From 3e8ced1397e9fb1b6d2420fbb843bf2823a5c062 Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Tue, 14 Nov 2023 12:54:32 -0500 Subject: [PATCH 01/45] First pass at training --- training.qmd | 353 +++++++++++++++++++++++++++++++++++++++++++-------- 1 file changed, 300 insertions(+), 53 deletions(-) diff --git a/training.qmd b/training.qmd index 90f5cefe..645a4b64 100644 --- a/training.qmd +++ b/training.qmd @@ -3,93 +3,340 @@ ::: {.callout-tip} ## Learning Objectives -* coming soon. +* Understand the mathematics behind deep learning (i.e: backpropagation, gradient descent) +* Identify the key operations and system bottlenecks in AI training +* Learn how CPUs and GPUs accelerate AI inference and training by speeding up key operations ::: ## Introduction -Explanation: An introductory section sets the stage for the reader, explaining what AI training is and why it's crucial, especially in the context of embedded systems. It helps to align the reader's expectations and prepares them for the upcoming content. +Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. In this section, we will explore the mathematics behind neural networks, specifically how neural networks work and how to train them, then dive into some of the key system challenges that underpin these models, and finally explore how to leverage CPUs and GPUs to accelerate them. -- Brief overview of what AI training entails -- Importance of training in the context of embedded AI +## Mathematics behind Neural Networks and Deep Learning -## Types of Training +At a high level, neural networks are mathematical models that consist of alternating linear and nonlinear operations, parameterized by a set of "weights" that are trained to minimize some loss function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value (i.e: the "loss") when evaluated on our model against the training data. Fundamentally, training neural networks involve evaluating the loss function to get a measure of how good our model is, then continuously tweaking the weights of our model so that the loss decreases, which ultimately optimizes the model to fit our training data. -Explanation: Understanding the different types of training methods is foundational. It allows the reader to appreciate the diversity of approaches and to select the most appropriate one for their specific embedded AI application. +### Neural Network Notation -- Supervised Learning -- Unsupervised Learning -- Reinforcement Learning -- Semi-supervised Learning +Diving into the details, the core of a neural network as introduced in section 3 can be viewed as a sequence of alternating linear and nonlinear operations: +$$ +L_i = F_{i}(W_i \times L_{i-1}) +$$ -## Data Preparation +:::{.callout-note} +Convolutions are also linear operators, and can be cast as a matrix multiplication. +::: + +where $L_{0}$ is a vector input to the neural network (i.e: an image that we want the neural network to classify, or some other data that the neural network operates on), $L_{n}$ (where $n$ is the number of layers of the network) is the vector output of the neural network (i.e: a vector of size 10 in the case of classifying pictures of handwritten digits), $W_i$s are the weights of the neural network that are tweaked at training time to fit our data, and $F_{i}$ is that layer's nonlinear activation function (i.e: ReLU, softmax, etc). As defined, the intermediate output of the neural network is a vector of real-valued numbers with dimensions: + +$$ +L_i \in \mathbb{R}^{d_{i}} +$$ + +where $d_{i}$ is the number of neurons at layer $i$; in the case of the first layer $i=0$, $d_{i}$ is the dimension of the input data, and in the last layer $i=n$, $d_{n}$ is the dimension of the output label, and anything in between can be set arbitrarily and may be viewed as the "architecture" of the neural network (i.e: dimensionality of the intermediate layers). The weights, which determine how each layer of the neural network interacts with each other, therefore are matrices of real numbers with shape + +$$ +W_i \in \mathbb{R}^{d_{i} \times d_{i-1}} +$$ + +Our neural network, as defined, performs a sequence of linear and nonlinear operations on the input data ($L_{0}$), to optain predictions ($L_{n}$) which hopefully is a good answer to what we want the neural network to do on the input (i.e: classify if the input image is a cat or not). Our neural network may then be represented succinctly as a function $N$ which takes in an input $x \in \mathbb{R}^{d_0}$ parameterized by $W_1, ..., W_n$: + +$$ +\begin{align*} +N(x; W_1, ... W_n) &= \text{Let } L_0 = x, \text{ then output } L_n +\end{align*} +$$ + +Next we will see how to evaluate this neural network against training data by introducing a loss function. + +### Loss Function as a Measure of Goodness of Fit against Training Data -Explanation: Data is the fuel for AI. This section is essential because it guides the reader through the initial steps of gathering and preparing data, which is a prerequisite for effective training. +After defining our neural network, we are given some training data, which is a set of points ${(x_j, y_j)}$ for $j=1..M$, and we want to evaluate how good our neural network is on fitting this data. To do this, we introduce a "loss function", which is a function that takes the output of the neural network on a particular datapoint ($N(x_j; W_1, ..., W_n)$), and compares it against the "label" of that particular datapoint (the corresponding $y_j$), and outputs a single numerical scalar (i.e: one real number) that represents how "good" the neural network fit that particular data point; the final measure of how good the neural network is on the entire dataset is therefore just the average of the losses across all datapoints. -- Data Collection -- Data Annotation -- Data Augmentation -- Data Preprocessing +There are many different types of loss functions, for example, in the case of image classification, we might use the cross-entropy loss function, which tells us how good two vectors that represent classification predictions compare (i.e: if our prediction predicts that an image is more likely a dog, but the label says it is a cat, it will return a high "loss" indicating a bad fit). -## Training Algorithms +Mathematically, this loss function is a function which takes in two real-valued vectors of the shape of the label, and outputs a single numerical scalar +$$ +L: \mathbb{R}^{d_{n}} \times \mathbb{R}^{d_{n}} \longrightarrow \mathbb{R} +$$ -Explanation: This section delves into the algorithms that power the training process. It's crucial for understanding how models learn from data and how to implement these algorithms efficiently in embedded systems. +and the loss across the entire dataset can be written as the average loss across all datapoints in the training data -- Gradient Descent -- Backpropagation -- Optimizers (SGD, Adam, RMSprop, etc.) +> Loss Function for Optimizing Neural Network Model on a Dataset +$$ +L_{full} = \frac{1}{M} \sum_{j=1}^{M} L(N(x_j; W_1,...W_n), y_j) +$$ -## Training Environments +### Training Neural Networks with Gradient Descent -Explanation: Different training environments have their own pros and cons. This section helps the reader make informed decisions about where to train their models, considering factors like computational resources and latency. +Now that we have a measure of how good our network fits the training data, we can optimize the weights of the neural network to minimize this loss. At a high level, we tweak the parameters of the real-valued matrices $W_i$s so that the loss function $L_{full}$ is minimized. Overall, our mathematical objective is -- Local vs. Cloud -- Specialized Hardware (GPUs, TPUs, etc.) +> Neural Network Training Objective +$$ +min_{W_1, ..., W_n} L_{full} +$$ +$$ += min_{W_1, ..., W_n} \frac{1}{M} \sum_{j=1}^{M} L(N(x_j; W_1,...W_n), y_j) +$$ + +So how do we optimize this objective? Recall from calculus that minimizing a function can be done by taking the derivative of the function with respect to the input parameters and tweaking the parameters in the direction of the gradient. This technique is called "gradient descent" and concretely involves calculating the derivative of the loss function $L_{full}$ with respect to $W_1, ..., W_n$ to obtain a gradient for these parameters to take a step in, then updating these parameters in the direction of the gradient. Thus, we can train our neural network using gradient descent which repeatedly applies the update rule + +> Gradient Descent Update Rule +$$ +W_i := W_i - \lambda \frac{\partial L_{full}}{\partial W_i} \mbox{ for } i=1..n +$$ + +:::{.callout-note} +In practice, the gradient is computed over a minibatch of datapoints, to improve computational efficiency. This is called stochastic gradient descent or batch gradient descent. +::: + +where $\lambda$ is the stepsize or learning rate of our tweaks. In training our neural network, we repeatedly perform the step above until convergence, or when the loss no longer decreases. This prior approach is known as "full gradient descent" since we are computing the derivative with respect to the entire training data, and only then taking a single gradient step; a more efficient approach is to calculate the gradient with respect to just a random "batch" of datapoints and then taking a step, a process known as "batch gradient descent" or "stochastic gradient descent", which is more efficient since now we are taking many more steps per pass of the entire training data. Next we will cover the mathematics behind computing the gradient of the loss function with respect to the $W_i$s, a process known as "backpropagation". + +### Backpropagation + +Training neural networks involve repeated applications of the gradient descent algorithm, which involves computing the derivative of the loss function with respect to the $W_i$s. How do we compute the derivative of the loss with respect to the $W_i$s given that the $W_i$s are nested functions of each other in a deep neural network? The trick is to leverage the "chain rule": we can compute the derivative of the loss with respect to the $W_i$s by repeatedly applying the chain rule, in a complete process known as "backpropagation". Specifically, we can calculate the gradients by computing the derivative of the loss with respect to the outputs of the last layer, then progressively use this to compute the derivative of the loss with respect to each prior layer, all the way to the input layer. This process starts from the end of the network (the layer closest to the output) and progresses backwards, and hence gets its name "backpropagation". + +Let's break this down. We can compute the derivative of the loss with respect to the _the outputs of each layer of the neural network_ by using repeated applications of the chain rule + +$$ +\frac{\partial L_{full}}{L_{n}} = \frac{\partial \frac{1}{M} \sum_{j=1}^{M} L(N(x_j; W_1, ..., W_n), y_j)}{\partial L_{n}} +$$ + +$$ +\frac{\partial L_{full}}{\partial L_{n-1}} = \frac{\partial L_{full}}{\partial L_{n}} \frac{\partial L_{n}}{\partial L_{n-1}} +$$ + +or more generally + +$$ +\frac{\partial L_{full}}{L_{i}} = \frac{\partial L_{full}}{\partial L_{n}} \frac{\partial L_{n}}{\partial L_{n-1}} \frac{\partial L_{n-1}}{\partial L_{n-2}} ... \frac{\partial L_{i+1}}{\partial L_{i}} +$$ + +After computing the derivative of the loss with respect to the _output of each layer_, we can easily obtain the derivative of the loss with respect to the _parameters_, again using the chain rule: + +$$ +\frac{\partial L_{full}}{W_{i}} = \frac{\partial L_{full}}{L_{i}} \frac{L_{i}}{W_{i}} +$$ + +And this is ultimately how the derivatives of the layers' weights are computed using backpropagation! What does this concretely look like in a specific example? Below we walk through a specific example on a simple 2 layer neural network, on a regression task using a MSE loss function, with 100-dimensional inputs and a 30-dimensional hidden layer: + +> Example of Backpropagation\ +Suppose we have a two-layer neural network +$$ +L_1 = ReLU(W_1 \times L_{0}) +$$ +$$ +L_2 = ReLU(W_2 \times L_{1}) +$$ +$$ +NN(x) = \mbox{Let } L_{0} = x \mbox{ then output } L_2 +$$ +where $W_1 \in \mathbb{R}^{30 \times 100}$ and $W_2 \in \mathbb{R}^{1 \times 30}$. Furthermore suppose we use the MSE loss function: +$$ +L(x, y) = (x-y)^2 +$$ +We wish to compute +$$ +\frac{\partial L(NN(x), y)}{\partial W_i} \mbox{ for } i=1,2 +$$ +We start by computing the gradient with respect to the final output: +$$ +\frac{\partial L(NN(x), y)}{\partial L_2} = \frac{\partial (L_2 - y)^2}{\partial L_2} = 2(L_2 - y) +$$ +With respect to the output $L_1$ +$$ +\frac{\partial L(NN(x), y)}{\partial L_1} = \frac{\partial L(NN(x), y)}{\partial L_2} \frac{\partial L_2}{\partial L_1} +$$ +$$ += [2(L_2 - y)] \times \frac{\partial ReLU(W_2 \times L_1)}{\partial L_1} = [2(L_2 - y)] \times W_2^T ReLU'(W_2 \times L_1) +$$ +where +$$ +ReLU'(x) = \begin{cases} + 0 & x\leq 0 \\ + 1 & x > 0 + \end{cases} +$$ +Then we can compute the gradients with respect to the weights +$$ +\frac{\partial L(NN(x), y)}{\partial W_2} = \frac{\partial L(NN(x), y)}{\partial L_2} \frac{\partial L_2}{\partial W_2} = [2(L_2 - y)] \times ReLU'(W_2 \times L_1) L_1^T +$$ +$$ +\frac{\partial L(NN(x), y)}{\partial W_1} = \frac{\partial L(NN(x), y)}{\partial L_1} \frac{\partial L_1}{\partial W_1} = [(2(L_2 - y)) \times W_2^T ReLU'(W_2 \times L_1)] \times ReLU'(W_1 \times L_0) +$$ +IMPORTANT: QUICKLY WRITTEN SO THIS IS NOT REALLY CORRECT THERE IS ACTUALLY A BUG HERE, REDO! + +## Optimizers + +Stochastic Gradient Descent involves updating the model's parameters by considering the gradient of the loss function with respect to the parameters for each training example. While the basic concept of SGD is straightforward, finding the optimal set of parameters that minimizes the overall loss across the entire dataset may be difficult as the loss landscape is nonconvex. + +To address the complexities of training neural networks, various optimization algorithms