import torch
import numpy as np
import matplotlib.pyplot as plt imports
로드맵
- 회귀분석 \(\to\) 로지스틱 \(\to\) 심층신경망(DNN) \(\to\) 합성곱신경망(CNN)
- 강의계획서
ref
- 넘파이 문법이 약하다면? (reshape, concatenate, stack)
- reshape: 아래 링크의 넘파이공부 2단계 reshape 참고
https://guebin.github.io/IP2022/2022/04/06/(6%EC%A3%BC%EC%B0%A8)-4%EC%9B%946%EC%9D%BC.html
- concatenate, stack: 아래 링크의 넘파이공부 4단계 참고
https://guebin.github.io/IP2022/2022/04/11/(6%EC%A3%BC%EC%B0%A8)-4%EC%9B%9411%EC%9D%BC.html
회귀모형 소개
- model: \(y_i= w_0+w_1 x_i +\epsilon_i = 2.5 + 4x_i +\epsilon_i, \quad i=1,2,\dots,n\)
- model: \({\bf y}={\bf X}{\bf W} +\boldsymbol{\epsilon}\)
- \({\bf y}=\begin{bmatrix} y_1 \\ y_2 \\ \dots \\ y_n\end{bmatrix}, \quad {\bf X}=\begin{bmatrix} 1 & x_1 \\ 1 & x_2 \\ \dots \\ 1 & x_n\end{bmatrix}, \quad {\bf W}=\begin{bmatrix} 2.5 \\ 4 \end{bmatrix}, \quad \boldsymbol{\epsilon}= \begin{bmatrix} \epsilon_1 \\ \dots \\ \epsilon_n\end{bmatrix}\)
회귀모형에서 데이터 생성
_rnt=torch.randn(100).sort() #100개의 난수 생성 .sort()는 정렬된 값 표현type(_rnt) #type해보니까 모르는 거네? lengh를 보면 2니까 리스트를 해볼수 있음!torch.return_types.sorta,_ = _rnt[0], _rnt[1] #첫번쨰 원소가 a에 들어가고 두번째 원소가 언더바에 들어가게 된다.x,_ = torch.randn(100).sort()
x # X벡터 안에 들어가는 x1, x2, x3 ... tensor([-2.6694e+00, -2.6132e+00, -2.2525e+00, -2.0763e+00, -1.9791e+00,
-1.8444e+00, -1.7486e+00, -1.7284e+00, -1.6991e+00, -1.6634e+00,
-1.6364e+00, -1.5948e+00, -1.5710e+00, -1.5043e+00, -1.5002e+00,
-1.4035e+00, -1.3328e+00, -1.3239e+00, -1.2964e+00, -1.2064e+00,
-1.1857e+00, -1.1184e+00, -1.0559e+00, -1.0148e+00, -1.0105e+00,
-9.7771e-01, -9.2156e-01, -8.9929e-01, -8.8333e-01, -7.6213e-01,
-6.8896e-01, -6.2386e-01, -6.0660e-01, -5.9161e-01, -5.7884e-01,
-4.4417e-01, -4.3631e-01, -3.8129e-01, -3.5062e-01, -3.4311e-01,
-3.1632e-01, -2.7753e-01, -2.7065e-01, -2.7020e-01, -2.6189e-01,
-2.2925e-01, -1.4359e-01, -1.2405e-01, -6.8853e-02, -5.1603e-02,
-4.9887e-02, -2.3798e-02, -1.6275e-03, 7.4200e-02, 1.6760e-01,
1.7279e-01, 2.3754e-01, 2.5730e-01, 2.6886e-01, 2.8250e-01,
2.9296e-01, 3.0017e-01, 3.1466e-01, 3.2627e-01, 3.5380e-01,
3.5664e-01, 3.6345e-01, 3.6429e-01, 4.3469e-01, 4.3551e-01,
4.6556e-01, 4.9491e-01, 4.9940e-01, 5.4481e-01, 6.4859e-01,
6.7236e-01, 6.8683e-01, 7.2763e-01, 7.3832e-01, 7.8508e-01,
8.0376e-01, 8.1716e-01, 8.2234e-01, 8.8814e-01, 9.1453e-01,
9.8436e-01, 1.0107e+00, 1.0332e+00, 1.0441e+00, 1.0577e+00,
1.1333e+00, 1.1406e+00, 1.2557e+00, 1.3057e+00, 1.3221e+00,
1.3361e+00, 1.6109e+00, 1.7063e+00, 1.8415e+00, 2.0672e+00])ones= torch.ones(100) #torch.ones(100) 1이 100개 들어간거. X벡터만들기 위해서 torch.stack([ones, x]) #stack 쌓는다!!! 근데 우리가 원하는 건 이거의 T (트랜스)를 가지고 싶으니까 벡터T 해주기!tensor([[ 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00],
[-2.6694e+00, -2.6132e+00, -2.2525e+00, -2.0763e+00, -1.9791e+00,
