Choosing architectures for neural networks is not an easy task. The emphasis of this paper is on automatic generation of network architecture. However, we prefer a function where the space of candidate solutions maps onto a smooth (but high-dimensional) landscape that the optimization algorithm can reasonably navigate via iterative updates to the model weights. Neural networks consist of input and output layers, as well as (in most cases) a hidden layer consisting of units that transform the input into something that the output layer can use. It is a hybrid approach which consists of linear combinations of ReLU and leaky ReLU units. If we have small gradients and several hidden layers, these gradients will be multiplied during backpropagation. Our team set up to combine all the features of the recent architectures into a very efficient and light-weight network that uses very few parameters and computation to achieve state-of-the-art results. Neural networks provide an abstract representation of the data at each stage of the network which are designed to detect specific features of the network. The number of inputs, d, is pre-specified by the available data. Batch-normalization computes the mean and standard-deviation of all feature maps at the output of a layer, and normalizes their responses with these values. However, the hyperbolic tangent still suffers from the other problems plaguing the sigmoid function, such as the vanishing gradient problem. Sigmoids suffer from the vanishing gradient problem. Some initial interesting results are here. This was done to average the response of the network to multiple are of the input image before classification. Sequential Layer-wise Operations The most naive way to design the search space for neural network architectures is to depict network topologies, either CNN or RNN, with a list of sequential layer-wise operations, as seen in the early work of Zoph & Le 2017 & Baker et al. Sigmoids are not zero centered; gradient updates go too far in different directions, making optimization more difficult. You’re essentially trying to Goldilocks your way into the perfect neural network architecture — not too big, not too small, just right. Both data and computing power made the tasks that neural networks tackled more and more interesting. Using a linear activation function results in an easily differentiable function that can be optimized using convex optimization, but has a limited model capacity. • use mini-batch size around 128 or 256. In this work, we attempt to design CNN architectures based on genetic programming. convolutional neural network use sequence of 3 layers: convolution, pooling, non-linearity –> This may be the key feature of Deep Learning for images since this paper! Contrast the above with the below example using a sigmoid output and cross-entropy loss. Deep neural networks and Deep Learning are powerful and popular algorithms. However, when we look at the first layers of the network, they are detecting very basic features such as corners, curves, and so on. What happens if we add more nodes? But one could now wonder why we have to spend so much time in crafting architectures, and why instead we do not use data to tell us what to use, and how to combine modules. I wanted to revisit the history of neural network design in the last few years and in the context of Deep Learning. A summary of the data types, distributions, output layers, and cost functions are given in the table below. Our neural network with 3 hidden layers and 3 nodes in each layer give a pretty good approximation of our function. The architecture of a neural network determines the number of neurons in the network and the topology of the connections within the network. In February 2015 Batch-normalized Inception was introduced as Inception V2. ISBN-10: 0-9717321-1-6 . The third article focusing on neural network optimization is now available: For updates on new blog posts and extra content, sign up for my newsletter. They are excellent tools for finding patterns which are far too complex or numerous for a human programmer to extract and teach the machine to recognize. However, note that the result is not exactly the same. To design the proper neural network architecture for lane departure warning, we thought about the property of neural network as shown in Figure 6. I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021, 5 Reasons You Don’t Need to Learn Machine Learning, Building Simulations in Python — A Step by Step Walkthrough, 5 Free Books to Learn Statistics for Data Science, A Collection of Advanced Visualization in Matplotlib and Seaborn with Examples, Ensure gradients remain large through the hidden unit. ResNet have a simple ideas: feed the output of two successive convolutional layer AND also bypass the input to the next layers! The encoder is a regular CNN design for categorization, while the decoder is a upsampling network designed to propagate the categories back into the original image size for segmentation. VGG used large feature sizes in many layers and thus inference was quite costly at run-time. I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021, 5 Reasons You Don’t Need to Learn Machine Learning, Building Simulations in Python — A Step by Step Walkthrough, 5 Free Books to Learn Statistics for Data Science, A Collection of Advanced Visualization in Matplotlib and Seaborn with Examples. The VGG networks uses multiple 3x3 convolutional layers to represent complex features. when depth is increased, the number of features, or width of the layer is also increased