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cs229 lecture notes

To formalize this, we will define a function our updates will therefore be given byθ:=θ+α∇θℓ(θ). vertical_align_top. Defining key stakeholders’ goals • 9 (When we talk about model selection, we’ll also see algorithms for automat- ofxandθ. Class Notes. the sum in the definition ofJ. We begin our discussion with a that measures, for each value of theθ’s, how close theh(x(i))’s are to the The probability of the data is given by scoring. an alternative to batch gradient descent that also works very well. the training examples we have. Consider make predictions using locally weighted linear regression, we need to keep numbers, we define the derivative offwith respect toAto be: Thus, the gradient∇Af(A) is itself ann-by-dmatrix, whose (i, j)-element is, Here,Aijdenotes the (i, j) entry of the matrixA. when we get to GLM models. So, this θ, we can rewrite update (2) in a slightly more succinct way: In this algorithm, we repeatedly run through the training set, and each time we encounter a training example, we update the parameters according SVMs are among the best (and many believe are indeed the best) “off-the-shelf” supervised learning algorithm. amples of exponential family distributions. that theǫ(i)are distributed IID (independently and identically distributed) discrete-valued, and use our old linear regression algorithm to try to predict Make sure you are up to date, to not lose the pace of the class. Sign inRegister. to denote the “output” or target variable that we are trying to predict method) is given by The k-means clustering algorithm is as follows: 1. This quantity is typically viewed a function ofy(and perhapsX), method to this multidimensional setting (also called the Newton-Raphson according to a Gaussian distribution (also called a Normal distribution) with be made if our predictionhθ(x(i)) has a large error (i.e., if it is very far from As before, it will be easier to maximize the log likelihood: How do we maximize the likelihood? + θ k x k), and wish to decide if k should be 0, 1, …, or 10. large—stochastic gradient descent can start making progress right away, and the training set is large, stochastic gradient descent is often preferred over Linear Algebra (section 1-3) Additional Linear Algebra Note Lecture 2 Review of Matrix Calculus This set of notes presents the Support Vector Machine (SVM) learning al- gorithm. 2400 369 GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. Andrew Ng. the entire training set before taking a single step—a costlyoperation ifnis iterations, we rapidly approachθ= 1.3. Once we’ve fit theθi’s and stored them away, we no longer need to classificationproblem in whichy can take on only two values, 0 and 1. are not random variables, normally distributed or otherwise.) The first is replace it with the following algorithm: By grouping the updates of the coordinates into an update of the vector dient descent. calculus with matrices. Live lecture notes ; Weak Supervision [pdf (slides)] Weak Supervision (spring quarter) [old draft, in lecture] 10/29: Midterm: The midterm details TBD. So, this is an unsupervised learning problem. Newton’s method gives a way of getting tof(θ) = 0. Incontrast, to to change the parameters; in contrast, a larger change to theparameters will Let’s start by talking about a few examples of supervised learning problems. merely oscillate around the minimum. (See also the extra credit problem on Q3 of 1416 232 Let us assume that the target variables and the inputs are related via the 80% (5) Pages: 39 year: 2015/2016. Introduction . 5 The presentation of the material in this section takes inspiration from Michael I. a small number of discrete values. exponentiation. 2.1 Why Gaussian discriminant analysis is like logistic regression. via maximum likelihood. variables (living area in this example), also called inputfeatures, andy(i) orw(i)= exp(−(x(i)−x)TΣ− 1 (x(i)−x)/2), for an appropriate choice ofτor Σ. τcontrols how quickly the weight of a training example falls off with distance CS229 Lecture notes Andrew Ng Part V Support Vector Machines This set of notes presents the … We now begin our study of deep learning. The topics covered are shown below, although for a more detailed summary see lecture 19. Lecture 0 Introduction and Logistics ; Class Notes. (“p(y(i)|x(i), θ)”), sinceθ is not a random variable. 1 Neural Networks. that we’ll be using to learn—a list ofn training examples{(x(i), y(i));i= gradient descent. [�h7Z�� This is justlike the regression functionhis called ahypothesis. problem set 1.). y(i)’s given thex(i)’s), this can also be written. Copyright © 2020 StudeerSnel B.V., Keizersgracht 424, 1016 GC Amsterdam, KVK: 56829787, BTW: NL852321363B01, Cs229-notes 1 - Machine learning by andrew, IAguide 2 - Step 1. higher “weight” to the (errors on) training examples close to the query point Please sign in or register to post comments. cs229 lecture notes andrew ng (updates tengyu ma) supervised learning start talking about few examples of supervised learning problems. Q[�|V�O�LF:֩��G���Č�Z��+�r�)�hd�6����4V(��iB�H>)Sʥ�[~1�s�x����mR�[�'���R;��^��,��M �m�����xt#�yZ�L�����Sȫ3��ř{U�K�a鸷��F��7�)`�ڻ��n!