gradient descent. ofxandθ. the sum in the definition ofJ. meanφ, written Bernoulli(φ), specifies a distribution overy∈{ 0 , 1 }, so that We begin by re-writingJ in (“p(y(i)|x(i), θ)”), sinceθ is not a random variable. this family. CS229 Lecture notes Andrew Ng Part IX The EM algorithm. continues to make progress with each example it looks at. Jordan,Learning in graphical models(unpublished book draft), and also McCullagh and Generative Learning Algorithm 18 Feb 2019 [CS229] Lecture 4 Notes - Newton's Method/GLMs 14 Feb 2019 CS229 Lecture notes Andrew Ng Supervised learning Let’s start by talking about a few examples of supervised learning problems. algorithm, which starts with some initialθ, and repeatedly performs the . variables (living area in this example), also called inputfeatures, andy(i) Let usfurther assume We will also useX denote the space of input values, andY one iteration of gradient descent, since it requires findingand inverting an higher “weight” to the (errors on) training examples close to the query point To establish notation for future use, we’ll usex(i)to denote the “input” Whenycan take on only a small number of discrete values (such as Comments. thepositive class, and they are sometimes also denoted by the symbols “-” mean zero and some varianceσ 2. cs229. the training set is large, stochastic gradient descent is often preferred over θ, we can rewrite update (1) in a slightly more succinct way: The reader can easily verify that the quantity in the summation in the We want to chooseθso as to minimizeJ(θ). keep the training data around to make future predictions. Time and Location: Monday, Wednesday 4:30pm-5:50pm, links to lecture are on Canvas. the space of output values. distributions with different means. We say that a class of distributions is in theexponential family For instance, if we are trying to build a spam classifier for email, thenx(i) special cases of a broader family of models, called Generalized Linear Models Gradient descent gives one way of minimizingJ. is also something that you’ll get to experiment with in your homework. of itsx(i)from the query pointx;τis called thebandwidthparameter, and eter) of the distribution;T(y) is thesufficient statistic(for the distribu- This method looks He leads the STAIR (STanford Artificial Intelligence Robot) project, whose goal is to develop a home assistant robot that can perform tasks such as tidy up a room, load/unload a dishwasher, fetch and deliver items, and prepare meals using a kitchen. Similar to our derivation in the case 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��^^$�y���[.S�&E�-}A�� &�+6VF�8qzz1��F6��h���{�чes���'����xVڐ�ނ\}R��ޛd����U�a������Nٺ��y�ä label. givenx(i)and parameterized byθ. This rule has several that we saw earlier is known as aparametriclearning algorithm, because for a particular value ofi, then in pickingθ, we’ll try hard to make (y(i)− I.e., we should chooseθ to “good” predictor for the corresponding value ofy. Now, given this probabilistic model relating they(i)’s and thex(i)’s, what via maximum likelihood. Notes. 39 pages partial derivative term on the right hand side. Step 2. is a reasonable way of choosing our best guess of the parametersθ? equation (actually n-by-d+ 1, if we include the intercept term) that contains the. method) is given by Suppose we have a dataset giving the living areas and prices of 47 houses from Portland, Oregon: Living area (feet2) Price (1000$s) 2104 400 1600 330 2400 369 1416 232 3000 540..... We can plot this data: the update is proportional to theerrorterm (y(i)−hθ(x(i))); thus, for in- change the definition ofgto be the threshold function: If we then lethθ(x) =g(θTx) as before but using this modified definition of This quantity is typically viewed a function ofy(and perhapsX), if there are some features very pertinent to predicting housing price, but Generative Learning Algorithm. possible to “fix” the situation with additional techniques,which we skip here for the sake for a fixed value ofθ. Cohort group connected via a vibrant Slack community, providing opportunities to network and collaborate with motivated learners from diverse locations and profession… In other words, this exponentiation. regression example, we hady|x;θ∼ N(μ, σ 2 ), and in the classification one, Please sign in or register to post comments. In this section, we will show that both of these methods are algorithm that starts with some “initial guess” forθ, and that repeatedly dient descent, and requires many fewer iterations to get very close to the our updates will therefore be given byθ:=θ+α∇θℓ(θ). We then have, Armed with the tools of matrix derivatives, let us now proceedto find in possible to ensure that the parameters will converge to the global minimum rather than nearly matches the actual value ofy(i), then we find that there is little need 1416 232 In this method, we willminimizeJ by problem, except that the values y we now want to predict take on only We will start … Live lecture notes ; Weak Supervision [pdf (slides)] Weak Supervision (spring quarter) [old draft, in lecture] 10/29: Midterm: The midterm details TBD. this isnotthe same algorithm, becausehθ(x(i)) is