7주차-4월 18일
빅데이터분석특강
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
import tensorflow.experimental.numpy as tnp
tnp.experimental_enable_numpy_behavior()
import graphviz
def gv(s): return graphviz.Source('digraph G{ rankdir="LR"'+s + '; }')
np.random.seed(43052)
N=100
x = np.linspace(-1,1,N)
lamb = lambda x: x*1+np.random.normal()*0.3 if x<0 else x*3.5+np.random.normal()*0.3
y= np.array(list(map(lamb,x)))
y
plt.plot(x,y,'.')
x= x.reshape(N,1)
y= y.reshape(N,1)
net = tf.keras.Sequential()
net.add(tf.keras.layers.Dense(1))
net.compile(optimizer=tf.optimizers.SGD(0.1),loss='mse')
net.fit(x,y,batch_size=N,epochs=1000,verbose=0) # numpy로 해도 돌아감
net.weights
yhat = x * 2.2616348 + 0.6069048
yhat = net.predict(x)
plt.plot(x,y,'.')
plt.plot(x,yhat,'--')
- 실패: 이 모형은 epoch을 10억번 돌려도 실패할 모형임
- 왜? 아키텍처 설계자체가 틀렸음
- 꺽인부분을 표현하기에는 아키텍처의 표현력이 너무 부족하다 -> under fit의 문제
gv('''
"x" -> "x*w, bias=True"[label="*w"] ;
"x*w, bias=True" -> "y"[label="indentity"] ''')
(수정후) hat은 생략
gv('''
"x" -> "x*w, bias=True"[label="*w"] ;
"x*w, bias=True" -> "y"[label="relu"] ''')
- 마지막에 $f(x)=x$ 라는 함수대신에 relu를 취하는 것으로 구조를 약간 변경
- 활성화함수(acitivation function)를 indentity에서 relu로 변경
- relu함수란?
_x = np.linspace(-1,1,100)
tf.nn.relu(_x)
plt.plot(_x,_x)
plt.plot(_x,tf.nn.relu(_x))
- 파란색을 주황색으로 바꿔주는 것이 렐루함수임
- $f(x)=\max(0,x)=\begin{cases} 0 & x\leq 0 \\ x & x>0 \end{cases}$
- 아키텍처: $\hat{y}_i=relu(\hat{w}_0+\hat{w}_1x_i)$, $relu(x)=\max(0,x)$
- 풀이시작
1단계
net2 = tf.keras.Sequential()
2단계
tf.random.set_seed(43053)
l1 = tf.keras.layers.Dense(1, input_shape=(1,))
a1 = tf.keras.layers.Activation(tf.nn.relu)
net2.add(l1)
net2.layers
net2.add(a1)
net2.layers
l1.get_weights()
net2.get_weights()
(네트워크 상황 확인)
u1= l1(x)
#u1= x@l1.weights[0] + l1.weights[1]
v1= a1(u1)
#v1= tf.nn.relu(u1)
plt.plot(x,x)
plt.plot(x,u1,'--r')
plt.plot(x,v1,'--b')
3단계
net2.compile(optimizer=tf.optimizers.SGD(0.1),loss='mse')
4단계
net2.fit(x,y,epochs=1000,verbose=0,batch_size=N)
- result
yhat = tf.nn.relu(x@l1.weights[0] + l1.weights[1])
yhat = net2.predict(x)
yhat = net2(x)
yhat = a1(l1(x))
yhat = net2.layers[1](net2.layers[0](x))
plt.plot(x,y,'.')
plt.plot(x,yhat,'--')
net3 = tf.keras.Sequential()
2단계
tf.random.set_seed(43053)
l1 = tf.keras.layers.Dense(2,input_shape=(1,))
a1 = tf.keras.layers.Activation(tf.nn.relu)
net3.add(l1)
net3.add(a1)
(네트워크 상황 확인)
l1(x).shape
# l1(x) : (100,1) -> (100,2)
plt.plot(x,x)
plt.plot(x,l1(x),'--')
plt.plot(x,x)
plt.plot(x,a1(l1(x)),'--')
- 이 상태에서는 yhat이 안나온다. 왜?
