594000e+01 2.811000e+01 2.811000e+01 texture_mean 427.0 1.924468e+01 4.113131e+00 10.38000 16.330000 18.84000 2.168000e+01 3.381000e+01 texture_mean texture_mean 427.0 427.0 1.924468e+01 1.924468e+01 4.113131e+00 4.113131e+00 10.38000 10.38000 16.330000 16.330000 18.84000 18.84000 2.168000e+01 2.168000e+01 3.381000e+01 3.381000e+01 perimeter_mean 427.0 9.206759e+01 2.431431e+01 43.79000 75.235000 86.87000 1.060000e+02 1.885000e+02 perimeter_mean perimeter_mean 427.0 427.0 9.206759e+01 9.206759e+01 2.431431e+01 2.431431e+01 43.79000 43.79000 75.235000 75.235000 86.87000 86.87000 1.060000e+02 1.060000e+02 1.885000e+02 1.885000e+02 area_mean 427.0 6.563190e+02 3.489106e+02 143.50000 420.050000 553.50000 7.908500e+02 2.499000e+03 area_mean area_mean 427.0 427.0 6.563190e+02 6.563190e+02 3.489106e+02 3.489106e+02 143.50000 143.50000 420.050000 420.050000 553.50000 553.50000 7.908500e+02 7.908500e+02 2.499000e+03 2.499000e+03 smoothness_mean 427.0 9.633618e-02 1.436820e-02 0.05263 0.085850 0.09566 1.050000e-01 1.634000e-01 smoothness_mean smoothness_mean 427.0 427.0 9.633618e-02 9.633618e-02 1.436820e-02 1.436820e-02 0.05263 0.05263 0.085850 0.085850 0.09566 0.09566 1.050000e-01 1.050000e-01 1.634000e-01 1.634000e-01 compactness_mean 427.0 1.036597e-01 5.351893e-02 0.02344 0.063515 0.09182 1.296500e-01 3.454000e-01 compactness_mean compactness_mean 427.0 427.0 1.036597e-01 1.036597e-01 5.351893e-02 5.351893e-02 0.02344 0.02344 0.063515 0.063515 0.09182 0.09182 1.296500e-01 1.296500e-01 3.454000e-01 3.454000e-01 concavity_mean 427.0 8.833008e-02 7.965884e-02 0.00000 0.029570 0.05999 1.297500e-01 4.268000e-01 concavity_mean concavity_mean 427.0 427.0 8.833008e-02 8.833008e-02 7.965884e-02 7.965884e-02 0.00000 0.00000 0.029570 0.029570 0.05999 0.05999 1.297500e-01 1.297500e-01 4.268000e-01 4.268000e-01 concave_poinits_mean 427.0 4.872688e-02 3.853594e-02 0.00000 0.019650 0.03390 7.409500e-02 2.012000e-01 concave_poinits_mean concave_poinits_mean 427.0 427.0 4.872688e-02 4.872688e-02 3.853594e-02 3.853594e-02 0.00000 0.00000 0.019650 0.019650 0.03390 0.03390 7.409500e-02 7.409500e-02 2.012000e-01 2.012000e-01 symmetry_mean 427.0 1.804597e-01 2.637837e-02 0.12030 0.161700 0.17840 1.947000e-01 2.906000e-01 symmetry_mean symmetry_mean 427.0 427.0 1.804597e-01 1.804597e-01 2.637837e-02 2.637837e-02 0.12030 0.12030 0.161700 0.161700 0.17840 0.17840 1.947000e-01 1.947000e-01 2.906000e-01 2.906000e-01 Given the inconsistent ranges, it is beneficial to standardize the data such that each feature has a zero mean and unit variance. This process is called normalization. normalization class Normalize(tf.Module): def __init__(self, x): # Initialize the mean and standard deviation for normalization self.mean = tf.Variable(tf.math.reduce_mean(x, axis=0)) self.std = tf.Variable(tf.math.reduce_std(x, axis=0)) def norm(self, x): # Normalize the input return (x - self.mean)/self.std def unnorm(self, x): # Unnormalize the input return (x * self.std) + self.mean norm_x = Normalize(x_train) x_train_norm, x_test_norm = norm_x.norm(x_train), norm_x.norm(x_test) class Normalize(tf.Module): def __init__(self, x): # Initialize the mean and standard deviation for normalization self.mean = tf.Variable(tf.math.reduce_mean(x, axis=0)) self.std = tf.Variable(tf.math.reduce_std(x, axis=0)) def norm(self, x): # Normalize the input return (x - self.mean)/self.std def unnorm(self, x): # Unnormalize the input return (x * self.std) + self.mean norm_x = Normalize(x_train) x_train_norm, x_test_norm = norm_x.norm(x_train), norm_x.norm(x_test) Logistic regression Before building a logistic regression model, it is crucial to understand the method's differences compared to traditional linear regression. Logistic regression fundamentals Linear regression returns a linear combination of its inputs; this output is unbounded. The output of a logistic regression is in the (0, 1) range. For each example, it represents the probability that the example belongs to the positive class. logistic regression (0, 1) positive Logistic regression maps the continuous outputs