have been developed. These optimizers are designed to enhance the efficiency and convergence speed of the training process. They achieve this by adjusting the learning rates, incorporating momentum, and implementing adaptive strategies, among other techniques. + +Some optimizers include: + +* ADAM +* AdaGrad +* RMSProp +* Momentum SGD + +Generally, from our experience Adam is the most popular optimizer and usually outperforms SGD in terms of training neural networks. ## Hyperparameter Tuning -Explanation: Hyperparameters can significantly impact the performance of a trained model. This section educates the reader on how to fine-tune these settings for optimal results, which is especially important for resource-constrained embedded systems. +The performance of the neural network model depends on a set of +crucial configurations known as hyperparameters. These +hyperparameters, which are external to the model and cannot be learned +during training, play a pivotal role in determining the model's +effectiveness, generalization capabilities, and overall performance on +diverse datasets. These hyperparameters include learning rate, batch +size, regularization strengths, and network architectures. + +Several techniques are employed to conduct hyperparameter search, +ranging from manual tuning by domain experts to automated methods such +as grid search, random search, and more sophisticated optimization +algorithms like Bayesian optimization. + +## Regularization + +Neural networks are optimized to fit the training data, however, we +would like the network to generalize and perform well to unseen data. + +Regularization stands as a key technique in achieving this. It serves +as a tool in preventing overfitting, a common pitfall where a model +becomes overly complex and tailored to the training data, losing its +ability to generalize. + +Common regularization techniques include + +* L1 and L2 Regularization + + These methods add a penalty term based on the magnitudes of the model's parameters to the loss function. L1 regularization encourages sparsity by introducing a penalty proportional to the absolute values of the parameters, while L2 regularization penalizes the square of the parameter values, promoting smaller weights. + +* Dropout + + A technique commonly applied to neural networks, dropout involves randomly "dropping out" a fraction of neurons during each training iteration. This helps prevent co-adaptation of neurons, promoting more robust and generalized learning. + +* Early Stopping + + By monitoring the model's performance on a validation set during training, early stopping interrupts the training process when the model's performance ceases to improve. This prevents the model from becoming overly specialized to the training data. + +## Weight Initialization + +Neural Networks are trained starting with weights initialized randomly, but how the weights are initialized may have a considerable impact on training convergence and feasibility. +Proper weight initialization helps in overcoming issues like vanishing or exploding gradients, which can hinder the learning process. Here are some commonly used neural network weight initialization techniques: + + + as orthogonal matrices. This helps in preserving the gradients during backpropagation and can be particularly useful in recurrent neural networks (RNNs). + Uniform and Normal Initialization: + +Weights are initialized with random values drawn from a uniform or normal distribution. The choice between uniform and normal depends on the specific requirements of the model and the activation functions used. +Choosing the right initialization method depends on the architecture of the neural network, the activation functions, and the specific problem being solved. Experimentation with different techniques is often necessary to find the most suitable initialization for a given scenario. + +* Xavier/Glorot Initialization + + Proposed by Xavier Glorot and Yoshua Bengio, this initialization is designed to work well with activation functions like tanh or logistic sigmoid. It initializes weights with random values drawn from a distribution with mean 0 and variance 2 / (number of input units + number of output units). + +* He Initialization + + Proposed by Kaiming He et al., this initialization is tailored for ReLU (Rectified Linear Unit) activation functions. It initializes weights with random values drawn from a distribution with mean 0 and variance 2 / number of input units. This helps to mitigate the vanishing gradient problem often associated with deep networks. -- Learning Rate -- Batch Size -- Number of Epochs -- Regularization Techniques +## Activation Functions -## Evaluation Metrics +Activation functions play a critical role in neural networks by introducing non-linearities into the model. These non-linearities enable neural networks to learn complex relationships and patterns in data, making them capable of solving a wide range of problems. Here are some commonly used activation functions: -Explanation: Knowing how to evaluate a model's performance is crucial. This section introduces metrics that help in assessing how well the model will perform in real-world embedded applications. +* Rectified Linear Unit (ReLU): -- Accuracy -- Precision and Recall -- F1 Score -- ROC and AUC + ReLU is a popular activation function that returns zero for + negative input values and passes positive input values + unchanged. It is computationally efficient and has been widely + used in deep learning models. -## Overfitting and Underfitting +* Sigmoid Function -Explanation: Overfitting and underfitting are common pitfalls in AI training. This section is vital for teaching strategies to avoid these issues, ensuring that the model generalizes well to new, unseen data. + The sigmoid function, also known as the logistic function, + squashes input values between 0 and 1. It is often used in the + output layer of binary classification models, where the goal is to + produce probabilities. -- Techniques to Avoid Overfitting (Dropout, Early Stopping, etc.) -- Understanding Underfitting and How to Address It +* Hyperbolic Tangent Function (tanh): -## Transfer Learning + The hyperbolic tangent function is similar to the sigmoid but + squashes input values between -1 and 1. It is often used in hidden + layers of neural networks, especially when zero-centered outputs + are desired. -Explanation: Transfer learning can save time and computational resources, which is particularly beneficial for embedded systems. This section explains how to leverage pre-trained models for new tasks. +## Key System Bottlenecks in AI inference & training -- Basics of Transfer Learning -- Applications in Embedded AI +As introduced, neural networks consist of alternating linear and nonlinear operations. The main performance bottleneck is the linear layer, a matrix multiplication that maps the previous activations inputs to the next layer's activation function. -## Challenges and Best Practices +### Runtime Complexity of Matrix Multiplication in Neural Networks -Explanation: Every technology comes with its own set of challenges. This section prepares the reader for potential hurdles in AI training, offering best practices to navigate them effectively. +Matrix multiplication is the main performance bottleneck in both +neural network inference and training since its runtime complexity is +the product of the dimensions of the input and output layers of the +neural network, as well as the batch size. Generally, using a batch +size of $B$ (i.e: training on batches of $B$ datapoints at a time), +with an input layer dimension of $M$ nodes, and an output layer +dimensions of $N$ nodes, the linear layer is a matrix-matrix +multiplication size $N \times M$ by a $M \times B$, leading to a +complexity of $O(NMB)$. This far dominates the computational +complexity of the nonlinear activation functions which only need to be +applied element-wise to the output vectors. -- Computational Constraints -- Data Privacy -- Ethical Considerations +### Compute vs Memory Bottleneck in Neural Network Training and Inference -## Conclusion +As previously outlined, matrix multiplication is the key operational +bottleneck in both neural network training and inference. However, +this operation may pose a bottleneck to either the memory or +computational capability of the underlying hardware system. +Concretely, batched matrix multiplication (i.e: matrix-multiplicaton +with a high batch-size $B$) exhibits a much higher computation to +memory ratio, and hence the underlying computational hardware must +have more arithmetic computing capabilities than memory-transfer +abilities to maximize performance in these scenarios (this is often +the case in both CPUs and GPUs). This is because matrix multiplication +performs $O(NMB)$ arithmetic operations, but only $O(NM + MB)$ memory +operations; hence when $B$ is large, neural network inference/training +requires more arithmetic operations, and when $B$ is small it requires +relatively more memory operations. This detail is important since most +of today's hardware is better at performing computation rather than +memory transfer, and hence from a computational perspective it is +better to maximize batch size $B$ at training time to fully utilize +the hardware. However, using a larger batch size $B$ at training time +means doing fewer updates per pass of the dataset, which might +decrease convergence time. Hence, selecting batch size $B$ has a +considerable impact on both the convergence, computational efficiency, +and runtime of neural networks, and must be tuned to attain high +performance. Generally, a good rule of thumb for batch size is between +8-128. -Explanation: A summary helps to consolidate the key points of the chapter, aiding in better retention and understanding of the material. +### Optimizing Matrix Multiplication -- Key Takeaways -- Future Trends in AI Training for Embedded Systems \ No newline at end of file +Matrix multiplication is essential to the performance of neural +network inference and training, and hence efficiently evaluating +matrix multiplication is crucial to performance. Broadly, various +methods to accelerate matrix-multiplication include carefully writing +the matrix-multiply routine to leverage the caches of the CPU, +leveraging hardware-level parallel operations (i.e: SIMD), using +hardware capabilities that perform multiply-adds simultaneously (i.e: +fused add-multiply, and others. Even better at performing matrix +multiplication than the CPU are GPUs, which have hardware that is +suitable for performing a mass amount of parallel operations, a +capability that is perfect for executing matrix-multiplication whose +individual steps are highly parallelizable. Using the GPU, matrix +multiplication can be accelerated by a significant factor over the GPU +since the matrix-multiplication can be efficiently parallelized on a +GPU, using standard techniques like blocking and tiling to maximize +resource utilization. Other alternative ways to accelerate +matrix-multiplication include using FPGAs, or dedicated hardware like +TPUs (i.e: systolic arrays). + +## Parallelizing AI training + +## Data Parallel + +## Model Parallel + + +## Distributed Training \ No newline at end of file From 5c266132521e85757da32e4f8907c793d1832760 Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Tue, 14 Nov 2023 16:30:42 -0500 Subject: [PATCH 02/45] Upd --- training.qmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/training.qmd b/training.qmd index 645a4b64..8c8a3829 100644 --- a/training.qmd +++ b/training.qmd @@ -11,7 +11,7 @@ ## Introduction -Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. In this section, we will explore the mathematics behind neural networks, specifically how neural networks work and how to train them, then dive into some of the key system challenges that underpin these models, and finally explore how to leverage CPUs and GPUs to accelerate them. +Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. In this section, we will explore the mathematics behind neural networks, specifically how neural networks work and how to train them, then dive into some of the key system challenges that underpin these models, and finally explore how to leverage CPUs and GPUs to accelerate them. ## Mathematics behind Neural Networks and Deep Learning From 5af74687e2348cf695555d00462251e229801348 Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Tue, 14 Nov 2023 19:52:15 -0500 Subject: [PATCH 03/45] Added section place holders --- training.qmd | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/training.qmd b/training.qmd index 8c8a3829..ff2f3280 100644 --- a/training.qmd +++ b/training.qmd @@ -167,7 +167,7 @@ $$ $$ IMPORTANT: QUICKLY WRITTEN SO THIS IS NOT REALLY CORRECT THERE IS ACTUALLY A BUG HERE, REDO! -## Optimizers +## Optimization Algorithms Stochastic Gradient Descent involves updating the model's parameters by considering the gradient of the loss function with respect to the parameters for each training example. While the basic concept of SGD is straightforward, finding the optimal set of parameters that minimizes the overall loss across the entire dataset may be difficult as the loss landscape is nonconvex. @@ -334,9 +334,11 @@ TPUs (i.e: systolic arrays). ## Parallelizing AI training -## Data Parallel +### Data Parallel -## Model Parallel +### Model Parallel -## Distributed Training \ No newline at end of file +## Efficient and Distributed Training + +## Debugging and Profiling \ No newline at end of file From 624e6c17609512fe797ab4a6da6380410d119918 Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Wed, 15 Nov 2023 08:51:37 -0500 Subject: [PATCH 04/45] Updated the introduction --- training.qmd | 10 +++++++++- 1 file changed, 9 insertions(+), 1 deletion(-) diff --git a/training.qmd b/training.qmd index ff2f3280..fa072617 100644 --- a/training.qmd +++ b/training.qmd @@ -11,7 +11,15 @@ ## Introduction -Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. In