- 차원이 안맞음.
a1(l1(x))의 차원은 (N,2)인데 최종적인 yhat의 차원은 (N,1)이어야 함. - 차원이 어찌저찌 맞다고 쳐도 relu를 통과하면 항상 yhat>0 임. 따라서 음수값을 가지는 y는 0으로 밖에 맞출 수 없음.
- 해결책: a1(l1(x))에 연속으로(Sequential하게!) 또 다른 레이어를 설계! (N,2) -> (N,1) 이 되도록!
yhat= bias + weight1 * a1(l1(x))[0] + weight2 * a1(l1(x))[1]
- 즉 a1(l1(x)) 를 새로운 입력으로 해석하고 출력을 만들어주는 선형모형을 다시태우면 된다.
- 입력차원: 2
- 출력차원: 1
net3.layers
tf.random.set_seed(43053)
l2 = tf.keras.layers.Dense(1, input_shape=(2,))
net3.add(l2)
net3.layers
net3.summary()
- 추정해야할 파라메터수가 4,0,3으로 나온다.
- 수식표현: $X \to X@W^{(1)}+b^{(1)} \to relu(X@W^{(1)}+b^{(1)}) \to relu(X@W^{(1)}+b^{(1)})@W^{(2)}+b^{(2)}=yhat$
- $X$: (N,1)
- $W^{(1)}$: (1,2) ==> 파라메터 2개 추정
- $b^{(1)}$: (2,) ==> 파라메터 2개가 추가 // 여기까지 추정할 파라메터는 4개
- $W^{(2)}$: (2,1) ==> 파라메터 2개 추정
- $b^{(2)}$: (1,) ==> 파라메터 1개가 추가 // 따라서 3개
- 참고: 추정할 파라메터수가 많다 = 복잡한 모형이다.
- 초거대AI: 추정할 파라메터수가 엄청 많은..
net3.weights
l1.weights
l2.weights
- 좀 더 간단한 수식표현: $X \to (u_1 \to v_1) \to (u_2 \to v_2) = yhat$
- $u_1= X@W^{(1)}+b^{(1)}$
- $v_1= relu(u_1)$
- $u_2= v_1@W^{(2)}+b^{(2)}$
- $v_2= indentity(u_2):=yhat$
gv('''
subgraph cluster_1{
style=filled;
color=lightgrey;
"X"
label = "Layer 0"
}
subgraph cluster_2{
style=filled;
color=lightgrey;
"X" -> "u1[:,0]"[label="*W1[0,0]"]
"X" -> "u1[:,1]"[label="*W1[0,1]"]
"u1[:,0]" -> "v1[:,0]"[label="relu"]
"u1[:,1]" -> "v1[:,1]"[label="relu"]
label = "Layer 1"
}
subgraph cluster_3{
style=filled;
color=lightgrey;
"v1[:,0]" -> "yhat"[label="*W2[0,0]"]
"v1[:,1]" -> "yhat"[label="*W2[1,0]"]
label = "Layer 2"
}
''')
gv('''
subgraph cluster_1{
style=filled;
color=lightgrey;
"X"
label = "Layer 0"
}
subgraph cluster_2{
style=filled;
color=lightgrey;
"X" -> "node1"
"X" -> "node2"
label = "Layer 1: relu"
}
subgraph cluster_3{
style=filled;
color=lightgrey;
"node1" -> "yhat"
"node2" -> "yhat"
label = "Layer 2"
}
''')
3단계
net3.compile(loss='mse',optimizer=tf.optimizers.SGD(0.1))
4단계
net3.fit(x,y,epochs=1000,verbose=0, batch_size=N)
- 결과확인
net3.weights
plt.plot(x,y,'.')
plt.plot(x,net3(x),'--')
- 분석
plt.plot(x,y,'.')
plt.plot(x,l1(x),'--')
plt.plot(x,y,'.')
plt.plot(x,a1(l1(x)),'--')
plt.plot(x,y,'.')
plt.plot(x,l2(a1(l1(x))),'--')
- 마지막 2개의 그림을 분석
l2.weights
fig, (ax1,ax2,ax3) = plt.subplots(1,3)
fig.set_figwidth(12)
ax1.plot(x,y,'.')
ax1.plot(x,a1(l1(x))[:,0],'--r')
ax1.plot(x,a1(l1(x))[:,1],'--b')
ax2.plot(x,y,'.')