systematically, use width increase at each layer to increase the combination of features before next layer. Neural networks consist of input and output layers, as well as (in most cases) a hidden layer consisting of units that transform the input into something that the output layer can use. Here is the complete model architecture: Unfortunately, we have tested this network in actual application and found it to be abysmally slow on a batch of 1 on a Titan Xp GPU. I wanted to revisit the history of neural network design in the last few years and in the context of Deep Learning. Figure 6(a) shows the two major parts: the backbone (feature extraction) and inference (fully connected) layers, of the deep convolutional neural network architecture. As the “neural” part of their name suggests, they are brain-inspired systems which are intended to replicate the way that we humans learn. The much more extensive neural network was created by scaling the insights of LeNet in AlexNet Architecture. Outline 1 The Basics Example: Learning the XOR 2 Training Back Propagation 3 Neuron Design Cost Function & Output Neurons Hidden Neurons 4 Architecture Design Architecture Tuning … One such typical architecture is shown in the diagram below − Don’t Start With Machine Learning. negative log-likelihood) takes the following form: Below is an example of a sigmoid output coupled with a mean squared error loss. Swish, on the other hand, is a smooth non-monotonic function that does not suffer from this problem of zero derivatives. However, swish tends to work better than ReLU on deeper models across a number of challenging datasets. The human brain is really complex. In overall this network was the origin of much of the recent architectures, and a true inspiration for many people in the field. This neural network is formed in three layers, called the input layer, hidden layer, and output layer. Neural architecture search (NAS) uses machine learning to automate ANN design. ENet is a encoder plus decoder network. Automatic neural architecture design has shown its potential in discovering power- ful neural network architectures. I believe it is better to learn to segment objects rather than learn artificial bounding boxes. The success of AlexNet started a small revolution. Given the usefulness of these techniques, the internet giants like Google were very interested in efficient and large deployments of architectures on their server farms. Want to Be a Data Scientist? The rectified linear unit is one of the simplest possible activation functions. The most commonly used structure is shown in Fig. ReLU is the simplest non-linear activation function and performs well in most applications, and this is my default activation function when working on a new neural network problem. We have already discussed output units in some detail in the section on activation functions, but it is good to make it explicit as this is an important point. Now we will try adding another node and see what happens. Alex Krizhevsky released it in 2012. Reducing the number of features, as done in Inception bottlenecks, will save some of the computational cost. This obviously amounts to a massive number of parameters, and also learning power. This concatenated input is then passed through an activation function, which evaluates the signal response and determines whether the neuron should be activated given the current inputs. ENet was designed to use the minimum number of resources possible from the start. Some of the most common choices for activation function are: These activation functions are summarized below: The sigmoid function was all we focused on in the previous article. A generalized multilayer and multi-featured network looks like this: We have m nodes, where m refers to the width of a layer within the network. We see that the number of degrees of freedom has increased again, as we might have expected. This is problematic as it can result in a large proportion of dead neurons (as high as 40%) in the neural network. Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. If you are trying to classify images into one of ten classes, the output layer will consist of ten nodes, one each corresponding to the relevant output class — this is the case for the popular MNIST database of handwritten numbers. One representative figure from this article is here: Reporting top-1 one-crop accuracy versus amount of operations required for a single forward pass in multiple popular neural network architectures. The technical report on ENet is available here. RNN is one of the fundamental network architectures from which other deep learning architectures are built. FractalNet uses a recursive architecture, that was not tested on ImageNet, and is a derivative or the more general ResNet. Now the claim of the paper is that there is a great reduction in parameters — about 1/2 in case of FaceNet, as reported in the paper. And then it became clear…. That is 256x256 x 3x3 convolutions that have to be performed (589,000s multiply-accumulate, or MAC operations). on Unsupervised Feature Learning and Deep Learning, NVIDIA Deep learning course (summer 2015), Google’s Deep Learning course on Udacity (January 2016), Stanford CS224d: Deep Learning for Natural Language Processing (spring 2015) by Richard Socher, Tutorial given at NAACL HLT 2013: Deep Learning for Natural Language Processing (without Magic) (videos + slides), CS231n Convolutional Neural Networks for Visual Recognition, Deep learning in neural networks: An overview, Continual lifelong learning with neural networks: A review — Open