The F=ma of Artificial Intelligence [Backpropagation]

Name: The F=ma of Artificial Intelligence [Backpropagation]
Uploaded: 2025-06-11T01:26:04.000Z
Duration: 1 h 32 s

The Evolution of Back Propagation in Neural Networks

Introduction to Back Propagation

In the early 1970s, Paul Worbos, a Harvard graduate student, discovered back propagation, a method for training multi-layer neural networks. He likened it to Newton's laws, suggesting its fundamental importance in understanding intelligence.

Initial Rejection and Subsequent Success

AI pioneer Marvin Minsky dismissed Worbos' discovery, arguing that back propagation could not learn complex tasks. Despite this skepticism, back propagation proved effective across various applications over the decades.

Impact on Modern AI Models

Today, nearly all modern AI models utilize back propagation for training. The video illustrates how Meta's Llama 3.2 model processes input data and updates its parameters through this algorithm.

Mechanism of Back Propagation

The algorithm adjusts the model's 1.2 billion parameters based on input text predictions. It identifies which connections within the model need updating to enhance accuracy in predicting subsequent tokens.

Importance of Attention Patterns

Back propagation effectively modifies weights associated with attention patterns in the model. For instance, it focuses on key tokens like "capital" and "France" when predicting "Paris," showcasing its ability to learn complex behaviors.

Understanding Loss Landscapes and Training Approaches

Transitioning from Visual to Mathematical Approaches

The series shifts from visual representations of loss landscapes to a mathematical approach for understanding back propagation. This transition aims to clarify how these algorithms function at a deeper level.

Simplifying Learning Problems

To illustrate back propagation’s mechanics, the discussion simplifies the learning problem by using GPS coordinates instead of text inputs. This allows for clearer demonstrations of how models predict outcomes based on numerical data.

Model Architecture Overview

A smaller model is introduced that predicts cities based on longitude coordinates using three neurons—one for each city (Paris, Madrid, Berlin). Each neuron operates as a simple linear equation involving weights and biases.

Understanding the Learning Algorithm

Adjusting Model Parameters

The learning algorithm's role is to adjust the parameters m and b to solve the overall task, with outputs labeled as h instead of y .

Outputs from neurons can be passed into various functions; final model outputs Y hat must represent probabilities between 0 and 1.

Softmax Function

The softmax function transforms neuron outputs into probabilities by exponentiating each output and normalizing them.

For example, if neuron outputs are 1, 2, and 1, softmax assigns a probability of 58% to Paris. If values change to 1, 10, and 1, Paris gets a probability of 99.98%.

Softmax amplifies differences in neuron outputs but is complex; calculus simplifies its differentiation.

Initial Model Predictions

Before training, model weights are initialized randomly; starting values for slope parameters are set (e.g., m_1 = 1 , m_2 = 0 , m_3 = -1 ).

An example prediction using coordinates leads to an incorrect high probability for Madrid (0.91).

Measuring Model Performance

Cross entropy loss is used as a performance metric: it calculates the negative logarithm of predicted probabilities.

For instance, if the predicted probability for Paris is only 8.6%, cross entropy loss equals approximately 2.45.

Gradient Descent Process

To improve predictions, adjustments are made to parameters based on computed slopes relative to loss.

By plotting loss against parameter changes (like increasing m_2 ), insights about improving model performance can be gained.

Optimizing Parameter Updates

Numerical Estimates vs Exact Solutions

While numerical estimates for slopes can guide updates in gradient descent, they are computationally intensive and potentially inaccurate.

Historical Context of Backpropagation

In the late '50s at Stanford, early neural network training relied on numerical slope estimates until backpropagation was discovered by Bernard Widrow and Ted Hoff.

Calculus Application in Backpropagation

Backpropagation efficiently computes slopes using calculus principles; this involves partial derivatives concerning model parameters like M_2 .

Loss Function Derivation

The relationship between cross entropy loss and model output probabilities allows for deriving complete equations linking inputs with all parameters.

Understanding Backpropagation in Neural Networks

The Chain Rule and Its Application

The process of backpropagation is complex but can be simplified by treating each layer of a neural network independently, allowing for efficient calculation of derivatives.

A simple example illustrates this: two compute blocks where the first maps input x to output y with y = 2x , and the second maps y to z with z = 4y . This leads to the overall equation z = 8x .

Instead of combining equations directly, we can compute derivatives separately for each block. For instance, the derivative from the first block is dy/dx = 2 , and from the second block is dz/dy = 4 . Thus, using the chain rule gives us dz/dx = dy/dx * dz/dy = 8 .

Simplifying Derivatives in Neural Networks

In neural networks, we apply similar principles by breaking down derivatives into manageable parts. Specifically, we separate out partial derivatives related to linear models and loss functions.

