Topics we covered. Machine Learning. Statistics. Optimization. Systems! Basics of probability Tail bounds Density Estimation Exponential Families
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- Nickolas Harrington
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1 Midterm Review
2 Topics we covered Machine Learning Optimization Basics of optimization Convexity Unconstrained: GD, SGD Constrained: Lagrange, KKT Duality Linear Methods Perceptrons Support Vector Machines Kernels Statistics Basics of probability Tail bounds Density Estimation Exponential Families Graphical Models Systems!
3 Basics of Machine Learning Supervised/Unsupervised Learning? Classification, Regression, Clustering Training error/test error? Model Complexity: Overfitting/Underfitting True error Bayes Optimal Error
4 Bias-Variance Tradeoff When estimating a quantity θ, we evaluate the performance of an estimator by computing its risk expected value of a loss function R θ, θ = E L(θ, θ), where L could be Mean Squared Error Loss 0/1 Loss Hinge Loss (used for SVMs) Bias-Variance Decomposition: Y = f x + ε Err x = E f x f x 2 = (E f x f(x)) 2 +E f x E f x 2 + σε 2 Bias Variance 3/3/2015 4
5 Copied from: Junier Oliva
6 Copied from: Junier Oliva
7 Copied from: Junier Oliva
8 Copied from: Junier Oliva
9 Copied from: Junier Oliva
10 Copied from: Junier Oliva
11 Copied from: Junier Oliva
12 Copied from: Junier Oliva
13 Copied from: Junier Oliva
14 Regression
15 Optimization Copied from: Xuezhi Wang
16 Convex Sets Copied from: Xuezhi Wang
17 Convex Functions Copied from: Xuezhi Wang
18 Convex Functions Examples
19 Useful Observations A function is convex if and only if its epigraph is a convex set. Below-Sets of Convex Functions is a convex set Convex functions cannot have local minima
20 Gradient Descent Copied from: Xuezhi Wang
21 Newton s Method Copied from: Prof Barnabas
22 Newton s Method Copied from: Prof Barnabas
23 Duality
24 Duality
25 KKT Conditions
26 Perceptrons
27 Convergence of Perceptrons
28 Back to Optimization
29 Gradient Descent
30 Stochastic Gradient Descent
31 SGD and Perceptron
32 SVM Primal Find maximum margin hyper-plane Hard Margin Copied from: Junier Oliva 3/3/
33 SVM Primal Find maximum margin hyper-plane Soft Margin Copied from: Junier Oliva 3/3/
34 SVM Dual Find maximum margin hyper-plane Dual for the hard margin SVM Copied from: Junier Oliva 3/3/
35 SVM Dual Find maximum margin hyper-plane Dual for the hard margin SVM Substituting α for w Copied from: Junier Oliva 3/3/
36 SVM Dual Find maximum margin hyper-plane Dual for the hard margin SVM The constraints are active for the support vectors Copied from: Junier Oliva 3/3/
37 SVM Dual Find maximum margin hyper-plane Dual for the hard margin SVM Copied from: Junier Oliva 3/3/
38 SVM Computing w Find maximum margin hyper-plane Dual for the hard margin SVM Copied from: Junier Oliva 3/3/
39 SVM Computing w Find maximum margin hyper-plane Dual for the soft margin SVM only difference from the separable case Copied from: Junier Oliva 3/3/
40 SVM the feature map Find maximum margin hyper-plane But data is not linearly separable We obtain a linear separator in the feature space.!! inputs feature map features is expensive to compute! Copied from: Junier Oliva 3/3/
41 Introducing the kernel The dual formulation no longer depends on w, only on a dot product! We obtain a linear separator in the feature space.!! is expensive to compute! But we don t have to! What we need is the dot product: Let s call this a kernel - 2-variable function - can be written as a dot product Copied from: Junier Oliva 3/3/
42 Kernel SVM The dual formulation no longer depends on w, only on a dot product! closed form This is the famous kernel trick. - never compute the feature map - learn using the closed form K - constant time for HD dot products Copied from: Junier Oliva 3/3/
43 Kernel SVM Run time What happens when we need to classify some x 0? Recall that w depends on α Our classifier for x 0 uses w Copied from: Junier Oliva 3/3/
44 Kernel SVM Run time What happens when we need to classify some x 0? Recall that w depends on α Our classifier for x 0 uses w Who needs w when we ve got dot products? Copied from: Junier Oliva 44
45 Kernel SVM Recap Pick kernel Solve the optimization to get α Classify as Compute b using the support vectors Copied from: Junier Oliva 45
46 Reminder on Kernels Remember Kernels are nothing but implicit feature maps Gram Matrix of a set of vectors x 1 x n in the inner product space defined by the kernel K Gram Matrix is always positive definite 46
47 Bayes Rule
48 Law of Large Numbers
49 Central Limit Theorem
50 Tail Bounds
51 More Tail Bounds
52
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