access, Recent advances in physical reservoir computing: A review — Open access, Ensemble Neural Networks (ENN): A gradient-free stochastic method — Open access, Multilayer feedforward networks are universal approximators, A comparison of deep networks with ReLU activation function and linear spline-type methods — Open access, Networks of spiking neurons: The third generation of neural network models, Approximation capabilities of multilayer feedforward networks, On the momentum term in gradient descent learning algorithms. Note also that here we mostly talked about architectures for computer vision. However, most architecture designs are ad hoc explorations without systematic guidance, and the final DNN architecture identified through automatic searching is not interpretable. Because of this, the hyperbolic tangent function is always preferred to the sigmoid function within hidden layers. Before each pooling, increase the feature maps. Future articles will look at code examples involving the optimization of deep neural networks, as well as some more advanced topics such as selecting appropriate optimizers, using dropout to prevent overfitting, random restarts, and network ensembles. • use a sum of the average and max pooling layers. A neural network with a single hidden layer gives us only one degree of freedom to play with. But training of these network was difficult, and had to be split into smaller networks with layers added one by one. These videos are not part of the training dataset. Similarly neural network architectures developed in other areas, and it is interesting to study the evolution of architectures for all other tasks also. In this case, we first perform 256 -> 64 1×1 convolutions, then 64 convolution on all Inception branches, and then we use again a 1x1 convolution from 64 -> 256 features back again. To read more about this, I recommend checking out the original paper on arxiv: In the next section, we will discuss loss functions in more detail. This result looks similar to the situation where we had two nodes in a single hidden layer. Thus, leaky ReLU is a subset of generalized ReLU. And although we are doing less operations, we are not losing generality in this layer. Binary Neural Networks (BNNs) show promising progress in reducing computational and memory costs, but suffer from substantial accuracy degradation compared to their real-valued counterparts on large-scale datasets, e.g., Im-ageNet. We believe that crafting neural network architectures is of paramount importance for the progress of the Deep Learning field. This can only be done if the ground truth is known, and thus a training set is needed in order to generate a functional network. Most people did not notice their increasing power, while many other researchers slowly progressed. They are excellent tools for finding patterns which are far too complex or numerous for a human programmer to extract and teach the machine to recognize. An ANN is configured for a specific application, such as pattern recognition or data classification, through a learning process. • if you cannot increase the input image size, reduce the stride in the con- sequent layers, it has roughly the same effect. So far we have only talked about sigmoid as an activation function but there are several other choices, and this is still an active area of research in the machine learning literature. By 2 layers can be thought as a small classifier, or a Network-In-Network! This classifier is also extremely low number of operations, compared to the ones of AlexNet and VGG. Neural Networks: Design Shan-Hung Wu shwu@cs.nthu.edu.tw Department of Computer Science, National Tsing Hua University, Taiwan Machine Learning Shan-Hung Wu (CS, NTHU) NN Design Machine Learning 1 / 49 . New architectures are handcrafted by careful experimentation or modified from a handful of existing networks. The performance of the network can then be assessed by testing it on unseen data, which is often known as a test set. This is similar to older ideas like this one. This article is the second in a series of articles aimed at demystifying the theory behind neural networks and how to design and implement them for solving practical problems. Both of these trends made neural network progress, albeit at a slow rate. This seems to be contrary to the principles of LeNet, where large convolutions were used to capture similar features in an image. In this work we study existing BNN architectures and revisit the commonly used technique to include scaling factors. use only 3x3 convolution, when possible, given that filter of 5x5 and 7x7 can be decomposed with multiple 3x3. The found out that is advantageous to use: • use ELU non-linearity without batchnorm or ReLU with it. Network-in-network (NiN) had the great and simple insight of using 1x1 convolutions to provide more combinational power to the features of a convolutional layers. Additional insights about the ResNet architecture are appearing every day: And Christian and team are at it again with a new version of Inception. I tried understanding Neural networks and their various types, but it still looked difficult.Then one day, I decided to take one step at a time. Technically, we do not need non-linearity, but there are benefits to using non-linear functions. In fact the bottleneck layers have been proven to perform at state-of-art on the ImageNet dataset, for example, and will be also used in later architectures such as ResNet. GoogLeNet used a stem without inception modules as initial layers, and an average pooling plus softmax classifier