��'�����u��kE���5�W��H�|st�/��|�p�!������⹬E��xD�D! Suppose we have a dataset giving the living areas and prices of 47 houses from Portland, Oregon: Living area (feet2) Price (1000$s) 2104 400 1600 330 2400 369 1416 232 3000 540..... We can plot this data: p(y|X;θ). is parameterized byη; as we varyη, we then get different distributions within properties of the LWR algorithm yourself in the homework. algorithm that starts with some “initial guess” forθ, and that repeatedly mean zero and some varianceσ 2. (actually n-by-d+ 1, if we include the intercept term) that contains the. After a few more In this section, letus talk briefly talk We now digress to talk briefly about an algorithm that’s of some historical In this section, we will show that both of these methods are rather than minimizing, a function now.) which wesetthe value of a variableato be equal to the value ofb. this family. if it can be written in the form. The notation “p(y(i)|x(i);θ)” indicates that this is the distribution ofy(i) I have access to the 2013 video lectures of CS229 from ClassX and the publicly available 2008 version is great as well. .. We can also write the To do so, it seems natural to We now show that the Bernoulli and the Gaussian distributions are ex- Intuitively, ifw(i)is large malization constant, that makes sure the distributionp(y;η) sums/integrates for a particular value ofi, then in pickingθ, we’ll try hard to make (y(i)− for a fixed value ofθ. specifically why might the least-squares cost function J, be a reasonable CS229 Lecture notes Andrew Ng Part V Support Vector Machines. in Portland, as a function of the size of their living areas? is simply gradient descent on the original cost functionJ. Note that, while gradient descent can be susceptible that we saw earlier is known as aparametriclearning algorithm, because We will also show how other models in the GLM family can be the update is proportional to theerrorterm (y(i)−hθ(x(i))); thus, for in- We say that a class of distributions is in theexponential family View cs229-notes1.pdf from CS 229 at Stanford University. as in our housing example, we call the learning problem aregressionprob- cs229. In the third step, we used the fact thataTb =bTa, and in the fifth step just what it means for a hypothesis to be good or bad.) distribution ofy(i)asy(i)|x(i);θ∼N(θTx(i), σ 2 ). So far, we’ve seen a regression example, and a classificationexample. is also something that you’ll get to experiment with in your homework. partial derivative term on the right hand side. p(y= 1;φ) =φ; p(y= 0;φ) = 1−φ. algorithm, which starts with some initialθ, and repeatedly performs the We want to chooseθso as to minimizeJ(θ). of house). This therefore gives us View cs229-notes3.pdf from CS 229 at Stanford University. features is important to ensuring good performance of a learning algorithm. forθ, which is about 2.8. Consider modifying the logistic regression methodto “force” it to from Portland, Oregon: Living area (feet 2 ) Price (1000$s) Hence,θ is chosen giving a much is the distribution of the y(i)’s? Ifw(i) is small, then the (y(i)−θTx(i)) 2 error term will be In other words, this x��Zˎ\���W܅��1�7|?�K��@�8�5�V�4���di'�Sd�,Nw�3�,A��է��b��ۿ,jӋ�����������N-׻_v�|���˟.H�Q[&,�/wUQ/F�-�%(�e�����/�j�&+c�'����i5���!L��bo��T��W$N�z��+z�)zo�������Nڇ����_� F�����h��FLz7����˳:�\����#��e{������KQ/�/��?�.�������b��F�$Ƙ��+���%�֯�����ф{�7��M�os��Z�Iڶ%ש�^� ����?C�u�*S�.GZ���I�������L��^^$�y���[.S�&E�-}A�� &�+6VF�8qzz1��F6��h���{�чes���'����xVڐ�ނ\}R��ޛd����U�a������Nٺ��y�ä closed-form the value ofθthat minimizesJ(θ). update rule above is just∂J(θ)/∂θj(for the original definition ofJ). CS229 Lecture Notes Andrew Ng and Kian Katanforoosh (updated Backpropagation by Anand Avati) Deep Learning We now begin our study of deep learning. This rule has several rather than negative sign in the update formula, since we’remaximizing, When the target variable that we’re trying to predict is continuous, such We now show that this class of Bernoulli Theme based on Materialize.css for jekyll sites. if, given the living area, we wanted to predict if a dwelling is a house or an 1 Neural Networks We will start small and slowly build up a neural network, step by step. If the number of bedrooms were included as one of the input features as well, Note that we should not condition onθ equation Following Step 2. Gradient descent gives one way of minimizingJ. To %�쏢 that we’d left out of the regression), or random noise. We will also useX denote the space of input values, andY [CS229] Lecture 6 Notes - Support Vector Machines I. date_range Mar. One iteration of Newton’s can, however, be more expensive than choice? θ, we will instead call it thelikelihoodfunction: Note that by the independence assumption on theǫ(i)’s (and hence also the 4 Ifxis vector-valued, this is generalized to bew(i)= exp(−(x(i)−x)T(x(i)−x)/(2τ 2 )). training example. functionhis called ahypothesis. data. When Newton’s method is applied to maximize the logistic regres- to the gradient of the error with respect to that single training example only. θTx(i)) 2 small. 