-1.8444e+00, -1.7486e+00, -1.7284e+00, -1.6991e+00, -1.6634e+00,
-1.6364e+00, -1.5948e+00, -1.5710e+00, -1.5043e+00, -1.5002e+00,
-1.4035e+00, -1.3328e+00, -1.3239e+00, -1.2964e+00, -1.2064e+00,
-1.1857e+00, -1.1184e+00, -1.0559e+00, -1.0148e+00, -1.0105e+00,
-9.7771e-01, -9.2156e-01, -8.9929e-01, -8.8333e-01, -7.6213e-01,
-6.8896e-01, -6.2386e-01, -6.0660e-01, -5.9161e-01, -5.7884e-01,
-4.4417e-01, -4.3631e-01, -3.8129e-01, -3.5062e-01, -3.4311e-01,
-3.1632e-01, -2.7753e-01, -2.7065e-01, -2.7020e-01, -2.6189e-01,
-2.2925e-01, -1.4359e-01, -1.2405e-01, -6.8853e-02, -5.1603e-02,
-4.9887e-02, -2.3798e-02, -1.6275e-03, 7.4200e-02, 1.6760e-01,
1.7279e-01, 2.3754e-01, 2.5730e-01, 2.6886e-01, 2.8250e-01,
2.9296e-01, 3.0017e-01, 3.1466e-01, 3.2627e-01, 3.5380e-01,
3.5664e-01, 3.6345e-01, 3.6429e-01, 4.3469e-01, 4.3551e-01,
4.6556e-01, 4.9491e-01, 4.9940e-01, 5.4481e-01, 6.4859e-01,
6.7236e-01, 6.8683e-01, 7.2763e-01, 7.3832e-01, 7.8508e-01,
8.0376e-01, 8.1716e-01, 8.2234e-01, 8.8814e-01, 9.1453e-01,
9.8436e-01, 1.0107e+00, 1.0332e+00, 1.0441e+00, 1.0577e+00,
1.1333e+00, 1.1406e+00, 1.2557e+00, 1.3057e+00, 1.3221e+00,
1.3361e+00, 1.6109e+00, 1.7063e+00, 1.8415e+00, 2.0672e+00]])type(torch.stack([ones, x]))torch.Tensor# 역슬래시 하고 입실론 쓰고 탭 누르면 입실론 수학기호생김 신기하군W = torch.tensor([2.5,4])
W
W.shape #원래 shape이 매트릭스여야 하는데 벡터네? 그럼 X@W하면 매트릭스가 아닌 벡터가 된다. ~~~~~~~~~~ 2차원 1차원,,, 훔 torch.Size([2])torch.manual_seed(43052) #이건 원래 난수가 봅히는건뎅 교수님이 설명하기 편하게 ,, 숫자 정해논고
ones= torch.ones(100) #torch.ones(100) 1이 100개 들어간거. X벡터만들기 위해서
x,_ = torch.randn(100).sort()
X = torch.stack([ones,x]).T # torch.stack([ones,x],axis=1) T:트랜스포 해주는거. 근데 그렇게 안하고 axis=1해도 된당!
W = torch.tensor([2.5,4])
ϵ = torch.randn(100)*0.5
y = X@W + ϵplt.plot(x,y,'o')
plt.plot(x,2.5+4*x,'--') #트루펑션 점선으로 찍어보기
#언더라인펑션~= Y=4x+2.5
# w0, w1를 추정하면 언더라인 펑션을 잘 추정했다고 확인 할 수 잇어염 _9월21일_ipynb의_사본_files/figure-html/cell-13-output-1.png)
# 파란점은 기본 데이터. 입실론을 뺀거(오차항 뺸거)=TRUU FUNCTION을 찾고 싶어!회귀모형에서 학습이란?
# x에서 y로가는 맵핑
# 리니어 맵핑,, 2.5랑 4에 가깝게 맞추는거!!- 파란점만 주어졌을때, 주황색 점선을 추정하는것. 좀 더 정확하게 말하면 given data로 \(\begin{bmatrix} \hat{w}_0 \\ \hat{w}_1 \end{bmatrix}\)를 최대한 \(\begin{bmatrix} 2.5 \\ 4 \end{bmatrix}\)와 비슷하게 찾는것.
given data : \(\big\{(x_i,y_i) \big\}_{i=1}^{n}\)
parameter: \({\bf W}=\begin{bmatrix} w_0 \\ w_1 \end{bmatrix}\)
estimated parameter: \({\bf \hat{W}}=\begin{bmatrix} \hat{w}_0 \\ \hat{w}_1 \end{bmatrix}\)
- 더 쉽게 말하면 아래의 그림을 보고 적당한 추세선을 찾는것이다.
# "적당한" 추세선이 뭐냐? 하면 웱,,
# 숫자로 만드는게 제일 편하다!!
# 적당한게 정도가 있어서 일단 안적당한 거 먼저 해볼게용plt.plot(x,y,'o')
plt.plot(x, -5+10*x, '--')
# 원데이터가 y1, y2 되고.. 언더바에 잇는게 y1 hat, y2 hat ..... _9월21일_ipynb의_사본_files/figure-html/cell-17-output-1.png)
- 시도: \((\hat{w}_0,\hat{w}_1)=(-5,10)\)을 선택하여 선을 그려보고 적당한지 판단.
