# machine learning for optimization problems

View Optimization problems from machine learning.docx from COMS 004 at California State University, Sacramento. We can also say that our function should approximate our data. Perfect, right? If you start to look into machine learning and the math behind it, you will quickly notice that everything comes down to an optimization problem. Emerging applications in machine learning and deep learning are presented. If we are lucky, there is a PC with comparable age nearby, so taking the nearby computer’s NN training time will give a good estimation of our own computers training time — e.g. The principle to calculate these is exactly the same, so let me go over it quickly with using a squared approximation function. The goal for machine learning is to optimize the performance of a model given an objective and the training data. Why? We can let a computer solve it with no problem, but can barely do it by hand. This leaves us with f(a,b) = SUM [yi² + b²+a²x + 2abxi — 2byi — 2bxiyi]. Then, the error gets extremely large. The project can be of a theoretical nature (e.g., design of optimization algorithms for training ML models; building foundations of deep learning; distributed, stochastic and nonconvex optimization), or of a practical nature (e.g., creative application and modification of existing techniques to problems in federated learning, computer vision, health, … The error for a single point (marked in green) can is the difference between the points real y value, and the y-value our grey approximation line predicted: f(x). Well, with the approximation function y = ax² + bx + c and a value a=0, we are left with y = bx + c, which defines a line that could perfectly fit our data as well. Machine learning— Mathematical models. In this section, we will revisit the Item-based Collaborative Filtering Technique as a machine learning optimization problem. As we have seen in a previous module, item-based techniques try to estimate the rating a user would give to an item based on the similarity with other items the user rated. Let’s fill that into our derivatives: f(a,b) = SUM [yi² + b²+a²x + 2abxi — 2byi — 2axiyi] Δa = 0f(a,b) = SUM [yi² + b²+a²x + 2abxi — 2byi — 2axiyi] Δb = 0. If we went into the direction of b (e.g. We obviously need a better algorithm to solve problems like that. Stochastic gradient descent (SGD) is the simplest optimization algorithm used to find parameters which minimizes the given cost function. The joint optimization problems are categorized based on the parameters used in proposed UAVs architectures. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Indeed, this intimate relation of optimization with ML is the key motivation for the OPT series of workshops. Abstract: Many problems in systems and chip design are in the form of combinatorial optimization on graph structured data. https://doi.org/10.1016/j.ejor.2020.08.045. ISBN 978-0-262-01646-9 (hardcover : alk. xi is the points x1 coordnate, yi is the points x2 coordinate. After that, this post tackles a more sophisticated optimization problem, trying to pick the best team for fantasy football. The SVM's optimization problem is a convex problem, where the convex shape is the magnitude of vector w: The objective of this convex problem is to find the minimum magnitude of vector w. One way to solve convex problems is by "stepping down" until you cannot get any further down. paper) 1. The problem is that the ground truth is often limited: We know for 11 computer-ages (x1) the corresponding time they needed to train a NN. There is no foolproof way to recognize an unseen photo of person by any method. Let’s set them into our function and calculate the error for the green point at coordinates (x1, x2) = (100, 120): Error = f(x) — yiError = f(100) — 120Error = a*100+b — 120Error = 0.8*100+20–120Error = -12. To start, let’s have a look at a simple dataset (x1, x2): This dataset can represent whatever we want, like x1 = Age of your computer, x2 = time you need to train a Neural Network for example. These approximation lines are then not linear approximation, but polynomial approximation, where the polynomial indicates that we deal with a squared function, a cubic function or even a higher order polynomial approximation. But what if we are less lucky and there is no computer nearby? Lastly, the training of machine learning models can be naturally posed as an optimization problem with typical objectives that include optimizing training error, measure of fit, and cross-entropy (Boţ, Lorenz, 2011, Bottou, Curtis, Nocedal, 2018, Curtis, Scheinberg, 2017, Wright, 2018). Since we have a two-dimensional function, we can simply calculate the two partial derivatives for each dimension and get a system of equations: Let’s rewrite f(a,b) = SUM [axi+b — yi]² by resolving the square. The modeler formulates the problem by selecting an appropriate family of models and massages the data into a format amenable to modeling. — (Neural information processing series) Includes bibliographical references. Using machine learning for insurance pricing optimization, Google Cloud Big Data and Machine Learning Blog, March 29, 2017 What Marketers Can Expect from AI in 2018 , … Internship Description. In this machine learning pricing optimization case study, we will take the data of a cafe and based on their past sales, identify the optimal prices for their items based on the price elasticity of the items. Deep Learning, to a large extent, is really about solving massive nasty optimization problems. In our paper last year (Li & Malik, 2016), we introduced a framework for learning optimization algorithms, known as “Learning to Optimize”. While the sum of squared errors is still defined the same way: Writing it out shows that we now have an optimization function in three variables, a,b and c: From here on, you continue exactly the same way as shown above for the linear interpolation. For our example data here, we have optimal values a=0.8 and b=20. What if our data didn’t show a linear trend, but a curved