Machine Learning & Gradient Descent

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🗂️ Course Assets (Google Drive)

🎯 Key Objectives

Understand empirical risk minimization.
Formalize Gradient Descent and convergence conditions.
Connect theoretical optimization to applied deep learning pipelines.

📐 Mathematical Formulation

Mean Squared Error (MSE) Loss

For a dataset of $N$ training points $(x_i, y_i)$ , the loss is defined as:

L(\theta) = \frac{1}{2N} \sum_{i=1}^N \left( h_\theta(x_i) - y_i \right)^2

Parameter Update Rule

In standard gradient descent with learning rate $\eta > 0$ :

\theta_{t+1} = \theta_t - \eta \nabla_\theta L(\theta_t)

Where the gradient vector is given by:

\nabla_\theta L(\theta) = \begin{bmatrix} \frac{\partial L}{\partial \theta_0} \\ \frac{\partial L}{\partial \theta_1} \\ \vdots \\ \frac{\partial L}{\partial \theta_d} \end{bmatrix}

💻 Python Implementation

import numpy as np

def gradient_descent(X, y, lr=0.01, epochs=1000):
    m, n = X.shape
    theta = np.zeros((n, 1))

    for epoch in range(epochs):
        predictions = X @ theta
        errors = predictions - y
        gradient = (1 / m) * (X.T @ errors)
        theta -= lr * gradient

    return theta

🎙️ Applied Deep Learning: Voice Synthesis & Checkpoints

Mathematical optimization and backpropagation directly power the neural models trained in the Voice ML Project:

Acoustic Modeling: Adapting pre-trained HuBERT representations to extract speaker latent features from reference clips.
Text-to-Semantic Synthesis: Autoregressive transformers trained to map phoneme sequences into semantic speech tokens.
Diffusion / VITS Vocoder: Inverting semantic tokens and mel-spectrogram representations into natural time-domain audio waveforms.
Checkpoint Archival: The synchronized 17.5 GB ml_models.zip archive contains fine-tuned checkpoints alongside UVR5 vocal separation models and Whisper ASR transcription models.