- \(\hat{y}_i=-5 +10 x_i\) 와 같이 \(y_i\)의 값을 적합시키겠다는 의미
- 벡터표현으로 주황색점선을 계산
What= torch.tensor([-5.0, 10.0])
Whattensor([-5., 10.])plt.plot(x,y,'o')
plt.plot(x,X@What,'--')
# 모델링: 데이터를 보고 아키텍처 설정,, _9월21일_ipynb의_사본_files/figure-html/cell-19-output-1.png)
파라메터를 학습하는 방법 (적당한 선으로 업데이트 하는 방법)
- 이론적으로 추론 <- 회귀분석시간에 배운것
- 컴퓨터의 반복계산을 이용하여 추론 (손실함수도입 + 경사하강법) <- 우리가 오늘 파이토치로 실습해볼 내용.
- 전략: 아래와 같은 3단계 전략을 취한다.
- stage1: 아무 점선이나 그어본다..
- stage2: stage1에서 그은 점선보다 더 좋은 점선으로 바꾼다.
- stage3: stage1 - 2 를 반복한다.
Stage1: 첫번째 점선 – 임의의 선을 일단 그어보자
- \(\hat{w}_0=-5, \hat{w}_1 = 10\) 으로 설정하고 (왜? 그냥) 임의의 선을 그어보자.
What= torch.tensor([-5.0, 10.0], requires_grad=True)
What # 나중에 미분하기 위해서 requires_grad 필요한 옵션
# 뒤에 꼬리표가 붙어있음! 따라다녀,,
#텐서플로우 패키지에서 tf.variable이랑 tf. 어ㅓㅉ고 랑 선언하는데 tf.variable로 설정한거tensor([-5., 10.], requires_grad=True)What + 1
# 뒤에 grad_fn 어쩌고가 따라오지만 신경쓰지 않아도 된당 ,, 벡터처럼 계산도 가능해!!
# 꼬리표를 빼고싶을땐......
What.detach()
What.datatensor([-5., 10.])처음에는 ${}=
\[\begin{bmatrix} \hat{w}_0 \\ \hat{w}_1 \end{bmatrix}\]=
\[\begin{bmatrix} -5 \\ 10 \end{bmatrix}\]$ 를 대입해서 주황색 점선을 적당히 그려보자는 의미
끝에 requires_grad=True는 나중에 미분을 위한 것
그려보자!
Stage2: 첫번째 수정 – 최초의 점선에 대한 ‘적당한 정도’를 판단하고 더 ’적당한’ 점선으로 업데이트 한다.
- ’적당한 정도’를 판단하기 위한 장치: loss function 도입!
\(loss=\sum_{i=1}^{n}(y_i-\hat{y}_i)^2=\sum_{i=1}^{n}(y_i-(\hat{w}_0+\hat{w}_1x_i))^2\)
\(=({\bf y}-{\bf\hat{y}})^\top({\bf y}-{\bf\hat{y}})=({\bf y}-{\bf X}{\bf \hat{W}})^\top({\bf y}-{\bf X}{\bf \hat{W}})\)
- loss 함수의 특징 - \(y_i \approx \hat{y}_i\) 일수록 loss값이 작다. - \(y_i \approx \hat{y}_i\) 이 되도록 \((\hat{w}_0,\hat{w}_1)\)을 잘 찍으면 loss값이 작다. - (중요) 주황색 점선이 ‘적당할 수록’ loss값이 작다.
# y와 yhat의 값이 비슷하면 loss값이 0에 가까워진다.loss=torch.sum((y-X@What)**2) #이 값이 주황색 점선에 대한 loss!!!
losstensor(8587.6875, grad_fn=<SumBackward0>)- 우리의 목표: 이 loss(=8587.6875)을 더 줄이자. - 궁극적으로는 아예 모든 조합 \((\hat{w}_0,\hat{w}_1)\)에 대하여 가장 작은 loss를 찾으면 좋겠다. (stage2에서 할일은 아님)
- 문제의 치환: 생각해보니까 우리의 문제는 아래와 같이 수학적으로 단순화 되었다. - 적당해보이는 주황색 선을 찾자 \(\to\) \(loss(w_0,w_1)\)를 최소로하는 \((w_0,w_1)\)의 값을 찾자.
- 수정된 목표: \(loss(w_0,w_1)\)를 최소로 하는 \((w_0,w_1)\)을 구하라. - 단순한 수학문제가 되었다. 마치 \(loss(w)=w^2-2w+3\) 을 최소화하는 \(w\)를 찾으라는 것과 같음. - 즉 “적당한 선으로 업데이트 하라 = 파라메터를 학습 하라 = 손실함수를 최소화 하라”
# 그 function을 minimize하는 정의역의 세트를 찾으면 된다. = 적당한 선으로 업데이트 하라= ...=- 우리의 무기: 경사하강법, 벡터미분
Stage2를 위한 경사하강법 복습
경사하강법 아이디어 (1차원)
(step 1) 임의의 점을 찍는다.
(step 2) 그 점에서 순간기울기를 구한다. (접선) <– 미분
(step 3) 순간기울기(=미분계수)의 부호를 살펴보고 부호와 반대방향으로 움직인다.
