Summary of The spelled-out intro to neural networks and backpropagation: building micrograd
00:00:00In the lecture, Andre, who has years of experience in training deep neural networks, introduces the process of neural network training with a focus on building Micrograd - an autograd engine that implements backpropagation. He explains that backpropagation helps in efficiently evaluating the gradient of a loss function with respect to the weights of a network, allowing for iterative weight adjustments to enhance network accuracy. Micrograd enables the creation of mathematical expressions using value objects, showcasing functions such as addition, multiplication, and division to build an expression graph.
00:02:39The value 'c' is obtained from an addition operation, with child nodes 'a' and 'b' maintaining pointers to their respective value objects. We can perform a forward pass to calculate the value of 'g', accessing it using the dot data attribute. Backward pass (backpropagation) initializes at 'g', recursively applying the chain rule to evaluate derivatives with respect to internal nodes and inputs, such as 'a' and 'b'. This derivative information is crucial as it indicates how 'a' and 'b' affect 'g'. Neural networks, while being mathematical expressions like the demonstrated one, take input data and weights to produce predictions or loss functions.
00:05:19The speaker emphasizes that micrograd, a scalar-valued autograd engine, simplifies understanding and teaching of backpropagation and chain rule for neural network training, by breaking down neural nets into individual scalars. This process is not typically used in production as in modern deep learning libraries tensors are used instead for efficiency, though the underlying math remains the same. Micrograd is highlighted as a simple and essential tool for training neural networks, with the autograd engine consisting of only 100 lines of simple Python code and the neural network library built on top of it being straightforward to understand.
00:07:50Neural networks are powerful with just 150 lines of code. Understanding derivatives is crucial for neural network training. Introduction shows defining and plotting a basic scalar function for insight into derivatives. Calculating derivatives manually would be impractical for neural networks due to their complexity.
00:10:26The speaker discusses the concept of derivatives in calculus, explaining that it measures how a function responds to a slight increase at a specific point. By evaluating the derivative numerically using a small increment (e.g., 0.001), one can determine the slope of the function at that point. The process involves calculating the change in the function's value when the input is adjusted incrementally and then normalizing it by the increment size to find the slope. It is noted that using extremely small increments can lead to inaccuracies due to limitations in floating-point arithmetic used by computers. Ultimately, at point 3, the slope of the function provided (3x squared minus 4x plus 5) is determined to be 14.
00:13:00The text discusses the process of finding derivatives for different values of a function. It starts by explaining how to calculate the slope of a function at specific points using differentiation. It then delves into a more complex scenario where a function has multiple input variables, and the derivative of the output variable with respect to each input variable is calculated. The process involves tweaking the inputs by a small value (h) and observing the change in the output, which helps determine the slope or derivative at that point.
00:15:57The text describes the process of calculating the slope of a function using small perturbations in the input variables a, b, and c. By increasing a, a slightly positive value is added to the function, causing a slight decrease due to the negative value of b. The resulting slope is calculated as negative 3. Similarly, adjusting b positively increases the function output by a slope of 2, reflecting the positive value of a. Finally, increasing c results in a one-unit increase in the function output, as c directly adds to the function output. The process involves sensitivity analysis of each variable's impact on the function output and finding the corresponding slopes for each variable.
00:19:05In this section, the focus is on building a value object in Python to handle mathematical expressions for neural networks. This involves defining special methods (e.g., \_\_add\_\_ for addition, \_\_mul\_\_ for multiplication) to enable operations between value objects. For example, with a value object representing 2.0 (a) and another representing -3.0 (b), addition (a + b) and multiplication (a * b) operations can be performed to obtain results like -1 and -6, respectively. The wrapper function is crucial for displaying results neatly.
00:22:03In Python, the explanation focuses on creating a more visually appealing expression that computes a value of -6 and a result of 4 when addition and multiplication operations are applied sequentially. The code introduces a way to track the relationships between different values by defining variables "children" and "_prev" to store the children and the operation that created the new value. This allows for more organized expression graphs to be maintained, helping to understand the flow of operations and values.
00:24:46The speaker explains how they are building a data structure to create and visualize mathematical expressions in a clear way. They demonstrate the code that creates a function called drawdot to visualize the expression graphs, using a tool called Graphis API for graph visualization. They use helper functions like trace to enumerate nodes and edges in the graph, creating special operation nodes to represent mathematical operations. Labels are added to the graphs to indicate variables, with actual value objects shown in squares while operation nodes are added for visual clarity.