of traditional linear regression, (-∞, ∞), to probabilities, (0, 1). This transformation is also symmetric so that flipping the sign of the linear output results in the inverse of the original probability. (-∞, ∞) (0, 1) Let Y denote the probability of being in class 1 (the tumor is malignant). The desired mapping can be achieved by interpreting the linear regression output as the log odds ratio of being in class 1 as opposed to class 0: 1 log odds 1 0 ln⁡(Y1−Y)=wX+b ln⁡(Y1−Y)=wX+b By setting wX+b=z, this equation can then be solved for Y: Y=ez1+ez=11+e−z Y=ez1+ez=11+e−z The expression 11+e−z is known as the sigmoid function σ(z). Hence, the equation for logistic regression can be written as Y=σ(wX+b). sigmoid function The dataset in this tutorial deals with a high-dimensional feature matrix. Therefore, the above equation must be rewritten in a matrix vector form as follows: Y=σ(Xw+b) Y=σ(Xw+b) where: Ym×1: a target vector Xm×n: a feature matrix wn×1: a weight vector b: a bias σ: a sigmoid function applied to each element of the output vector Ym×1: a target vector Xm×n: a feature matrix wn×1: a weight vector b: a bias σ: a sigmoid function applied to each element of the output vector Start by visualizing the sigmoid function, which transforms the linear output, (-∞, ∞), to fall between 0 and 1. The sigmoid function is available in tf.math.sigmoid. (-∞, ∞) 0 1 tf.math.sigmoid x = tf.linspace(-10, 10, 500) x = tf.cast(x, tf.float32) f = lambda x : (1/20)*x + 0.6 plt.plot(x, tf.math.sigmoid(x)) plt.ylim((-0.1,1.1)) plt.title("Sigmoid function"); x = tf.linspace(-10, 10, 500) x = tf.cast(x, tf.float32) f = lambda x : (1/20)*x + 0.6 plt.plot(x, tf.math.sigmoid(x)) plt.ylim((-0.1,1.1)) plt.title("Sigmoid function"); The log loss function The log loss, or binary cross-entropy loss, is the ideal loss function for a binary classification problem with logistic regression. For each example, the log loss quantifies the similarity between a predicted probability and the example's true value. It is determined by the following equation: log loss L=−1m∑i=1myi⋅log⁡(y^i)+(1−yi)⋅log⁡(1−y^i) L=−1m∑i=1myi⋅log⁡(y^i)+(1−yi)⋅log⁡(1−y^i) where: y^: a vector of predicted probabilities y: a vector of true targets y^: a vector of predicted probabilities y: a vector of true targets You can use the tf.nn.sigmoid_cross_entropy_with_logits function to compute the log loss. This function automatically applies the sigmoid activation to the regression output: tf.nn.sigmoid_cross_entropy_with_logits def log_loss(y_pred, y): # Compute the log loss function ce = tf.nn.sigmoid_cross_entropy_with_logits(labels=y, logits=y_pred) return tf.reduce_mean(ce) def log_loss(y_pred, y): # Compute the log loss function ce = tf.nn.sigmoid_cross_entropy_with_logits(labels=y, logits=y_pred) return tf.reduce_mean(ce) The gradient descent update rule The TensorFlow Core APIs support automatic differentiation with tf.GradientTape. If you are curious about the mathematics behind the logistic regression gradient updates, here is a short explanation: tf.GradientTape gradient updates In the above equation for the log loss, recall that each y^i can be rewritten in terms of the inputs as σ(Xiw+b). The goal is to find a w∗ and b∗ that minimize the log loss: L=−1m∑i=1myi⋅log⁡(σ(Xiw+b))+(1−yi)⋅log⁡(1−σ(Xiw+b)) L=−1m∑i=1myi⋅log⁡(σ(Xiw+b))+(1−yi)⋅log⁡(1−σ(Xiw+b)) By taking the gradient L with respect to w, you get the following: ∂L∂w=1m(σ(Xw+b)−y)X ∂L∂w=1m(σ(Xw+b)−y)X By taking the gradient L with respect to b, you get the following: ∂L∂b=1m∑i=1mσ(Xiw+b)−yi ∂L∂b=1m∑i=1mσ(Xiw+b)−yi Now, build the logistic regression model. class LogisticRegression(tf.Module): def __init__(self): self.built = False def __call__(self, x, train=True): # Initialize the model parameters on the first call if not self.built: # Randomly generate the weights and the bias term rand_w = tf.random.uniform(shape=[x.shape[-1], 1], seed=22) rand_b = tf.random.uniform(shape=[], seed=22) self.w = tf.Variable(rand_w) self.b = tf.Variable(rand_b) self.built = True # Compute the model output z = tf.add(tf.matmul(x, self.w), self.b) z = tf.squeeze(z, axis=1) if train: return z return tf.sigmoid(z) class LogisticRegression(tf.Module): def __init__(self): self.built = False def __call__(self, x, train=True): # Initialize the model parameters on the first call if not self.built: # Randomly generate the weights and the bias term rand_w = tf.random.uniform(shape=[x.shape[-1], 1], seed=22) rand_b = tf.random.uniform(shape=[], seed=22) self.w = tf.Variable(rand_w) self.b = tf.Variable(rand_b) self.built = True # Compute the model output z = tf.add(tf.matmul(x, self.w), self.b) z = tf.squeeze(z, axis=1) if train: return z return tf.sigmoid(z) To validate, make sure the untrained model outputs values in the range of (0, 1) for a small subset of the training data. (0, 1) log_reg = LogisticRegression() log_reg = LogisticRegression() y_pred = log_reg(x_train_norm[:5], train=False) y_pred.numpy() y_pred = log_reg(x_train_norm[:5], train=False) y_pred.numpy() array([0.9994985 , 0.9978607 , 0.29620072, 0.01979049, 0.3314926 ], dtype=float32) array([0.9994985 , 0.9978607 , 0.29620072, 0.01979049, 0.3314926 ], dtype=float32) Next, write an accuracy function to calculate the proportion of correct classifications during training. In order to retrieve the classifications from the predicted probabilities, set a threshold for which all probabilities higher than the threshold belong to class 1. This is a configurable hyperparameter that can be set to 0.5 as a default. 1 0.5 def predict_class(y_pred, thresh=0.5): # Return a tensor with `1` if `y_pred` > `0.5`, and `0` otherwise return tf.cast(y_pred > thresh, tf.float32) def accuracy(y_pred, y): # Return the proportion of matches between `y_pred` and `y` y_pred = tf.math.sigmoid(y_pred) y_pred_class = predict_class(y_pred) check_equal = tf.cast(y_pred_class == y,tf.float32) acc_val = tf.reduce_mean(check_equal) return acc_val def predict_class(y_pred, thresh=0.5): # Return a tensor with `1` if `y_pred` > `0.5`, and `0` otherwise return tf.cast(y_pred > thresh, tf.float32) def accuracy(y_pred, y): # Return the proportion of matches between `y_pred` and `y` y_pred = tf.math.sigmoid(y_pred) y_pred_class = predict_class(y_pred) check_equal = tf.cast(y_pred_class == y,tf.float32) acc_val = tf.reduce_mean(check_equal) return acc_val Train the model Using mini-batches for training provides both memory efficiency and faster convergence. The tf.data.Dataset API has useful functions for batching and shuffling. The API enables you to build complex input pipelines from simple, reusable pieces. tf.data.Dataset batch_size = 64 train_dataset = tf.data.Dataset.from_tensor_slices((x_train_norm, y_train)) train_dataset = train_dataset.shuffle(buffer_size=x_train.shape[0]).batch(batch_size) test_dataset = tf.data.Dataset.from_tensor_slices((x_test_norm, y_test)) test_dataset = test_dataset.shuffle(buffer_size=x_test.shape[0]).batch(batch_size) batch_size = 64 train_dataset = tf.data.Dataset.from_tensor_slices((x_train_norm, y_train)) train_dataset = train_dataset.shuffle(buffer_size=x_train.shape[0]).batch(batch_size) test_dataset = tf.data.Dataset.from_tensor_slices((x_test_norm, y_test)) test_dataset = test_dataset.shuffle(buffer_size=x_test.shape[0]).batch(batch_size) Now write a training loop for the logistic regression model. The loop utilizes the log loss function and its gradients with respect to the input in order to iteratively update the model's parameters. # Set training parameters epochs = 200 learning_rate = 0.01 train_losses, test_losses = [], [] train_accs, test_accs = [], [] # Set up the training loop and begin training for epoch in range(epochs): batch_losses_train, batch_accs_train = [], [] batch_losses_test, batch_accs_test = [], [] # Iterate over the training data for x_batch, y_batch in train_dataset: with tf.GradientTape() as tape: y_pred_batch = log_reg(x_batch) batch_loss = log_loss(y_pred_batch, y_batch) batch_acc = accuracy(y_pred_batch, y_batch) # Update the parameters with respect to the gradient calculations grads = tape.gradient(batch_loss, log_reg.variables) for