(팁) 기울기의 절대값 크기와 비례하여 보폭(=움직이는 정도)을 조절한다.
# 접선의 기울기가 만약 -4 라는 음수가 나오면 양수값으로 가면 된다..
# 미분계수의 절대값이 작아지는 정도로.. 보폭 조절!!경사하강법 아이디어 (2차원)
(step 1) 임의의 점을 찍는다.
(step 2) 그 점에서 순간기울기를 구한다. (접평면) <– 편미분
(step 3) 순간기울기(=미분계수)의 부호를 살펴보고 부호와 반대방향으로 각각 움직인다.
(팁) 기울기의 절대값 크기와 비례하여 보폭(=움직이는 정도)을 각각 조절한다.
# 왼쪽으로 갈래? 오른쪽으로 갈래?
# 위로 갈래? 아래로 갈래? ->점에서 한쪽방향을 고정되어있다 생각하고 왼오, 또는 위아래로만 -> 편미분으로 가넝!!
# 2차원, 3차원,, 원리는 결국 1차원과 똑같당!loss를 줄이도록 \({\bf W}\)를 개선하는 방법
- $수정값 원래값 - 기울어진크기(=미분계수) $
- 여기에서 \(\alpha\)는 전체적인 보폭의 크기를 결정한다. 즉 \(\alpha\)값이 클수록 한번의 update에 움직이는 양이 크다.
# 반대방향으로 가야하니까 - 를 붙인다.
# a 알파값은 .. 만약 미분계수가 -8이 나왔엉. 근데 그렇다고 8곱해버리면 너무 크니까 0.8 이든 0.08이든.. 그런 알파값을 곱해줘야해!!
# a 값은 양수여야함!!! 음수면 방향이 바꾸기 때문에 a는 정답이 없어. 0.0001 이렇게 걍 맞춰가면 뎀..- \({\bf W} \leftarrow {\bf W} - \alpha \times \frac{\partial}{\partial {\bf W}}loss(w_0,w_1)\)
마이너스의 의미: 기울기의 부호를 보고 반대방향으로 움직여라.
\(\frac{\partial}{\partial {\bf W}}loss(w_0,w_1):\) 기울기의 절대값 크기와 비례하여 움직이는 정도를 조정하라.
\(\alpha\)의 의미: 전체적인 보폭의 속도를 조절, \(\alpha\)가 크면 전체적으로 빠르게 움직인다. 다리의 길이로 비유할 수 있다.
losstensor(8587.6875, grad_fn=<SumBackward0>)- 우리의 목표: loss=8587.6875 인데, 이걸 줄이는 것이 목표라고 했었음. 이것을 줄이는 방법이 경사하강법이다.
- 경사하강법으로 loss를 줄이기 위해서는 \(\frac{\partial}{\partial {\bf W}}loss(w_0,w_1)\)의 계산이 필요한데, 이를 위해서 벡터미분이 필요하다. (loss.backward()로 하면된다)
What.gradloss.backward()
# 자기 혼자 미분하고 결과값을 안보여죵
# 뭘로 미분하라는거야? ->꼬리표 추적... What까지 가서 얘를 미분하라는 거구나 하고 미분해줌X@What #꼬리표가 있어!!!tensor([-29.8211, -28.6215, -24.9730, -21.2394, -19.7919, -19.6354, -19.5093,
-19.4352, -18.7223, -18.0793, -16.9040, -16.0918, -16.0536, -15.8746,
-14.4690, -14.3193, -13.6426, -12.8578, -12.5486, -12.4213, -11.9484,
-11.1034, -10.8296, -10.6210, -10.5064, -10.0578, -9.8063, -9.7380,
-9.7097, -9.6756, -8.8736, -8.7195, -8.6880, -8.1592, -7.7752,
-7.7716, -7.7339, -7.7208, -7.6677, -7.1551, -7.0004, -6.8163,
-6.7081, -6.5655, -6.4480, -6.3612, -6.0566, -5.6031, -5.5589,
-5.2137, -4.3446, -4.3165, -3.8047, -3.5801, -3.4793, -3.4325,
-2.3545, -2.3440, -1.8434, -1.7799, -1.5386, -1.0161, -0.8103,
0.4426, 0.5794, 0.9125, 1.1483, 1.4687, 1.4690, 1.5234,
1.6738, 2.0592, 2.1414, 2.8221, 3.1536, 3.6682, 4.2907,
4.8037, 4.8531, 4.9414, 5.3757, 5.3926, 5.6973, 6.0239,
6.1261, 6.5317, 7.2891, 8.4032, 8.4936, 9.2794, 9.9943,
10.0310, 10.4369, 11.7886, 15.8323, 17.4440, 18.9350, 21.0560,
21.0566, 21.6324], grad_fn=<MvBackward0>)y-X@What #꼬리표가 있는 걸로 파생된 모든 것들은 다 꼬리표가 있땅tensor([21.2791, 22.0448, 19.0234, 16.7600, 15.5403, 16.5028, 15.4853, 15.2491,
15.3820, 15.8766, 14.8778, 13.5299, 14.7183, 13.8280, 14.0026, 12.9680,
11.9954, 12.7489, 12.2415, 11.7914, 11.9046, 11.5198, 11.2462, 10.5267,
10.7726, 10.5168, 10.6967, 10.6377, 10.3411, 11.0601, 9.6820, 9.9789,
9.8090, 10.0825, 8.8370, 9.1268, 9.8500, 8.8644, 9.2922, 8.9190,
8.6027, 8.5628, 7.6912, 8.3478, 8.5596, 9.2232, 8.1731, 7.1257,
8.1160, 8.0498, 7.7402, 6.3844, 6.6187, 7.0653, 7.0851, 6.0290,
5.2399, 6.2613, 5.4961, 5.8827, 5.8510, 4.4186, 4.0283, 4.1260,
3.7978, 3.3949, 3.3412, 3.0141, 3.8481, 3.9753, 3.7895, 3.9736,
3.1428, 2.2318, 2.3002, 2.3654, 1.4343, 0.9550, 1.3489, 1.6579,
1.0864, 1.1214, 0.9873, 1.3258, 1.9648, 0.5476, -0.4224, -0.9803,
-1.2392, -2.0827, -0.4937, -0.9971, -2.9482, -2.7127, -4.7377, -7.1180,
-6.6685, -7.9578, -8.5098, -7.7984], grad_fn=<SubBackward0>)loss.backward()의 의미: loss를 미분해라! 뭘로?