ax2.plot(x,a1(l1(x))[:,0]*1.6328746,'--r')
ax2.plot(x,a1(l1(x))[:,1]*(-1.2001747)+1.0253307,'--b')
ax3.plot(x,y,'.')
ax3.plot(x,a1(l1(x))[:,0]*1.6328746+a1(l1(x))[:,1]*(-1.2001747)+1.0253307,'--')
tf.random.set_seed(43054)
## 1단계
net3 = tf.keras.Sequential()
## 2단계
net3.add(tf.keras.layers.Dense(2))
net3.add(tf.keras.layers.Activation('relu'))
net3.add(tf.keras.layers.Dense(1))
## 3단계
net3.compile(optimizer=tf.optimizers.SGD(0.1),loss='mse')
## 4단계
net3.fit(x,y,epochs=1000,verbose=0,batch_size=N)
plt.plot(x,y,'.')
plt.plot(x,net3(x),'--')
- 엥? 에폭이 부족한가?
net3.fit(x,y,epochs=10000,verbose=0,batch_size=N)
plt.plot(x,y,'.')
plt.plot(x,net3(x),'--')
- 실패분석
l1,a1,l2 = net3.layers
l2.weights
fig, (ax1,ax2,ax3,ax4) = plt.subplots(1,4)
fig.set_figwidth(16)
ax1.plot(x,y,'.')
ax1.plot(x,l1(x)[:,0],'--r')
ax1.plot(x,l1(x)[:,1],'--b')
ax2.plot(x,y,'.')
ax2.plot(x,a1(l1(x))[:,0],'--r')
ax2.plot(x,a1(l1(x))[:,1],'--b')
ax3.plot(x,y,'.')
ax3.plot(x,a1(l1(x))[:,0]*0.65121335,'--r')
ax3.plot(x,a1(l1(x))[:,1]*(1.8592643)+(-0.60076195),'--b')
ax4.plot(x,y,'.')
ax4.plot(x,a1(l1(x))[:,0]*0.65121335+a1(l1(x))[:,1]*(1.8592643)+(-0.60076195),'--')
- 보니까 빨간색선이 하는 역할을 없음
- 그런데 생각해보니까 이 상황에서는 빨간색선이 할수 있는 일이 별로 없음
- 왜? 지금은 나름 파란색선에 의해서 최적화가 된 상태임 $\to$ 빨간선이 뭔가 하려고하면 최적화된 상태가 깨질 수 있음 (loss 증가)
- 즉 이 상황 자체가 나름 최적회된 상태이다. 이러한 현상을 "global minimum을 찾지 못하고 local minimum에 빠졌다"라고 표현한다.
확인:
net3.weights
W1= tf.Variable(tnp.array([[-0.03077251, 1.8713338 ]]))
b1= tf.Variable(tnp.array([-0.04834982, 0.3259186 ]))
W2= tf.Variable(tnp.array([[0.65121335],[1.8592643 ]]))
b2= tf.Variable(tnp.array([-0.60076195]))
with tf.GradientTape() as tape:
u = tf.constant(x) @ W1 + b1
v = tf.nn.relu(u)
yhat = v@W2 + b2
loss = tf.losses.mse(y,yhat)
tape.gradient(loss,[W1,b1,W2,b2])
gv('''
subgraph cluster_1{
style=filled;
color=lightgrey;
"X"
label = "Layer 0"
}
subgraph cluster_2{
style=filled;
color=lightgrey;
"X" -> "node1"
"X" -> "node2"
"X" -> "..."
"X" -> "node512"
label = "Layer 1: relu"
}
subgraph cluster_3{
style=filled;
color=lightgrey;
"node1" -> "yhat"
"node2" -> "yhat"
"..." -> "yhat"
"node512" -> "yhat"
label = "Layer 2"
}
''')
tf.random.set_seed(43056)
net4= tf.keras.Sequential()
net4.add(tf.keras.layers.Dense(512,activation='relu')) # 이렇게 해도됩니다.
net4.add(tf.keras.layers.Dense(1))
net4.compile(loss='mse',optimizer=tf.optimizers.SGD(0.1))
net4.fit(x,y,epochs=1000,verbose=0,batch_size=N)
plt.plot(x,y,'.')
plt.plot(x,net4(x),'--')
- 잘된다..
