61A Extra Lecture 13
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- Eustacia Mills
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1 61A Extra Lecture 13
2 Announcements
3 Prediction
4 Regression 4
5 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values 4
6 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] 4
7 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Data from a health survey of 6342 adults 4
8 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Height in inches: 5 feet 4¾ inches Data from a health survey of 6342 adults 4
9 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Height in inches: 5 feet 4¾ inches Weight in pounds Data from a health survey of 6342 adults 4
10 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Height in inches: 5 feet 4¾ inches Weight in pounds Data from a health survey of 6342 adults Measuring error: y-f(x) or (f(x)-y) 2 are both typical 4
11 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Height in inches: 5 feet 4¾ inches Weight in pounds Data from a health survey of 6342 adults Measuring error: y-f(x) or (f(x)-y) 2 are both typical Over the whole set of (x, y) pairs, we can average this "squared error" 4
12 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Height in inches: 5 feet 4¾ inches Weight in pounds Data from a health survey of 6342 adults Measuring error: y-f(x) or (f(x)-y) 2 are both typical Over the whole set of (x, y) pairs, we can average this "squared error" Squared error has the wrong units, so it's common to take the square root 4
13 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Height in inches: 5 feet 4¾ inches Weight in pounds Data from a health survey of 6342 adults Measuring error: y-f(x) or (f(x)-y) 2 are both typical Over the whole set of (x, y) pairs, we can average this "squared error" Squared error has the wrong units, so it's common to take the square root The result is the "root mean squared error" of a predictor f on a set of (x, y) pairs 4
14 Regression Given a set of (x, y) pairs, find a function f(x) that returns good y values pairs = [(64.75, 163.5), (63.75, 147.5), (72.5, 224),...] Height in inches: 5 feet 4¾ inches Weight in pounds Data from a health survey of 6342 adults Measuring error: y-f(x) or (f(x)-y) 2 are both typical Over the whole set of (x, y) pairs, we can average this "squared error" Squared error has the wrong units, so it's common to take the square root The result is the "root mean squared error" of a predictor f on a set of (x, y) pairs (Demo) 4
15 Purpose of Newton's Method Quickly finds accurate approximations to zeroes of differentiable functions! 5
16 Purpose of Newton's Method Quickly finds accurate approximations to zeroes of differentiable functions! f(x) = x 2-2 5
17 Purpose of Newton's Method Quickly finds accurate approximations to zeroes of differentiable functions! 2.5 f(x) = x
18 Purpose of Newton's Method Quickly finds accurate approximations to zeroes of differentiable functions! 2.5 f(x) = x 2-2 A "zero" of a function f is an input x such that f(x)=
19 Purpose of Newton's Method Quickly finds accurate approximations to zeroes of differentiable functions! 2.5 f(x) = x 2-2 A "zero" of a function f is an input x such that f(x)= x=
20 Purpose of Newton's Method Quickly finds accurate approximations to zeroes of differentiable functions! 2.5 f(x) = x 2-2 A "zero" of a function f is an input x such that f(x)= x= Application: Find the minimum of a function by finding the zero of its derivative 5
21 Approximate Differentiation 6
22 Approximate Differentiation 6
23 Approximate Differentiation Differentiation can be performed symbolically or numerically 6
24 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x
25 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x 6
26 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x f'(2) = 4 6
27 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x f'(2) = 4 6
28 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x f'(2) = 4 f 0 (x) =lim a!0 f(x + a) f(x) a 6
29 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x f'(2) = 4 f 0 (x) =lim a!0 f(x + a) f(x) a f 0 (x) f(x + a) f(x) a 6
30 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x f'(2) = 4 f 0 (x) =lim a!0 f(x + a) f(x) a f 0 (x) f(x + a) f(x) a (if a is small) 6
31 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x f'(2) = 4 f 0 (x) =lim a!0 f(x + a) f(x) a f 0 (x) f(x + a) f(x) a (if a is small) 6
32 Approximate Differentiation Differentiation can be performed symbolically or numerically f(x) = x 2-16 f'(x) = 2x f'(2) = 4 f 0 (x) =lim a!0 f(x + a) f(x) a f 0 (x) f(x + a) f(x) a (if a is small) 6
33 Critical Points and Inverses 7
34 Critical Points and Inverses Maxima, minima, and inflection points of a differentiable function occur when the derivative is 0 7
35 Critical Points and Inverses Maxima, minima, and inflection points of a differentiable function occur when the derivative is 0 7