requires_grad=True를 가진 텐서로!!
loss=torch.sum((y-yhat)**2)= torch.sum((y-X@What)**2)
# 이었고
What=torch.tensor([-5.0,10.0],requires_grad=True)
# 이므로 결국 What으로 미분하라는 의미.
# 미분한 식이 나오는 것이 아니고,
# 그 식에 (-5.0, 10.0)을 대입한 계수값이 계산됨. - 위에서 loss.backward()의 과정은 미분을 활용하여 \((-5,10)\)에서의 순간기울기를 구했다는 의미임.
What.gradtensor([-1342.2522, 1188.9305])- (-5,10)에서 loss의 순간기울기 값은 What.grad로 확인가능하다.
- 이것이 의미하는건 \((-5,10)\)에서의 \(loss(w_0,w_1)\)의 순간기울기가 \((-1342.2523, 1188.9307)\) 이라는 의미
- (확인1) loss.backward()가 미분을 잘 계산해 주는 것이 맞는가? 손계산으로 검증하여 보자.
\(loss(w_0,w_1)=({\bf y}-\hat{\bf y})^\top ({\bf y}-\hat{\bf y})=({\bf y}-{\bf XW})^\top ({\bf y}-{\bf XW})\)
\(\frac{\partial}{\partial {\bf W} }loss(w_0,w_1)=-2{\bf X}^\top {\bf y}+2{\bf X}^\top {\bf X W}\)
- 2 * X.T @ y + 2 * X.T @ X @ Whattensor([-1342.2523, 1188.9305], grad_fn=<AddBackward0>)- (확인2) loss.backward()가 미분을 잘 계산해 주는 것이 맞는가? 편미분을 간단히 구현하여 검증하여 보자.
\(\frac{\partial}{\partial {\bf W} } loss(w_0,w_1)=\begin{bmatrix}\frac{\partial}{\partial w_0} \\ \frac{\partial}{\partial w_1} \end{bmatrix}loss(w_0,w_1) =\begin{bmatrix}\frac{\partial}{\partial w_0}loss(w_0,w_1) \\ \frac{\partial}{\partial w_1}loss(w_0,w_1) \end{bmatrix}\)
\(\frac{\partial}{\partial w_0}loss(w_0,w_1) \approx \frac{loss(w_0+h,w_1)-loss(w_0,w_1)}{h}\)
\(\frac{\partial}{\partial w_1}loss(w_0,w_1) \approx \frac{loss(w_0,w_1+h)-loss(w_0,w_1)}{h}\)
_lossfn = lambda w0,w1: torch.sum((y-w0-w1*x)**2)
_lossfn(-5,10)tensor(8587.6875)h=0.001
(_lossfn(-5+h,10) - _lossfn(-5,10))/h, (_lossfn(-5,10+h) - _lossfn(-5,10))/h(tensor(-1341.7968), tensor(1190.4297))- 약간 오차가 있지만 얼추비슷 \(\to\) 잘 계산했다는 소리임
- 수정전, 수정하는폭, 수정후의 값은 차례로 아래와 같다.
What.datatensor([-5., 10.])str(What.data) #문자열이 됨! 'tensor([-5., 10.])'alpha=0.001
print('수정전: ' + str(What.data)) # What 에서 미분꼬리표를 떼고 싶다면? What.data or What.detach()
print('수정하는폭: ' +str(-alpha * What.grad)) #1341*0.001, 1190*0.001
print('수정후: ' +str(What.data-alpha * What.grad))
print('*참값: (2.5,4)' )수정전: tensor([-5., 10.])
