Stochastic Gradient Descent: The Workhorse of Machine Learning. CS6787 Lecture 1 Fall 2017
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1 Stochastic Gradient Descent: The Workhorse of Machine Learning CS6787 Lecture 1 Fall 2017
2 Fundamentals of Machine Learning? Machine Learning in Practice this course
3 What s missing in the basic stuff? Efficiency!
4 Motivation: Machine learning applications involve large amounts of data More data à Better services Better systems à More data
5 How do practitioners make their systems better?
6 How do we improve our systems? The use methods for accelerating convergence Make ML algorithms converge in fewer iterations They use metaparameter optimization How to set parameters of these techniques Techniques for improving runtime Making each iteration of the algorithm run faster Machine learning hardware and frameworks Course outline Part 1 Part 2 Part 3 Part 4
7 Course Format One half One half Traditional lectures Broad description of techniques Important papers Presentations by you In-class discussions Reviews of each paper
8 Prerequisites Basic ML knowledge (CS 4780) Math/statistics knowledge At the level of the entrance exam for CS 4780
9 Grading Paper presentations Paper quizzes Discussion participation Paper reviews Final project
10 Paper presentations Presenting in groups of three-to-four Papers listed on the website Signups on Wednesday
11 Final Project Open-ended Work on what you think is interesting! Project should use some of the ideas from the course to speed up an implementation Groups of up to three
12 Late Policy This is a graduate level course Two free late days for any of the paper reviews I will also drop your lowest paper review score And lowest pop quiz score No late days on the final project
13 Questions?
14 Stochastic Gradient Descent: The Workhorse of Machine Learning CS6787 Lecture 1 Fall 2017
15 Optimization Much of machine learning can be written as an optimization problem loss function model min x NX i=1 f(x; y i ) training examples Example loss functions: logistic regression, linear regression, principle component analysis, neural network loss
16 Types of Optimization Convex optimization The easy case Includes logistic regression, linear regression, SVM Non-convex optimization NP-hard in general Includes deep learning
17 An Abridged Introduction to Convex Functions
18 Convex Functions 8 2 [0, 1], f( x +(1 )y) apple f(x)+(1 )f(y) f(x) =x 2
19 Example: Quadratic f(x) =x 2 ( x +(1 )y) 2 = 2 x 2 +2 (1 )xy +(1 ) 2 y 2 = x 2 +(1 )y 2 (1 )(x 2 +2xy + y 2 ) apple x 2 +(1 )y 2
20 Example: Abs f(x) = x x +(1 )y apple x + (1 )y = x +(1 ) y
21 Example: Exponential f(x) =e x e x+(1 )y = e y e (x y) = e y 1 X apple e y 1+ 1X n=1 n=0 1 n! n (x y) n 1 (x y)n n! = e y (1 )+ e x y =(1 )e y + e x! (if x>y)
22 Properties of convex functions Any line segment we draw between two points lies above the curve Corollary: every local minimum is a global minimum Why? This is what makes convex optimization easy It suffices to find a local minimum, because we know it will be global
23 Properties of convex functions (continued) Non-negative combinations of convex functions are convex h(x) =af(x)+bg(x) Affine scalings of convex functions are convex h(x) =f(ax + b) Compositions of convex functions are NOT generally convex Neural nets are like this h(x) =f(g(x))
24 Convex Functions: Alternative Definitions First-order condition Second-order condition hx y, rf(x) rf(y)i 0 r 2 f(x) 0 This means that the matrix of second derivatives is positive semidefinite A 0,8x, hx, Axi 0
25 Example: Quadratic f(x) =x 2 f 00 (x) =2 0
26 Example: Exponential f(x) =e x f 00 (x) =e x 0
27 Example: Logistic Loss f(x) = log(1 + e x ) f 0 (x) = ex 1+e x = 1 1+e x f 00 (x) = e x (1 + e x ) 2 = 1 (1 + e x )(1 + e x ) 0.