g,v in zip(grads, log_reg.variables): v.assign_sub(learning_rate * g) # Keep track of batch-level training performance batch_losses_train.append(batch_loss) batch_accs_train.append(batch_acc) # Iterate over the testing data for x_batch, y_batch in test_dataset: y_pred_batch = log_reg(x_batch) batch_loss = log_loss(y_pred_batch, y_batch) batch_acc = accuracy(y_pred_batch, y_batch) # Keep track of batch-level testing performance batch_losses_test.append(batch_loss) batch_accs_test.append(batch_acc) # Keep track of epoch-level model performance train_loss, train_acc = tf.reduce_mean(batch_losses_train), tf.reduce_mean(batch_accs_train) test_loss, test_acc = tf.reduce_mean(batch_losses_test), tf.reduce_mean(batch_accs_test) train_losses.append(train_loss) train_accs.append(train_acc) test_losses.append(test_loss) test_accs.append(test_acc) if epoch % 20 == 0: print(f"Epoch: {epoch}, Training log loss: {train_loss:.3f}") # Set training parameters epochs = 200 learning_rate = 0.01 train_losses, test_losses = [], [] train_accs, test_accs = [], [] # Set up the training loop and begin training for epoch in range(epochs): batch_losses_train, batch_accs_train = [], [] batch_losses_test, batch_accs_test = [], [] # Iterate over the training data for x_batch, y_batch in train_dataset: with tf.GradientTape() as tape: y_pred_batch = log_reg(x_batch) batch_loss = log_loss(y_pred_batch, y_batch) batch_acc = accuracy(y_pred_batch, y_batch) # Update the parameters with respect to the gradient calculations grads = tape.gradient(batch_loss, log_reg.variables) for g,v in zip(grads, log_reg.variables): v.assign_sub(learning_rate * g) # Keep track of batch-level training performance batch_losses_train.append(batch_loss) batch_accs_train.append(batch_acc) # Iterate over the testing data for x_batch, y_batch in test_dataset: y_pred_batch = log_reg(x_batch) batch_loss = log_loss(y_pred_batch, y_batch) batch_acc = accuracy(y_pred_batch, y_batch) # Keep track of batch-level testing performance batch_losses_test.append(batch_loss) batch_accs_test.append(batch_acc) # Keep track of epoch-level model performance train_loss, train_acc = tf.reduce_mean(batch_losses_train), tf.reduce_mean(batch_accs_train) test_loss, test_acc = tf.reduce_mean(batch_losses_test), tf.reduce_mean(batch_accs_test) train_losses.append(train_loss) train_accs.append(train_acc) test_losses.append(test_loss) test_accs.append(test_acc) if epoch % 20 == 0: print(f"Epoch: {epoch}, Training log loss: {train_loss:.3f}") Epoch: 0, Training log loss: 0.661 Epoch: 20, Training log loss: 0.418 Epoch: 40, Training log loss: 0.269 Epoch: 60, Training log loss: 0.178 Epoch: 80, Training log loss: 0.137 Epoch: 100, Training log loss: 0.116 Epoch: 120, Training log loss: 0.106 Epoch: 140, Training log loss: 0.096 Epoch: 160, Training log loss: 0.094 Epoch: 180, Training log loss: 0.089 Epoch: 0, Training log loss: 0.661 Epoch: 20, Training log loss: 0.418 Epoch: 40, Training log loss: 0.269 Epoch: 60, Training log loss: 0.178 Epoch: 80, Training log loss: 0.137 Epoch: 100, Training log loss: 0.116 Epoch: 120, Training log loss: 0.106 Epoch: 140, Training log loss: 0.096 Epoch: 160, Training log loss: 0.094 Epoch: 180, Training log loss: 0.089 Performance evaluation Observe the changes in your model's loss and accuracy over time. plt.plot(range(epochs), train_losses, label = "Training loss") plt.plot(range(epochs), test_losses, label = "Testing loss") plt.xlabel("Epoch") plt.ylabel("Log loss") plt.legend() plt.title("Log loss vs training iterations"); plt.plot(range(epochs), train_losses, label = "Training loss") plt.plot(range(epochs), test_losses, label = "Testing loss") plt.xlabel("Epoch") plt.ylabel("Log loss") plt.legend() plt.title("Log loss vs training iterations"); plt.plot(range(epochs), train_accs, label = "Training accuracy") plt.plot(range(epochs), test_accs, label = "Testing accuracy") plt.xlabel("Epoch") plt.ylabel("Accuracy (%)") plt.legend() plt.title("Accuracy vs training iterations"); plt.plot(range(epochs), train_accs, label = "Training