수정하는폭: tensor([ 1.3423, -1.1889])
수정후: tensor([-3.6577, 8.8111])
*참값: (2.5,4)- Wbefore, Wafter 계산
Wbefore = What.data
Wafter = What.data- alpha * What.grad
Wbefore, Wafter(tensor([-5., 10.]), tensor([-3.6577, 8.8111]))- Wbefore, Wafter의 시각화
plt.plot(x,y,'o')
plt.plot(x,X@Wbefore,'--') #주황색
plt.plot(x,X@Wafter,'--') #초록색_9월21일_ipynb의_사본_files/figure-html/cell-43-output-1.png)
Stage3: Learn (=estimate \(\bf\hat{W})\)
- 이 과정은 Stage1,2를 반복하면 된다.
What= torch.tensor([-5.0,10.0],requires_grad=True)
Whattensor([-5., 10.], requires_grad=True)alpha=0.001
for epoc in range(30): ## 30번 반복합니다!!
yhat=X@What
loss=torch.sum((y-yhat)**2)
loss.backward() #미분.. what.grad로 미분이 된 것을 볼 수 있따->편미분계수값
What.data = What.data-alpha * What.grad #경사하강법~~
What.grad=None # 파이토치 특징.. 미분된 grad값에 새 미분값이 안들어가있고.. 이전미분값에 그다음미분값이 더해져서 들어가기 때문에->defalut기 대문에 초기상태로 바꿔줘야햄....Whattensor([2.4290, 4.0144], requires_grad=True)- 원래 철자는 epoch이 맞아요
- 반복결과는?! (최종적으로 구해지는 What의 값은?!) - 참고로 true
What.data ## true인 (2.5,4)와 상당히 비슷함tensor([2.4290, 4.0144])- 반복결과를 시각화하면?
plt.plot(x,y,'o')
plt.plot(x,X@What.data,'--') #그림을 그리기위해서 꼬리표를 떼준당.._9월21일_ipynb의_사본_files/figure-html/cell-49-output-1.png)
파라메터의 학습과정 음미 (학습과정 모니터링)
학습과정의 기록
- 기록을 해보자.
loss_history = [] # 기록하고 싶은것 1 .
yhat_history = [] # 기록하고 싶은것 2 yhat이 어떻게 변하는지
What_history = [] # 기록하고 싶은것 3 #loss_history #처음엔 비어있지만.. 값을 점점 넣고 싶어!!!#loss_history.append(loss)
#loss_history#loss.item()#loss_history.append(loss.item())
#loss_history # 텐서를 리스트로 바꿔줘서 넣어주기..What= torch.tensor([-5.0,10.0],requires_grad=True)
alpha=0.001
for epoc in range(30):
yhat=X@What ; yhat_history.append(yhat.data.tolist()) #세미콜론만 주석처리하면.. 위랑 똑같은 코드..
loss=torch.sum((y-yhat)**2); loss_history.append(loss.item())
loss.backward()
What.data = What.data-alpha * What.grad; What_history.append(What.data.tolist())
What.grad=Noneloss_history[8587.6875,
5675.2109375,
3755.637451171875,
2489.581787109375,
1654.0390625,
1102.3206787109375,
737.8441162109375,
496.96514892578125,
337.7142028808594,
232.39694213867188,
162.72906494140625,
116.63263702392578,
86.1263656616211,
65.93397521972656,
52.566444396972656,
43.71583557128906,
37.855220794677734,
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31.403636932373047,
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27.826366424560547,
27.331483840942383,
27.003639221191406,
26.78643798828125,
26.642536163330078,
26.547197341918945,
26.48402976989746,
26.442174911499023,
26.414440155029297]- \(\hat{y}\) 관찰 (epoch=3, epoch=10, epoch=15)
yhat_history[[-29.821128845214844,
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8.447975158691406,
8.462722778320312,
8.625760078430176,
9.168725967407227,
10.792997360229492,
11.440369606018066,
12.039276123046875,
12.89124870300293,
12.891464233398438,
13.122757911682129]]len(yhat_history)30len(yhat_history[0]) #0에 100개... [1]에 100개.. 100plt.plot(x,y,'o')
plt.plot(x,yhat_history[2],'--')_9월21일_ipynb의_사본_files/figure-html/cell-60-output-1.png)
plt.plot(x,y,'o')
plt.plot(x,yhat_history[9],'--')_9월21일_ipynb의_사본_files/figure-html/cell-61-output-1.png)
plt.plot(x,y,'o')
plt.plot(x,yhat_history[14],'--')_9월21일_ipynb의_사본_files/figure-html/cell-62-output-1.png)
- \(\hat{\bf W}\) 관찰
What_history[[-3.657747745513916, 8.81106948852539],
[-2.554811477661133, 7.861191749572754],
[-1.649186134338379, 7.101552963256836],
[-0.9060712456703186, 6.49347448348999],
[-0.2966785430908203, 6.006272315979004],
[0.20277434587478638, 5.615575313568115],
[0.6119105815887451, 5.302003383636475],
[0.9469034671783447, 5.050129413604736],