- 한두개의 노드가 역할을 못해도 다른노드들이 잘 보완해주는듯!
- 노드수가 많으면 무조건 좋다? -> 대부분 나쁘지 않음. 그런데 종종 맞추지 말아야할것도 맞춤.. (overfit)
np.random.seed(43052)
N=100
_x = np.linspace(0,1,N).reshape(N,1)
_y = np.random.normal(loc=0,scale=0.001,size=(N,1))
plt.plot(_x,_y)
tf.random.set_seed(43052)
net4 = tf.keras.Sequential()
net4.add(tf.keras.layers.Dense(512,activation='relu'))
net4.add(tf.keras.layers.Dense(1))
net4.compile(loss='mse',optimizer=tf.optimizers.SGD(0.5))
net4.fit(_x,_y,epochs=1000,verbose=0,batch_size=N)
plt.plot(_x,_y)
plt.plot(_x,net4(_x),'--')
- 현실에서 이런 경우가 많음
- $x$가 커질수록 (혹은 작아질수록) 성공확률이 올라간다.
- 이러한 모형은 아래와 같이 설계할 수 있음 <-- 외우세요!!
$y_i \sim Ber(\pi_i)$, where $\pi_i=\frac{\exp(w_0+w_1x_i)}{1+\exp(w_0+w_1x_i)}$
$\hat{y}_i =\frac{\exp(\hat{w}_0+\hat{w}_1x_i)}{1+\exp(\hat{w}_0+\hat{w}_1x_i)}=\frac{1}{1+exp(-\hat{w}_0-\hat{w}_1x_i)}$
$loss=-\frac{1}{n}\sum_{i=1}^{n}\big(y_i\log(\hat{y}_i)+(1-y_i)\log(1-\hat{y}_i)\big)$
N = 2000
x = tnp.linspace(-1,1,N).reshape(N,1)
w0 = -1
w1 = 5
u = w0 + x*w1
#v = tf.constant(np.exp(u)/(1+np.exp(u))) # v=πi
v = tf.nn.sigmoid(u)
y = tf.constant(np.random.binomial(1,v),dtype=tf.float64)
plt.plot(x,y,'.',alpha=0.02)
plt.plot(x,v,'--r')
- 이 아키텍처(yhat을 얻어내는 과정)를 다어어그램으로 나타내면 아래와 같다.
gv('''
subgraph cluster_1{
style=filled;
color=lightgrey;
"x"
label = "Layer 0"
}
subgraph cluster_2{
style=filled;
color=lightgrey;
"x" -> "x*w, bias=True"[label="*w"]
"x*w, bias=True" -> "yhat"[label="sigmoid"]
label = "Layer 1"
}
''')
- 또는 간단하게 아래와 같이 쓸 수 있다.
gv('''
subgraph cluster_1{
style=filled;
color=lightgrey;
x
label = "Layer 0"
}
subgraph cluster_2{
style=filled;
color=lightgrey;
x -> "node1=yhat"
label = "Layer 1: sigmoid"
}
''')
- 케라스를 이용하여 적합을 해보면
- $loss=-\frac{1}{n}\sum_{i=1}^{n}\big(y_i\log(\hat{y}_i)+(1-y_i)\log(1-\hat{y}_i)\big)$
tf.random.set_seed(43052)
net = tf.keras.Sequential()
net.add(tf.keras.layers.Dense(1,activation='sigmoid'))
bceloss_fn = lambda y,yhat: -tf.reduce_mean(y*tnp.log(yhat) + (1-y)*tnp.log(1-yhat))
net.compile(loss=bceloss_fn, optimizer=tf.optimizers.SGD(0.1))
net.fit(x,y,epochs=1000,verbose=0,batch_size=N)
net.weights
plt.plot(x,y,'.',alpha=0.1)
plt.plot(x,v,'--r')
plt.plot(x,net(x),'--b')
tf.random.set_seed(43052)
net = tf.keras.Sequential()
net.add(tf.keras.layers.Dense(1,activation='sigmoid'))
mseloss_fn = lambda y,yhat: tf.reduce_mean((y-yhat)**2)
net.compile(loss=mseloss_fn, optimizer=tf.optimizers.SGD(0.1))
net.fit(x,y,epochs=1000,verbose=0,batch_size=N)
plt.plot(x,y,'.',alpha=0.1)
plt.plot(x,v,'--r')
plt.plot(x,net(x),'--b')
w0, w1 = np.meshgrid(np.arange(-10,3,0.2), np.arange(-1,10,0.2), indexing='ij')
w0, w1 = w0.reshape(-1), w1.reshape(-1)
