For Your Information You Need To Implement Multiclass Multi

For Your Information You Need To Implement Multi Class Multi Nomial

For your information, you need to implement Multi-Class (multi-nomial) Logistic Regression using one-vs-all method in Assignment-2. Please see the lecture slides for more information on one-vs-all method. All three datasets of UCI Datasets have more than two classes, so datasets are applicable for Multi-Class Logistic Regression. Another reminder (as announced several times in the class): please do not use any library functions (other than Numpy and Pandas) for your implementation. Please let me know if you have any questions. Thank you.

Paper For Above instruction

The assignment requires the implementation of multi-class (multinomial) logistic regression using the one-vs-all classification strategy. This approach involves training multiple binary classifiers, each distinguishing one class from all the others, to effectively classify multi-class datasets. Given that the datasets from the UCI Repository contain more than two classes, they are suitable for applying this method.

Logistic regression is a fundamental machine learning algorithm used for classification tasks. In binary classification, it models the probability that an instance belongs to a particular class. Extending this to multi-class scenarios involves either using multinomial logistic regression directly or, as in this task, implementing a one-vs-all approach. The one-vs-all method simplifies the multi-class problem into multiple binary classification tasks, each trained to recognize a single class versus the rest. During prediction, the class with the highest probability across all classifiers is selected as the final class label.

The primary challenge in this assignment is implementing the algorithm from scratch, restricting the use of any external library functions outside of Numpy and Pandas. This restriction emphasizes understanding the underlying mathematics and computational procedures of logistic regression. The implementation involves creating functions for the sigmoid activation, cost calculation, gradient descent optimization, and prediction, all formulated using Numpy arrays for efficiency.

The implementation process can be summarized in the following steps: First, load and preprocess the dataset, including normalizing features and encoding class labels if necessary. Second, for each class in the dataset, create a binary target vector that marks the current class as 1 and all others as 0. Third, train a logistic regression classifier by minimizing the cost function via gradient descent, updating weights iteratively until convergence. Fourth, repeat this process for every class, storing the trained weights for each classifier. Finally, during prediction, compute the probabilities for each classifier and assign the class corresponding to the classifier with the highest predicted probability.

In terms of mathematical formulation, logistic regression uses the sigmoid function to model probabilities:

\( \sigma(z) = \frac{1}{1 + e^{-z}} \)

where \( z = Xw \), with \( X \) being the feature matrix and \( w \) the weight vector. The cost function to minimize is the logistic loss or cross-entropy loss:

\( J(w) = - \frac{1}{m} \sum_{i=1}^{m} \left[ y^{(i)} \log (h^{(i)}) + (1 - y^{(i)}) \log (1 - h^{(i)}) \right] \)

where \( h^{(i)} = \sigma(z^{(i)}) \). Gradient descent updates are performed using the derivative of this cost function with respect to the weights, enabling iterative optimization.

Furthermore, proper data handling and feature scaling improve the training process, ensuring stable convergence. The lack of external libraries for this purpose necessitates writing custom functions for norm calculation, matrix operations, and other mathematical procedures. Once training is completed, the model's accuracy should be evaluated on a held-out test set, and confusion matrices can be generated to analyze misclassification patterns.

In summary, this task deepens understanding of multi-class classification, the logistic regression algorithm, and the importance of implementation from fundamental principles. Successfully completing this project demonstrates proficiency in both theoretical concepts and practical coding skills in machine learning, especially within restrictions that require manual computation and algorithm design without reliance on pre-built functions.

References

  • Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.
  • Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press.
  • Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. MIT Press.
  • Ng, A. Y. (2012). Machine Learning Yearning. Stanford University.
  • Scikit-learn documentation. (2020). Multi-class and multi-label classification. Retrieved from https://scikit-learn.org/stable/
  • UCI Machine Learning Repository. (2023). Datasets. Retrieved from https://archive.ics.uci.edu/
  • Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning. Springer.
  • Zhang, H. (2004). The optimality of Naive Bayes. Proceedings of the 17th International Florida Artificial Intelligence Research Society Conference.
  • Andrew Ng. (2018). Machine Learning. Stanford University Course Notes.
  • Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of statistical software.

Related Assignments