Cocalc Optimization Ipynb

This class, Optimization, is the eighth of eight classes in the Machine Learning Foundations series. It builds upon the material from each of the other classes in the series -- on linear algebra, calculus, probability, statistics, and algorithms -- in order to provide a detailed introduction to training machine... Through the measured exposition of theory paired with interactive examples, you’ll develop a working understanding of all of the essential theory behind the ubiquitous gradient descent approach to optimization as well as how to... You’ll also learn about the latest optimizers, such as Adam and Nadam, that are widely-used for training deep neural networks. Over the course of studying this topic, you'll: Discover how the statistical and machine learning approaches to optimization differ, and why you would select one or the other for a given problem you’re solving.

Understand exactly how the extremely versatile (stochastic) gradient descent optimization algorithm works, including how to apply it This notebook contains Part 4 from the main SageMath_Calculus_Derivatives_Optimization notebook. For the complete course, please refer to the main notebook: SageMath_Calculus_Derivatives_Optimization.ipynb Identify the quantity to optimize and constraints Set up variables and express the objective function Find the domain of the objective function

This notebook contains Part 2 from the main SageMath_Calculus_Derivatives_Optimization notebook. For the complete course, please refer to the main notebook: SageMath_Calculus_Derivatives_Optimization.ipynb Power Rule: (xn)′=nxn−1(x^n)' = nx^{n-1}(xn)′=nxn−1 Constant Multiple: (cf)′=cf′(cf)' = cf'(cf)′=cf′ Sum/Difference: (f±g)′=f′±g′(f \pm g)' = f' \pm g'(f±g)′=f′±g′ A critical task in most machine learning or probabilistic programming pipelines is the optimization of model hyperparameters.

Several strategies can be used for function optimization, such as randomly sampling the parameter space (random search) or systematically evaluating the parameter space (grid search). This is often not trivial, because the loss function for a particular parameter can be noisy and non-linear, and for most problems we are omptimizing a set of parameters simultaneously, which can result in... Moreover, for large problems and complex models (e.g. deep neural networks) a single model run can be expensive and time-consuming. As a result, doing systematic searches over the hyperparameter space is infeasible, and random searches are usually ineffective. To circumvent this, Bayesian optimization offers a principled and efficient approach for directing a search of arbitrary global optimization problems.

It involves constructing a probabilistic model of the objective function, and then using an auxiliary function, called an acquisition function, to obtain candidate values for evaluation using the true objective function. Bayesian Optimization is often used in applied machine learning to tune the hyperparameters of a given model on a validation dataset. Global function optimization involves finding the minimum (maximum) of a function of interest. Samples are drawn from the domain and evaluated by the objective function to give a score or cost. These samples are candidate optimal values, which are compared to previous samples based on their cost. While the objective function may be simple to specify mathematically and in code, it can be computationally challenging to compute, and its form may be non-linear and multi-dimensional.

Moreover, its solution may be non-convex, implying that a discovered mimimum value may not be a global minimum. Specific to data science, many machine learning algorithms involve the optimization of weights, coefficients, and hyperparameters based on information contained in training data. We seek a principled method for evaluating the parmaeter space, such that consecutive samples are taken from regions of the search space that are more likely to contain minima. The methods learned in Chapter 4 of the text for finding extreme values have practical applications in many areas of life. In this lab, we will use SageMath to help with solving several optimization problems. The following strategy for solving optimization problems is outlined on Page 264 of the text.

Read and understand the problem. What is the unknown? What are the given quantities and conditions? Draw a picture. In most problems it is useful to draw a picture and identify the given and required quantities in the picture. Introduce variables.

Asign a symbol for the quantity, let us call it QQQ, that is to be maximized or minimized. Also, select symbols for other unknown quantities. Use suggestive notation whenever possible: AAA for area, hhh for height, rrr for radius, etc. This notebook explores numerical optimization techniques available in SageMath, from finding minima and maxima of functions to solving linear and integer programming problems. The history of optimization is rich and spans millennia. Ancient Greeks studied isoperimetric problems (finding shapes with maximum area for fixed perimeter).

Isaac Newton and Gottfried Leibniz developed calculus in the 17th century, providing tools for finding extrema via derivatives. The simplex algorithm, revolutionary for linear programming, was developed by George Dantzig in 1947. Leonid Kantorovich and Tjalling Koopmans won the Nobel Prize in Economics (1975) for their work on optimal resource allocation. Modern optimization combines classical analysis, linear algebra, and computational algorithms. Optimization problems generally take the form: Unconstrained optimization: No constraints on xxx

Linear programming (LP): fff, gig_igi​, hjh_jhj​ all linear Until now, you've always used Gradient Descent to update the parameters and minimize the cost. In this notebook, you'll gain skills with some more advanced optimization methods that can speed up learning and perhaps even get you to a better final value for the cost function. Having a good optimization algorithm can be the difference between waiting days vs. just a few hours to get a good result. By the end of this notebook, you'll be able to:

Apply optimization methods such as (Stochastic) Gradient Descent, Momentum, RMSProp and Adam Use random minibatches to accelerate convergence and improve optimization Gradient descent goes "downhill" on a cost function JJJ. Think of it as trying to do this: Some code in this notebook runs pretty slowly in CoCalc. If you have the ability, you might wish to run this notebook locally or on a more powerful remote machine.

Fitting a model in machine learning is an optimization problem. In a previous lesson we saw how logistic and linear regression use optimization to find the regression model coefficients to minimize the difference between observed and predicted values of the response variable. Most machine learning models also come with a bunch of parameters that need to be set which can alter the fit of the model. For example, here is the LogisticRegression class from scikit learn (sklearn): Some of these parameters have to do with exactly what model is fit. For instance, penalty changes the form of regularization added to the objective function to prevent overfitting while C changes the strength of the regularization (larger C is less regularization).

These extra parameters are usually called hyperparameters and to get the best model they often need to be tuned. This tuning is another kind of optimization and is usually called "hyperparameter optimization" or "hyperparameter tuning". This is a hot area and a little searching with Google will yield a ton of results. Here is one article that gives an overview of hyperparameter tuning methods (but gets a bit technical at the end). To keep everything straight it helps to remember that model parameters and hyperparameters are different. Hyperparameters are set or determined before the model is fit.

Model parameters are determined during the process of fitting the model to the data.

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