28 Strongly Convex Functions Basically the easiest class of functions for optimization First-order condition: Second-order condition: Equivalently: hx y, rf(x) rf(y)i µkx yk 2 h(x) =f(x) r 2 f(x) µi µ 2 kxk2 is convex
29 Which of the functions we ve looked at are strongly convex?
30 Which of the functions we ve looked at are strongly convex?
31 Concave functions A function is concave if its negation is convex f is convex, h(x) = f(x) is concave Example: f(x) = log(x) f 00 (x) = 1 x 2 apple 0
32 Why care about convex functions?
33 Convex Optimization Goal is to minimize a convex function
34 Gradient Descent x x rf(x)
35 Gradient Descent Converges Iterative definition of gradient descent Assumptions/terminology: x t+1 = x t rf(x t ) Global optimum is x Bounded second derivative µi r 2 f(x) LI
36 Gradient Descent Converges (continued) x t+1 x = x t x (rf(x t ) rf(x )) = x t x r 2 f(z t )(x t x ) =(I r 2 f(z t ))(x t x ). Taking the norm kx t+1 x kapple I r 2 f(z t ) kx t x k apple max ( 1 µ, 1 L ) kx t x k
37 Gradient Descent Converges (continued) So if we set =2/(L + µ) then kx t+1 x kapple L µ L + µ kx t x k And recursively L µ T kx 0 kx T x kapple L + µ x k Called convergence at a linear rate or sometimes (confusingly) exponential rate
38 The Problem with Gradient Descent Large-scale optimization h(x) = 1 N Computing the gradient takes O(N) time rh(x) = 1 N NX f(x; y i ) i=1 NX rf(x; y i ) i=1
39 Gradient Descent with More Data Suppose we add more examples to our training set For simplicity, imagine we just add an extra copy of every training example rh(x) = 1 2N NX i=1 rf(x; y i )+ 1 2N Same objective function But gradients take 2x the time to compute (unless we cheat) NX rf(x; y i ) i=1 We want to scale up to huge datasets, so how can we do this?
40 Stochastic Gradient Descent Idea: rather than using the full gradient, just use one training example Super fast to compute x t+1 = x t rf(x t ; yĩt ) In expectation, it s just gradient descent: E [x t+1 ]=E[x t ] E [rf(x t ; y it )] = E [x t ] 1 NX rf(x t ; y i ) N i=1 This is an example selected uniformly at random from the dataset.
41 Stochastic Gradient Descent Convergence Can SGD converge using just one example to estimate the gradient? x t+1 x = x t x (rh(x t ) rh(x )) (rf(x t ; y it ) rh(x t )) = I r 2 h(z t ) (x t x ) (rf(x t ; y it ) rh(x t )) How do we handle this extra noise term? Answer: bound it using the variance! Variance of a constant plus a random variable is just the variance of that random variable, so we don t need to think about the rest of the expression.
42 Stochastic Gradient Descent Convergence Var (x t+1 x x t ) = Var I r 2 h(z t ) (x t x ) (rf(x t ; y it ) rh(x t )) x t = Var ( (rf(x t ; y it ) rh(x t )) x t ) = 2 Var (rf(x t ; y it ) rh(x t ) x t ) = 2 E krf(x t ; y it ) rh(x t )k 2 x t apple 2 M.
43 Stochastic Gradient Descent Convergence E kx t+1 x k 2 x t = ke [x t+1 x x t ]k 2 + Var (x t+1 x x t ) apple (1 µ) 2 kx t x k M (for 1) So by the law of total expectation E kx t+1 x k 2 apple (1 µ) 2 E kx t x k M
44 Stochastic Gradient Descent Convergence Already we can see that this converges to a fixed point of E kx t+1 x k 2 = M 2µ µ 2 This phenomenon is called converging to a noise ball Rather than approaching the optimum, SGD (with a constant step size) converges to a region of low variance around the optimum This is okay for a lot of applications that only need approximate solutions
45 Demo
46 Stochastic gradient descent is super popular.
47 What Does SGD Power? Everything!
48 But how SGD is implemented in practice is not exactly what I ve just shown you and we ll see how it s different in the upcoming lectures.
49 Questions? Upcoming things Paper presentation signups on Wednesday
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