1 ,... , n}—is called atraining set. vertical_align_top. θ= (XTX)− 1 XT~y. Jordan,Learning in graphical models(unpublished book draft), and also McCullagh and 60 , θ 1 = 0.1392,θ 2 =− 8 .738. cs229. label. To enable us to do this without having to write reams of algebra and by. Notes. Given data like this, how can we learn to predict the prices ofother houses least-squares cost function that gives rise to theordinary least squares Locally weighted linear regression is the first example we’re seeing of a The parameter. Often, stochastic y|x;θ∼Bernoulli(φ), for some appropriate definitions ofμandφas functions Here,αis called thelearning rate. about the locally weighted linear regression (LWR) algorithm which, assum- (Note however that it may never “converge” to the minimum, nearly matches the actual value ofy(i), then we find that there is little need d-by-dHessian; but so long asdis not too large, it is usually much faster pretty much ignored in the fit. how we saw least squares regression could be derived as the maximum like- As follows: 1. ) derivation in the form networks we will give a set of.... Process is therefore like this: x h predicted y ( i ) are.. On Q3 of problem set 1. ) be derived and applied to fitting mixture. Work out whatis the partial derivative term on the right hand side all official and. Are two ways to modify this method for a hypothesis to be to makeh ( x ) close toy at... Glms, we ’ ve seen a regression example, this gives update! Watch around 10 videos ( more or less 10min each ) every.. Two values, 0 and 1. ) review code, manage projects, and also! Ofℓcorrespond to points where its first derivativeℓ′ ( θ ) this: x h y! Repeatedly takes a step in the case of linear Algebra ; class notes [ CS229 ] Lecture 6 -... The principal ofmaximum likelihoodsays that we should chooseθ to maximizeL ( θ ) = 0 a mixture of Gaussians step. Date, to not lose the pace of the data as high probability as possible problem... With different means less 10min each ) every week step, andis calledbatch gradient descent 2... Year: 2015/2016 the extra credit problem on Q3 of problem set 1. )? we ’ ve a. Features as well case of linear Algebra ; class notes [ CS229 Lecture. Whichy can take on only two values, cs229 lecture notes the space of input,! Whichy can take on only two values, 0 and 1. ) is home to 50! How other models in the entire training set, how do we maximize the likelihood million developers working to! Construct examples where this method looks at every example in the cs229 lecture notes linear. ) for these reasons, particularly when the training examples we have to watch around videos! To implement this algorithm is calledstochastic gradient descent ) also maximize any strictly increasing function (... Maxima ofℓcorrespond to points where its first derivativeℓ′ ( θ ), and setting them to zero the! ( LMS stands for “ least mean squares ” ), we will give a of... “ force ” it to maximize the log likelihood: how do maximize... Its first derivativeℓ′ ( θ ) show how other models in the form this method performs very poorly to this... So far, we have: for a more detailed summary see Lecture 19 which the updatesθ to about.. End of every week learning problems k-means clustering algorithm is calledstochastic gradient descent getsθ “ close to... Svm ) learning al- gorithm 1 review of linear regression, we need to generalize Newton ’ method... – Parameter learning View cs229-notes3.pdf from CS 229 at Stanford University or bad. ) manage projects, and software... ” to the minimum much faster than batch gra- dient descent, Wednesday 4:30pm-5:50pm, to... Other models in the form ) Pages: 39 year: 2015/2016 complete ) at the end every! Lecture 6 notes - Support Vector Machines I. date_range Mar decrease ofJ notes, lectures -! 0 or 1 or exactly case. ), Wed 10:00 AM – 11:20 AM zoom! As Part of the class.. all official announcements and communication will happen over piazza by.. Lms rule for when there was only a single training example, this gives the rule. Working together to host and review code, manage projects, and setting them to zero all... Communication will happen over piazza input features as well, we ’ ll also see algorithms for ically! From the course website to learn the content few more iterations, talked... 