accuracy") plt.plot(range(epochs), test_accs, label = "Testing accuracy") plt.xlabel("Epoch") plt.ylabel("Accuracy (%)") plt.legend() plt.title("Accuracy vs training iterations"); print(f"Final training log loss: {train_losses[-1]:.3f}") print(f"Final testing log Loss: {test_losses[-1]:.3f}") print(f"Final training log loss: {train_losses[-1]:.3f}") print(f"Final testing log Loss: {test_losses[-1]:.3f}") Final training log loss: 0.089 Final testing log Loss: 0.077 Final training log loss: 0.089 Final testing log Loss: 0.077 print(f"Final training accuracy: {train_accs[-1]:.3f}") print(f"Final testing accuracy: {test_accs[-1]:.3f}") print(f"Final training accuracy: {train_accs[-1]:.3f}") print(f"Final testing accuracy: {test_accs[-1]:.3f}") Final training accuracy: 0.968 Final testing accuracy: 0.979 Final training accuracy: 0.968 Final testing accuracy: 0.979 The model demonstrates a high accuracy and a low loss when it comes to classifying tumors in the training dataset and also generalizes well to the unseen test data. To go one step further, you can explore error rates that give more insight beyond the overall accuracy score. The two most popular error rates for binary classification problems are the false positive rate (FPR) and the false negative rate (FNR). For this problem, the FPR is the proportion of malignant tumor predictions amongst tumors that are actually benign. Conversely, the FNR is the proportion of benign tumor predictions among tumors that are actually malignant. Compute a confusion matrix using sklearn.metrics.confusion_matrix, which evaluates the accuracy of the classification, and use matplotlib to display the matrix: sklearn.metrics.confusion_matrix def show_confusion_matrix(y, y_classes, typ): # Compute the confusion matrix and normalize it plt.figure(figsize=(10,10)) confusion = sk_metrics.confusion_matrix(y.numpy(), y_classes.numpy()) confusion_normalized = confusion / confusion.sum(axis=1, keepdims=True) axis_labels = range(2) ax = sns.heatmap( confusion_normalized, xticklabels=axis_labels, yticklabels=axis_labels, cmap='Blues', annot=True, fmt='.4f', square=True) plt.title(f"Confusion matrix: {typ}") plt.ylabel("True label") plt.xlabel("Predicted label") y_pred_train, y_pred_test = log_reg(x_train_norm, train=False), log_reg(x_test_norm, train=False) train_classes, test_classes = predict_class(y_pred_train), predict_class(y_pred_test) def show_confusion_matrix(y, y_classes, typ): # Compute the confusion matrix and normalize it plt.figure(figsize=(10,10)) confusion = sk_metrics.confusion_matrix(y.numpy(), y_classes.numpy()) confusion_normalized = confusion / confusion.sum(axis=1, keepdims=True) axis_labels = range(2) ax = sns.heatmap( confusion_normalized, xticklabels=axis_labels, yticklabels=axis_labels, cmap='Blues', annot=True, fmt='.4f', square=True) plt.title(f"Confusion matrix: {typ}") plt.ylabel("True label") plt.xlabel("Predicted label") y_pred_train, y_pred_test = log_reg(x_train_norm, train=False), log_reg(x_test_norm, train=False) train_classes, test_classes = predict_class(y_pred_train), predict_class(y_pred_test) show_confusion_matrix(y_train, train_classes, 'Training') show_confusion_matrix(y_train, train_classes, 'Training') show_confusion_matrix(y_test, test_classes, 'Testing') show_confusion_matrix(y_test, test_classes, 'Testing') Observe the error rate measurements and interpret their significance in the context of this example. In many medical testing studies such as cancer detection, having a high false positive rate to ensure a low false negative rate is perfectly acceptable and in fact encouraged since the risk of missing a malignant tumor diagnosis (false negative) is a lot worse than misclassifying a benign tumor as malignant (false positive). In order to control for the FPR and FNR, try changing the threshold hyperparameter before classifying the probability predictions. A lower threshold increases the model's overall chances of making a malignant tumor classification. This inevitably increases the number of false positives and the FPR but it also helps to decrease the