The spelled-out intro to neural networks and backpropagation: building micrograd
Andrej Karpathy・122 minutes read
Andre demonstrates neural network training processes by utilizing Micrograd, focusing on backpropagation and mathematical expressions to efficiently evaluate gradients and build neural networks, with the discussion transitioning to more complex operations in Pytorch for designing and implementing neural networks effectively.
Insights
- Neural networks are essentially mathematical expressions that process input data and weights to generate predictions or loss functions, with backpropagation being a fundamental algorithm applicable beyond neural networks.
- Micrograd, an autograd engine, simplifies understanding neural network training by enabling the evaluation of gradients efficiently, crucial for optimizing network weights.
- Backpropagation recursively applies the chain rule to calculate derivatives, highlighting how inputs influence the final output and the importance of understanding local gradients.
- The process of backpropagation involves iteratively adjusting inputs based on gradients to enhance the final outcome, showcasing the significance of properly accumulating gradients to ensure correct results.
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Recent questions
What is backpropagation in neural networks?
Backpropagation is a crucial algorithm for efficiently evaluating gradients of a loss function with respect to neural network weights. It involves recursively applying the chain rule from calculus to compute derivatives of internal nodes and inputs, essential for understanding how inputs affect the output in neural networks.
How does Micrograd simplify neural network training?
Micrograd simplifies neural network training by operating on scalar values for pedagogical reasons, making it easier to understand before transitioning to tensor operations for efficiency in larger networks. It allows building mathematical expressions using value objects for inputs and operations like addition and multiplication, offering a concise yet powerful tool for efficiently training neural networks.
What are the key components of a neural network?
A neural network consists of interconnected neurons with weights and biases, modeled mathematically with inputs, weights, biases, and activation functions. Neurons in a neural network take input data and weights to produce predictions or loss functions, with activation functions like tanh squashing input values to generate neuron outputs.
How does PyTorch differ from Micrograd in neural network training?
PyTorch, a modern deep neural network library, simplifies the implementation of neural networks by using tensors, n-dimensional arrays of scalars, for operations that can be more complex than scalar values. It allows for parallel operations on tensors, making it efficient for building complex mathematical expressions and neural networks compared to Micrograd.
What is the purpose of the forward pass in neural networks?
The forward pass in neural networks involves evaluating the output value of a mathematical expression, which is then followed by a loss function to measure prediction accuracy. This process ensures that the network behaves as desired, with low loss indicating accurate predictions, setting the stage for subsequent backpropagation to tune parameters and decrease loss through iterative gradient descent.
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Summary
00:00
"Training Neural Networks with Micrograd"
- Andre has been training deep neural networks for over a decade and aims to demonstrate neural network training processes in this lecture.
- The lecture will involve starting with a blank Jupiter notebook and concluding with defining and training a neural network to showcase the inner workings.
- Micrograd, an autograd engine released on GitHub two years ago, will be explored step by step in this lecture to understand its functionality.
- Micrograd implements backpropagation, a crucial algorithm for efficiently evaluating gradients of a loss function with respect to neural network weights.
- Micrograd allows building mathematical expressions using value objects for inputs and operations like addition, multiplication, and more.
- The forward pass in Micrograd involves evaluating the output value of a mathematical expression, while the backward pass initiates backpropagation to compute derivatives.
- Backpropagation recursively applies the chain rule from calculus to evaluate derivatives of internal nodes and inputs, crucial for understanding how inputs affect the output.
- Neural networks are essentially mathematical expressions that take input data and weights to produce predictions or loss functions, with backpropagation being a general algorithm applicable beyond neural networks.
- Micrograd operates on scalar values for pedagogical reasons, simplifying understanding before transitioning to tensor operations for efficiency in larger networks.
- Micrograd, comprising a simple autograd engine and a neural network library, offers a concise yet powerful tool for understanding and training neural networks efficiently.
14:14
Exploring Derivatives and Neural Network Structures
- The function has an output variable d dependent on three scalar inputs a, b, and c.
- Printing d results in the value of four.
- Derivatives of d with respect to a, b, and c are explored to understand their significance.
- To evaluate the derivatives, a small value of h is used, and inputs are fixed at specific values.
- The derivative of d with respect to a is calculated by bumping a by h and observing the change in the function.
- The slope of the derivative indicates the rate of change in the function.
- Similar calculations are done for derivatives with respect to b and c, revealing their impact on the function.
- The process of building a data structure to maintain mathematical expressions for neural networks is initiated.
- A value object class is created to handle scalar values and operations like addition and multiplication.
- Visualization of expression graphs using Graphviz API is demonstrated, showcasing the forward pass of a mathematical expression.
30:39
"Visualizing Backpropagation: Derivatives and Chain Rule"
- The data nodes hold the data, while the weights are iterated using gradient information.
- A variable "grad" in the value class maintains the derivative of the loss function with respect to that value.
- Initially, "grad" is set to zero, indicating no impact on the output.
- Visualizing the "grad" after data shows it as 0.4f.
- Manual backpropagation starts by setting "l.grad" to one.
- Derivatives of the loss function with respect to "d" and "f" are calculated.
- The derivative of "l" with respect to "a" is found to be six.
- The derivative of "l" with respect to "d" is "f," and with respect to "f" is "d."
- The derivative of "l" with respect to "c" is calculated using the chain rule, resulting in a value of negative two.
- Incrementing "c" and "e" by a small amount confirms the calculated derivatives.
47:52
Understanding Backpropagation in Neural Networks
- The local gradient is determined by the derivative of the output with respect to the input variables.
- The local gradient is calculated by considering the derivative of the output with respect to each input variable.
- The derivative of the output with respect to a specific input variable is determined by the value of another variable.