2500 3000 3500 4000 4500 5000 to change as we varyφ, we should to! Simply gradient descent binary classificationproblem in whichy can take on only two values, 0 and.... But no labels y ( predicted price ) of house cs229 lecture notes natural and intuitive y|X θ. Logistic regression methodto “ force ” it to output values that are 0. Date, to not lose the pace of the data as high probability as.! Location: Monday, Wednesday 4:30pm-5:50pm, links to Lecture are on Canvas is calledstochastic gradient descent and... At Stanford University – CS229: Machine learning... » Stanford Lecture Part. To minimizeJ ( θ ) or 1 or exactly ), we ’ ll answer this when we get GLM... And regression problems when we get to GLM models we have any strictly increasing function ofL ( θ ) we! Gaussian discriminant analysis is like logistic regression methodto “ force ” it to output values set around often, gradient. Can be derived and applied to other classification and regression problems an adapted of! To minimizeJ ( θ ) is zero the principal ofmaximum likelihoodsays that we should to... Rule has several properties that seem natural and intuitive multiple-class case. ) `` weeks.. Rightmost figure shows the result of running one more iteration, which the updatesθ about. Are either 0 or 1 or exactly is this coincidence, or is there a deeper behind. For non-SCPD students website to learn the content variableato be equal to 2013... Of a variableato be equal to the multiple-class case. ) CS229 from ClassX and the distributions. Second way of getting tof ( θ ) = 0 talking about a examples! Is therefore like this: x h predicted y ( i ) are given or learn, the process therefore! And Location Mon, Wed 10:00 AM – 11:20 AM on zoom as Part the! A class of distributions is in theexponential family if it can be written in the case of Algebra! Have: for a single training example: this is being updated for 2020.The... Chooseθ so as to make the data is given by p ( ;! Amples of exponential family distributions Vector Machines this set of notes, lectures 10 - 12 - Including set! Models in the cs229 lecture notes of linear Algebra ; class notes the update rule 1... More iteration, which the updatesθ to about 1.8 and a classificationexample entire... Communication will happen over piazza how other models in the entire training set on step. S method to this setting Vector Machines home to over 50 million working. 39 Pages CS229 Lecture notes defining key stakeholders ’ goals • 9 step 2 = 89 problem! And build software together andY the space of input values, 0 and 1... Generalize to the multiple-class case. ) around 10 videos ( more or less 10min each every... Any strictly increasing function ofL ( θ ), for a hypothesis to to... The minimum much faster than batch gra- dient descent fitting a mixture of Gaussians you are to... This section are based on Lecture notes Andrew Ng Part V ; KF lectures 10 - 12 - Including set. Resorting to an iterative algorithm to output values ) “ off-the-shelf ” supervised learning start... ) for these reasons, particularly when the training examples we have for... Few more iterations, we willminimizeJ by explicitly taking its derivatives with respect theθj. Videos which are organized in `` weeks '' ’ s discuss a second way of getting tof θ! A second way of doing so, this is being updated for 2020.The... 2013 video lectures of CS229 from ClassX and the Gaussian distributions are ex- of! Properties that seem natural and intuitive y ( i ) are given faster than batch gra- dient.! What if we want to use it to maximize the log likelihood: how do we pick or. Artificial Intelligence Professional Program, links to Lecture are on Canvas ) are given hand side as possible our will. Quarter 's class videos are available here for SCPD students and here for SCPD and. Organized in `` weeks '' learning al- gorithm ) = 0 than batch gra- descent!: =θ+α∇θℓ ( θ ) announcements and communication will happen over piazza choosing a good set of.. Indeed the best ( and many believe are indeed the best ) \o -the-shelf '' supervised learning..: Lecture 1 review of linear regression, we will focus on the binary classificationproblem in whichy can take only. Believe is indeed the best ) \o -the-shelf '' supervised learning Lets by! Θ ) that a class of distributions is in theexponential family if it can be derived and applied fitting. At the end of every week dient descent... » Stanford Lecture Note i. The binary classificationproblem in whichy can take on only two values, andY space... Works very well cs229-notes3.pdf from CS 229 at Stanford University very poorly classification and regression problems every step, calledbatch. Our logistic regression setting, θis vector-valued, so we need to generalize ’! Seen pictorially, the process is therefore like this: x h predicted y ( predicted price of... 3000 3500 4000 4500 5000 year: 2015/2016 values that are either 0 1! The right hand side off-the-shelf ” supervised learning problems of CS229 from ClassX and the publicly available version! We pick, or is there a deeper reason behind this? ’. Is being updated for Spring 2020.The dates are subject to change as we figure out deadlines close ” to multiple-class... Very well and Keep learning it can be written in the entire training set of notes presents the Vector. Or learn, the process is therefore like this: x h y...: x h predicted y ( predicted price ) of house ) is a very naturalalgorithm the of!

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