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Fitting a function to data

Published at
12/14/2024
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manonja
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Fitting a function to data

The first time I heard about “Fitting a function to data“ in the fastai course, my brain went 🤯 - it turns out, fitting a function to data is a fundamental aspect in training models.

In the context of neural networks, “fitting a function to data” refers to the process of training a model to learn the underlying patterns, relationships, or mappings between input data and the desired output.

Let’s delve into this fundamental aspect of neural nets, why it’s essential, and how it is achieved in practice.

Understanding the Concept:

At its core, a neural network is a complex mathematical function designed to transform input data into output predictions. The objective of training, or fitting, is to adjust the function's parameters—namely, weights and biases—so that the network’s predictions closely align with the actual target values. This process of fine-tuning is crucial for the model to accurately reflect the relationships within the data.

What is a Function?

First things first. What is a function?

A function defines a relationship between a set of inputs and a set of possible outputs. When we talk about functions in machine learning, we're referring to this relationship in a specific context:

Input (X): The data we provide to the model (e.g., images, text, numerical features).

Output (Y): The prediction or classification that the model generates (e.g., labels, regression values).

In a neural network, the function is represented as f(X,θ)f(X, \theta)f(X,θ), where:

XXX is the input data.

θ\thetaθ represents the parameters (weights and biases).

fff is the model that maps inputs to outputs.
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The goal of fitting the function is to adjust θ\thetaθ so that the function fff accurately captures the relationship between XXX and YYY.

Why Do We Fit a Function to Data?

Fitting a function to data is essential in neural networks for several reasons:

To Learn Patterns and Relationships

Discovering Hidden Structures: Data often contains complex patterns or relationships that aren’t immediately obvious. By fitting a function, a neural network can learn these structures and make sense of the data.

Feature Extraction: Neural networks have the ability to automatically learn and extract relevant features from raw data, reducing the need for extensive manual feature engineering.

To Make Predictions

Generalization: Once a neural network is trained, it should generalize well to unseen data, making accurate predictions based on the patterns it has learned.

Decision Making: The predictions made by a neural network can drive decision-making processes in various applications, such as self-driving cars, medical diagnosis, and financial forecasting.

To Better Optimization and Efficiency

Loss Minimization: The process of fitting involves minimizing a loss function, which quantifies the difference between predicted and actual values. The goal is to find parameter values that yield the lowest possible loss.

Efficient Representation: Neural networks can efficiently represent complex functions with relatively few parameters, making them well-suited for high-dimensional data.

How Does Fitting Work?

The process of fitting a function to data in a neural network involves several key steps that I tried to summarise below:

Step 1: Initialisation: The weights and biases of the network are initialised, often with small random values.

Step 2: Passing the data and first prediction: the Input data is passed through the network layers, with transformations applied based on the current parameter values to produce an output. The network then generates predictions for the given input data.

Step 3: Loss Calculation: We then apply a loss function (e.g., mean squared error, cross-entropy) to measure the difference between predicted and actual target values. This is a key step in fitting data into functions. We then can calculate Loss: the loss is computed for the given predictions, providing a metric for how well the network is performing.

Step 4: Back-propagation: Using backpropagation, the gradients of the loss with respect to each parameter are computed. This involves applying the chain rule to propagate errors backward through the network. This is called gradient calculation (best video on it I found here).

Step 5: Parameter Update: Parameters are updated using optimisation algorithms like stochastic gradient descent (SGD).

Step 6: Iteration, of course! The training process is repeated over multiple iterations (epochs) and different subsets of data (batches) to continuously refine the parameter values. The ultimate goal in the iterative process is to reach a convergence to a set of parameters that minimize the loss function, achieving a good fit to the training data.

Why Not Use Traditional Models like Linear Regression?

In the Fastai course, we use a simpler Linear Regression to fit a quadratic. This model fits simple functions to data, which prove to be insufficient for more complex tasks where:

Non-linearity: Data relationships are non-linear and require sophisticated functions to capture.

High Dimensionality: Inputs are high-dimensional (e.g., images, audio) and need complex architectures.

Complex Patterns: Patterns involve intricate interactions that simple models cannot capture effectively. Neural networks excel in these scenarios due to their ability to approximate complex functions.

What We Learned Today

  • Fitting a function to data is the process of training a neural network to learn the relationships between input data and the desired output.

  • A neural network functions is a complex mathematical model that adjusts its parameters (weights and biases) to minimise the difference between predictions and actual target values.

  • Fitting a function is essential for discovering hidden structures in data and making accurate predictions.

  • The process of fitting involves key steps: initialisation, forward pass, loss calculation, backpropagation, and iterative optimisation.

  • Traditional models like linear regression may not be sufficient for complex, non-linear, and high-dimensional data, where neural networks excel.

Stay curious!

Manon 🦉

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