this section, we will explore the mathematics behind neural networks, specifically how neural networks work and how to train them, then dive into some of the key system challenges that underpin these models, and finally explore how to leverage CPUs and GPUs to accelerate them. +Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. + +Briefly, neural networks are computing systems loosely inspired by how the brain works. They learn to perform tasks like image recognition, language translation, and more by analyzing examples, rather than being explicitly programmed with rules. + +At a high level, neural networks are made up of simple functions ("neurons") layered on top of each other. Each layer takes in some data, performs a calculation on it, and passes it to the next layer. For example, in an image recognition network, the input layer may take in pixel values. The next layers detect simple shapes like edges, then larger shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. + +The "network" in a neural network refers to how these neurons are connected. Each neuron is connected to many others in the layers above and below it. The neurons have numeric weights associated with them, kind of like the synaptic strengths in a brain neuron. The network is trained by adjusting these weights. Initially the weights are set randomly. An input is fed in, and the output is compared to the desired result using a loss function. This loss function outputs a number indicating how far off the network's prediction was. The weights are then tweaked slightly to reduce the loss. This process is repeated with many training examples until the network reliably minimizes the loss, indicating it has learned the patterns in the data. This training technique, called backpropagation, allows neural networks to learn without explicit programming. Once trained, the network can apply what it's learned to new unseen data. + +In this chapter, we will explore the mathematics behind neural networks, specifically how neural networks work and how to train them, then we dive into some of the key system challenges that underpin these models, and finally explore how to leverage CPUs and GPUs to accelerate them. ## Mathematics behind Neural Networks and Deep Learning From bf96aca101d4e7acfac41c6c23f85bb800000d40 Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Wed, 15 Nov 2023 10:12:51 -0500 Subject: [PATCH 05/45] Add overview --- training.qmd | 1 + 1 file changed, 1 insertion(+) diff --git a/training.qmd b/training.qmd index fa072617..d9d2f0a2 100644 --- a/training.qmd +++ b/training.qmd @@ -1,4 +1,5 @@ # AI Training +The process of training is central to developing accurate and useful AI systems using machine learning techniques. At a high level, training involves feeding data into machine learning algorithms so they can learn patterns and make predictions. However, effectively training models requires tackling a variety of challenges around data, algorithms, optimization of model parameters, and enabling generalization. In this chapter, we will dive into the nuances and considerations around training machine learning models. ::: {.callout-tip} ## Learning Objectives From ac1f6598625bd104a8648c03ca1a230b50dcd39d Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Wed, 15 Nov 2023 10:13:53 -0500 Subject: [PATCH 06/45] Updated introduction --- training.qmd | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) diff --git a/training.qmd b/training.qmd index d9d2f0a2..93dc25ef 100644 --- a/training.qmd +++ b/training.qmd @@ -1,4 +1,5 @@ # AI Training + The process of training is central to developing accurate and useful AI systems using machine learning techniques. At a high level, training involves feeding data into machine learning algorithms so they can learn patterns and make predictions. However, effectively training models requires tackling a variety of challenges around data, algorithms, optimization of model parameters, and enabling generalization. In this chapter, we will dive into the nuances and considerations around training machine learning models. ::: {.callout-tip} @@ -12,6 +13,22 @@ The process of training is central to developing accurate and useful AI systems ## Introduction +Training is a critical process for developing accurate and useful AI systems using machine learning. The goal of training is to create a machine learning model that can generalize to new, unseen data, rather than memorizing the training examples. This is done by feeding **training data** into algorithms that learn patterns from these examples by adjusting internal parameters. + +The algorithms minimize a **loss function**, which compares their predictions on the training data to the known labels or solutions, guiding the learning. Effective training often requires high-quality, representative training data sets that are large enough to capture variability in the real-world use cases. + +It also requires choosing an **algorithm** suited to the task, whether that be a neural network for computer vision, a reinforcement learning algorithm for robotic control, or a tree-based method for categorical prediction. Careful tuning is needed for the model structure, such as neural network depth and width, and learning parameters like step size and regularization strength. + +Techniques to prevent **overfitting** like regularization penalties and validation with held-out data are also important. Overfitting can occur when a model fits the training data too closely, failing to generalize to new data. This can happen if the model is too complex or trained for too long. + +To avoid overfitting **regularization** techniques can help constrain the model. One regularization method is adding a penalty term to the loss function that discourages complexity, like the L2 norm of the weights. This penalizes large parameter values. Another technique is dropout, where a percentage of neurons are randomly set to zero during training. This reduces co-adaptation of neurons. + +**Validation** methods also help detect and avoid overfitting. Part of the training data is held out from the training loop as a validation set. The model is evaluated on this data. If validation error increases while training error decreases, overfitting is occurring. The training can then be stopped early or regularized more strongly. Careful use of regularization and validation enables models to train to maximum capability without overfitting the training data. + +Training takes significant **computing resources**, especially for deep neural networks used in computer vision, natural language processing, and other areas. These networks have millions of adjustable weights that must be tuned through extensive training. Hardware improvements and distributed training techniques have enabled training ever larger neural nets that can achieve human-level performance on some tasks. + +If some of the bold terms sound new, then that's good! Don't worry, we will walk you through these details in the rest of the sections. Understanding how to effectively leverage data, algorithms, parameter optimization, and generalization through thorough training is essential for developing capable, deployable AI systems that work robustly in the real world. + Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. Briefly, neural networks are computing systems loosely inspired by how the brain works. They learn to perform tasks like image recognition, language translation, and more by analyzing examples, rather than being explicitly programmed with rules. From aab70188f7cd7ec003f6304a94445095634823c9 Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Wed, 15 Nov 2023 10:14:35 -0500 Subject: [PATCH 07/45] updated NN intro --- training.qmd | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/training.qmd b/training.qmd index 93dc25ef..ca8b54e5 100644 --- a/training.qmd +++ b/training.qmd @@ -29,11 +29,13 @@ Training takes significant **computing resources**, especially for deep neural n If some of the bold terms sound new, then that's good! Don't worry, we will walk you through these details in the rest of the sections. Understanding how to effectively leverage data, algorithms, parameter optimization, and generalization through thorough training is essential for developing capable, deployable AI systems that work robustly in the real world. +## Mathematics behind Neural Networks and Deep Learning + Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. -Briefly, neural networks are computing systems loosely inspired by how the brain works. They learn to perform tasks like image recognition, language translation, and more by analyzing examples, rather than being explicitly programmed with rules. +Briefly, neural networks are mathematical models that consist of alternating linear and nonlinear operations, parameterized by a set of "weights" that are trained to minimize some loss function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value (i.e: the "loss") when evaluated on our model against the training data. Fundamentally, training neural networks involve evaluating the loss function to get a measure of how good our model is, then continuously tweaking the weights of our model so that the loss decreases, which ultimately optimizes the model to fit our training data. -At a high level, neural networks are made up of simple functions ("neurons") layered on top of each other. Each layer takes in some data, performs a calculation on it, and passes it to the next layer. For example, in an image recognition network, the input layer may take in pixel values. The next layers detect simple shapes like edges, then larger shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. +Neural networks are made up of simple functions ("neurons") layered on top of each other. Each layer takes in some data, performs a calculation on it, and passes it to the next layer. For example, in an image recognition network, the input layer may take in pixel values. The next layers detect simple shapes like edges, then larger shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. The "network" in a neural network refers to how these neurons are connected. Each neuron is connected to many others in the layers above and below it. The neurons have numeric weights associated with them, kind of like the synaptic strengths in a brain neuron. The network is trained by adjusting these weights. Initially the weights are set randomly. An input is fed in, and the output is compared to the desired result using a loss function. This loss function outputs a number indicating how far off the network's prediction was. The weights are then tweaked slightly to reduce the loss. This process is repeated with many training examples until the network reliably minimizes the loss, indicating it has learned the patterns in the data. This training technique, called backpropagation, allows neural networks to learn without explicit programming. Once trained, the network can apply what it's learned to new unseen data. From fb27e3888ef795ad81d198181a623d1b2a217d71 Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Wed, 15 Nov 2023 10:17:48 -0500 Subject: [PATCH 08/45] Minor rework of the writing --- training.qmd | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/training.qmd b/training.qmd index ca8b54e5..f0f273bc 100644 --- a/training.qmd +++ b/training.qmd @@ -31,19 +31,17 @@ If some of the bold terms sound new, then that's good! Don't worry, we will walk ## Mathematics behind Neural Networks and Deep Learning -Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. +Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. Briefly, neural networks are mathematical models that consist of alternating linear and nonlinear operations, parameterized by a set of "weights" that are trained to minimize some loss function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value (i.e: the "loss") when evaluated on our model against the training data. Fundamentally, training neural networks involve evaluating the loss function to get a measure of how good our model is, then continuously tweaking the weights of our model so that the loss decreases, which ultimately optimizes the model to fit our training data. Neural networks are made up of simple functions ("neurons") layered on top of each other. Each layer takes in some data, performs a calculation on it, and passes it to the next layer. For example, in an image recognition network, the input layer may take in pixel values. The next layers detect simple shapes like edges, then larger shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. -The "network" in a neural network refers to how these neurons are connected. Each neuron is connected to many others in the layers above and below it. The neurons have numeric weights associated with them, kind of like the synaptic strengths in a brain neuron. The network is trained by adjusting these weights. Initially the weights are set randomly. An input is fed in, and the output is compared to the desired result using a loss function. This loss function outputs a number indicating how far off the network's prediction was. The weights are then tweaked slightly to reduce the loss. This process is repeated with many training examples until the network reliably minimizes the loss, indicating it has learned the patterns in the data. This training technique, called backpropagation, allows neural networks to learn without explicit programming. Once trained, the network can apply what it's learned to new unseen data. +The "network" in a neural network refers to how these neurons are connected. Each neuron is connected to many others in the layers above and below it. The neurons have numeric weights associated with them, kind of like the synaptic strengths in a brain neuron. The network is trained by adjusting these weights. -In this chapter, we will explore the mathematics behind neural networks, specifically how neural networks work and how to train them, then