Crandi

Cocalc Advanced Optimization Methods Ipynb

Leo Migdal
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cocalc advanced optimization methods ipynb

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: 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 No need for learning rate hyper-parameter (α\alphaα). Usually converge much faster than gradient descent.

The cost function is the function which we need to minimize. After defining the cost function, we can use the minimize function from scipy.optimize to minimize the cost function. To use the minimize function, we need to provide the following three arguments: 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. Until now, you've always used Gradient Descent to update the parameters and minimize the cost. In this notebook, you will learn 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. Gradient descent goes "downhill" on a cost function JJJ.

Think of it as trying to do this: Notations: As usual, ∂J∂a=\frac{\partial J}{\partial a } = ∂a∂J​= da for any variable a. To get started, run the following code to import the libraries you will need. A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all mmm examples on each step, it is also called Batch Gradient Descent. This notebook uses the Optim.jl package which has general purpose routines for optimization.

(As alternatives, consider the NLopt.jl and JuMP.jl) For linear-quadratic problems (mean-variance, least squares, etc), it is probably more efficient use specialized routines. This is discussed in another notebook. finds the x value (in the interval [a,b]) that minimizes fn1(x,0.5). The x->fn1(x,0.5) syntax makes this a function of x only. The output (Sol) contains a lot of information.

If you prefer to give a starting guess c instead of an interval, then supply it as as a vector [c]. of linear-quadratic problems, using the OSQP.jl package. The example is (for pedagogical reasons) the same as in the other notebooks on optimization. Otherwise, the methods illustrated here are well suited for cases when the objective involves the portfolio variance (w′Σw w'\Sigma w w′Σw) or when the estimation problem is based on minimizing the sum of squared... The OSQP.jl package is tailor made for solving linear-quadratic problems (with linear restrictions). It solves problems of the type

min⁡0.5θ′Pθ+q′θ\min 0.5\theta' P \theta + q' \thetamin0.5θ′Pθ+q′θ subject to l≤Aθ≤ul \leq A \theta \leq ul≤Aθ≤u. To get an equality restriction in row i, set l[i]=u[i]. Notice that (P,A)(P,A)(P,A) to should be Sparse matrices and (q,l,u)(q,l,u)(q,l,u) vectors with Float64 numbers. This notebook contains Part 3 from the main SageMath_Calculus_Derivatives_Optimization notebook. For the complete course, please refer to the main notebook: SageMath_Calculus_Derivatives_Optimization.ipynb Critical Point: Where f′(x)=0f'(x) = 0f′(x)=0 or f′(x)f'(x)f′(x) is undefined

Local Maximum: f(c)≥f(x)f(c) \geq f(x)f(c)≥f(x) for all xxx near ccc Local Minimum: f(c)≤f(x)f(c) \leq f(x)f(c)≤f(x) for all xxx near ccc

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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: 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 ...

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