def loss_fn1(w0,w1):
u = w0+w1*x
yhat = np.exp(u)/(np.exp(u)+1)
return mseloss_fn(y,yhat)
def loss_fn2(w0,w1):
u = w0+w1*x
yhat = np.exp(u)/(np.exp(u)+1)
return bceloss_fn(y,yhat)
loss1 = list(map(loss_fn1,w0,w1))
loss2 = list(map(loss_fn2,w0,w1))
fig = plt.figure()
fig.set_figwidth(9)
fig.set_figheight(9)
ax1=fig.add_subplot(1,2,1,projection='3d')
ax2=fig.add_subplot(1,2,2,projection='3d')
ax1.elev=15
ax2.elev=15
ax1.azim=75
ax2.azim=75
ax1.scatter(w0,w1,loss1,s=0.1)
ax2.scatter(w0,w1,loss2,s=0.1)
- 왼쪽곡면(MSEloss)보다 오른쪽곡면(BCEloss)이 좀더 예쁘게 생김 -> 오른쪽 곡면에서 더 학습이 잘될것 같음
- 파라메터학습과정 시각화 // 옵티마이저: SGD, 초기값: (w0,w1) = (-3.0,-1.0)
(1) 데이터정리
X = tf.concat([tf.ones(N,dtype=tf.float64).reshape(N,1),x],axis=1)
X
(2) 1ter돌려봄
net_mse = tf.keras.Sequential()
net_mse.add(tf.keras.layers.Dense(1,use_bias=False,activation='sigmoid'))
net_mse.compile(optimizer=tf.optimizers.SGD(0.1),loss=mseloss_fn)
net_mse.fit(X,y,epochs=1,batch_size=N)
net_bce = tf.keras.Sequential()
net_bce.add(tf.keras.layers.Dense(1,use_bias=False,activation='sigmoid'))
net_bce.compile(optimizer=tf.optimizers.SGD(0.1),loss=bceloss_fn)
net_bce.fit(X,y,epochs=1,batch_size=N)
net_mse.get_weights(), net_bce.get_weights()
net_mse.set_weights([tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)])
net_bce.set_weights([tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)])
net_mse.get_weights(), net_bce.get_weights()
(4) 학습과정기록: 15에폭마다 기록
What_mse = tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)
What_bce = tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)
for k in range(29):
net_mse.fit(X,y,epochs=15,batch_size=N,verbose=0)
net_bce.fit(X,y,epochs=15,batch_size=N,verbose=0)
What_mse = tf.concat([What_mse,net_mse.weights[0]],axis=1)
What_bce = tf.concat([What_bce,net_bce.weights[0]],axis=1)
(5) 시각화
from matplotlib import animation
plt.rcParams["animation.html"] = "jshtml"
fig = plt.figure()
fig.set_figwidth(6)
fig.set_figheight(6)
fig.suptitle("SGD, Winit=(-3,-1)")
ax1=fig.add_subplot(2,2,1,projection='3d')
ax2=fig.add_subplot(2,2,2,projection='3d')
ax1.elev=15;ax2.elev=15;ax1.azim=75;ax2.azim=75
ax3=fig.add_subplot(2,2,3)
ax4=fig.add_subplot(2,2,4)
ax1.scatter(w0,w1,loss1,s=0.1);ax1.scatter(-1,5,loss_fn1(-1,5),color='red',marker='*',s=200)
ax2.scatter(w0,w1,loss2,s=0.1);ax2.scatter(-1,5,loss_fn2(-1,5),color='red',marker='*',s=200)
ax3.plot(x,y,','); ax3.plot(x,v,'--r');
line3, = ax3.plot(x,1/(1+np.exp(-X@What_mse[:,0])),'--b')
ax4.plot(x,y,','); ax4.plot(x,v,'--r')
line4, = ax4.plot(x,1/(1+np.exp(-X@What_bce[:,0])),'--b')
def animate(i):
_w0_mse,_w1_mse = What_mse[:,i]
_w0_bce,_w1_bce = What_bce[:,i]