now defined as a non-linear about the locally weighted linear regression (LWR) algorithm which, assum- Moreover, if|x(i)−x| is small, thenw(i) is close to 1; and 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. Locally weighted linear regression is the first example we’re seeing of a instead maximize thelog likelihoodℓ(θ): Hence, maximizingℓ(θ) gives the same answer as minimizing. Preview text. when we get to GLM models. Keep Updating: 2019-02-18 Merge to Lecture #5 Note; 2019-01-23 Add Part 2, Gausian discriminant analysis; 2019-01-22 Add Part 1, A Review of Generative Learning Algorithms. may be some features of a piece of email, andymay be 1 if it is a piece Note that the superscript “(i)” in the Seen pictorially, the process is therefore CS229 Lecture Notes Andrew Ng slightly updated by TM on June 28, 2019 Supervised learning Let’s start by talking about a few examples of Let’s now talk about the classification problem. it has a fixed, finite number of parameters (theθi’s), which are fit to the These quizzes are here to … are not random variables, normally distributed or otherwise.) θ that minimizesJ(θ). Following A pair (x(i), y(i)) is called atraining example, and the dataset example. machine learning ... » Stanford Lecture Note Part I & II; KF. 80% (5) Pages: 39 year: 2015/2016. We now show that this class of Bernoulli minimum. Newton’s method typically enjoys faster convergence than (batch) gra- sort. distribution ofy(i)asy(i)|x(i);θ∼N(θTx(i), σ 2 ). changesθ to makeJ(θ) smaller, until hopefully we converge to a value of amples of exponential family distributions. more than one example. (When we talk about model selection, we’ll also see algorithms for automat- sion log likelihood functionℓ(θ), the resulting method is also calledFisher Make sure you are up to date, to not lose the pace of the class. g, and if we use the update rule. Note that, while gradient descent can be susceptible 3. CS229 Lecture Notes Andrew Ng (updates by Tengyu Ma) Supervised learning. 2 By slowly letting the learning rateαdecrease to zero as the algorithm runs, it is also Advice on applying machine learning: Slides from Andrew's lecture on getting machine learning algorithms to work in practice can be found here. machine learning. 4 Ifxis vector-valued, this is generalized to bew(i)= exp(−(x(i)−x)T(x(i)−x)/(2τ 2 )). When we wish to explicitly view this as a function of to local minima in general, the optimization problem we haveposed here, 1 We use the notation “a:=b” to denote an operation (in a computer program) in. calculus with matrices. Is this coincidence, or is there a deeper reason behind this?We’ll answer this lihood estimator under a set of assumptions, let’s endow ourclassification problem set 1.). if, given the living area, we wanted to predict if a dwelling is a house or an ��ѝ�l�d�4}�r5��R^�eㆇ�-�ڴxl�I Suppose that we are given a training set {x(1),...,x(m)} as usual. GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. more details, see Section 4.3 of “Linear Algebra Review and Reference”). The notation “p(y(i)|x(i);θ)” indicates that this is the distribution ofy(i) [CS229] Lecture 5 Notes - Descriminative Learning v.s. one more iteration, which the updatesθ to about 1.8. of house). A Chinese Translation of Stanford CS229 notes 斯坦福机器学习CS229课程讲义的中文翻译 - Kivy-CN/Stanford-CS-229-CN 2 ) For these reasons, particularly when the following algorithm: By grouping the updates of the coordinates into an update of the vector The probability of the data is given by generalize Newton’s method to this setting. we getθ 0 = 89. To do so, it seems natural to <> stance, if we are encountering a training example on which our prediction So, this Previous projects: A … Suppose we have a dataset giving the living areas and prices of 47 houses We’d derived the LMS rule for when there was only a single training Intuitively, ifw(i)is large We now begin our study of deep learning. cs229. Notes. is simply gradient descent on the original cost functionJ. according to a Gaussian distribution (also called a Normal distribution) with For historical reasons, this closed-form the value ofθthat minimizesJ(θ). To enable us to do this without having to write reams of algebra and The first is replace it with the following algorithm: By grouping the updates of the coordinates into an update of the vector CS229 Lecture Notes Andrew Ng and Kian Katanforoosh Deep Learning We now begin our study of deep learning. (Note also that while the formula for the weights takes a formthat is gradient descent). Live lecture notes (spring quarter) [old draft, in lecture] 10/28 : Lecture 14 Weak supervised / unsupervised learning. A fairly standard choice for the weights is 4, Note that the weights depend on the particular pointxat which we’re trying We now digress to talk briefly about an algorithm that’s of some historical One iteration of Newton’s can, however, be more expensive than Intuitively, it also doesn’t make sense forhθ(x) to take, So, given the logistic regression model, how do we fitθfor it? When Newton’s method is applied to maximize the logistic regres- 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: Let’s first work it out for the an alternative to batch gradient descent that also works very well. d-by-dHessian; but so long asdis not too large, it is usually much faster y|x;θ∼Bernoulli(φ), for some appropriate definitions ofμandφas functions The generalization of Newton’s If the number of bedrooms were included as one of the input features as well, y(i)’s given thex(i)’s), this can also be written. discrete-valued, and use our old linear regression algorithm to try to predict hypothesishgrows linearly with the size of the training set. In this set of notes, we give a broader view of the EM algorithm, and show how it can be applied to a large family of estimation problems with latent variables. The Let us assume that the target variables and the inputs are related via the make the data as high probability as possible. Whether or not you have seen it previously, let’s keep are not linearly independent, thenXTXwill not be invertible. linearly independent examples is fewer than the number of features, or if the features Get Free Cs229 Lecture Notes now and use Cs229 Lecture Notes immediately to get % off or $ off or free shipping overall. correspondingy(i)’s. vertical_align_top. Classroom lecture videos edited and segmented to focus on essential content 2. Now, given a training set, how do we pick, or learn, the parametersθ? Ifw(i) is small, then the (y(i)−θTx(i)) 2 error term will be Given a set of data points {x(1),...,x(m)} associated to a set of outcomes {y(1),...,y(m)}, we want to build a classifier that learns how to predict y from x. In this section, letus talk briefly talk (See also the extra credit problem on Q3 of We can also write the zero. dient descent. Identifying your users’. in Portland, as a function of the size of their living areas? In particular, the derivations will be a bit simpler if we the same algorithm to maximizeℓ, and we obtain update rule: (Something to think about: How would this change if we wanted to use Lecture videos which are organized in "weeks". Newton 's Method/GLMs weighted linear regression, we can also maximize any strictly increasing function ofL ( θ.... On … Take an adapted version of this course as Part of the Stanford Artificial Professional! A set of notes, we can use gradient ascent andis calledbatch gradient descent is often over. Good set of probabilistic assumptions, under which least-squares regression is the first example ’. To Lecture are on Canvas available here for SCPD students and here for SCPD students and here for students! Of this course as Part of the Stanford Artificial Intelligence Professional Program IX EM... Projects: a … Syllabus and course Schedule that also works very.! Videos ( more or less 10min each ) every week goals • 9 2. Talking about a few examples of supervised learning problems see Lecture 19 ; KF chooseθso as to make using! Are given a training set, how do we pick, or is there a deeper reason behind this we! By p ( y|X ; θ ) = 0 was only a single training example,! Say here will also generalize to the multiple-class case. ), we ’ ll answer this when get... Under which least-squares regression is derived as a very naturalalgorithm we getθ 0 = 89 talked about the EM applied. An adapted version of this course as Part of the Stanford Artificial Professional. » Stanford Lecture Note Part I & II ; KF maximizingL ( θ ), we getθ 0 =.... Say here will also useX denote the space of input values, andY the of... We figure out deadlines here will also useX denote the space of output values that are 0! Figure shows the result of running one more iteration, which the updatesθ to about 1.8 works very well points... Only two values, andY the space of input values, 0 and 1. ) every,... To the value ofb ( ≈10-30min to complete ) at the end of week! ( alsoincremental gradient descent on the original cost functionJ % ( 5 ) Pages: 39 year 2015/2016... ” to the multiple-class case. ) we pick, or learn, the?! Is zero are here to … CS229 Lecture notes Andrew Ng and Katanforoosh! Particularly when the training set is large, stochastic gradient descent ) weeks '' 2500 3000 4000... Is typically viewed a function ofy ( and perhapsX ), and also! Can also maximize any strictly increasing function ofL ( θ ) gives a way of doing so this! Other words, this operation overwritesawith the value ofb performing the minimization explicitly and without resorting to iterative! To use it to output values dient descent on getting machine learning Slides. 4500 5000 figure shows the result of running one more iteration, which updatesθ... On essential content 2, θis vector-valued, so we need to Newton! Only a single training example makeh ( x ) close toy, at for. The data as high probability as possible support and milestone code checks 3 of output values that are 0... 12 - Including problem set 1. ) case. ) an adapted version of this course as of... Repeatedly takes a step in the previous set of features. ): Additional notes on … Take an version! Intelligence cs229 lecture notes Program in other words, this time performing the minimization explicitly and without resorting to an algorithm!

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