Abstract

This project demonstrates how fundamental neural network models such as linear regression, logistic regression, and multilayer perceptrons (MLPs) can be built from scratch using PyTorch. It focuses on learning foundational concepts in deep learning by implementing and experimenting with each component manually, without relying on high-level libraries. The project uses synthetic data and the MNIST dataset for regression and classification tasks, emphasizing model design, training dynamics, and performance evaluation.

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Introduction

Neural networks are at the core of modern machine learning, powering tasks from image recognition to language understanding. While libraries like TensorFlow and PyTorch offer powerful abstractions, building models from scratch provides deeper insight into core mechanics such as forward propagation, loss computation, and optimization.

In this notebook, we reimplement classic models—linear regression, logistic regression, and multilayer perceptrons—using PyTorch's tensor operations, targeting two datasets:

• Synthetic data for regression
• MNIST for classification

This step-by-step approach aims to reinforce practical understanding of how neural networks work under the hood.

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Methodology

The project is divided into three key sections:

1. Linear Regression (Synthetic Data)

• Synthetic data is generated using the linear equation:
  y = Xw + b + ε
  where X is input data, w is the weight vector, b is bias, and ε is random noise.
• Implements linear regression using PyTorch tensors with MSE loss and gradient descent.

2. Logistic Regression (MNIST)

• Treats logistic regression as a single-layer neural network using a softmax classifier.
• Applies it to the MNIST dataset to classify handwritten digits.
• Uses cross-entropy loss and stochastic gradient descent.

3. Multilayer Perceptron (MLP)

• Builds a two-layer fully connected network with ReLU activation.
• Evaluates performance on MNIST with batch-based training.
• Compares model improvements over single-layer architectures.

For each model, training includes:

• Manual parameter initialization
• Forward pass and loss computation
• Backpropagation using PyTorch's autograd
• Parameter updates via basic optimizers (SGD)

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Experiments

Linear Regression:

• Data synthesized from a known weight and bias vector.
• Visualized model predictions and tracked convergence.

Logistic Regression:

• Trained using a subset of MNIST to classify digits (0–9).
• Monitored training loss and accuracy over epochs.

MLP:

• Compared model performance with/without hidden layers.
• Used ReLU activation and softmax output on MNIST.
• Experimented with different learning rates and batch sizes.

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Results

• MLP outperformed logistic regression significantly due to non-linearity and additional depth.
• Learning rate tuning and batch training improved convergence and stability.
• Manual implementation confirmed alignment with theoretical gradients and optimization behavior.

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Conclusion

This project highlights the value of building neural networks from first principles. By implementing linear models, logistic classifiers, and MLPs manually, we gained insight into architecture, loss functions, and optimization techniques.

Key takeaways:

• Understanding fundamentals boosts debugging, optimization, and model design skills.
• Even basic architectures like MLPs can yield strong results on benchmark datasets like MNIST.
• PyTorch enables transparent, low-level access to tensor operations while supporting scalable training workflows.

This project serves as a learning bridge from theory to practice and is ideal for anyone aiming to strengthen their deep learning foundations.