00:27:52The overview video demonstrates building mathematical expressions utilizing addition and multiplication operations with scalar values to create a single output. The process involves conducting a forward pass to calculate the output, in this case, negative eight. Following this, backpropagation is initiated to determine the gradient of each intermediate value with respect to the final output. This process aims to evaluate how the weights in a neural network impact the loss function by computing the derivatives of the output with respect to the neural network weights.
00:30:46The loss function is calculated with respect to the data, as the weights will be adjusted using gradient information during iterations. A variable named grad is created in the value class to maintain the derivative of the loss function with respect to the value. The gradient is initialized as zero, indicating no impact on the output. Back propagation is then manually performed by filling in the gradients starting from the end, calculating derivatives of the output with respect to different values. Numerical gradients can be estimated using a local function to avoid altering the global scope.
00:33:46The passage discusses the concept of derivatives in neural network backpropagation, specifically focusing on the calculation of the derivative of a function 'l' with respect to variables 'a', 'd', and 'f'. It explains that the derivative of 'l' with respect to 'a' is six, the derivative of 'l' with respect to 'd' is 'f', and the derivative of 'l' with respect to 'f' is 'd'. The explanation includes a breakdown of the calculus involved in calculating these derivatives. Lastly, it mentions setting manual values for the gradients of 'd' and 'f', with 'f' having a value of 4 and 'd' having a value of -2.
00:37:03To summarize, the speaker demonstrates the process of numerical verification in gradient checking and explains the importance of understanding backpropagation, specifically focusing on deriving the derivative of the loss function with respect to intermediate variables like 'c.' They explain the concept of evaluating how changes in one variable (c) affect the loss function through another variable (d), highlighting the necessity to combine these impacts to determine the effect of c on the overall loss. The differentiation process is shown, emphasizing the key role of understanding these relationships in effectively implementing backpropagation for training neural networks.
00:40:01The discussion focuses on using the chain rule to calculate derivatives in neural networks. By applying the chain rule, the derivative of a sum expression simplifies to 1.0 due to the cancellation of like terms. The concept of local derivatives is introduced, highlighting how a small node in a network can determine the impact of individual variables on the final output. The chain rule is essential for combining these local derivatives to calculate the overall impact of specific variables on the output. The rule states that if a variable (z) depends on another variable (y), which in turn depends on a third variable (x), then the derivative of z with respect to x is the product of the derivatives of z with respect to y and y with respect to x, demonstrating how these derivatives are chained together.
00:42:58The explanation discusses the chain rule in calculus, demonstrating how knowing the rates of change between variables allows for calculating the overall rate of change. Using the example of car speed related to bicycle and walking man speeds, it illustrates how intermediate rates of change are multiplied together in the chain rule. The process involves determining local derivatives and applying them to obtain the final derivative, showing how gradients are routed through the nodes in a neural network during backpropagation.
00:46:13The concept of backpropagation involves propagating a signal carrying derivative information backwards through a neural network graph. This signal distributes derivatives to all connected nodes, following the chain rule. The example calculates derivatives for internal nodes using this method, showing how the local gradients of nodes contribute to determining overall derivatives at various points in the network. In this specific calculation, the derivative of the loss function with respect to a particular node 'a' is determined by multiplying the derivative of the loss with respect to an intermediate node 'e' by the local gradient of the connection between 'a' and 'e'.
00:49:20The value of a in um db is 2.2 and the claimed derivatives are a dot grad equals 6 and b dot grad equals -4. Manual backpropagation involved verifying these claims by iteratively applying the chain rule from output to leaf nodes. By nudging inputs in the direction of the gradient, such as increasing a.data, we can expect the output l to increase, aiming for a less negative value. This process involves recursively multiplying local derivatives to propagate back through the computation graph, demonstrating the power of backpropagation in optimizing neural networks.
00:52:33The speaker performs one step of optimization, highlighting the power of gradients in influencing outcomes and aiding in training neural networks. They introduce the concept of manual backpropagation through a two-layer neural network comprised of neurons with weights, biases, and activation functions like tanh. A basic explanation of how neurons interact and produce output through the activation function is provided, using tanh as an example.