- The backpropagation process involves iteratively applying the chain rule to compute derivatives.
- The backpropagation process entails recursively multiplying local derivatives through the computation graph.
- Adjusting inputs based on gradients can influence the final outcome positively.
- The process of backpropagation involves understanding the impact of inputs on the final output.
- A neural network model consists of interconnected neurons with weights and biases.
- Neurons in a neural network are modeled mathematically with inputs, weights, biases, and activation functions.
- The activation function, such as tanh, squashes the input values to produce the neuron's output.
01:04:47
"Automated Back Propagation Enhances Gradient Calculations"
- Back propagation process continues with a plus node receiving a gradient of 0.5, distributing it equally to both connected nodes due to a local derivative of 1 for each node.
- The gradient for the node connected to the plus node is determined as 0.5, with the same value assigned to the connected node's gradient.
- Moving forward, another plus node receives a gradient of 0.5, distributing it equally to its connected nodes, setting their gradients to 0.5 as well.
- Back propagating through a times node, the local derivative is the other term, leading to the calculation of x2.grad and w2.grad based on the data and gradients involved.
- The gradient on weight 2 is determined as 0 due to the input x2 being 0, while x2's gradient is 0.5 due to the data being 1.
- Exploring the impact of the times operation, it is noted that the gradient is zero when the input is zero, aligning with the concept of derivatives indicating the influence on the final output.
- Proceeding with the back propagation, a gradient of 0.5 flows through a times operation, determining x1.grad based on the local derivative of the operation.
- The final derivatives are calculated as 0, 0.5, -1.5, and 1 for the respective nodes, showcasing the impact of weights on neuron outputs.
- The manual back propagation process is deemed impractical, leading to the introduction of a more automated backward pass implementation.
- Implementing a backward function for addition, multiplication, and tanh operations, the process of propagating gradients through nodes is streamlined, ensuring accurate gradient calculations.
01:23:45
Accumulating Gradients with += Operator in Backpropagation
- When a variable is used more than once in an expression, issues arise during the backward pass in backpropagation.
- Backward pass involves backpropagating from F to E to D, but when D is called backward, it overwrites gradients at A and B.
- To resolve the issue of overwritten gradients, accumulation of gradients is necessary by using the `+=` operator instead of setting gradients.
- Accumulating gradients ensures correct results as contributions flowing backward add up, fixing the issue.
- Implementing the `+=` operator ensures that gradients accumulate correctly, starting from zero and adding contributions as needed.
- Breaking down the tanh function into its explicit atoms allows for a deeper understanding and the ability to implement more expressions.
- To enable operations like addition and multiplication with constants, non-Value objects need to be wrapped in a Value object for compatibility.
- Implementing the power function allows for exponentiation, with the chain rule applied to propagate gradients through the operation.
- Subtraction is achieved by implementing it as addition with negation, ensuring all basic arithmetic operations are covered.
- Breaking down complex functions into simpler expressions for computation aids in understanding and ensures equivalent results in both forward and backward passes.
01:39:03
Efficient Neural Network Implementation with Pytorch
- The concept of a neural network is a composite operation of atomic operations, focusing on inputs and outputs.
- The ability to perform forward and backward passes allows for flexibility in designing functions within neural networks.
- Pytorch, a modern deep neural network library, simplifies the process of implementing neural networks.
- Pytorch uses tensors, n-dimensional arrays of scalars, for operations, which can be more complex than scalar values.
- Explicitly setting tensors to require gradients in Pytorch is necessary for efficient computation.
- Arithmetic operations in Pytorch are similar to those in micrograd, with the addition of tensor attributes like data and grad.
- Pytorch's efficiency stems from its ability to perform parallel operations on tensors.
- Building complex mathematical expressions and neural networks is achievable using Pytorch's tensor operations.
- A neural network consists of interconnected neurons, with layers of neurons evaluated independently.
- An MLP (multi-layer perceptron) is a sequence of layers of neurons, with each layer feeding into the next sequentially.
01:54:49
Optimizing Neural Net Parameters for Better Predictions
- Lower loss indicates better predictions, with zero loss being ideal.
- After performing a backward pass, examining end.layers.neuron reveals neuron details.
- Each layer in the MLP contains neurons with weights denoted as 'w'.
- Gradients on weights influence loss; negative gradients decrease loss.
- The loss graph includes four forward passes per example, ending at a loss value of 7.12.
- Weight parameters are inputs to the neural net, while input data remains fixed.
- Convenience code gathers all neural net parameters for simultaneous operation.
- Parameters of the neural net, totaling 41, can be adjusted based on gradient information.
- Adjusting parameters using gradient descent involves small steps in the direction of the negative gradient.
- Iterative forward and backward passes, along with parameter updates, lead to improved predictions and reduced loss.
02:12:37
"Neural Nets: Principles, Operations, and Training"
- Neural nets are mathematical expressions used in multi-layer perceptrons, taking input as data, weights, and parameters.
- Forward pass in neural nets involves a mathematical expression followed by a loss function to measure prediction accuracy.
- Loss function manipulation ensures the network behaves as desired, with low loss indicating accurate predictions.
- Backpropagation is used to get gradients for tuning parameters to decrease loss, followed by iterative gradient descent.
- Neural nets can be complex, with billions of parameters, but operate on the same principles.
- Micrograd, a neural networks library, includes operations like addition, multiplication, and non-linearities like ReLU.
- PyTorch, a production-grade library, showcases backward pass implementation for functions like tanh, demonstrating complexity.
- PyTorch allows users to register new functions by subclassing existing classes and defining forward and backward passes.
- Learning rate decay, batching, loss functions like max margin, and L2 regularization are key concepts in training neural nets.