ax1.scatter(_w0_mse, _w1_mse, loss_fn1(_w0_mse, _w1_mse),color='gray')
ax2.scatter(_w0_bce, _w1_bce, loss_fn2(_w0_bce, _w1_bce),color='gray')
line3.set_ydata(1/(1+np.exp(-X@What_mse[:,i])))
line4.set_ydata(1/(1+np.exp(-X@What_bce[:,i])))
ani = animation.FuncAnimation(fig, animate, frames=30)
plt.close()
ani
</input>
X = tf.concat([tf.ones(N,dtype=tf.float64).reshape(N,1),x],axis=1)
X
(2) 1ter돌려봄
net_mse = tf.keras.Sequential()
net_mse.add(tf.keras.layers.Dense(1,use_bias=False,activation='sigmoid'))
net_mse.compile(optimizer=tf.optimizers.Adam(0.1),loss=mseloss_fn)
net_mse.fit(X,y,epochs=1,batch_size=N)
net_bce = tf.keras.Sequential()
net_bce.add(tf.keras.layers.Dense(1,use_bias=False,activation='sigmoid'))
net_bce.compile(optimizer=tf.optimizers.Adam(0.1),loss=bceloss_fn)
net_bce.fit(X,y,epochs=1,batch_size=N)
net_mse.get_weights(), net_bce.get_weights()
net_mse.set_weights([tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)])
net_bce.set_weights([tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)])
net_mse.get_weights(), net_bce.get_weights()
(4) 학습과정기록: 15에폭마다 기록
What_mse = tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)
What_bce = tnp.array([[-3.0 ],[ -1.0]],dtype=tf.float32)
for k in range(29):
net_mse.fit(X,y,epochs=15,batch_size=N,verbose=0)
net_bce.fit(X,y,epochs=15,batch_size=N,verbose=0)
What_mse = tf.concat([What_mse,net_mse.weights[0]],axis=1)
What_bce = tf.concat([What_bce,net_bce.weights[0]],axis=1)
(5) 시각화
from matplotlib import animation
plt.rcParams["animation.html"] = "jshtml"
fig = plt.figure()
fig.set_figwidth(6)
fig.set_figheight(6)
fig.suptitle("Adam, Winit=(-3,-1)")
ax1=fig.add_subplot(2,2,1,projection='3d')
ax2=fig.add_subplot(2,2,2,projection='3d')
ax1.elev=15;ax2.elev=15;ax1.azim=75;ax2.azim=75
ax3=fig.add_subplot(2,2,3)
ax4=fig.add_subplot(2,2,4)
ax1.scatter(w0,w1,loss1,s=0.1);ax1.scatter(-1,5,loss_fn1(-1,5),color='red',marker='*',s=200)
ax2.scatter(w0,w1,loss2,s=0.1);ax2.scatter(-1,5,loss_fn2(-1,5),color='red',marker='*',s=200)
ax3.plot(x,y,','); ax3.plot(x,v,'--r');
line3, = ax3.plot(x,1/(1+np.exp(-X@What_mse[:,0])),'--b')
ax4.plot(x,y,','); ax4.plot(x,v,'--r')
line4, = ax4.plot(x,1/(1+np.exp(-X@What_bce[:,0])),'--b')
def animate(i):
_w0_mse,_w1_mse = What_mse[:,i]
_w0_bce,_w1_bce = What_bce[:,i]
ax1.scatter(_w0_mse, _w1_mse, loss_fn1(_w0_mse, _w1_mse),color='gray')
ax2.scatter(_w0_bce, _w1_bce, loss_fn2(_w0_bce, _w1_bce),color='gray')
line3.set_ydata(1/(1+np.exp(-X@What_mse[:,i])))
line4.set_ydata(1/(1+np.exp(-X@What_bce[:,i])))
ani = animation.FuncAnimation(fig, animate, frames=30)
plt.close()
ani
</input>