00:55:32In a two-dimensional neuron, inputs x1 and x2 are multiplied by weights w1 and w2 and added to bias b to calculate the raw activation n. This value is then passed through an activation function, such as tanh, to produce the output o. However, since tanh requires exponentiation, which is not yet implemented in the model, it cannot be directly calculated. One option is to implement exponentiation directly or modify the approach to use x values instead of tanh for the time being. This showcases the flexibility in creating functions at different levels of abstraction within the model.
00:58:30In this segment, the focus is on creating and implementing the tanh function within a neural network. The process involves taking inputs and producing outputs through a complex function, as long as the local derivative can be determined. The implementation of the tanh function involves creating a node for it and defining its operation, allowing for backpropagation through it. The changes in the bias values affect the output of the tanh function, which plays a crucial role in the upcoming backpropagation process.
01:01:39In the typical neural network setting, the focus is on calculating the derivatives of neurons with respect to the weights (like w2 and w1) for optimization. Neural networks usually consist of multiple connected neurons, forming a larger system. Backpropagation involves adjusting these weights to improve the accuracy measured by a loss function.
To backpropagate through a tanh function, the derivative is calculated as 1 - (output squared). For instance, if the output is 'o', the local derivative would be 1 - o^2, which simplifies to 0.5 if o equals 0.5. This derivative helps adjust the weights during backpropagation to enhance network performance.
01:05:02In this passage, the concept of distributing gradients through additions and multiplications in a neural network using backpropagation is explained. The process involves calculating the local derivatives at each node and updating the gradients accordingly. The derivatives indicate the influence of each input on the final output, with multiplication by zero resulting in no change and therefore a gradient of zero. Through the chain rule, the gradients for each parameter (such as weights) are computed, ultimately guiding the adjustments needed to increase the neuron's output.
01:08:50In the given text, the process of implementing the backward pass in neural networks is discussed. The focus is on automating the backpropagation process to avoid manual calculations. By defining functions like '_backward', the gradient propagation through different operations such as addition and multiplication is explained. The concept of chain rule is emphasized in updating the gradients of nodes based on the operation and its local derivative. The text describes the implementation of these functions to handle gradient computations more efficiently in neural network training.
01:12:07The process involves chaining outgrad into self.grad and others.grad, applying the chain rule for multiplication. The local derivatives are calculated as others.data times grad, and self.data times grad respectively. This involves setting out gradients to be equal to backward, back propagating outgrad into self.grad. The local derivative depends on the derivative of the output function with respect to its input, which is then multiplied by the global derivative due to the chain rule. By utilizing the dot backward function in the correct order and initializing gradients appropriately, the propagation through the network can be automated efficiently.
01:16:03The video demonstrates the process of manually calling the backward function in neural networks to update gradients. Through this process, the speaker ensures that the gradients are correctly propagated through the network, ultimately achieving the desired outputs. They explain the importance of properly ordering nodes in the graph using topological sort to ensure that dependencies are resolved before updating each node's gradients. This method involves marking visited nodes and iteratively traversing through children nodes to lay them out from left to right, allowing for an efficient update of gradients in the backward pass of backpropagation.
01:19:30The function ensures that a node is added to the list only after all its children have been processed. This maintains the invariant that a node is included in the list only after its children. The topological graph is built with the values ordered prior to the output. The backward process sets gradients to zero, sets the base case, computes derivatives in reverse topological order, and hides this functionality within the value class for cleaner code. A bug is identified when considering a specific case involving nodes a and b and implementing the necessary corrections.
01:23:04In the given explanation, the issue with calculating gradients in neural networks during backpropagation is discussed. When a variable is used more than once in an expression, the gradients get overwritten, leading to incorrect values. The proposed solution is to accumulate the gradients by using the "+=" operation instead of setting them directly. By initializing gradients at zero and accumulating contributions as they flow backward, correct gradients can be obtained. This approach ensures that the gradients are correctly calculated even when variables are reused multiple times in the network.
01:26:22The process involves depositing gradients from each branch during backpropagation, accumulating them by adding on top of each other to fix issues. Intermediate work is cleaned up prior to moving forward, keeping only essential parts. The function tanh is discussed, with the ability to break it down into explicit atoms expressed in terms of other functions for better understanding and implementation. Addition of constants to values is enabled by wrapping non-Value instances like integers or floats in a Value object to make expressions coherent.
