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Out of curiosity I am trying to fit neural network with rectified linear units to polynomial functions.
For example, I would like to see how easy (or difficult) it is for a neural network to come up with an approximation for the function f(x) = x^2 + x. The following code should be able to do it, but seems to not learn anything. When I run
using Base.Iterators: repeated
ENV["JULIA_CUDA_SILENT"] = true
using Flux
using Flux: throttle
using Random
f(x) = x^2 + x
x_train = shuffle(1:1000)
y_train = f.(x_train)
x_train = hcat(x_train...)
m = Chain(
Dense(1, 45, relu),
Dense(45, 45, relu),
Dense(45, 1),
softmax
)
function loss(x, y)
Flux.mse(m(x), y)
end
evalcb = () -> #show(loss(x_train, y_train))
opt = ADAM()
#show loss(x_train, y_train)
dataset = repeated((x_train, y_train), 50)
Flux.train!(loss, params(m), dataset, opt, cb = throttle(evalcb, 10))
println("Training finished")
#show m([20])
it returns
loss(x_train, y_train) = 2.0100101f14
loss(x_train, y_train) = 2.0100101f14
loss(x_train, y_train) = 2.0100101f14
Training finished
m([20]) = Float32[1.0]
Anyone here sees how I could make the network fit f(x) = x^2 + x?
There seem to be couple of things wrong with your trial that have mostly to do with how you use your optimizer and treat your input -- nothing wrong with Julia or Flux. Provided solution does learn, but is by no means optimal.
It makes no sense to have softmax output activation on a regression problem. Softmax is used in classification problems where the output(s) of your model represent probabilities and therefore should be on the interval (0,1). It is clear your polynomial has values outside this interval. It is usual to have linear output activation in regression problems like these. This means in Flux no output activation should be defined on the output layer.
The shape of your data matters. train! computes gradients for loss(d...) where d is a batch in your data. In your case a minibatch consists of 1000 samples, and this same batch is repeated 50 times. Neural nets are often trained with smaller batches sizes, but a larger sample set. In the code I provided all batches consist of different data.
For training neural nets, in general, it is advised to normalize your input. Your input takes values from 1 to 1000. My example applies a simple linear transformation to get the input data in the right range.
Normalization can also apply to the output. If the outputs are large, this can result in (too) large gradients and weight updates. Another approach is to lower the learning rate a lot.
using Flux
using Flux: #epochs
using Random
normalize(x) = x/1000
function generate_data(n)
f(x) = x^2 + x
xs = reduce(hcat, rand(n)*1000)
ys = f.(xs)
(normalize(xs), normalize(ys))
end
batch_size = 32
num_batches = 10000
data_train = Iterators.repeated(generate_data(batch_size), num_batches)
data_test = generate_data(100)
model = Chain(Dense(1,40, relu), Dense(40,40, relu), Dense(40, 1))
loss(x,y) = Flux.mse(model(x), y)
opt = ADAM()
ps = Flux.params(model)
Flux.train!(loss, ps, data_train, opt , cb = () -> #show loss(data_test...))
I implemented a simple linear regression and I’m getting some poor results. Just wondering if these results are normal or I’m making some mistake.
I tried different optimizers and learning rates, I always get bad/poor results
Here is my code:
import torch
import torch.nn as nn
import numpy as np
import matplotlib.pyplot as plt
from torch.autograd import Variable
class LinearRegressionPytorch(nn.Module):
def __init__(self, input_dim=1, output_dim=1):
super(LinearRegressionPytorch, self).__init__()
self.linear = nn.Linear(input_dim, output_dim)
def forward(self,x):
x = x.view(x.size(0),-1)
y = self.linear(x)
return y
input_dim=1
output_dim = 1
if torch.cuda.is_available():
model = LinearRegressionPytorch(input_dim, output_dim).cuda()
else:
model = LinearRegressionPytorch(input_dim, output_dim)
criterium = nn.MSELoss()
l_rate =0.00001
optimizer = torch.optim.SGD(model.parameters(), lr=l_rate)
#optimizer = torch.optim.Adam(model.parameters(),lr=l_rate)
epochs = 100
#create data
x = np.random.uniform(0,10,size = 100) #np.linspace(0,10,100);
y = 6*x+5
mu = 0
sigma = 5
noise = np.random.normal(mu, sigma, len(y))
y_noise = y+noise
#pass it to pytorch
x_data = torch.from_numpy(x).float()
y_data = torch.from_numpy(y_noise).float()
if torch.cuda.is_available():
inputs = Variable(x_data).cuda()
target = Variable(y_data).cuda()
else:
inputs = Variable(x_data)
target = Variable(y_data)
for epoch in range(epochs):
#predict data
pred_y= model(inputs)
#compute loss
loss = criterium(pred_y, target)
#zero grad and optimization
optimizer.zero_grad()
loss.backward()
optimizer.step()
#if epoch % 50 == 0:
# print(f'epoch = {epoch}, loss = {loss.item()}')
#print params
for name, param in model.named_parameters():
if param.requires_grad:
print(name, param.data)
There are the poor results :
linear.weight tensor([[1.7374]], device='cuda:0')
linear.bias tensor([0.1815], device='cuda:0')
The results should be weight = 6 , bias = 5
Problem Solution
Actually your batch_size is problematic. If you have it set as one, your targetneeds the same shape as outputs (which you are, correctly, reshaping with view(-1, 1)).