number of false negatives and the FNR. Save the model Start by making an export module that takes in raw data and performs the following operations: Normalization Probability prediction Class prediction Normalization Probability prediction Class prediction class ExportModule(tf.Module): def __init__(self, model, norm_x, class_pred): # Initialize pre- and post-processing functions self.model = model self.norm_x = norm_x self.class_pred = class_pred @tf.function(input_signature=[tf.TensorSpec(shape=[None, None], dtype=tf.float32)]) def __call__(self, x): # Run the `ExportModule` for new data points x = self.norm_x.norm(x) y = self.model(x, train=False) y = self.class_pred(y) return y class ExportModule(tf.Module): def __init__(self, model, norm_x, class_pred): # Initialize pre- and post-processing functions self.model = model self.norm_x = norm_x self.class_pred = class_pred @tf.function(input_signature=[tf.TensorSpec(shape=[None, None], dtype=tf.float32)]) def __call__(self, x): # Run the `ExportModule` for new data points x = self.norm_x.norm(x) y = self.model(x, train=False) y = self.class_pred(y) return y log_reg_export = ExportModule(model=log_reg, norm_x=norm_x, class_pred=predict_class) log_reg_export = ExportModule(model=log_reg, norm_x=norm_x, class_pred=predict_class) If you want to save the model at its current state, you can do so with the tf.saved_model.save function. To load a saved model and make predictions, use the tf.saved_model.load function. tf.saved_model.save tf.saved_model.load models = tempfile.mkdtemp() save_path = os.path.join(models, 'log_reg_export') tf.saved_model.save(log_reg_export, save_path) models = tempfile.mkdtemp() save_path = os.path.join(models, 'log_reg_export') tf.saved_model.save(log_reg_export, save_path) INFO:tensorflow:Assets written to: /tmpfs/tmp/tmp9k_sar52/log_reg_export/assets INFO:tensorflow:Assets written to: /tmpfs/tmp/tmp9k_sar52/log_reg_export/assets INFO:tensorflow:Assets written to: /tmpfs/tmp/tmp9k_sar52/log_reg_export/assets INFO:tensorflow:Assets written to: /tmpfs/tmp/tmp9k_sar52/log_reg_export/assets log_reg_loaded = tf.saved_model.load(save_path) test_preds = log_reg_loaded(x_test) test_preds[:10].numpy() log_reg_loaded = tf.saved_model.load(save_path) test_preds = log_reg_loaded(x_test) test_preds[:10].numpy() array([1., 1., 1., 1., 0., 1., 1., 1., 1., 1.], dtype=float32) array([1., 1., 1., 1., 0., 1., 1., 1., 1., 1.], dtype=float32) Conclusion This notebook introduced a few techniques to handle a logistic regression problem. Here are a few more tips that may help: The TensorFlow Core APIs can be used to build machine learning workflows with high levels of configurability Analyzing error rates is a great way to gain more insight about a classification model's performance beyond its overall accuracy score. Overfitting is another common problem for logistic regression models, though it wasn't a problem for this tutorial. Visit the Overfit and underfit tutorial for more help with this. The TensorFlow Core APIs can be used to build machine learning workflows with high levels of configurability TensorFlow Core APIs Analyzing error rates is a great way to gain more insight about a classification model's performance beyond its overall accuracy score. Overfitting is another common problem for logistic regression models, though it wasn't a problem for this tutorial. Visit the Overfit and underfit tutorial for more help with this. Overfit and underfit For more examples of using the TensorFlow Core APIs, check out the guide. If you want to learn more about loading and preparing data, see the tutorials on image data loading or CSV data loading. guide image data loading CSV data loading Originally published on the TensorFlow website, this article appears here under a new headline and is licensed under CC BY 4.0. Code samples shared under the Apache 2.0 License Originally published on the TensorFlow website, this article appears here under a new headline and is licensed under CC BY 4.0. Code samples shared under the Apache 2.0 License TensorFlow TensorFlow