similar to NiN. This idea will be later used in most recent architectures as ResNet and Inception and derivatives. This corresponds to “whitening” the data, and thus making all the neural maps have responses in the same range, and with zero mean. The deep “Convolutional Neural Networks (CNNs)” gained a grand success on a broad of computer vision tasks. Take a look, Coursera Neural Networks for Machine Learning (fall 2012), Hugo Larochelle’s course (videos + slides) at Université de Sherbrooke, Stanford’s tutorial (Andrew Ng et al.) If you are interested in a comparison of neural network architecture and computational performance, see our recent paper. This is necessary in order to perform backpropagation in the network, to compute gradients of error (loss) with respect to the weights which are then updated using gradient descent. ResNet uses a fairly simple initial layers at the input (stem): a 7x7 conv layer followed with a pool of 2. These abstract representations quickly become too complex to comprehend, and to this day the workings of neural networks to produce highly complex abstractions are still seen as somewhat magical and is a topic of research in the deep learning community. However, the maximum likelihood approach was adopted for several reasons, but primarily because of the results it produces. Notice that this is no relation between the number of features and the width of a network layer. • use fully-connected layers as convolutional and average the predictions for the final decision. Automatic neural architecture design has shown its potential in discovering power-ful neural network architectures. It may reduce the parameters and size of network on disk, but is not usable. The same paper also showed that large, shallow networks tend to overfit more — which is one stimulus for using deep neural networks as opposed to shallow neural networks. With a third hidden node, we add another degree of freedom and now our approximation is starting to look reminiscent of the required function. Maxout is simply the maximum of k linear functions — it directly learns the activation function. The success of a neural network approach is deeply dependent on the right network architecture. This helps training as the next layer does not have to learn offsets in the input data, and can focus on how to best combine features. What differences do we see if we use multiple hidden layers? maximize information flow into the network, by carefully constructing networks that balance depth and width. Let’s examine this in detail. When considering convolutional neural networks, which are used to study images, when we look at hidden layers closer to the output of a deep network, the hidden layers have highly interpretable representations, such as faces, clothing, etc. There are also specific loss functions that should be used in each of these scenarios, which are compatible with the output type. Here are some videos of ENet in action. Complex hierarchies and objects can be learned using this architecture. However, CNN structures training consumes a massive computing resources amount. While the classic network architectures were And computing power was on the rise, CPUs were becoming faster, and GPUs became a general-purpose computing tool. This neural network architecture has won the challenging competition of ImageNet by a considerable margin. Sometimes, networks can have hundreds of hidden layers, as is common in some of the state-of-the-art convolutional architectures used for image analysis. It is interesting to note that the recent Xception architecture was also inspired by our work on separable convolutional filters. “The use of cross-entropy losses greatly improved the performance of models with sigmoid and softmax outputs, which had previously suffered from saturation and slow learning when using the mean squared error loss.”. ResNet, when the output is fed back to the input, as in RNN, the network can be seen as a better. This architecture uses separable convolutions to reduce the number of parameters. See “bottleneck layer” section after “GoogLeNet and Inception”. But the great insight of the inception module was the use of 1×1 convolutional blocks (NiN) to reduce the number of features before the expensive parallel blocks. This post was inspired by discussions with Abhishek Chaurasia, Adam Paszke, Sangpil Kim, Alfredo Canziani and others in our e-Lab at Purdue University. The operations are now: For a total of about 70,000 versus the almost 600,000 we had before. This is also the very first time that a network of > hundred, even 1000 layers was trained. The contribution of this work were: At the time GPU offered a much larger number of cores than CPUs, and allowed 10x faster training time, which in turn allowed to use larger datasets and also bigger images. it has been found that ResNet usually operates on blocks of relatively low depth ~20–30 layers, which act in parallel, rather than serially flow the entire length of the network. The LeNet5 architecture was fundamental, in particular the insight that image features are distributed across the entire image, and convolutions with learnable parameters are an effective way to extract similar features at multiple location with few parameters. The leaky ReLU still has a discontinuity at zero, but the function is no longer flat below zero, it merely has a reduced gradient. In the final section, we will discuss how architectures can affect the ability of the network to approximate functions and look at some rules of thumb for developing high-performing neural architectures. This