[1.2210699319839478, 4.847657680511475],
[1.4453645944595337, 4.684779167175293],
[1.6287915706634521, 4.553659439086914],
[1.778746247291565, 4.448036193847656],
[1.90129816532135, 4.3628973960876465],
[2.0014259815216064, 4.294229507446289],
[2.0832109451293945, 4.238814353942871],
[2.149996757507324, 4.194070339202881],
[2.204521894454956, 4.157923698425293],
[2.249027729034424, 4.128708839416504],
[2.285348415374756, 4.105085849761963],
[2.31498384475708, 4.0859761238098145],
[2.339160442352295, 4.070511341094971],
[2.3588807582855225, 4.057991027832031],
[2.3749637603759766, 4.0478515625],
[2.3880786895751953, 4.039637088775635],
[2.3987717628479004, 4.032979965209961],
[2.40748929977417, 4.027583599090576],
[2.414595603942871, 4.023208141326904],
[2.4203879833221436, 4.019659042358398],
[2.4251089096069336, 4.016779899597168],
[2.4289560317993164, 4.014443874359131]]- loss 관찰
loss_history[8587.6875,
5675.2109375,
3755.637451171875,
2489.581787109375,
1654.0390625,
1102.3206787109375,
737.8441162109375,
496.96514892578125,
337.7142028808594,
232.39694213867188,
162.72906494140625,
116.63263702392578,
86.1263656616211,
65.93397521972656,
52.566444396972656,
43.71583557128906,
37.855220794677734,
33.974090576171875,
31.403636932373047,
29.701112747192383,
28.57339096069336,
27.826366424560547,
27.331483840942383,
27.003639221191406,
26.78643798828125,
26.642536163330078,
26.547197341918945,
26.48402976989746,
26.442174911499023,
26.414440155029297]plt.plot(loss_history)_9월21일_ipynb의_사본_files/figure-html/cell-65-output-1.png)
학습과정을 animation으로 시각화
from matplotlib import animationplt.rcParams['figure.figsize'] = (7.5,2.5)
plt.rcParams["animation.html"] = "jshtml" - 왼쪽에는 \((x_i,y_i)\) and \((x_i,\hat{y}_i)\) 을 그리고 오른쪽에는 \(loss(w_0,w_1)\) 을 그릴것임
fig = plt.figure()
ax1 = fig.add_subplot(1, 2, 1)
ax2 = fig.add_subplot(1, 2, 2, projection='3d')_9월21일_ipynb의_사본_files/figure-html/cell-68-output-1.png)
# 왼쪽 2d 오른쪽 3d 축만 생긴다- 왼쪽그림!
ax1.plot(x,y,'o')
line, = ax1.plot(x,yhat_history[0]) # 나중에 애니메이션 할때 필요해요..fig_9월21일_ipynb의_사본_files/figure-html/cell-71-output-1.png)
- 오른쪽 그림1: \(loss(w_0,w_1)\)
_w0 = np.arange(-6, 11, 0.5) ## 파란색곡면을 그리는 코드 (시작)
_w1 = np.arange(-6, 11, 0.5)
w1,w0 = np.meshgrid(_w1,_w0)
lss=w0*0
for i in range(len(_w0)):
for j in range(len(_w1)):
lss[i,j]=torch.sum((y-_w0[i]-_w1[j]*x)**2)
ax2.plot_surface(w0, w1, lss, rstride=1, cstride=1, color='b',alpha=0.35) ## 파란색곡면을 그리는 코드(끝)
ax2.azim = 40 ## 3d plot의 view 조절
ax2.dist = 8 ## 3d plot의 view 조절
ax2.elev = 5 ## 3d plot의 view 조절 fig_9월21일_ipynb의_사본_files/figure-html/cell-73-output-1.png)
- 오른쪽 그림2: \((w_0,w_1)=(2.5,4)\) 와 \(loss(2.5,4)\) 값 <- loss 함수가 최소가 되는 값 (이거 진짜야? ㅋㅋ)
ax2.scatter(2.5,4,torch.sum((y-2.5-4*x)**2),s=200,color='red',marker='*') ## 최소점을 표시하는 코드 (붉은색 별) <mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x7fac8c960610>fig_9월21일_ipynb의_사본_files/figure-html/cell-75-output-1.png)
- 오른쪽 그림3: \((w_0,w_1)=(-3.66, 8.81)\) 와 \(loss(-3.66,8.81)\) 값
What_history[0][-3.657747745513916, 8.81106948852539]ax2.scatter(What_history[0][0],What_history[0][1],loss_history[0],color='grey') ## 업데이트되는 What을 표시하는 점 (파란색 동그라미)
#처음 값!! 오른쪽 그림에서 저 동그라미 회색점이 별표로 가야해~~ 경사하강법생각하자!<mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x7fac8c7e5710>fig_9월21일_ipynb의_사본_files/figure-html/cell-78-output-1.png)
- 애니메이션
def animate(epoc):
line.set_ydata(yhat_history[epoc])
ax2.scatter(What_history[epoc][0],What_history[epoc][1],loss_history[epoc],color='grey')
return line
ani = animation.FuncAnimation(fig, animate, frames=30)
plt.close()
ani- 함수로 만들자..
