# Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach These concepts are taught in the Academia University degrees and Masters , together with laboratory lessons, where students learn to model, design, and plan, transportation networks intersections, road capacity, traffic demands, bus frequencies, etc. Transportation researchers, practitioners and private companies have developed professional models and simulator packages to be used in the planning, design, and operations of such systems, but most of them are not open-source, and their trial versions do not offer the necessary tools to meet the Academia requirements.

## A Modern Network Design Course using Python

In this paper, we present Net2plan-UTN, a planning and optimization tool for transportation networks built up as an extension to the Net2Plan planning tool. As a teaching material, Net2Plan is being used since in several University courses in UPCT and other Universities, and is the selected software tool for the practical exercises in the accompanying book . The internal parameters of a Model play a very important role in efficiently and effectively training a Model and produce accurate results.

Types of optimization algorithms? Optimization Algorithm falls in 2 major categories -. What is a Gradient of a function? The difference is that to calculate a derivative of a function which is dependent on more than one variable or multiple variables, a Gradient takes its place. And a gradient is calculated using Partial Derivatives.

### Evaluation Copy

Also another major difference between the Gradient and a derivative is that a Gradient of a function produces a Vector Field. A Gradient is represented by a Jacobian Matrix — which is simply a Matrix consisting of first order partial Derivatives Gradients. Hence summing up, a derivative is simply defined for a function dependent on single variables , whereas a Gradient is defined for function dependent on multiple variables.

Second Order Optimization Algorithms — Second-order methods use the second order derivative which is also called Hessian to minimize or maximize the Loss function. Since the second derivative is costly to compute, the second order is not used much. Second Order Derivative provide us with a quadratic surface which touches the curvature of the Error Surface.

## NET2PLAN-UTN: AN EDUCATIONAL TOOL FOR MODELING AND PLANNING URBAN TRANSPORTATION NETWORKS

Although the Second Order Derivative may be a bit costly to find and calculate, but the advantage of a Second order Optimization Technique is that is does not neglect or ignore the curvature of Surface. Secondly, in terms of Step-wise Performance they are better.

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Best way to know which one converges fast is to try it out yourself. Gradient Descent is the most important technique and the foundation of how we train and optimize Intelligent Systems. What is does is —. It is the most popular Optimization algorithms used in optimizing a Neural Network. Now gradient descent is majorly used to do Weights updates in a Neural Network Model , i. Now we all know a Neural Network trains via a famous technique called Backpropagation , in which we first propagate forward calculating the dot product of Inputs signals and their corresponding Weights and then apply a activation function to those sum of products, which transforms the input signal to an output signal and also is important to model complex Non-linear functions and introduces Non-linearities to the Model which enables the Model to learn almost any arbitrary functional mappings.

After this we propagate backwards in the Network carrying Error terms and updating Weights values using Gradient Descent, in which we calculate the gradient of Error E function with respect to the Weights W or the parameters , and update the parameters here Weights in the opposite direction of the Gradient of the Loss function w. The image on above shows the process of Weight updates in the opposite direction of the Gradient Vector of Error w.

The U-Shaped curve is the Gradient slope. As one can notice if the Weight W values are too small or too large then we have large Errors , so want to update and optimize the weights such that it is neither too small nor too large , so we descent downwards opposite to the Gradients until we find a local minima. Another thing while using Standard batch Gradient descent is that it computes redundant updates for large data sets. Stochastic Gradient Descent SGD on the other hand performs a parameter update for each training example. It is usually much faster technique. It performs one update at a time.

Now due to these frequent updates ,parameters updates have high variance and causes the Loss function to fluctuate to different intensities.

Optimization of Computer Networks Modeling and Algorithms A Hands On Approach

This is actually a good thing because it helps us discover new and possibly better local minima , whereas Standard Gradient Descent will only converge to the minimum of the basin as mentioned above. But the problem with SGD is that due to the frequent updates and fluctuations it ultimately complicates the convergence to the exact minimum and will keep overshooting due to the frequent fluctuations.

The problems of high variance parameter updates and unstable convergence can be rectified in another variant called Mini-Batch Gradient Descent. An improvement to avoid all the problems and demerits of SGD and standard Gradient Descent would be to use Mini Batch Gradient Descent as it takes the best of both techniques and performs an update for every batch with n training examples in each batch. S —Actually the term SGD is used also when mini-batch gradient descent is used. Now we will discuss about the various algorithms which are used to further optimize Gradient Descent.

Stochastic gradient descent maintains a single learning rate termed alpha for all weight updates and the learning rate does not change during training. A learning rate is maintained for each network weight parameter and separately adapted as learning unfolds. The method computes individual adaptive learning rates for different parameters from estimates of first and second moments of the gradients.

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Instead of adapting the parameter learning rates based on the average first moment the mean as in RMSProp, Adam also makes use of the average of the second moments of the gradients the uncentered variance. Specifically, the algorithm calculates an exponential moving average of the gradient and the squared gradient, and the parameters beta1 and beta2 control the decay rates of these moving averages.

The initial value of the moving averages and beta1 and beta2 values close to 1. This bias is overcome by first calculating the biased estimates before then calculating bias-corrected estimates. The paper is quite readable and I would encourage you to read it if you are interested in the specific implementation details. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. In the original paper, Adam was demonstrated empirically to show that convergence meets the expectations of the theoretical analysis.

They conclude:. Using large models and datasets, we demonstrate Adam can efficiently solve practical deep learning problems. Insofar, RMSprop, Adadelta, and Adam are very similar algorithms that do well in similar circumstances. Insofar, Adam might be the best overall choice. In practice Adam is currently recommended as the default algorithm to use, and often works slightly better than RMSProp.

Further, learning rate decay can also be used with Adam.

### Online optimizer

The default value of 1e-8 for epsilon might not be a good default in general. For example, when training an Inception network on ImageNet a current good choice is 1. We can see that the popular deep learning libraries generally use the default parameters recommended by the paper.

Do you have any questions? Ask your questions in the comments below and I will do my best to answer. It provides self-study tutorials on topics like: weight decay , batch normalization , dropout , model stacking and much more…. Click to learn more. The name thing is a little strange. What was so wrong with AdaMomE? The abbreviated name is only useful if it encapsulates the name, adaptive moment estimation. I think part of the process of writing useful papers is coming up with an abbreviation that will not irritate others in the field, such as anyone named Adam.

My main issue with deep learning remains the fact that a lot of efficiency is lost due to the fact that neural nets have a lot of redundant symmetry built in that leads to multiple equivalent local optima. There must be a way to address this mathematically. It puzzles me that nobody had done anything about.

## Optimization of Computer Networks - Modeling and Algorithms: Hands-On Approach, A

If you did this in combinatorics Traveling Salesman Problems type of problems , this would qualify as a horrendous model formulation. It would be great to see what you can dig up on the topic. Neural nets have been studied for a long time by some really bright people. Those bright people may excel in statistics , but non linear non convex optimization is a very specialized field where other very bright people excel. The same applies to Integer and Combinatorial optimization : very specialized field. By the way , although I am impressed by recent results in deep learning , I am not so deeply impressed by the technology. Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach Optimization of Computer Networks: Modeling and Algorithms: A Hands-On Approach

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