01:28:53The explanation covers defining a multiplication operation in Python for different operand orders. It explains how to use an rmod method as a fallback for when Python cannot directly handle certain operations like 2 times a. By introducing a new function x for exponentiation, the text outlines the process of calculating the local derivative of e to the x, which is e to the x itself, using the chain rule to propagate the gradient.
01:31:37The text outlines the implementation of a more powerful operation than division, which involves redefining division as multiplication by the reciprocal. It introduces the concept of implementing x to the power of k, where k is an integer or float, and outlines the differentiation process for this operation. By applying the chain rule, the text describes the backpropagation process for calculating the derivative of the power function. The focus is on redefining division and implementing the pow function to handle raising a value to a specific power, ensuring the power is restricted to being an integer or float for mathematical consistency.
01:34:16The video introduces the concept of the power rule in calculus for finding local derivatives of functions. It explains how to implement the power rule in Python by utilizing the chain rule and walks through the process of computing the backward pass for a neural network neuron. The subtraction operation is implemented as the addition of a negation, achieved by multiplying by -1. Finally, the video demonstrates the computation of gradients for a two-dimensional neuron using the implemented formula involving exponential functions.
01:37:09In the text, the writer explains the process of breaking down a function into multiple operations and verifying its correctness through forward and backward passes. The purpose of this exercise is to demonstrate that the level of operation implementation is flexible and can be tailored to individual preferences. The writer also mentions the option of using modern deep learning libraries like PyTorch to simplify the process. Overall, the key points include breaking down functions, verifying correctness through forward and backward passes, flexibility in operation implementation, and the possibility of using deep learning libraries for ease.
01:39:47The process involves importing PyTorch and defining initial value objects, taking into account that PyTorch operates with tensors, which are n-dimensional arrays of scalars. Manipulating tensors in PyTorch involves handling more complex arrays compared to micrograd's scalar values. Leaf nodes in PyTorch default to not requiring gradients, so specifying that these nodes need gradients is necessary. Arithmetic operations can be performed similarly to micrograd, with PyTorch also providing tensor objects with data and grad attributes that can be accessed using `.item()`.
01:42:38The speaker demonstrates a neural network and backpropagation process using PyTorch. They show how tensors work in PyTorch and how to access individual elements in a tensor. A single-neuron model implementing PyTorch's neural network module API is also discussed. The efficiency of PyTorch in handling mathematical operations on tensors is emphasized, making it suitable for building complex neural networks like multilayer perceptrons. The goal is to replicate PyTorch's API while constructing neural network components.
01:45:24In this section, the code is explained for forwarding a single neuron within a neural network. It involves multiplying the weights and inputs, summing them together, adding a bias, passing it through an activation function, and producing an output. The process is demonstrated with the use of Python functions and iterators to efficiently calculate these steps. Additionally, the concept of a layer of neurons in a neural network is introduced, where each neuron in the layer acts independently and is fully connected to the input. The layer is essentially a collection of neurons evaluated separately.
01:48:34In neural networks, the number of neurons in a layer and the number of outputs are defined independently. Neurons are initialized with given dimensionality and evaluated independently. Layers in multi-layer perceptron (MLP) feed into each other sequentially, with each layer having its own set of neurons. An MLP is defined by a list of output sizes for each layer, and inputs pass through the layers sequentially during a forward pass. To backpropagate through the network and update weights, the process can be simplified with tools like Micrograd, allowing for efficient differentiation and training.
01:51:29The video discusses a simple binary classifier neural network and the process of tuning its weights to better predict desired targets using backpropagation. It explains how the loss function measures the network's performance by calculating the mean squared error between predictions and ground truth values. By iteratively pairing up predictions and ground truths, subtracting and squaring their differences, the loss is determined for each example. The goal is to minimize this loss to improve the model's performance.
01:54:20In summary, the goal is to minimize the loss in neural networks to improve predictions. The loss is the sum of errors, where high loss indicates inaccurate predictions. By performing backpropagation, the network adjusts weights to reduce the loss. Each neuron's weight affects the loss, with negative gradients indicating a potential decrease in loss. The process involves running forward passes for each example and calculating the mean squared error. The loss propagates backwards through the network's values to update the weights and ultimately improve prediction accuracy.