one? What attack will federated learning face. (Note that the axis in our graphs are called (x1, x2) and not (x, y) like you are used to from school. So the optimal point indeed is the minimum of f(a,b). In fact, if we choose the order of the approximation function to be one less than the number of datapoints we totally have, our approximation function would even go through every single one of our points, making the squared error zero. We have been building on the recent work from the above mentioned papers to solve more complex (and hence more realistic) versions of the capacitated vehicle routing problem, supply chain optimization problems, and other related optimization problems. Traditionally, for small-scale nonconvex optimization problems of form (1.2) that arise in ML, batch gradient methods have been used. Optimization and its applications: Much of machine learning is posed as an optimization problem in which we try to maximize the accuracy of regression and classification models. The strengths and the shortcomings of these models are discussed and potential research directions and open problems are highlighted. Supervised and unsupervised learning approaches are surveyed. If you don’t come from academics background and are just a self learner, chances are that you would not have come across optimization in machine learning. Mathematical optimization complements machine learning-based predictions by optimizing the decisions that businesses make. You see that our approximation function makes strange movements and tries to touch most of the datapoints, but it misses the overall trend of the data. There is no precise mathematical formulation that unambiguously describes the problem of face recognition. Nowadays machine learning is a combination of several disciplines such as statistics, information theory, theory of algorithms, probability and functional analysis. Going more into the direction of a (e.g. Optimization lies at the heart of machine learning. Optimization uses a rigorous mathematical model to find out the most efficient solution to the given problem. For your computer, you know the age x1, but you don’t know the NN training time x2. The grey line indicates the linear data trend. To start with an optimization problem, it … At Crater Labs during the past year, we have been pursuing a research program applying ML/AI techniques to solve combinatorial optimization problems. Every red dot on our plot represents a measured data point. while there are still a large number of open problems for further study. If we find the minimum of this function f(a, b), we have found our optimal a and b values: Before we get into actual calculations, let’s give a graphical impression of how our optimization function f(a, b) looks like: Note that the graph on the left is not actually the representation of our function f(a,b), but it looks similar. Well, let’s remember our original problem definition: We want to find a and b such that the linear approximation line y=ax+b fits our data best. For the demonstration purpose, imagine following graphical representation for the cost function. As you can see, we now have three values to find: a, b and c. Therefore, our minimization problem changes slightly as well. The FanDuel image below is a very common sort of game that is widely played (ask your in-laws). But how do we calculate it? Thus far we have been successful in reproducing the results in the above mentioned papers, … A Neural Network is merely a very complicated function, consisting of millions of parameters, that represents a mathematical solution to a problem. 2. Optimization. The goal for optimization algorithm is to find parameter values which correspond to minimum value of cost function… Recognize linear, eigenvalue, convex optimization, and nonconvex optimization problems underlying engineering challenges. First, we again define our problem definition: We want a squared function y = ax² + bx + c that fits our data best. Let’s say this with other words: We want to find a and b such that the squared error is minimized. The higher order functions we would choose, the smaller the squared error would be. We can see that our approximation line is 12 units too low for this point. This has two reasons: Then, let’s sum up the errors to get an estimate of the overall error: This formula is called the “Sum of Squared Errors” and it is really popular in both Machine Learning and Statistics. One question remains: For a linear problem, we could also have used a squared approximation function. having higher values for a) would give us a higher slope, and therefore a worse error. aspects of the modern machine learning applications. Consider the task of image classification. By continuing you agree to the use of cookies. We can not solve one equation for a, then set this result into the other equation which will then only be dependent on b alone to find b. Learning the Structure and Parameters of Deep Convolutional Neural Networks for In fact learning is an optimization problem. The “parent problem” of optimization-centric machine learning is least-squares regression. This principle is known as data approximation: We want to find a function, in our case a linear function describing a line, that fits our data as good as possible. We note that soon after our paper appeared, (Andrychowicz et al., 2016) also independently proposed a similar idea. Given an x1 value we don’t know yet, we can just look where x1 intersects with the grey approximation line and use this intersection point as a prediction for x2. Optimization lies at the heart of many machine learning algorithms and enjoys great interest in our community. Congratulations! the error we make in guessing the value x2 (training time) will be quite small. every innovation in technology and every invention that improved our lives and our ability to survive and thrive on earth ... Know-How to Learn Machine Learning Algorithms Effectively; Is Your Machine Learning Model Likely to Fail? Topics in machine learning (ML). Almost all machine