The logarithm from cross-entropy loss and exponentials from softmax operations effectively cancel out during calculations, simplifying our results significantly.

The resulting partial derivative for our model's output probabilities becomes straightforward: it equals dL/dh = y_hat - y , where y_hat represents predicted probabilities.

One-Hot Encoding and Its Implications

Using one-hot encoding for ground truth labels allows us to define vectors such that when predicting Paris as correct, we have values like y_1 = 0, y_2 = 1, and so forth.

Calculating partial derivatives reveals insights about how changes in outputs affect loss; e.g., increasing certain outputs will either increase or decrease loss based on their respective values.

Understanding Model Parameters' Impact

To derive further insights into model training, we need to understand how intermediate outputs relate to model parameters through equations like h_2 = m_2 * x + b_2.

Here, considering parameters as variables rather than constants helps clarify how learning occurs—specifically how changes in parameters influence neuron outputs based on input values.

Finalizing Gradient Computation

By replacing terms in our expressions with derived results (e.g., substituting for slopes), we can compute complete expressions for gradients needed during training.

For example, calculating gradient values shows that if parameter adjustments are made (like increasing parameter m2), expected changes in loss can be quantified accurately.

Gradient Descent and Back Propagation in Model Training

Understanding Gradient Descent

To minimize loss in model training, parameters are adjusted opposite to the gradient direction. Specifically, reducing M1 and increasing M2 helps decrease loss.

The process of gradient descent is visualized as moving downhill on a loss landscape. The learning rate, a scaling factor for gradients, controls the step size taken during this process.

Gradients provide local slope information; however, loss landscapes can be complex with rapidly changing slopes. This complexity makes simple downhill visualization incomplete.

Visualizing Learning Process

Gradients will be visualized as bars around connections in the model, where thicker bars indicate larger gradients. A heat map will show model outputs overlaid on a geographical map.

Initial training points are plotted on the map with probabilities for cities represented by colors: cyan for Madrid, yellow for Paris, and green for Berlin.

For an initial input from Paris, high output probability is incorrectly assigned to Madrid. This results in significant gradients affecting M1 and M2 parameters.

Adjusting Model Parameters

As training progresses through multiple steps (about 40), the model begins to correctly classify cities like Madrid and Berlin while adjusting its predictions for Paris.

Gradients decrease as errors reduce; thus, adjustments become smaller as the model learns more accurately over time.

Analyzing Neuron Behavior

Each neuron's linear models evolve during training; initially set slopes change based on learning outcomes—Madrid neuron shifts downhill while others adjust accordingly.

The height of each neuron's line corresponds to its output value at given inputs. The softmax function amplifies the highest values leading to final predictions.

Scaling Complexity with Additional Inputs

Back propagation effectively scales to larger problems such as adding new cities (e.g., Barcelona), requiring both longitude and latitude inputs (X1 and X2).

With additional parameters introduced (two slope values per neuron), each neuron now represents a plane rather than a line. This increases total parameters needed for updates significantly.

Training New Models

The updated two-input model quickly adapts by fitting planes corresponding to four cities accurately across the map regions designated for each city.

Understanding Complex Patterns in Language Models

The Challenge of Complex Borders

The discussion begins with the analogy of planes representing different regions, highlighting the limitations of simple models in understanding complex patterns.

A specific example is given regarding the intricate border between Belgium and the Netherlands, illustrating how a single model for each country fails to accurately represent these complexities.

Token Representation in Language Models

The llama model represents tokens (e.g., city names) as vectors composed of 48 floating-point numbers, which are crucial for processing language.

When inputting text like "the capital of France is," the model generates a vector that progressively aligns with the vector for "Paris" through its layers.

High-Dimensional Language Mapping

By analyzing various training examples leading to next tokens like Paris or Madrid, a set of 248 vectors is created, representing coordinates in a high-dimensional language space.

Utilizing UMAP algorithms allows projection from 248 dimensions down to two, revealing interesting clusters based on context (e.g., references to treaties or cultural works).

Learning and Model Adaptation

Just as disconnected regions must map correctly within their respective countries, language models must adapt their internal representations to ensure accurate predictions for next tokens.

Historical context is provided by referencing Marvin Minsky's skepticism towards backpropagation due to its slow convergence and perceived limitations; however, this underestimates its potential for solving complex problems today.

Personal Journey and Educational Goals

The speaker shares personal experiences transitioning from machine learning engineering back into content creation focused on education after leaving Welch Labs in 2019 due to financial challenges.

Reflecting on past educational experiences reveals a desire to improve math and science education through engaging content while acknowledging the need for sustainable business practices first.

Building Support Through Community Engagement

In an effort to support his educational initiatives financially, the speaker discusses strategies such as sponsorships and Patreon support aimed at achieving full-time engagement with Welch Labs projects.