we dive into some of the key system challenges that underpin these models, and finally explore how to leverage CPUs and GPUs to accelerate them. +Initially the weights are set randomly. An input is fed in, and the output is compared to the desired result using a loss function. This loss function outputs a number indicating how far off the network's prediction was. The weights are then tweaked slightly to reduce the loss. -## Mathematics behind Neural Networks and Deep Learning - -At a high level, neural networks are mathematical models that consist of alternating linear and nonlinear operations, parameterized by a set of "weights" that are trained to minimize some loss function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value (i.e: the "loss") when evaluated on our model against the training data. Fundamentally, training neural networks involve evaluating the loss function to get a measure of how good our model is, then continuously tweaking the weights of our model so that the loss decreases, which ultimately optimizes the model to fit our training data. +This process is repeated with many training examples until the network reliably minimizes the loss, indicating it has learned the patterns in the data. This training technique, called backpropagation, allows neural networks to learn without explicit programming. Once trained, the network can apply what it's learned to new unseen data. ### Neural Network Notation From 65b4812dc66f29006636219f5077a8b557ae4bbb Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Wed, 15 Nov 2023 10:53:27 -0500 Subject: [PATCH 09/45] Added training data content --- training.qmd | 175 +++++++++++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 171 insertions(+), 4 deletions(-) diff --git a/training.qmd b/training.qmd index f0f273bc..3fb4cf3e 100644 --- a/training.qmd +++ b/training.qmd @@ -193,6 +193,172 @@ $$ $$ IMPORTANT: QUICKLY WRITTEN SO THIS IS NOT REALLY CORRECT THERE IS ACTUALLY A BUG HERE, REDO! + +## Training Data + +To enable effective training of neural networks, the available data must be split into training, validation, and test sets. The training set is used to train the model parameters. The validation set evaluates the model during training to tune hyperparameters and prevent overfitting. The test set provides an unbiased final evaluation of the trained model's performance. + +Maintaining clear splits between train, validation, and test sets with representative data in each is crucial to properly training, tuning, and evaluating models to achieve the best real-world performance. To this end, we will learn about the common pitfalls or mistakes that people make in creating these data splits. + +Here is a summary table for training, validation, and test data splits: + +| Data Split | Purpose | Typical Size | +|-|-|-| +| Training Set | Train the model parameters | 60-80% of total data | +| Validation Set | Evaluate model during training to tune hyperparameters and prevent overfitting | ∼20% of total data | +| Test Set | Provide unbiased evaluation of final trained model | ∼20% of total data | + + +### Dataset Splits + +#### Training Set + +The training set is used to actually train the model. It is the largest subset consisting of typically 60-80% of the total data. The model sees and learns from the training data in order to make predictions. A sufficiently large and representative training set is required for the model to effectively learn the underlying patterns. + +#### Validation Set + +The validation set is used to evaluate the model during training, usually after each epoch. Typically 20% of the data is allocated for the validation set. The model does not learn or update its parameters based on the validation data. It is used to tune hyperparameters and make other tweaks to improve training. Monitoring metrics like loss and accuracy on the validation set prevents overfitting on just the training data. + +#### Test Set + +The test set acts as a completely unseen dataset that the model did not see during training. It is used to provide an unbiased evaluation of the final trained model. Typically 20% of the data is reserved for testing. Maintaining a hold-out test set is vital for obtaining an accurate estimate of how the trained model would perform on real world unseen data. Data leakage from the test set must be avoided at all costs. + +The relative proportions of the training, validation and test sets can vary based on data size and application. But following the general guideline of a 60/20/20 split is a good starting point. Careful splitting of data ensures models are properly trained, tuned and evaluated to achieve the best performance. + +### Common Pitfalls and Mistakes + +#### Insufficient Training Data + +Allocating too little data to the training set is a common mistake when splitting data that can severely impact model performance. If the training set is too small, the model will not have enough samples to effectively learn the true underlying patterns in the data. This leads to high variance and causes the model to fail to generalize well to new data. + +For example, if you are training an image classification model to recognize handwritten digits, providing only 10 or 20 images per digit class would be completely inadequate. The model would struggle to capture the wide variances in writing styles, rotations, stroke widths and other variations with so few examples. + +As a rule of thumb, the training set size should be at least in the hundreds or thousands of examples for most machine learning algorithms to work effectively. For deep neural networks, especially those using convolutional layers, the training set often needs to be in the tens or hundreds of thousands due to the large number of parameters. + +Insufficient training data typically manifests in symptoms like high error rates on validation/test sets, low model accuracy, high variance, and overfitting on the small training set samples. Collecting more quality training data is the solution. Data augmentation techniques can also help virtually increase training data size for images, audio etc. + +Carefully factoring in the model complexity and problem difficulty when allocating training samples is important to ensure sufficient data is available for the model to learn successfully. Following guidelines on minimum training set sizes for different algorithms is also recommended. Insufficient training data is a fundamental issue that will undermine the overall success of any machine learning application. + +#### Data Leakage Between Sets + +Data leakage refers to the unintentional transfer of information between the training, validation, and test sets. This violates the fundamental assumption that the splits are completely separated. Data leakage leads to seriously compromised evaluation results and inflated performance metrics. + +A common way data leakage can occur is if some samples from the test set inadvertently get included in the training data. Now when evaluating on the test set, the model has already seen some of the data which gives overly optimistic scores. For example, if 2% of the test data leaks into the training set of a binary classifier, it can result in a accuracy boost of up to 20%! + +More subtle forms of leakage can happen if the data splits are not done carefully. If the splits are not properly randomized and shuffled, samples close to each other in the dataset may end up across different splits. This creates information bleed through based on proximity in the dataset. Time series data is especially vulnerable unless special cross validation techniques are used. + +Preventing data leakage requires creating solid separation between splits - no sample should exist in more than one split. Shuffling and randomized splitting help create robust divisions. Cross validation techniques can be used for more rigorous evaluation. Detecting leakage is difficult buttelltale signs include models doing way better on test vs. validation data. + +Data leakage severely compromises the validity of evaluation because the model has already partially seen the test data. No amount of tuning or complex architectures can substitute for clean data splits. It is better to be conservative and create complete separation between splits to avoid this fundamental mistake in machine learning pipelines. + +#### Small or Unrepresentative Validation Set + +The validation set is used to evaluate models during training and for hyperparameter tuning. If the validation set is too small or not representative of the real data distribution, it will not provide reliable or stable evaluations during training. This makes model selection and tuning more difficult. + +For example, if the validation set only contains 100 samples, metrics calculated on it will have high variance. The accuracy may fluctuate up to 5-10% between epochs just due to noise. This makes it difficult to know if a drop in validation accuracy is due to overfitting or natural variance. With a larger validation set of say 1000 samples, the metrics will be much more stable. + +Additionally, if the validation set is not representative, perhaps missing certain subclasses, the estimated skill of the model may be inflated. This could lead to poor choices of hyperparameters or stopping training prematurely. Models selected based on such biased validation sets do not generalize well to real data. + +A good rule of thumb is the validation set size should be at least several hundred samples, and up to 10-20% size of the training set. The splits should also be stratified, especially if working with imbalanced datasets. A larger validation set that well represents the original data characteristics is essential for proper model selection and tuning. + +Care should be taken that the validation set is also not too large, leaving insufficient samples for training. Overall, the validation set is a critical piece of the data splitting process and care should be taken to avoid the pitfalls of small, inadequate samples that negatively impact model development. + +#### Reusing the Test Set Multiple Times + +The test set is designed to provide an unbiased evaluation of the fully-trained model only once at the end of the model development process. Reusing the test set multiple times during development for model evaluation, hyperparameter tuning, model selection etc. can result in overfitting on the test data. + +If the test set is reused as part of the validation process, the model may start to see and learn from the test samples. This coupled with intentionally or unintentionally optimizing model performance on the test set can artificially inflate metrics like accuracy. + +For example, if the test set is used repeatedly for model selection out of 5 architectures, the model may achieve 99% test accuracy just by memorizing the samples rather than learning generalizable patterns. However, deployed in the real world, the accuracy could drop to 60% on new data. + +Best practice is to interact with the test set only once at the very end to report unbiased metrics on how the final tuned model would perform in the real world. The validation set should be used for all parameter tuning, model selection, early stopping etc. while developing the model. + +Maintaining the complete separation of training/validation from the test set is essential to obtain accurate estimates of model performance. Even minor deviations from single use of the test set could positively bias results and metrics, providing an overly optimistic view of real world efficacy. + +#### Same Data Splits Across Experiments + +When comparing different machine learning models or experimenting with various architectures and hyperparameters, using the same data splits for training, validation and testing across the different experiments can introduce bias and invalidate the comparisons. + +If the same splits are reused, the evaluation results may be overly correlated and not provide an accurate measure of which model performs better. For example, a certain random split of the data may happen to favor model A over model B irrespective of the algorithms. Reusing this split will then be biased towards model A. + +Instead, the data splits should be randomized or shuffled for each experimental iteration. This ensures that randomness in the sampling of the splits does not confer an unfair advantage to any model. + +With different splits per experiment, the evaluation becomes more robust. Each model is tested on a wide range of test sets drawn randomly from the overall population. This smoothens out variation and removes correlation between results. + +Proper practice is to set a random seed before splitting the data for each experiment. Splitting should be carried out after any shuffling/resampling as part of the experimental pipeline. Carrying out comparisons on the same splits violates the i.i.d (independent and identically distributed) assumption required for statistical validity. + +Unique splits are essential for fair model comparisons. Though more compute intensive, randomized allocation per experiment removes sampling bias and enables valid benchmarking. This highlights the true differences in model performance irrespective of a particular split's characteristics. + +#### Information Leakage Between Sets + +Information leakage between the training, validation and test sets occurs when information from one set inadvertently bleeds into another set. This could happen due to flaws in the data splitting process and violates the assumption that the sets are mutually exclusive. + +For example, consider a dataset sorted chronologically. If a simple random split is performed, samples close to each other in the dataset may end up in different splits. Models could then learn from 'future' data if test samples are leaked into the training set. + +Similarly, if the splits are not properly shuffled, distribution biases may persist across sets. The training set may not contain certain