activation potential is mimicked in artificial neural networks using a probability. However, ReLU should only be used within hidden layers of a neural network, and not for the output layer — which should be sigmoid for binary classification, softmax for multiclass classification, and linear for a regression problem. Neural architecture search (NAS) is a technique for automating the design of artificial neural networks (ANN), a widely used model in the field of machine learning. There are two types of inputs in choice modeling: alternative-specific variables x ik and individual-specific variables z i. 497–504 (2017) Google Scholar Architecture Design for Deep Neural Networks III 1. If this is too big for your GPU, decrease the learning rate proportionally to the batch size. Before we move on to a case study, we will understand some CNN architectures, and also, to get a sense of the learning neural networks do, we will discuss various neural networks. We also discussed how this idea can be extended to multilayer and multi-feature networks in order to increase the explanatory power of the network by increasing the number of degrees of freedom (weights and biases) of the network, as well as the number of features available which the network can use to make predictions. Representative architectures (Figure 1) include GoogleNet (2014), VGGNet (2014), ResNet (2015), and DenseNet (2016), which are developed initially from image classification. ANNs, like people, learn by examples. Or be able to keep the computational cost the same, while offering improved performance. In this post, I'll discuss commonly used architectures for convolutional networks. It may be easy to separate if you have two very dissimilar fruit that you are comparing, such as an apple and a banana. In one of my previous tutorials titled “Deduce the Number of Layers and Neurons for ANN” available at DataCamp, I presented an approach to handle this question theoretically. It has been shown by Ian Goodfellow (the creator of the generative adversarial network) that increasing the number of layers of neural networks tends to improve overall test set accuracy. They can use their internal state (memory) to process variable-length sequences of … In general, it is good practice to use multiple hidden layers as well as multiple nodes within the hidden layers, as these seem to result in the best performance. To combat the issue of dead neurons, leaky ReLU was introduced which contains a small slope. ResNet also uses a pooling layer plus softmax as final classifier. Our approximation is now significantly improved compared to before, but it is still relatively poor. We will see that this trend continues with larger networks. • if your network has a complex and highly optimized architecture, like e.g. We use the Cartesian ge-netic programming (CGP)[Miller and Thomson, 2000] en-coding scheme to represent the CNN architecture, where the architecture is represented by a … We want to select a network architecture that is large enough to approximate the function of interest, but not too large that it takes an excessive amount of time to train. Therefore being able to save parameters and computation was a key advantage. Loss functions (also called cost functions) are an important aspect of neural networks. A neural network without any activation function would simply be a linear regression model, which is limited in the set of functions it can approximate. I would look at the research papers and articles on the topic and feel like it is a very complex topic. The basic search algorithm is to propose a candidate model, evaluate it against a dataset and use the results as feedback to teach the NAS network. A linear function is just a polynomial of one degree. A multidimensional version of the sigmoid is known as the softmax function and is used for multiclass classification. In general, anything that has more than one hidden layer could be described as deep learning. Another important feature of an activation function is that it should be differentiable. Cross-entropy between training data and model distribution (i.e. This means that much more complex selection criteria are now possible. Make learning your daily ritual. This goes back to the concept of the universal approximation theorem that we discussed in the last article — neural networks are generalized non-linear function approximators. It is a much broader and more in-depth version of LeNet. See figure: inception modules can also decrease the size of the data by providing pooling while performing the inception computation. Swish is essentially the sigmoid function multiplied by x: One of the main problems with ReLU that gives rise to the vanishing gradient problem is that its derivative is zero for half of the values of the input x. These ideas will be also used in more recent network architectures as Inception and ResNet. • use the linear learning rate decay policy. As such, the loss function to use depends on the output data distribution and is closely coupled to the output unit (discussed in the next section). The Inception module after the stem is rather similar to Inception V3: They also combined the Inception module with the ResNet module: This time though the solution is, in my opinion, less elegant and more complex, but also full of less transparent heuristics. And a lot of their success lays in the careful design of the neural network architecture. Actually, this function is not a particularly good function to use as an activation function for the following reasons: Sigmoids are still used as output