01:57:17The input data in a neural network has gradients, but they are not useful as the input data is fixed and not changeable. The gradients we are interested in are for the neural network parameters like weights and biases, which we want to change. To simplify gathering all the parameters of the neural network, we can create a function that collects and returns all parameter tensors in a single list using list comprehensions. This makes it easier to operate on all parameters simultaneously and update them based on gradient information.
02:00:50In this section, the neural network is reinitialized which changes some numbers. The Multi-Layer Perceptron (MLP) has a total of 41 parameters which can be updated by calculating the loss and adjusting the weights and biases accordingly. The goal is to iteratively update each parameter by a small step size in the direction opposite to the gradient of the loss to minimize it. By ensuring the correct direction of the updates, the network aims to decrease the loss and improve predictions.
02:04:04The data for the neuron has been slightly updated, leading to a small increase in its value. This change results in a lower loss according to the gradient calculation. Through iteratively performing forward and backward passes and updating parameters using gradient descent, the neural network improves its predictions. It is important to be cautious with the learning rate to avoid overstepping and destabilizing the training process. The goal is to minimize the loss function, indicating that the predictions are closer to the targets as the training progresses.
02:07:15The speaker highlights the importance of setting the learning rate correctly in neural network training to avoid slow convergence or unstable updates. They implement a training loop involving forward and backward passes, gradient descent, and updating the neural network parameters. By optimizing the learning rate and number of steps, they show controlled convergence to a low loss. However, they also mention encountering a common bug in their code.
02:10:42Working with neural nets can be challenging due to common mistakes like failing to zero the gradients before the backward pass, causing them to accumulate and lead to incorrect results. Resetting the gradients to zero before the backward pass ensures proper updates and optimal optimization. Failure to address such bugs can lead to misleadingly fast convergence on simple problems but require more steps for complex tasks, emphasizing the importance of thorough code debugging in neural network development.
02:13:51Neural networks are mathematical expressions that take input data, weights, and parameters to make predictions. The process involves a forward pass through the network followed by a loss function to measure accuracy. Backpropagation is used to adjust parameters and minimize the loss through gradient descent. This iterative process allows neural networks to learn and adapt to complex tasks, with the ability to scale up to billions or trillions of parameters. The principles of training remain the same, even for larger networks, but may involve different updates and loss functions for more challenging problems like language modeling with massive datasets.
02:16:34The video discusses the concept of cross-entropy loss and neural network training, highlighting the similarities and fundamental aspects of the process. It also introduces micrograd and its code structure, mentioning various operations like addition, multiplication, and relu non-linearity. The video stresses the equivalence of different non-linearities like relu, tanh, and sigmoid, explaining the choice of tanh for added complexity. The video showcases the implementation of neural network layers in micrograd, aligning with PyTorch's nn.module class. A test is conducted comparing micrograd's operations with PyTorch, ensuring consistency in forward and backward passes. Additionally, a demo illustrates a binary classification scenario with more complexity than earlier examples.
02:19:12In this example, a binary classifier using a more complex Multi-Layer Perceptron (MLP) is built to classify two-dimensional points as red or blue. The loss function employed supports batching for larger datasets by processing random subsets (batches) during training. Various loss functions like max margin, mean squared error, and binary cross-entropy are discussed for binary classification. L2 regularization is mentioned for generalizing the neural network and controlling overfitting. The training loop involves forward pass, backward pass with zero grad, and updates, with learning rate decay applied for improved convergence. Finally, the neural net's decision surface effectively separates the red and blue areas based on the input data points. The video also touches on implementing this concept in a production-grade library like PyTorch, showing an example of the backward pass for the hyperbolic tangent function (tanh).
02:21:45The speaker spent time searching through a complex codebase on Github, trying to understand the numerous references to "10h" without success. They eventually found code for the backward pass of "10h" function, which varied depending on whether PyTorch was running on a CPU or GPU. Despite the complexity of the code, they highlighted the potential to register new functions with PyTorch by defining forward and backward passes, enabling the integration of custom functions within PyTorch's existing architecture.
02:24:32The speaker wraps up the lecture by explaining that in Pytorch, you only need to specify the desired functionality and everything else will work smoothly. They mention that new types of functions can be added similarly. They provide information on related links in the video description and mention the possibility of a follow-up video to address common questions. Finally, they ask viewers to like and subscribe to support the channel. Additionally, the speaker briefly references discussing the problem with dl and mentions moving on to multiplying.