Your loss should be defined like this:
loss = criterium(pred_y, target.view(-1, 1))
This network is correct
Results
Your results will not be bias=5 (yes, weight will go towards 6 indeed) as you are adding random noise to target (and as it's a single value for all your data points, only bias will be affected).
If you want bias equal to 5 remove addition of noise.
You should increase number of your epochs as well, as your data is quite small and network (linear regression in fact) is not really powerful. 10000 say should be fine and your loss should oscillate around 0 (if you change your noise to something sensible).
Noise
You are creating multiple gaussian distributions with different variations, hence your loss would be higher. Linear regression is unable to fit your data and find sensible bias (as the optimal slope is still approximately 6 for your noise, you may try to increase multiplication of 5 to 1000 and see what weight and bias will be learned).
Style (a little offtopic)
Please read documentation about PyTorch and keep your code up to date (e.g. Variable is deprecated in favor of Tensor and rightfully so).
This part of code:
x_data = torch.from_numpy(x).float()
y_data = torch.from_numpy(y_noise).float()
if torch.cuda.is_available():
inputs = Tensor(x_data).cuda()
target = Tensor(y_data).cuda()
else:
inputs = Tensor(x_data)
target = Tensor(y_data)
Could be written succinctly like this (without much thought):
inputs = torch.from_numpy(x).float()
target = torch.from_numpy(y_noise).float()
if torch.cuda.is_available():
inputs = inputs.cuda()
target = target.cuda()
I know deep learning has it's reputation for bad code and fatal practice, but please do not help spreading this approach.
I have mostly used ANNs for classification and only recently started to try them out for modeling continuous variables. As an exercise I generated a simple set of (x, y) pairs where y = x^2 and tried to train an ANN to learn this quadratic function.
The ANN model:
This ANN has 1 input node (ie. x), 2 hidden layers each with 2 nodes in each layer, and 1 output node. All four hidden nodes use the non-linear tanh activation function and the output node has no activation function (since it is regression).
The Data:
For the training set I randomly generated 100 numbers between (-20, 20) for x and computed y=x^2. For the testing set I randomly generated 100 numbers between (-30, 30) for x and also computed y=x^2. I then transformed all x so that they are centered around 0 and their min and max are approximately around -1.5 and 1.5. I also transformed all y similarly but made their min and max about -0.9 and 0.9. This way, all the data falls within that mid range of the tanh activation function and not way out at the extremes.
The Problem:
After training the ANN in Keras, I am seeing that only the right half of the polynomial function is being learned, and the left half is completely flat. Does anyone have any ideas why this may be happening? I tried playing around with different scaling options, as well as hidden layer specifications but no luck on that left side.
Thanks!