36 Critical Points and Inverses Maxima, minima, and inflection points of a differentiable function occur when the derivative is 0 The global minimum of convex functions that are (mostly) twice-differentiable can be computed numerically using techniques that are similar to Newton's method 7
37 Critical Points and Inverses Maxima, minima, and inflection points of a differentiable function occur when the derivative is 0 The global minimum of convex functions that are (mostly) twice-differentiable can be computed numerically using techniques that are similar to Newton's method (Demo) 7
38 Multiple Linear Regression Given a set of (xs, y) pairs, find a linear function f(xs) that returns good y values 8
39 Multiple Linear Regression Given a set of (xs, y) pairs, find a linear function f(xs) that returns good y values A linear function has the form w xs + b for vectors w and xs and scalar b 8
40 Multiple Linear Regression Given a set of (xs, y) pairs, find a linear function f(xs) that returns good y values A linear function has the form w xs + b for vectors w and xs and scalar b (Demo) 8
41 Multiple Linear Regression Given a set of (xs, y) pairs, find a linear function f(xs) that returns good y values A linear function has the form w xs + b for vectors w and xs and scalar b (Demo) Note: Root mean squared error can be optimized through linear algebra alone, but numerical optimization works for a much larger class of related error measures 8
42 Classification
43 Classification 10
44 Classification When the output of prediction is a category instead of a quantity, prediction is called classification instead of regression 10
45 Classification When the output of prediction is a category instead of a quantity, prediction is called classification instead of regression The approach is basically the same, except that the output is coerced into a category 10
46 Classification When the output of prediction is a category instead of a quantity, prediction is called classification instead of regression The approach is basically the same, except that the output is coerced into a category Error is measured by accuracy: the proportion of categorical predictions that were correct 10
47 Classification When the output of prediction is a category instead of a quantity, prediction is called classification instead of regression The approach is basically the same, except that the output is coerced into a category Error is measured by accuracy: the proportion of categorical predictions that were correct (Demo) 10
48 Loss Functions 11
49 Loss Functions (2*y-1) * (w xs + b - 0.5) How close to being right/wrong 11
50 Loss Functions Changes (0, 1) to (-1, 1) (2*y-1) * (w xs + b - 0.5) How close to being right/wrong 11
51 Loss Functions Changes (0, 1) to (-1, 1) Shifts choice to 0 from 0.5 (2*y-1) * (w xs + b - 0.5) How close to being right/wrong 11
52 Loss Functions Loss used for accuracy evaluation Changes (0, 1) to (-1, 1) Shifts choice to 0 from 0.5 (2*y-1) * (w xs + b - 0.5) How close to being right/wrong 11
53 Loss Functions Loss used for accuracy evaluation Correct, but almost wrong Changes (0, 1) to (-1, 1) Shifts choice to 0 from 0.5 (2*y-1) * (w xs + b - 0.5) How close to being right/wrong 11
54 Loss Functions Linear regression (min squared error) Loss used for accuracy evaluation Correct, but almost wrong Changes (0, 1) to (-1, 1) Shifts choice to 0 from 0.5 (2*y-1) * (w xs + b - 0.5) How close to being right/wrong 11
55 Language Models
56 Natural Language Can Be Predicted 13
57 Natural Language Can Be Predicted Programs should be written fur people two read. 13
58 Natural Language Can Be Predicted for Programs should be written fur people two read. 13
59 Natural Language Can Be Predicted for to Programs should be written fur people two read. 13
60 Natural Language Can Be Predicted for Programs should be written fur people two read. to X 13
61 Natural Language Can Be Predicted for Programs should be written fur people two read. to X Y 13
62 Natural Language Can Be Predicted for Programs should be written fur people two read. to X Y "should be written for": 1 13
63 Natural Language Can Be Predicted for Programs should be written fur people two read. to X Y "should be written for": 1 "be written for": 1 13
64 Natural Language Can Be Predicted for Programs should be written fur people two read. to X Y "should be written for": 1 "be written for": 1 "written for": 1 13
65 Natural Language Can Be Predicted for Programs should be written fur people two read. to X Y "should be written for": 1 "be written for": 1 "written for": 1 "for": 1 13
66 Natural Language Can Be Predicted for Programs should be written fur people two read. to X Y "should be written for": 1 "be written for": 1 "written for": 1 "for": 1 "nice computer": 0 13
67 Natural Language Can Be Predicted for Programs should be written fur people two read. to X Y "should be written for": 1 "be written for": 1 "written for": 1 "for": 1 "nice computer": 0 (Demo) 13
2 2 + x =
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