outliers that end up in the test set only, compromising generalization. Issues like class imbalance may also get amplified if splitting is not stratified. + +Another case is when datasets have linked samples that are inherently connected, such as graphs, networks or time series data. Naive splitting may isolate connected nodes or time steps into different sets. Models can make invalid assumptions based on partial information. + +Preventing information leakage requires awareness of the structure of the dataset and relationships between samples. Shuffling, stratification and grouped splitting of related samples can help mitigate leakage. Proper cross validation procedures should be followed, being mindful of temporal or sample proximity. + +Subtle leakage of information between sets undermines model evaluation and training. It creates misleading results on model effectiveness. Data splitting procedures should account for sample relationships and distribution differences to ensure mutual exclusivity between sets. + +#### Failing to Stratify Splits + +When splitting data into training, validation and test sets, failing to stratify the splits can result in uneven representation of the target classes across the splits and introduce sampling bias. This is especially problematic for imbalanced datasets. + +Stratified splitting involves sampling data points such that the proportion of output classes is approximately preserved in each split. For example, if performing a 70/30 train-test split on a dataset with 60% negative and 40% positive samples, stratification ensures ~60% negative and ~40% positive examples in both training and test sets. + +Without stratification, due to random chance, the training split could end up with 70% positive while test has 30% positive samples. The model trained on this skewed training distribution will not generalize well. Class imbalance also compromises model metrics like accuracy. + +Stratification works best when done using the labels though proxies like clustering can be used for unsupervised learning. It becomes essential for highly skewed datasets with rare classes that could easily get omitted from splits. + +Libraries like Scikit-Learn have stratified splitting methods inbuilt. Failing to use them could inadvertently introduce sampling bias and hurt model performance on minority groups. The overall class balance should be examined after performing the splits to ensure even representation across the splits. + +Stratification provides a balanced dataset for both model training and evaluation. Though simple random splitting is easy, being mindful of stratification needs, especially for real-world imbalanced data, results in more robust model development and evaluation. + +#### Ignoring Time Series Dependencies + +Time series data has an inherent temporal structure with observations depending on past context. Naively splitting time series data into train and test sets without accounting for this dependency leads to data leakage and lookahead bias. + +For example, simply splitting a time series into the first 70% training and last 30% as test data will contaminate the training data with future data points. The model can use this information to "peek" ahead during training. + +This results in overly optimistic evaluation of the model's performance. The model may appear to forecast the future accurately but has actually implicitly learned based on future data. This does not translate to real world performance. + +Proper time series cross validation techniques should be used to preserve order and dependency, such as forward chaining. The test set should only contain data points from a future time window that the model did not get exposed to for training. + +Failing to account for temporal relationships leads to invalid assumptions of causality. The model may also not learn how to extrapolate forecasts further into the future if the training data contains future points. + +Maintaining the temporal flow of events and avoiding lookahead bias is key for properly training and testing time series models to ensure they can truly predict future patterns and not just memorize past training data. + +#### No Unseen Data for Final Evaluation + +A common mistake when splitting data is failing to keep aside some portion of the data just for final evaluation of the completed model. All of the data is used for training, validation and test sets during development. + +This leaves no unseen data to get an unbiased estimate of how the final tuned model would perform in the real world. The metrics on the test set used during development may not fully reflect actual model skill. + +For example, choices like early stopping and hyperparameter tuning are often optimized based on performance on the test set. This couples the model to the test data. An unseen dataset is needed to break this coupling and get true real-world metrics. + +Best practice is to reserve a portion like 20-30% of the full dataset solely for final model evaluation. This data should not be used for any validation, tuning or model selection during development. + +Saving some unseen data allows evaluating the completely trained model as a black box on real-world like data. This provides reliable metrics to decide if the model is truly ready for production deployment. + +Failing to keep an unseen hold-out set for final validation risks optimistically biasing results and overlooking potential failures before model release. Having some fresh data provides a final sanity check on real-world efficacy. + +#### Overoptimizing on the Validation Set + +The validation set is meant to guide the model training process, not serve as additional training data. Overoptimizing on the validation set to maximize performance metrics treats it more like a secondary training set and leads to inflated metrics and poor generalization. + +For example, techniques like extensively tuning hyperparameters or adding data augmentations targeted to boost validation accuracy can cause the model to fit too closely to the validation data. The model may achieve 99% validation accuracy but only 55% test accuracy. + +Similarly, reusing the validation set for early stopping can also optimize the model specifically for that data. Stopping at the best validation performance overfits to noise and fluctuations caused by the small validation size. + +The validation set serves as a proxy to tune and select models. But the end goal remains maximizing performance on real-world data, not the validation set. Minimizing the loss or error on validation data does not automatically translate to good generalization. + +A good approach is to keep the validation set use minimal - hyperparameters can be tuned coarsely first on training data for example. The validation set guides the training, but should not influence or alter the model itself. It is a diagnostic, not an optimization tool. + +Care should be taken to not overfit when assessing performance on the validation set. Tradeoffs are needed to build models that perform well on the overall population, not overly tuned to the validation samples. + ## Optimization Algorithms Stochastic Gradient Descent involves updating the model's parameters by considering the gradient of the loss function with respect to the parameters for each training example. While the basic concept of SGD is straightforward, finding the optimal set of parameters that minimizes the overall loss across the entire dataset may be difficult as the loss landscape is nonconvex. @@ -292,7 +458,7 @@ Activation functions play a critical role in neural networks by introducing non- layers of neural networks, especially when zero-centered outputs are desired. -## Key System Bottlenecks in AI inference & training +## System Bottlenecks As introduced, neural networks consist of alternating linear and nonlinear operations. The main performance bottleneck is the linear layer, a matrix multiplication that maps the previous activations inputs to the next layer's activation function. @@ -358,13 +524,14 @@ resource utilization. Other alternative ways to accelerate matrix-multiplication include using FPGAs, or dedicated hardware like TPUs (i.e: systolic arrays). -## Parallelizing AI training +## Training Parallelization ### Data Parallel ### Model Parallel +## Distributed Training -## Efficient and Distributed Training +## Debugging and Profiling -## Debugging and Profiling \ No newline at end of file +## Conclusion \ No newline at end of file From 3c67582910333cc4d257b8b2329209739fa3db38 Mon Sep 17 00:00:00 2001 From: Vijay Janapa Reddi Date: Wed, 15 Nov 2023 10:53:48 -0500 Subject: [PATCH 10/45] Focusing on the big items we cover in the chapter --- training.qmd | 13 +++++++++++-- 1 file changed, 11 insertions(+), 2 deletions(-) diff --git a/training.qmd b/training.qmd index 3fb4cf3e..859c837b 100644 --- a/training.qmd +++ b/training.qmd @@ -27,9 +27,18 @@ To avoid overfitting **regularization** techniques can help constrain the model. Training takes significant **computing resources**, especially for deep neural networks used in computer vision, natural language processing, and other areas. These networks have millions of adjustable weights that must be tuned through extensive training. Hardware improvements and distributed training techniques have enabled training ever larger neural nets that can achieve human-level performance on some tasks. -If some of the bold terms sound new, then that's good! Don't worry, we will walk you through these details in the rest of the sections. Understanding how to effectively leverage data, algorithms, parameter optimization, and generalization through thorough training is essential for developing capable, deployable AI systems that work robustly in the real world. +In summary, some key points about training: -## Mathematics behind Neural Networks and Deep Learning +* **Data is crucial:** Machine learning models learn from examples in training data. More high-quality, representative data leads to better model performance. Data needs to be processed and formatted for training. +* **Algorithms learn from data:** Different algorithms (neural networks, decision trees, etc.) have different approaches to finding patterns in data. Choosing the right algorithm for the task is important. +* **Training refines model parameters:** Model training adjusts internal parameters to find patterns in data. Advanced models like neural networks have many adjustable weights. Training iteratively adjusts weights to minimize a loss function. +* **Generalization is the goal:** A model that overfits to the training data will not generalize well. Regularization techniques (dropout, early stopping, etc.) reduce overfitting. Validation data is used to evaluate generalization. +* **Training takes compute resources:** Training complex models requires significant processing power and time. Hardware improvements and distributed training across GPUs/TPUs have enabled advances. + +We will walk you through these details in the rest of the sections. Understanding how to effectively leverage data, algorithms, parameter optimization, and generalization through thorough training is essential for developing capable, deployable AI systems that work robustly in the real world. + + +## Basic Mathematics Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. From d3902eeb8cf1d9c7a8475e5bce7b80d85b0cde09 Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Sun, 19 Nov 2023 22:27:36 -0500 Subject: [PATCH 11/45] Update backprop --- training.qmd | 70 +++++++++++++++++++++++++++++++++------------------- 1 file changed, 44 insertions(+), 26 deletions(-) diff --git a/training.qmd b/training.qmd index 859c837b..a4059289 100644 --- a/training.qmd +++ b/training.qmd @@ -54,19 +54,22 @@ This process is repeated with many training examples until the network reliably ### Neural Network Notation -Diving into the details, the core of a neural network as introduced in section 3 can be viewed as a sequence of alternating linear and nonlinear operations: +Diving into the details, the core of a neural network can be viewed as a sequence of alternating linear and nonlinear operations: $$ -L_i = F_{i}(W_i \times L_{i-1}) +L_i = W_i A_{i-1} +$$ +$$ +A_i = F_i(L_{i}) $$ :::{.callout-note} Convolutions are also linear operators, and can be cast as a matrix multiplication. ::: -where $L_{0}$ is a vector input to the neural network (i.e: an image that we want the neural network to classify, or some other data that the neural network operates on), $L_{n}$ (where $n$ is the number of layers of the network) is the vector output of the neural network (i.e: a vector of size 10 in the case of classifying pictures of handwritten digits), $W_i$s are the weights of the neural network that are tweaked at training time to fit our data, and $F_{i}$ is that layer's nonlinear activation function (i.e: ReLU, softmax, etc). As defined, the intermediate output of the neural network is a vector of real-valued numbers with dimensions: +where $A_{0}$ is a vector input to the neural network (i.e: an image that we want the neural network to classify, or some other data that the neural network operates on), $A_{n}$ (where $n$ is the number of layers of the network) is the vector output of the neural network (i.e: a vector of size 10 in the case of classifying pictures of handwritten digits), $W_i$s are the weights of the neural network that are tweaked at training time to fit our data, and $F_{i}$ is that layer's nonlinear activation function (i.e: ReLU, softmax, etc). As defined, the intermediate output of the neural network is a vector of real-valued numbers with dimensions: $$ -L_i \in \mathbb{R}^{d_{i}} +L_i, A_i \in \mathbb{R}^{d_{i}} $$ where $d_{i}$ is the number of neurons at layer $i$; in the case of the first layer $i=0$, $d_{i}$ is the dimension of the input data, and in the last layer $i=n$, $d_{n}$ is the dimension of the output label, and anything in between can be set arbitrarily and may be viewed as the "architecture" of the neural network (i.e: dimensionality of the intermediate layers). The weights, which determine how each layer of the neural network interacts with each other, therefore are matrices of real numbers with shape @@ -79,7 +82,7 @@ Our neural network, as defined, performs a sequence of linear and nonlinear oper $$ \begin{align*} -N(x; W_1, ... W_n) &= \text{Let } L_0 = x, \text{ then output } L_n +N(x; W_1, ... W_n) &= \text{Let } A_0 = x, \text{ then output } A_n \end{align*} $$ @@ -135,23 +138,23 @@ Training neural networks involve repeated applications of the gradient descent a Let's break this down. We can compute the derivative of the loss with respect to the _the outputs of each layer of the neural network_ by using repeated applications of the chain rule $$ -\frac{\partial L_{full}}{L_{n}} = \frac{\partial \frac{1}{M} \sum_{j=1}^{M} L(N(x_j; W_1, ..., W_n), y_j)}{\partial L_{n}} +\frac{\partial L_{full}}{\partial L_{n}} = \frac{\partial A_{n}}{\partial L_{n}} \frac{\partial L_{full}}{\partial A_{n}} $$ $$ -\frac{\partial L_{full}}{\partial L_{n-1}} = \frac{\partial L_{full}}{\partial L_{n}} \frac{\partial L_{n}}{\partial L_{n-1}} +\frac{\partial L_{full}}{\partial L_{n-1}} = \frac{\partial A_{n-1}}{\partial L_{n-1}} \frac{\partial L_{n}}{\partial A_{n-1}} \frac{\partial A_{n}}{\partial L_{n}} \frac{\partial L_{full}}{\partial A_{n}} $$ or more generally $$ -\frac{\partial L_{full}}{L_{i}} = \frac{\partial L_{full}}{\partial L_{n}} \frac{\partial L_{n}}{\partial L_{n-1}} \frac{\partial L_{n-1}}{\partial L_{n-2}} ... \frac{\partial L_{i+1}}{\partial L_{i}} +\frac{\partial L_{full}}{\partial L_{i}} = \frac{\partial A_{i}}{\partial L_{i}} \frac{\partial L_{i+1}}{\partial A_{i}} ... \frac{\partial A_{n}}{\partial L_{n}} \frac{\partial L_{full}}{\partial A_{n}} $$ After computing the derivative of the loss with respect to the _output of each layer_, we can easily obtain the derivative of the loss with respect to the _parameters_, again using the chain rule: $$ -\frac{\partial L_{full}}{W_{i}} = \frac{\partial L_{full}}{L_{i}} \frac{L_{i}}{W_{i}} +\frac{\partial L_{full}}{W_{i}} = \frac{\partial L_{i}}{\partial W_{i}} \frac{\partial L_{full}}{\partial L_{i}} $$ And this is ultimately how the derivatives of the layers' weights are computed using backpropagation! What does this concretely look like in a specific example? Below we walk through a specific example on a simple 2 layer neural network, on a regression task using a MSE loss function, with 100-dimensional inputs and a 30-dimensional hidden layer: @@ -159,13 +162,19 @@ And this is ultimately how the derivatives of the layers' weights are computed u > Example of Backpropagation\ Suppose we have a two-layer neural network $$ -L_1 = ReLU(W_1 \times L_{0}) +L_1 = W_1 A_{0} +$$ +$$ +A_1 = ReLU(L_1) $$ $$ -L_2 = ReLU(W_2 \times L_{1}) +L_2 = W_2 A_{1} $$ $$ -NN(x) = \mbox{Let } L_{0} = x \mbox{ then output } L_2 +A_2 = ReLU(L_2) +$$ +$$ +NN(x) = \mbox{Let } A_{0} = x \mbox{ then output } A_2 $$ where $W_1 \in \mathbb{R}^{30 \times 100}$ and $W_2 \in \mathbb{R}^{1 \times 30}$. Furthermore suppose we use the MSE loss function: $$ @@ -175,33 +184,42 @@ We wish to compute $$ \frac{\partial L(NN(x), y)}{\partial W_i} \mbox{ for } i=1,2 $$ -We start by computing the gradient with respect to the final output: +Note the following: +$$ +\frac{\partial L(x, y)}{\partial x} = 2 \times (x-y) $$ -\frac{\partial L(NN(x), y)}{\partial L_2} = \frac{\partial (L_2 - y)^2}{\partial L_2} = 2(L_2 - y) $$ -With respect to the output $L_1$ +\frac{\partial ReLU(x)}{\partial x} \delta = \left\{\begin{array}{lr} + 0 & \text{for } x \leq 0 \\ + 1 & \text{for } x \geq 0 \\ + \end{array}\right\} \odot \delta $$ -\frac{\partial L(NN(x), y)}{\partial L_1} = \frac{\partial L(NN(x), y)}{\partial L_2} \frac{\partial L_2}{\partial L_1} $$ +\frac{\partial WA}{\partial A} \delta = W^T \delta $$ -= [2(L_2 - y)] \times \frac{\partial ReLU(W_2 \times L_1)}{\partial L_1} = [2(L_2 - y)] \times W_2^T ReLU'(W_2 \times L_1) $$ -where +\frac{\partial WA}{\partial W} \delta = \delta A^T $$ -ReLU'(x) = \begin{cases} - 0 & x\leq 0 \\ - 1 & x > 0 - \end{cases} +Then we have $$ -Then we can compute the gradients with respect to the weights +\frac{\partial L(NN(x), y)}{\partial W_2} = \frac{\partial L_2}{\partial W_2} \frac{\partial A_2}{\partial L_2} \frac{\partial L(NN(x), y)}{\partial A_2} $$ -\frac{\partial L(NN(x), y)}{\partial W_2} = \frac{\partial L(NN(x), y)}{\partial L_2} \frac{\partial L_2}{\partial W_2} = [2(L_2 - y)] \times ReLU'(W_2 \times L_1) L_1^T $$ += (2L(NN(x) - y) \odot ReLU'(L_2)) A_1^T $$ -\frac{\partial L(NN(x), y)}{\partial W_1} = \frac{\partial L(NN(x), y)}{\partial L_1} \frac{\partial L_1}{\partial W_1} = [(2(L_2 - y)) \times W_2^T ReLU'(W_2 \times L_1)] \times ReLU'(W_1 \times L_0) +and $$ -IMPORTANT: QUICKLY WRITTEN SO THIS IS NOT REALLY CORRECT THERE IS ACTUALLY A BUG HERE, REDO! +\frac{\partial L(NN(x), y)}{\partial W_1} = \frac{\partial L_1}{\partial W_1} \frac{\partial A_1}{\partial L_1} \frac{\partial L_2}{\partial A_1} \frac{\partial A_2}{\partial L_2} \frac{\partial L(NN(x), y)}{\partial A_2} +$$ +$$ += [ReLU'(L_1) \odot (W_2^T [2L(NN(x) - y) \odot ReLU'(L_2)])] A_0^T +$$ + +::: {.callout-tip} +Double check your work by making sure that the shapes are correct! +::: +The entire backpropagation process can be complex, especially for networks that are very deep. Fortunately, frameworks like PyTorch support automatic differentiation, which does this for us; in this frameworks we simply need to specify the forward pass, and the derivatives will be automatically calculated for us. Nevertheless, it is beneficial to understand the theoretical process that is going on under the hood in some of these machine-learning frameworks. ## Training Data From 307dafbe49bbc1b00bba26609245c587d241daa8 Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Sun, 19 Nov 2023 22:44:25 -0500 Subject: [PATCH 12/45] Markdown --- training.qmd | 24 ++++++++++++++++-------- 1 file changed, 16 insertions(+), 8 deletions(-) diff --git a/training.qmd b/training.qmd index a4059289..3a8c0106 100644 --- a/training.qmd +++ b/training.qmd @@ -42,9 +42,9 @@ We will walk you through these details in the rest of the sections. Understandin Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. -Briefly, neural networks are mathematical models that consist of alternating linear and nonlinear operations, parameterized by a set of "weights" that are trained to minimize some loss function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value (i.e: the "loss") when evaluated on our model against the training data. Fundamentally, training neural networks involve evaluating the loss function to get a measure of how good our model is, then continuously tweaking the weights of our model so that the loss decreases, which ultimately optimizes the model to fit our training data. +Briefly, neural networks are mathematical models that consist of alternating linear and nonlinear operations, parameterized by a set of **weights** that are trained to minimize some loss function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value (i.e: the loss) when evaluated on our model against the training data. Fundamentally, training neural networks involve evaluating the loss function to get a measure of how good our model is, then continuously tweaking the weights of our model so that the loss decreases, which ultimately optimizes the model to fit our training data. -Neural networks are made up of simple functions ("neurons") layered on top of each other. Each layer takes in some data, performs a calculation on it, and passes it to the next layer. For example, in an image recognition network, the input layer may take in pixel values. The next layers detect simple shapes like edges, then larger shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. +Neural networks are made up of simple functions (neurons) layered on top of each other. Each layer takes in some data, performs a calculation on it, and passes it to the next layer. For example, in an image recognition network, the input layer may take in pixel values. The next layers detect simple shapes like edges, then larger shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. The "network" in a neural network refers to how these neurons are connected. Each neuron is connected to many others in the layers above and below it. The neurons have numeric weights associated with them, kind of like the synaptic strengths in a brain neuron. The network is trained by adjusting these weights. @@ -72,7 +72,7 @@ $$ L_i, A_i \in \mathbb{R}^{d_{i}} $$ -where $d_{i}$ is the number of neurons at layer $i$; in the case of the first layer $i=0$, $d_{i}$ is the dimension of the input data, and in the last layer $i=n$, $d_{n}$ is the dimension of the output label, and anything in between can be set arbitrarily and may be viewed as the "architecture" of the neural network (i.e: dimensionality of the intermediate layers). The weights, which determine how each layer of the neural network interacts with each other, therefore are matrices of real numbers with shape +where $d_{i}$ is the number of neurons at layer $i$; in the case of the first layer $i=0$, $d_{i}$ is the dimension of the input data, and in the last layer $i=n$, $d_{n}$ is the dimension of the output label, and anything in between can be set arbitrarily and may be viewed as the **architecture** of the neural network (i.e: dimensionality of the intermediate layers). The weights, which determine how each layer of the neural network interacts with each other, therefore are matrices of real numbers with shape $$ W_i \in \mathbb{R}^{d_{i} \times d_{i-1}} @@ -90,7 +90,7 @@ Next we will see how to evaluate this neural network against training data by in ### Loss Function as a Measure of Goodness of Fit against Training Data -After defining our neural network, we are given some training data, which is a set of points ${(x_j, y_j)}$ for $j=1..M$, and we want to evaluate how good our neural network is on fitting this data. To do this, we introduce a "loss function", which is a function that takes the output of the neural network on a particular datapoint ($N(x_j; W_1, ..., W_n)$), and compares it against the "label" of that particular datapoint (the corresponding $y_j$), and outputs a single numerical scalar (i.e: one real number) that represents how "good" the neural network fit that particular data point; the final measure of how good the neural network is on the entire dataset is therefore just the average of the losses across all datapoints. +After defining our neural network, we are given some training data, which is a set of points ${(x_j, y_j)}$ for $j=1..M$, and we want to evaluate how good our neural network is on fitting this data. To do this, we introduce a **loss function**, which is a function that takes the output of the neural network on a particular datapoint ($N(x_j; W_1, ..., W_n)$), and compares it against the "label" of that particular datapoint (the corresponding $y_j$), and outputs a single numerical scalar (i.e: one real number) that represents how "good" the neural network fit that particular data point; the final measure of how good the neural network is on the entire dataset is therefore just the average of the losses across all datapoints. There are many different types of loss functions, for example, in the case of image classification, we might use the cross-entropy loss function, which tells us how good two vectors that represent classification predictions compare (i.e: if our prediction predicts that an image is more likely a dog, but the label says it is a cat, it will return a high "loss" indicating a bad fit). @@ -118,7 +118,7 @@ $$ = min_{W_1, ..., W_n} \frac{1}{M} \sum_{j=1}^{M} L(N(x_j; W_1,...W_n), y_j) $$ -So how do we optimize this objective? Recall from calculus that minimizing a function can be done by taking the derivative of the function with respect to the input parameters and tweaking the parameters in the direction of the gradient. This technique is called "gradient descent" and concretely involves calculating the derivative of the loss function $L_{full}$ with respect to $W_1, ..., W_n$ to obtain a gradient for these parameters to take a step in, then updating these parameters in the direction of the gradient. Thus, we can train our neural network using gradient descent which repeatedly applies the update rule +So how do we optimize this objective? Recall from calculus that minimizing a function can be done by taking the derivative of the function with respect to the input parameters and tweaking the parameters in the direction of the gradient. This technique is called **gradient descent** and concretely involves calculating the derivative of the loss function $L_{full}$ with respect to $W_1, ..., W_n$ to obtain a gradient for these parameters to take a step in, then updating these parameters in the direction of the gradient. Thus, we can train our neural network using gradient descent which repeatedly applies the update rule > Gradient Descent Update Rule $$ @@ -129,11 +129,11 @@ $$ In practice, the gradient is computed over a minibatch of datapoints, to improve computational efficiency. This is called stochastic gradient descent or batch gradient descent. ::: -where $\lambda$ is the stepsize or learning rate of our tweaks. In training our neural network, we repeatedly perform the step above until convergence, or when the loss no longer decreases. This prior approach is known as "full gradient descent" since we are computing the derivative with respect to the entire training data, and only then taking a single gradient step; a more efficient approach is to calculate the gradient with respect to just a random "batch" of datapoints and then taking a step, a process known as "batch gradient descent" or "stochastic gradient descent", which is more efficient since now we are taking many more steps per pass of the entire training data. Next we will cover the mathematics