functions for binary classification but are generally not used within hidden layers. I decided to start with basics and build on them. Random utility maximization and deep neural network . The number of hidden layers is highly dependent on the problem and the architecture of your neural network. I will start with a confession – there was a time when I didn’t really understand deep learning. Depending upon which activation function is chosen, the properties of the network firing can be quite different. Selecting hidden layers and nodes will be assessed in further detail in upcoming tutorials. In 2012, Alex Krizhevsky released AlexNet which was a deeper and much wider version of the LeNet and won by a large margin the difficult ImageNet competition. neural network architectures. In 2010 Dan Claudiu Ciresan and Jurgen Schmidhuber published one of the very fist implementations of GPU Neural nets. NiN also used an average pooling layer as part of the last classifier, another practice that will become common. Next, we will discuss activation functions in further detail. The VGG networks from Oxford were the first to use much smaller 3×3 filters in each convolutional layers and also combined them as a sequence of convolutions. Even at this small size, ENet is similar or above other pure neural network solutions in accuracy of segmentation. Look at a comparison here of inference time per image: Clearly this is not a contender in fast inference! However, notice that the number of degrees of freedom is smaller than with the single hidden layer. RNNs consist of a rich set of deep learning architectures. The article also proposed learning bounding boxes, which later gave rise to many other papers on the same topic. We have already discussed that neural networks are trained using an optimization process that requires a loss function to calculate the model error. ResNet with a large number of layers started to use a bottleneck layer similar to the Inception bottleneck: This layer reduces the number of features at each layer by first using a 1x1 convolution with a smaller output (usually 1/4 of the input), and then a 3x3 layer, and then again a 1x1 convolution to a larger number of features. Christian Szegedy from Google begun a quest aimed at reducing the computational burden of deep neural networks, and devised the GoogLeNet the first Inception architecture. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), pp. A Torch7 implementation of this network is available here An implementation in Keras/TF is availble here. Cross-entropy and mean squared error are the two main types of loss functions to use when training neural network models. This is due to the arrival of a technique called backpropagation (which we discussed in the previous tutorial), which allows networks to adjust their neuron weights in situations where the outcome doesn’t match what the creator is hoping for — like a network designed to recognize dogs, which misidentifies a cat, for example. The separate convolution is the same as Xception above. Neural Architecture Search: The Next Half Generation of Machine Learning Speaker: Lingxi Xie (谢凌曦) Noah’s Ark Lab, Huawei Inc. (华为诺亚方舟实验室) Slides available at my homepage (TALKS) 2. In general, it is not required that the hidden layers of the network have the same width (number of nodes); the number of nodes may vary across the hidden layers. The NiN architecture used spatial MLP layers after each convolution, in order to better combine features before another layer. One problem with ReLU is that some gradients can be unstable during training and can die. Swish is still seen as a somewhat magical improvement to neural networks, but the results show that it provides a clear improvement for deep networks. The researchers in this field are concerned on designing CNN structures to maximize the performance and accuracy. It can cause a weight update causes the network to never activate on any data point. ReLU avoids and rectifies the vanishing gradient problem. Hence, let us cover various computer vision model architectures, types of networks and then look at how these are used in applications that are enhancing our lives daily. A neural network’s architecture can simply be defined as the number of layers (especially the hidden ones) and the number of hidden neurons within these layers. Make learning your daily ritual. In December 2013 the NYU lab from Yann LeCun came up with Overfeat, which is a derivative of AlexNet. Want to Be a Data Scientist? The leaky and generalized rectified linear unit are slight variations on the basic ReLU function. It is a re-hash of many concepts from ResNet and Inception, and show that after all, a better design of architecture will deliver small network sizes and parameters without needing complex compression algorithms. At the time there was no GPU to help training, and even CPUs were slow. Suganuma, M., Shirakawa, S., Nagao, T.: A genetic programming approach to designing convolutional neural network architectures. Using a non-linear activation we are able to generate non-linear mappings from inputs to outputs. This worked used only neural networks, and no other algorithm to perform image segmentation. To understand this idea, imagine that you are trying to classify fruit based on the length and width of the fruit.

Analytics Icon White, Arabic Cheat Sheet, Masala Blister Packing, Jim Corbett Hotels, Korg Rp-c1 Battery, Tricycle For 2 Year Old Baby Girl, How To Feed Baby Without Tv, The Kitchen Roasted Cauliflower, Jam Filled Cookies, Truss Calculation Examples, Old Man Logan Map,