def show_lrpr(data,history):
x,y = data
loss_history,yhat_history,What_history = history
fig = plt.figure()
ax1 = fig.add_subplot(1, 2, 1)
ax2 = fig.add_subplot(1, 2, 2, projection='3d')
## ax1: 왼쪽그림
ax1.plot(x,y,'o')
line, = ax1.plot(x,yhat_history[0])
## ax2: 오른쪽그림
_w0 = np.arange(-6, 11, 0.5) ## 파란색곡면을 그리는 코드 (시작)
_w1 = np.arange(-6, 11, 0.5)
w1,w0 = np.meshgrid(_w1,_w0)
lss=w0*0
for i in range(len(_w0)):
for j in range(len(_w1)):
lss[i,j]=torch.sum((y-_w0[i]-_w1[j]*x)**2)
ax2.plot_surface(w0, w1, lss, rstride=1, cstride=1, color='b',alpha=0.35) ## 파란색곡면을 그리는 코드(끝)
ax2.scatter(2.5,4,torch.sum((y-2.5-4*x)**2),s=200,color='red',marker='*') ## 최소점을 표시하는 코드 (붉은색 별)
ax2.scatter(What_history[0][0],What_history[0][1],loss_history[0],color='b') ## 업데이트되는 What을 표시하는 점 (파란색 동그라미)
ax2.azim = 40 ## 3d plot의 view 조절
ax2.dist = 8 ## 3d plot의 view 조절
ax2.elev = 5 ## 3d plot의 view 조절
def animate(epoc):
line.set_ydata(yhat_history[epoc])
ax2.scatter(np.array(What_history)[epoc,0],np.array(What_history)[epoc,1],loss_history[epoc],color='grey')
return line
ani = animation.FuncAnimation(fig, animate, frames=30)
plt.close()
return anishow_lrpr([x,y],[loss_history,yhat_history,What_history])\(\alpha\)에 대하여 (\(\alpha\)는 학습률)
#머신러닝에서 a는 학습률...(1) \(\alpha=0.0001\): \(\alpha\) 가 너무 작다면? \(\to\) 비효율적이다.
loss_history = [] # 기록하고 싶은것 1
yhat_history = [] # 기록하고 싶은것 2
What_history = [] # 기록하고 싶은것 3 What= torch.tensor([-5.0,10.0],requires_grad=True)
alpha=0.0001
for epoc in range(30):
yhat=X@What ; yhat_history.append(yhat.data.tolist())
loss=torch.sum((y-yhat)**2); loss_history.append(loss.item())
loss.backward()
What.data = What.data-alpha * What.grad; What_history.append(What.data.tolist())
What.grad=Noneshow_lrpr([x,y],[loss_history,yhat_history,What_history])(2) \(\alpha=0.0083\): \(\alpha\)가 너무 크다면? \(\to\) 다른의미에서 비효율적이다 + 위험하다..
loss_history = [] # 기록하고 싶은것 1
yhat_history = [] # 기록하고 싶은것 2
What_history = [] # 기록하고 싶은것 3 What= torch.tensor([-5.0,10.0],requires_grad=True)
alpha=0.0083
for epoc in range(30):
yhat=X@What ; yhat_history.append(yhat.data.tolist())
loss=torch.sum((y-yhat)**2); loss_history.append(loss.item())
loss.backward()
What.data = What.data-alpha * What.grad; What_history.append(What.data.tolist())
What.grad=Noneshow_lrpr([x,y],[loss_history,yhat_history,What_history])(3) \(\alpha=0.0085\)
loss_history = [] # 기록하고 싶은것 1
yhat_history = [] # 기록하고 싶은것 2
What_history = [] # 기록하고 싶은것 3 What= torch.tensor([-5.0,10.0],requires_grad=True)
alpha=0.0085
for epoc in range(30):
yhat=X@What ; yhat_history.append(yhat.data.tolist())
loss=torch.sum((y-yhat)**2); loss_history.append(loss.item())
loss.backward()
What.data = What.data-alpha * What.grad.data; What_history.append(What.data.tolist())
What.grad=Noneshow_lrpr([x,y],[loss_history,yhat_history,What_history])(4) \(\alpha=0.01\)
loss_history = [] # 기록하고 싶은것 1
yhat_history = [] # 기록하고 싶은것 2
What_history = [] # 기록하고 싶은것 3 What= torch.tensor([-5.0,10.0],requires_grad=True)
alpha=0.01
for epoc in range(30):
yhat=X@What ; yhat_history.append(yhat.data.tolist())
loss=torch.sum((y-yhat)**2); loss_history.append(loss.item())
loss.backward()
What.data = What.data-alpha * What.grad; What_history.append(What.data.tolist())
What.grad=Noneshow_lrpr([x,y],[loss_history,yhat_history,What_history])숙제
- 학습률(\(\alpha\))를 조정하며 실습해보고 스크린샷 제출
# α=0.00912
loss_history = []
yhat_history = []
What_history = [] What= torch.tensor([-5.0,10.0],requires_grad=True)
alpha=0.00912
for epoc in range(30):
yhat=X@What ; yhat_history.append(yhat.data.tolist())
loss=torch.sum((y-yhat)**2); loss_history.append(loss.item())
loss.backward()
What.data = What.data-alpha * What.grad; What_history.append(What.data.tolist())
What.grad=Noneshow_lrpr([x,y],[loss_history,yhat_history,What_history])