behind computing the gradient of the loss function with respect to the $W_i$s, a process known as "backpropagation". +where $\lambda$ is the stepsize or learning rate of our tweaks. In training our neural network, we repeatedly perform the step above until convergence, or when the loss no longer decreases. This prior approach is known as full gradient descent since we are computing the derivative with respect to the entire training data, and only then taking a single gradient step; a more efficient approach is to calculate the gradient with respect to just a random batch of datapoints and then taking a step, a process known as batch gradient descent or stochastic gradient descent, which is more efficient since now we are taking many more steps per pass of the entire training data. Next we will cover the mathematics behind computing the gradient of the loss function with respect to the $W_i$s, a process known as backpropagation. ### Backpropagation -Training neural networks involve repeated applications of the gradient descent algorithm, which involves computing the derivative of the loss function with respect to the $W_i$s. How do we compute the derivative of the loss with respect to the $W_i$s given that the $W_i$s are nested functions of each other in a deep neural network? The trick is to leverage the "chain rule": we can compute the derivative of the loss with respect to the $W_i$s by repeatedly applying the chain rule, in a complete process known as "backpropagation". Specifically, we can calculate the gradients by computing the derivative of the loss with respect to the outputs of the last layer, then progressively use this to compute the derivative of the loss with respect to each prior layer, all the way to the input layer. This process starts from the end of the network (the layer closest to the output) and progresses backwards, and hence gets its name "backpropagation". +Training neural networks involve repeated applications of the gradient descent algorithm, which involves computing the derivative of the loss function with respect to the $W_i$s. How do we compute the derivative of the loss with respect to the $W_i$s given that the $W_i$s are nested functions of each other in a deep neural network? The trick is to leverage the **chain rule**: we can compute the derivative of the loss with respect to the $W_i$s by repeatedly applying the chain rule, in a complete process known as backpropagation. Specifically, we can calculate the gradients by computing the derivative of the loss with respect to the outputs of the last layer, then progressively use this to compute the derivative of the loss with respect to each prior layer, all the way to the input layer. This process starts from the end of the network (the layer closest to the output) and progresses backwards, and hence gets its name backpropagation. Let's break this down. We can compute the derivative of the loss with respect to the _the outputs of each layer of the neural network_ by using repeated applications of the chain rule @@ -151,6 +151,10 @@ $$ \frac{\partial L_{full}}{\partial L_{i}} = \frac{\partial A_{i}}{\partial L_{i}} \frac{\partial L_{i+1}}{\partial A_{i}} ... \frac{\partial A_{n}}{\partial L_{n}} \frac{\partial L_{full}}{\partial A_{n}} $$ +:::{.callout-note} +In our notation, we assume the intermediate activations $A_{i}$ are *column* vectors, rather than *row* vectors, hence the chain rule is $\frac{\partial L}{\partial L_{i}} = \frac{\partial L_{i+1}}{\partial L_{i}} ... \frac{\partial L}{\partial L_{n}}$ rather than $\frac{\partial L}{\partial L_{i}} = \frac{\partial L}{\partial L_{n}} ... \frac{\partial L_{i+1}}{\partial L_{i}}$ +::: + After computing the derivative of the loss with respect to the _output of each layer_, we can easily obtain the derivative of the loss with respect to the _parameters_, again using the chain rule: $$ @@ -217,9 +221,13 @@ $$ ::: {.callout-tip} Double check your work by making sure that the shapes are correct! + +* All hadamard products ($\odot$) should operate on tensors of the same shape +* All matrix multiplications should operate on matrices that share a common dimension (i.e: m by n, n by k) +* All gradients with respect to the weights should have the same shape as the weight matrices themselves ::: -The entire backpropagation process can be complex, especially for networks that are very deep. Fortunately, frameworks like PyTorch support automatic differentiation, which does this for us; in this frameworks we simply need to specify the forward pass, and the derivatives will be automatically calculated for us. Nevertheless, it is beneficial to understand the theoretical process that is going on under the hood in some of these machine-learning frameworks. +The entire backpropagation process can be complex, especially for networks that are very deep. Fortunately, machine learning frameworks like PyTorch support automatic differentiation, which performs backpropagation for us. In these machine learning frameworks we simply need to specify the forward pass, and the derivatives will be automatically computed for us. Nevertheless, it is beneficial to understand the theoretical process that is happening under the hood in these machine-learning frameworks. ## Training Data From cf3b67fac91fc98e74eb0b249c226e5896b6a7f0 Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Sun, 19 Nov 2023 23:01:50 -0500 Subject: [PATCH 13/45] Upd training --- training.qmd | 12 ++++-------- 1 file changed, 4 insertions(+), 8 deletions(-) diff --git a/training.qmd b/training.qmd index 3a8c0106..a03b2a2b 100644 --- a/training.qmd +++ b/training.qmd @@ -38,19 +38,15 @@ In summary, some key points about training: We will walk you through these details in the rest of the sections. Understanding how to effectively leverage data, algorithms, parameter optimization, and generalization through thorough training is essential for developing capable, deployable AI systems that work robustly in the real world. -## Basic Mathematics +## Mathematics of Neural Networks Deep learning has revolutionized the fields of machine learning and artificial intelligence, enabling computers to learn complex patterns and make intelligent decisions. At the heart of the deep learning revolution is the neural network, which, as discussed in section 3 "Deep Learning Primer", is a cornerstone in some of these advancements. -Briefly, neural networks are mathematical models that consist of alternating linear and nonlinear operations, parameterized by a set of **weights** that are trained to minimize some loss function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value (i.e: the loss) when evaluated on our model against the training data. Fundamentally, training neural networks involve evaluating the loss function to get a measure of how good our model is, then continuously tweaking the weights of our model so that the loss decreases, which ultimately optimizes the model to fit our training data. +Neural networks are made up of simple functions layered on top of each other. Each **layer** takes in some data, performs some computation, and passes it to the next layer. These layers learn progressively high level features that are useful for the task the network is trained to perform. For example, in a network trained for image recognition, the input layer may take in pixel values, while the next layers may detect simple shapes like edges, then the layers after that may detect more complex shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. -Neural networks are made up of simple functions (neurons) layered on top of each other. Each layer takes in some data, performs a calculation on it, and passes it to the next layer. For example, in an image recognition network, the input layer may take in pixel values. The next layers detect simple shapes like edges, then larger shapes like noses or eyes, and so on. The final output layer classifies the image as a whole. +The network in a neural network refers to how these layers are connected. Each layer's output is considered as a single neuron, and is connected to many other neurons in the layers preceding it, forming a "network". The way these neurons interact with each other is determined by the weights between them, which model synaptic strengths similar to that of a brain's neuron. The neural network is trained by adjusting these weights. Concretely, the weights are initially set randomly, then an input is fed in and the output is compared to the desired result, and finally the weights are then tweaked to make the network better. This process is repeated until the network reliably minimizes the loss, indicating it has learned the patterns in the data. -The "network" in a neural network refers to how these neurons are connected. Each neuron is connected to many others in the layers above and below it. The neurons have numeric weights associated with them, kind of like the synaptic strengths in a brain neuron. The network is trained by adjusting these weights. - -Initially the weights are set randomly. An input is fed in, and the output is compared to the desired result using a loss function. This loss function outputs a number indicating how far off the network's prediction was. The weights are then tweaked slightly to reduce the loss. - -This process is repeated with many training examples until the network reliably minimizes the loss, indicating it has learned the patterns in the data. This training technique, called backpropagation, allows neural networks to learn without explicit programming. Once trained, the network can apply what it's learned to new unseen data. +How is this process defined mathematically? Formally, neural networks are mathematical models that consist of alternating **linear** and **nonlinear** operations, parameterized by a set of learnable **weights** that are trained to minimize some **loss** function. This loss function is a measure of how good our model is with respect to fitting our training data, and it produces a numerical value when evaluated on our model against the training data. Training neural networks involve repeatedly evaluating the loss function on many different datapoints to get a measure of how good our model is, then continuously tweaking the weights of our model using backpropagation so that the loss decreases, which ultimately optimizes the model to fit our data. ### Neural Network Notation From 0d0c6138138a21b556b3f09d4d8aa64a55e3b129 Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Sun, 19 Nov 2023 23:32:40 -0500 Subject: [PATCH 14/45] Upd training --- training.qmd | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/training.qmd b/training.qmd index a03b2a2b..cbd7beab 100644 --- a/training.qmd +++ b/training.qmd @@ -147,6 +147,12 @@ $$ \frac{\partial L_{full}}{\partial L_{i}} = \frac{\partial A_{i}}{\partial L_{i}} \frac{\partial L_{i+1}}{\partial A_{i}} ... \frac{\partial A_{n}}{\partial L_{n}} \frac{\partial L_{full}}{\partial A_{n}} $$ +:::{.callout-note} +In what order should we perform this computation? It is preferrable from a computational perspective to perform the calculations from the end to the front +(i.e: first compute $\frac{\partial L_{full}}{\partial A_{n}}$ then the prior terms, rather than start with $\frac{\partial A_{0}}{\partial L_{0}}$) since this avoids materializing and computing large jacobians. This is because $\frac{\partial L_{full}}{\partial A_{n}}$ is a vector, hence any matrix operation that includes this term has an output that is squished to be a vector. +::: + + :::{.callout-note} In our notation, we assume the intermediate activations $A_{i}$ are *column* vectors, rather than *row* vectors, hence the chain rule is $\frac{\partial L}{\partial L_{i}} = \frac{\partial L_{i+1}}{\partial L_{i}} ... \frac{\partial L}{\partial L_{n}}$ rather than $\frac{\partial L}{\partial L_{i}} = \frac{\partial L}{\partial L_{n}} ... \frac{\partial L_{i+1}}{\partial L_{i}}$ ::: @@ -225,6 +231,10 @@ Double check your work by making sure that the shapes are correct! The entire backpropagation process can be complex, especially for networks that are very deep. Fortunately, machine learning frameworks like PyTorch support automatic differentiation, which performs backpropagation for us. In these machine learning frameworks we simply need to specify the forward pass, and the derivatives will be automatically computed for us. Nevertheless, it is beneficial to understand the theoretical process that is happening under the hood in these machine-learning frameworks. +:::{.callout-note} +As seen above, intermediate activations $A_i$ are re-used in backpropagation. To improve performance, these activations are cached from the forward pass to avoid recomputing them. However, this means that activations must be kept in memory between the forward and backward passes, leading to higher memory usage. If the network and batchsize is large, this may lead to memory issues. +::: + ## Training Data To enable effective training of neural networks, the available data must be split into training, validation, and test sets. The training set is used to train the model parameters. The validation set evaluates the model during training to tune hyperparameters and prevent overfitting. The test set provides an unbiased final evaluation of the trained model's performance. From 893b75032ea53400cf526de3906a4b5b58cdfafe Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Sun, 19 Nov 2023 23:37:36 -0500 Subject: [PATCH 15/45] Upd training --- training.qmd | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/training.qmd b/training.qmd index cbd7beab..a1622e6d 100644 --- a/training.qmd +++ b/training.qmd @@ -149,7 +149,7 @@ $$ :::{.callout-note} In what order should we perform this computation? It is preferrable from a computational perspective to perform the calculations from the end to the front -(i.e: first compute $\frac{\partial L_{full}}{\partial A_{n}}$ then the prior terms, rather than start with $\frac{\partial A_{0}}{\partial