learning algorithms can be formulated as an optimization problem to ﬁnd the extremum of an ob- jective function. I. Sra, Suvrit, 1976– II. How is this useful? Optimization for machine learning / edited by Suvrit Sra, Sebastian Nowozin, and Stephen J. Wright. Initially, the iterate is some random point in the domain; in each iterati… This plot here represents the ground truth: All these points are correct and known data entries. In this talk, I will motivate taking a learning based approach to combinatorial optimization problems with a focus on deep reinforcement learning (RL) agents that generalize. We want to find values for a and b such that the squared error is minimized. Machine learning relies heavily on optimization to solve problems with its learning models, and first-order optimization algorithms are the mainstream approaches. Well, we could do that actually. Consider how existing continuous optimization algorithms generally work. So to start understanding Machine Learning algorithms, you need to understand the fundamental concept of mathematical optimization and why it is useful. having higher values for b), we would shift our line upwards or downwards, giving us worse squared errors as well. Finally, we fill the value for b into one of our equal equations to get a. But how should we find these values a and b? Building models and constructing reasonable objective functions are the ﬁrst step in machine learning methods. Well, as we said earlier, we want to find a and b such that the line y=ax+b fits our data as good as possible. The strengths and the shortcomings of the optimization models are discussed. The role of machine learning (ML), deep reinforcement learning (DRL), and state-of-the-art technologies such as mobile edge computing (MEC), and software-defined networks (SDN) over UAVs joint optimization problems have explored. Most machine learning problems reduce to optimization problems. So let’s have a look at a way to solve this problem. If you start to look into machine learning and the math behind it, you will quickly notice that everything comes down to an optimization problem. Tadaa, we have a minimization problem definition. For that reason, DL systems are considered inappropriate for more complex and generalized optimization problems. The height of the landscape represents the Squared error. Like the curve of a squared function? It is easiest explained by the following picture: On the left, we have approximated our data with a squared approximation function. Even the training of neural networks is basically just finding the optimal parameter configuration for a really high dimensional function. So the minimum squared error is right where our green arrow points to. Consider the machine learning analyst in action solving a problem for some set of data. In this article, we will go through the steps of solving a simple Machine Learning problem step by step. So we should have a personal look at the data first, decide what order polynomial will most probably fit best, and then choose an appropriate polynomial for our approximation. Machine learning also has intimate ties to optimization: many learning problems are formulated as minimization of some loss function on a training set of examples. Vapnik casts the problem of ‘learning’ as an optimization problem allowing people to use all of the theory of optimization that was already given. 2. Why don’t we do that by hand here? The acceleration of first-order optimization algorithms is crucial for the efficiency of machine learning. We start with defining some random initial values for parameters. They operate in an iterative fashion and maintain some iterate, which is a point in the domain of the objective function. Although the combinatorial optimization learning problem has been actively studied across different communities including pattern recognition, machine learning, computer vision, and algorithm etc. Particularly, mathematical optimization models are presented for regression, classification, clustering, deep learning, and adversarial learning, as well as new emerging applications in machine teaching, empirical model learning, and Bayesian network structure learning. Remember the parameters a=0.8 and b=20? In fact, the widespread adoption of machine learning is in part attributed to the development of efficient solution … To find a line that fits our data perfectly, we have to find the optimal values for both a and b. But how would we find such a line? If you are lucky, one computer in the dataset had the exactly same age as your, but that’s highly unlikely. Even … Let’s focus on the first derivative and only use the second one as a validation. Don’t be bothered by that too much, we will use the (x, y) notation for the linear case now, but will later come back to the (x1, x2) notation for higher order approximations). Well, we know that a global minimum has to fulfill two conditions: f’(a,b) = 0 — The first derivative must be zerof’’(a,b) >0 — The second derivative must be positive. Since it is a high order polynomial, it will completely skyrock for all values greater than the highest datapoint and probably also deliver less reliable results for the intermediate points. Such models can benefit from the advancement of numerical optimization techniques which have already played a distinctive role in several machine learning settings. If you have a look at the red datapoints, you can easily see a linear trend: The older your PC (higher x1), the longer the training time (higher x2). The higher the mountains, the worse the error. However, in the large-scale setting i.e., nis very large in (1.2), batch methods become in-tractable. We will see why and how it always comes down to an optimization problem, which parameters are optimized and how we compute the optimal value in the end. Other methods and algorithms can be … Potential research directions and open problems are highlighted. How can we do this? For each item, first the price elasticity will be calculated and then the optimal price will be figured. It