X = tf.concat([tf.ones(N,dtype=tf.float64).reshape(N,1),x],axis=1)
X
(2) 1ter돌려봄
net_mse = tf.keras.Sequential()
net_mse.add(tf.keras.layers.Dense(1,use_bias=False,activation='sigmoid'))
net_mse.compile(optimizer=tf.optimizers.Adam(0.1),loss=mseloss_fn)
net_mse.fit(X,y,epochs=1,batch_size=N)
net_bce = tf.keras.Sequential()
net_bce.add(tf.keras.layers.Dense(1,use_bias=False,activation='sigmoid'))
net_bce.compile(optimizer=tf.optimizers.Adam(0.1),loss=bceloss_fn)
net_bce.fit(X,y,epochs=1,batch_size=N)
net_mse.get_weights(), net_bce.get_weights()
net_mse.set_weights([tnp.array([[-10.0 ],[ -1.0]],dtype=tf.float32)])
net_bce.set_weights([tnp.array([[-10.0 ],[ -1.0]],dtype=tf.float32)])
net_mse.get_weights(), net_bce.get_weights()
(4) 학습과정기록: 15에폭마다 기록
What_mse = tnp.array([[-10.0 ],[ -1.0]],dtype=tf.float32)
What_bce = tnp.array([[-10.0 ],[ -1.0]],dtype=tf.float32)
for k in range(29):
net_mse.fit(X,y,epochs=15,batch_size=N,verbose=0)
net_bce.fit(X,y,epochs=15,batch_size=N,verbose=0)
What_mse = tf.concat([What_mse,net_mse.weights[0]],axis=1)
What_bce = tf.concat([What_bce,net_bce.weights[0]],axis=1)
(5) 시각화
from matplotlib import animation
plt.rcParams["animation.html"] = "jshtml"
fig = plt.figure()
fig.set_figwidth(6)
fig.set_figheight(6)
fig.suptitle("Adam, Winit=(-10,-1)")
ax1=fig.add_subplot(2,2,1,projection='3d')
ax2=fig.add_subplot(2,2,2,projection='3d')
ax1.elev=15;ax2.elev=15;ax1.azim=75;ax2.azim=75
ax3=fig.add_subplot(2,2,3)
ax4=fig.add_subplot(2,2,4)
ax1.scatter(w0,w1,loss1,s=0.1);ax1.scatter(-1,5,loss_fn1(-1,5),color='red',marker='*',s=200)
ax2.scatter(w0,w1,loss2,s=0.1);ax2.scatter(-1,5,loss_fn2(-1,5),color='red',marker='*',s=200)
ax3.plot(x,y,','); ax3.plot(x,v,'--r');
line3, = ax3.plot(x,1/(1+np.exp(-X@What_mse[:,0])),'--b')
ax4.plot(x,y,','); ax4.plot(x,v,'--r')
line4, = ax4.plot(x,1/(1+np.exp(-X@What_bce[:,0])),'--b')
def animate(i):
_w0_mse,_w1_mse = What_mse[:,i]
_w0_bce,_w1_bce = What_bce[:,i]
ax1.scatter(_w0_mse, _w1_mse, loss_fn1(_w0_mse, _w1_mse),color='gray')
ax2.scatter(_w0_bce, _w1_bce, loss_fn2(_w0_bce, _w1_bce),color='gray')
line3.set_ydata(1/(1+np.exp(-X@What_mse[:,i])))
line4.set_ydata(1/(1+np.exp(-X@What_bce[:,i])))
ani = animation.FuncAnimation(fig, animate, frames=30)
plt.close()
ani
</input>
- 아무리 아담이라고 해도 이건 힘듬
- discussion
- mse_loss는 경우에 따라서 엄청 수렴속도가 느릴수도 있음.
- 근본적인 문제점: mse_loss일 경우 loss function의 곡면이 예쁘지 않음. (전문용어로 convex가 아니라고 말함)
- 좋은 옵티마지어를 이용하면 mse_loss일 경우에도 수렴속도를 올릴 수 있음 (학습과정 시각화예시2).그렇지만 이는 근본적인 해결책은 아님. (학습과정 시각화예시3)
- 요약: 왜 logistic regression에서 mse loss를 쓰면 안되는가?
- mse loss를 사용하면 손실함수가 convex하지 않으니까!
- 그리고 bce loss를 사용하면 손실함수가 convex하니까!