L_{0}}$) since this avoids materializing and computing large jacobians. This is because $\frac{\partial L_{full}}{\partial A_{n}}$ is a vector, hence any matrix operation that includes this term has an output that is squished to be a vector. +(i.e: first compute $\frac{\partial L_{full}}{\partial A_{n}}$ then the prior terms, rather than start in the middle) since this avoids materializing and computing large jacobians. This is because $\frac{\partial L_{full}}{\partial A_{n}}$ is a vector, hence any matrix operation that includes this term has an output that is squished to be a vector. Thus performing the computation from the end avoids large matrix-matrix multiplications by ensuring that the intermediate products are vectors. ::: @@ -232,7 +232,7 @@ Double check your work by making sure that the shapes are correct! The entire backpropagation process can be complex, especially for networks that are very deep. Fortunately, machine learning frameworks like PyTorch support automatic differentiation, which performs backpropagation for us. In these machine learning frameworks we simply need to specify the forward pass, and the derivatives will be automatically computed for us. Nevertheless, it is beneficial to understand the theoretical process that is happening under the hood in these machine-learning frameworks. :::{.callout-note} -As seen above, intermediate activations $A_i$ are re-used in backpropagation. To improve performance, these activations are cached from the forward pass to avoid recomputing them. However, this means that activations must be kept in memory between the forward and backward passes, leading to higher memory usage. If the network and batchsize is large, this may lead to memory issues. +As seen above, intermediate activations $A_i$ are re-used in backpropagation. To improve performance, these activations are cached from the forward pass to avoid recomputing them. However, this means that activations must be kept in memory between the forward and backward passes, leading to higher memory usage. If the network and batchsize is large, this may lead to memory issues. Similarly, the derivatives with respect to each layer's outputs are cached to avoid recomputation. ::: ## Training Data From fe2a42449d8af241a7f0c2b44524b36f34a2550b Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Mon, 20 Nov 2023 00:02:32 -0500 Subject: [PATCH 16/45] Upd training --- training.qmd | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/training.qmd b/training.qmd index a1622e6d..c33796fa 100644 --- a/training.qmd +++ b/training.qmd @@ -62,6 +62,10 @@ $$ Convolutions are also linear operators, and can be cast as a matrix multiplication. ::: +:::{.callout-note} +Why are the nonlinear operations necessary? If we only had linear layers the entire network is equivalent to just a single linear layer consisting of the product of the linear operators. Hence, the nonlinear functions play a key role in the power of neural networks as they enhance the neural network's ability to fit functions. +::: + where $A_{0}$ is a vector input to the neural network (i.e: an image that we want the neural network to classify, or some other data that the neural network operates on), $A_{n}$ (where $n$ is the number of layers of the network) is the vector output of the neural network (i.e: a vector of size 10 in the case of classifying pictures of handwritten digits), $W_i$s are the weights of the neural network that are tweaked at training time to fit our data, and $F_{i}$ is that layer's nonlinear activation function (i.e: ReLU, softmax, etc). As defined, the intermediate output of the neural network is a vector of real-valued numbers with dimensions: $$ From 79757d177e43b8f37d51eb599bc8ce2dbb5fcdfc Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Mon, 20 Nov 2023 00:02:51 -0500 Subject: [PATCH 17/45] Upd training --- training.qmd | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/training.qmd b/training.qmd index c33796fa..9d7a7d37 100644 --- a/training.qmd +++ b/training.qmd @@ -59,13 +59,14 @@ A_i = F_i(L_{i}) $$ :::{.callout-note} -Convolutions are also linear operators, and can be cast as a matrix multiplication. +Why are the nonlinear operations necessary? If we only had linear layers the entire network is equivalent to just a single linear layer consisting of the product of the linear operators. Hence, the nonlinear functions play a key role in the power of neural networks as they enhance the neural network's ability to fit functions. ::: :::{.callout-note} -Why are the nonlinear operations necessary? If we only had linear layers the entire network is equivalent to just a single linear layer consisting of the product of the linear operators. Hence, the nonlinear functions play a key role in the power of neural networks as they enhance the neural network's ability to fit functions. +Convolutions are also linear operators, and can be cast as a matrix multiplication. ::: + where $A_{0}$ is a vector input to the neural network (i.e: an image that we want the neural network to classify, or some other data that the neural network operates on), $A_{n}$ (where $n$ is the number of layers of the network) is the vector output of the neural network (i.e: a vector of size 10 in the case of classifying pictures of handwritten digits), $W_i$s are the weights of the neural network that are tweaked at training time to fit our data, and $F_{i}$ is that layer's nonlinear activation function (i.e: ReLU, softmax, etc). As defined, the intermediate output of the neural network is a vector of real-valued numbers with dimensions: $$ From 733ec36f4c475d0ab187c7f245117c70c292f82c Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Mon, 20 Nov 2023 13:17:20 -0500 Subject: [PATCH 18/45] Upd training --- training.qmd | 33 ++++++++++++++++++++++++++++++--- 1 file changed, 30 insertions(+), 3 deletions(-) diff --git a/training.qmd b/training.qmd index 9d7a7d37..09fe672f 100644 --- a/training.qmd +++ b/training.qmd @@ -572,12 +572,39 @@ TPUs (i.e: systolic arrays). ## Training Parallelization +Training can be computationally intensive. As outlined above, backpropagation can be expensive both in terms of computation and memory. In terms of computation, backpropagation requires many large matrix multiplications which require considerable arithmetic operation. In terms of memory, backpropagation requires storing the model parameters and intermediate activations in memory, which can be significant if training large models. Handling these difficult system challenges is key to training large models. To mitigate these computational and memory challenges, parallelization is necessary. Broadly, there are two different approaches to parallelizing training: Data Parallel and Model Parallel, discussed below. + ### Data Parallel +Data parallelization is a common approach to parallelize machine learning (ML) training across multiple processing units, such as GPUs or distributed computing resources. In data parallelism, the training dataset is divided into batches, and each batch is processed by a separate processing unit. The model parameters are then updated based on the gradients computed from the processing of each batch. + +Here's a step-by-step description of data parallel parallelization for ML training: + +* Dividing the Dataset + The entire training dataset is divided into smaller batches. Each batch contains a subset of the training examples. + +* Replicating the Model + The ML model (neural network, for example) is replicated across all processing units. Each processing unit has its copy of the model. + +* Parallel Computation + Each processing unit takes a different batch and computes the forward and backward passes independently. During the forward pass, the model makes predictions on the input data. During the backward pass, gradients are computed for the model parameters using the loss function. + +* Gradient Aggregation + After processing their respective batches, the gradients from each processing unit are aggregated. Common aggregation methods include summation or averaging of the gradients. + +* Parameter Update + The aggregated gradients are used to update the model parameters. The update can be performed using optimization algorithms like Stochastic Gradient Descent (SGD) or variants like Adam. + +* Synchronization + All processing units synchronize their model parameters after the update. This ensures that each processing unit has the latest version of the model. + +The prior steps are repeated for a certain number of iterations or until convergence. +Data parallelism is effective when the model is large, and the dataset is substantial, as it allows for parallel processing of different parts of the data. It is widely used in deep learning frameworks and libraries that support distributed training, such as TensorFlow and PyTorch. However, care must be taken to handle issues like communication overhead, load balancing, and synchronization to ensure efficient parallelization. + ### Model Parallel -## Distributed Training +The counterpart to data parallelism is model parallelism, which is an approach to parallelize training by distributing the model across multiple processing units. This is particularly useful when a model is too large to fit into the memory of a single device (e.g., a GPU). In model parallelism, different parts or layers of the model are assigned to different devices, and data is passed between these devices during the training process. By splitting the model across different devices, memory and computational burdens are distributed across individual devices. However, as these devices are separate from each other, they must communicate during the forward and backward passes to share gradient updates, and hence care must be taken with regards to communication and coordination to ensure high performance. -## Debugging and Profiling +## Conclusion -## Conclusion \ No newline at end of file +In this section we have introduced the basics of AI training, including the fundamental mathematics behind neural networks, loss functions, backpropagation, as well as key concepts such as training and validation, and key system challenges. \ No newline at end of file From c7be42a47e0cafe14518364f0bba8393e2fd3271 Mon Sep 17 00:00:00 2001 From: Maximilian Lam Date: Mon, 27 Nov 2023 13:48:35 -0500 Subject: [PATCH 19/45] aitraining images --- images/aitraincover.png | Bin 0 -> 1926477 bytes images/aitrainingfit.png | Bin 0 -> 6152 bytes images/aitrainingnn.png | Bin 0 -> 45341 bytes images/aitrainingpara.png | Bin 0 -> 82057 bytes images/aitrainingroof.png | Bin 0 -> 49356 bytes images/aitrainingsgd.png | Bin 0 -> 24445 bytes training.qmd | 14 ++++++++++++++ 7 files changed, 14 insertions(+) create mode 100644 images/aitraincover.png create mode 100644 images/aitrainingfit.png create mode 100644 images/aitrainingnn.png create mode 100644 images/aitrainingpara.png create mode 100644 images/aitrainingroof.png create mode 100644 images/aitrainingsgd.png diff --git a/images/aitraincover.png b/images/aitraincover.png new file mode 100644 index 0000000000000000000000000000000000000000..81cfa2dce9f091ead4446f48124c43ee88e6c8bc GIT binary patch literal 1926477 zcmV((K;XZLP)Hrb{&R|G1orl-23i*@73$7V->54MUi4BQ4K|Pkw_3?z_c^^FBJrbRV0Td|5{l~~=Ys@LA;oU_-Q`7y>^ zd!Kt>l}a>W3CWT8>Ynd>-~RSqYpyxR9J6B}2FI+KNCE);fBjPcfr9^5r$6*@r-^;squ)aeds{+W=U(avUmCmoAN^d7f z)bSnyr3)X+Mgmm?2}XY*d;zff+6DG4_`h}MYu9a5e@ov$f9G?Hsn?5SomaiTrj85O zv3mWLJpEc#=j)QJs?(nFykW;G5#L4!vBS^{Pk zP?eR*D&zzRL;*yF_PrlTo5TVsiKrl}FnZnfK&NeHWoeO$OKge~yeh1_t={%}_TlMO z>c#fWu#P_!$zzqZjaA@j-}|xzDw9EN1ib!)Zw4SLi+X~SIMn*BO5a{Y^%hs5$n|zo z3r^p|y4@CdOPy~5>mqCGtI9IBq@}9=&`W?jz&QexSqP`r7rX#UmrW2i#lsFy|PZ)`RV+u^qN2tNP4Zx z*U^GyW#5MuDvuMrM8elS*7i9$rIkaqwj@|ZGONhw-EFa7JFIhEC}az(k43FzT#|U# z3>lU)nOMuZ)@n(z_UoiM&o@H}CGTG9wB24P9ggFP zS{uWsu^(Pb*M_MN_u8!fQdDJ~zAiFTVy#WAOj{X$J*#plHGin;w6p6LfLJF17HcvK zagvU<3*Pb2mR2tHvQEFhvOtz}<6&T|nQ;;%SWYDpRZAwz3)nZYc4_^|pKd$jjk~jJ zysI~?*Su^etEBB~ZM|0MIUsQC`kg-auV? zUqGIg%bR}Koi^%+YV_N_AN+_^n;dPsN)U20G7{w{doj(UbxfePm1~OPuTU18Jin<0N&8yqW7oVcB%-i%0*p*-=AN-Mamb6h~IuKn#- zmwrA0lap!ge|)eKSl@iwfZ|$QwA#^Po@CnyyVfG>*yi0^Iqq!P1fM#F;UEEnsJ<>{91uF!d1PygWY zuy@|}{10_hmvo)4%jf_3?5Cf*_93|+bQyhNon#NE7w>#sE#GX}-Ln=vI}L}xp5JhK zkrE9Y$Ge}JK=FfD=lK({ZjeARD2dFhR@Y1a|Nr#Q9lZfz#5*7mOSxEK^_p;DhCa(Y zF0>5k$ted)z9m2~TB5m(rEZ~?0qL8lBE$9@3{v(E3m-Oixce}^rkq|oiVb6Gd6j;) z_6ZM{auEm1k{jox9*tyHvdzp|2m7%`Oui>$g{+7`Rm9LGV+_k4gAq6oy1$Q0@e-Ly zGIQ!(nN#|)vQSksiIk78Z$Tx4CZ6R%I}%7y+al``+gQ$V4&PpO>3N z6ihLoynyT38L`$=`zN@f??OZsN1On^mNX9RTLlCwD*~C?iHyFZ)2%^BG4*PoGJ{ch z=*ui{FlXv+-twBlti zEQzdTc^cuF-8h{)9B#zgnLb#{d>3%YUVBsUHADV+`_&7KozA>Ln-^k9V76qQJo=>~ zxa`luRp<+rCG%{q4>mAs31BWT>ia)!$n)ORQW{I->h!@%31+ll%r@sd-B)unhiL|R z5{BbJ78Aik6QZ~AtF4GIQw= zZ&0zoB9O_D?v1L7h|C!bEen@T$yPhkUCXhM=C>fO4f5jT0cdnTy8fZ(mmF5BwJHRW zHG^T}p#MRk5QwUhqtiB90T{f35o@t139#8^pIA|4~{+R7kT%6t(Z?1P+McgZvMvhq76aX z(UPQP3wti7zV)bmiA<1zO5JlG_R90Vl$W4YbqkX|(xTA^KiW}euL%fXS@X6344_Wy z!|Eq#v5_3uy=%V7FiCr4KJ*N$+Y0qBHWb1_cfAWC1l01oCDC^rdVrZT8BDKU63l}D z^08t7Rn<$}*IM>?-Bt5ILIHQ|B;|05Xo?s(4jdR!IT?|SflIbbD{_ELWO>(YYL`Km zh74vVBQkR>J5DavIyHSR*0HMU*iWB`6Z|+#v$X;mrC@^9ORmo@Cq zv@%@x!w{Un-J9njWkX;X&FY&*AxUIHFbYe$5XLjn4=ABPv=9 zv#gXvONnP2tZJxMy<$f6%K)C(QSh`Uca5ue?2fj!*YJT}^qr6YMt}NC@XXeJ>#bfY z_0Bn^@4aDYgGUQwe{fy=nV;$xYw523rf1)L2=)n(EV}6KbDcl49E~Kf{U$BnZ|2W4 z8;?7Wdgrg}=b2r4e#h3bEV`Ar^8uselR@D6+=6K+A{M~R)(aAJGRh(my%BBi`*gIS zV1Ln_ry>s>Bd@iuZ~YULnOszOo$SiVp37w;(X1T$DbQhCVyRaz#aXMd1w}F#;Cc?6 zzooUN9$Wti1mqGTswx=l_&Xq-f)$Q8j$}iwrGqnR-Gx}dN#PCr3l4?J8!oJE zBnbw>X&CMBuP67u6YHU>^CJ%1y##}dJ0@8Bbucoqa7GnCrF&Rm`&h7bPd~W+E&vCN z<-Opf9SXhL669=|JS~#eK?oX*enlkz=_Y+-;QKoiB<&FPDPc!qtqr6JX}r$R0ncNP zX{aq*XV`&0GIrE>9RKPk>Ni`(X(8Dmj_QT4vxQS=C^ID$%dFC92huhvrKty6P;~9z zNgp12j35qzW0 zdee^$js?LMD#v2MRWYZI^mX9GTB>DZ3gYzTQT?9!$NkgROOZWF}X}c6{_k)tD!aX zT4bQax;AlulWHz5TY$=5SguRKhOnQv;Sdg++!XGDrF~~*#07NpqCZLWI9%%pa{X~fh2o+N`0RcVyk^W*Fo6zF(9KE zB>-gA%81f67uEx6q2KDrgzhpTM8SlS@WOReC@P~hx$Y)y;Ewa!lEu#cJy58=BP++8(@*Dp7ja`AyJB#?n{|3}R7q&wf+; 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