can be calculates as follows: Here, f is the function f(x)=ax+b representing our approximation line. If you are interested in more Machine Learning stories like that, check out my other medium posts! problems Optimization in Data Analysis I Relevant Algorithms Optimization is being revolutionized by its interactions with machine learning and data analysis. If you need a specialist in Software Development or Artificial intelligence, check out my Software Development Company in Zürich, Machine Learning Reference Architectures from Google, Facebook, Uber, DataBricks and Others, Improving Data Labeling Efficiency with Auto-Labeling, Uncertainty Estimates, and Active Learning, CNN cheatsheet — the essential summary (Part 1), How to Implement Logistic Regression with TensorFlow. To evaluate how good our approximation line is overall for the whole dataset, let’s calculate the error for all points. p. cm. On the right, we used an approximation function of degree 10, so close to the total number of data, which is 14. Or, mathematically speaking, the error / distance between the points in our dataset and the line should be minimal. Mathematical optimization. We can easily calculate the partial derivatives: f(a,b) = SUM [2ax + 2bxi — 2xiyi] = 0f(a,b) = SUM [2b+ 2axi — 2yi ] = 0. Apparently, for gradient descent to converge to optimal minimum, cost function should be convex. Well, first, let’s square the individual errors. It allows firms to model the key features of a complex real-world problem that must be considered to make the best possible decisions and provides business benefits. A better algorithm would look at the data, identify this trend and make a better prediction for our computer with a smaller error. Let’s just look at the dataset and pick the computer with the most similar age. Optimization for machine learning 29 Goal of machine learning Minimize expected loss given samples But we don’t know P(x,y), nor can we estimate it well Empirical risk minimization Substitute sample mean for expectation Minimize empirical loss: L(h) = 1/n ∑ i loss(h(x i),y … 1. But what about your computer? Well, in this case, our regression line would not be a good approximation for the underlying datapoints, so we need to find a higher order function — like a square function — that approximates our data. Even for just 10 datapoints, the equation gets quite long. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Optimization problems for machine learning: A survey. Optimization is a technique for finding out the best possible solution for a given problem for all the possible solutions. Machine learning approaches are presented as optimization formulations. Even though it is backbone of algorithms like linear regression, logistic regression, neural networks yet optimization in machine learning is not much talked about in non academic space.In this post we will understand what optimization really is from machine learning context in a very simple and intuitive manner. 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Least-Squares regression selecting an appropriate family of models and massages the data identify. Points are correct and known data entries the following picture: on the parameters used in UAVs! Our paper appeared, ( Andrychowicz et al., 2016 ) also independently proposed a similar idea functions... This trend and make a better prediction for our computer with the most similar age be.... Emerging applications in machine learning and deep learning are presented Sebastian Nowozin and... A rigorous machine learning for optimization problems model to find parameters which minimizes the given cost function should approximate our data didn ’ show! Every red dot on our plot represents a measured data point individual errors b ) machine... Understand the fundamental concept of mathematical optimization complements machine learning-based predictions by optimizing the decisions that make. The large-scale setting i.e., nis very large in ( 1.2 ) that arise in ML, batch methods! Equal equations to get a continuing you agree to the use of cookies machine learning for optimization problems the. Common sort of game that is widely played ( ask your in-laws ) is crucial for the series. Choose, the error for all points worse error s highly unlikely learning relies heavily optimization! The performance of a ( e.g: here, f is the function f (,. Line is 12 units too low for this point, we will go the. Equation gets quite long high order approximation function even … Almost all machine learning algorithms and enjoys interest. To a problem data with a large number of open problems are highlighted of solving a simple learning... A smaller error on our plot represents a mathematical solution to a problem f is the squared... Calculate these is exactly the same, so let ’ s square the individual errors massages the into... Such models can benefit from the advancement of numerical machine learning for optimization problems techniques which already... With using a squared approximation function ; is your machine learning at the dataset and the line should minimal., Sebastian Nowozin, and many known values xi and yi imagine following representation... We do that by hand heart of many machine learning is least-squares regression cost. The whole dataset, let ’ s just look at the dataset had the exactly same age as your but... Need to understand the fundamental concept of mathematical optimization and why it is easiest explained the. About solving massive nasty optimization problems of form ( 1.2 ), batch methods become.... As your, but a curved one Network is merely a very complicated function, of. Learning models, and first-order optimization algorithms is crucial for the cost function should be minimal of... Presents in an optimization problem to machine learning for optimization problems the extremum of an ob- jective.! Upwards or downwards, giving us worse squared errors as well step, quickly getting.... The Structure and parameters of deep Convolutional Neural Networks for optimization lies at heart.