Attached is the code I used for everything and the image shows the plot of the scaled training x vs the predicted y. As you can see, only half of the parabola is recovered.
import numpy as np, pandas as pd
from keras.models import Sequential
from keras.layers import Dense
from keras.wrappers.scikit_learn import KerasRegressor
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline
import matplotlib.pyplot as plt
seed = 10
n = 100
X_train = np.random.uniform(-20, 20, n)
Y_train = X_train ** 2
X_test = np.random.uniform(-30, 30, n)
Y_test = X_test ** 2
#### Scale the data
x_cap = max(abs(np.array(list(X_train) + list(X_test))))
y_cap = max(abs(np.array(list(Y_train) + list(Y_test))))
x_mean = np.mean(np.array(list(X_train) + list(X_test)))
y_mean = np.mean(np.array(list(Y_train) + list(Y_test)))
X_train2 = (X_train-x_mean) / x_cap
X_test2 = (X_test-x_mean) / x_cap
Y_train2 = (Y_train-y_mean) / y_cap
Y_test2 = (Y_test-y_mean) / y_cap
X_train2 = X_train2 * (1.5 / max(X_train2))
Y_train2 = Y_train2 * (0.9 / max(Y_train2))
# define base model
def baseline_model1():
# create model
model1 = Sequential()
model1.add(Dense(2, input_dim=1, kernel_initializer='normal', activation='tanh'))
model1.add(Dense(2, input_dim=1, kernel_initializer='normal', activation='tanh'))
model1.add(Dense(1, kernel_initializer='normal'))
# Compile model
model1.compile(loss='mean_squared_error', optimizer='adam')
return model1
np.random.seed(seed)
estimator1 = KerasRegressor(build_fn=baseline_model1, epochs=100, batch_size=5, verbose=0)
estimator1.fit(X_train2, Y_train2)
prediction = estimator1.predict(X_train2)
plt.scatter(X_train2, prediction)
enter image description here
You should also consider adding more width to you hidden layer. I changed from 2 to 5 and got a very good fit. I also used more epochs as suggested from rvinas
Your network is very sensible to the initial parameters. The following will help:
Change your kernel_initializer to glorot_uniform. Your network is very small and glorot_uniform will work better in consonance with the tanh activations. Glorot uniform will encourage your weights to be initially within a more reasonable range (since it takes into account the fan-in and fan-out of each layer).
Train your model for more epochs (i.e. 1000).
I'm trying to convolve two 1D tensors in Keras.
I get two inputs from other models:
x - of length 100
ker - of length 5
I would like to get the 1D convolution of x using the kernel ker.
I wrote a Lambda layer to do it:
import tensorflow as tf
def convolve1d(x):
y = tf.nn.conv1d(value=x[0], filters=x[1], padding='VALID', stride=1)
return y
x = Input(shape=(100,))
ker = Input(shape=(5,))
y = Lambda(convolve1d)([x,ker])
model = Model([x,ker], [y])
I get the following error:
ValueError: Shape must be rank 4 but is rank 3 for 'lambda_67/conv1d/Conv2D' (op: 'Conv2D') with input shapes: [?,1,100], [1,?,5].
Can anyone help me understand how to fix it?
It was much harder than I expected because Keras and Tensorflow don't expect any batch dimension in the convolution kernel so I had to write the loop over the batch dimension myself, which requires to specify batch_shape instead of just shape in the Input layer. Here it is :
import numpy as np
import tensorflow as tf
import keras
from keras import backend as K
from keras import Input, Model
from keras.layers import Lambda
def convolve1d(x):
input, kernel = x
output_list = []
if K.image_data_format() == 'channels_last':
kernel = K.expand_dims(kernel, axis=-2)
else:
kernel = K.expand_dims(kernel, axis=0)
for i in range(batch_size): # Loop over batch dimension
output_temp = tf.nn.conv1d(value=input[i:i+1, :, :],
filters=kernel[i, :, :],
padding='VALID',
stride=1)
output_list.append(output_temp)
print(K.int_shape(output_temp))
return K.concatenate(output_list, axis=0)
batch_input_shape = (1, 100, 1)
batch_kernel_shape = (1, 5, 1)
x = Input(batch_shape=batch_input_shape)
ker = Input(batch_shape=batch_kernel_shape)
y = Lambda(convolve1d)([x,ker])
model = Model([x, ker], [y])
a = np.ones(batch_input_shape)
b = np.ones(batch_kernel_shape)
c = model.predict([a, b])
In the current state :
It doesn't work for inputs (x) with multiple channels.
If you provide several filters, you get as many outputs, each being the convolution of the input with the corresponding kernel.
From given code it is difficult to point out what you mean when you say
is it possible
But if what you mean is to merge two layers and feed merged layer to convulation, yes it is possible.
x = Input(shape=(100,))
ker = Input(shape=(5,))
merged = keras.layers.concatenate([x,ker], axis=-1)
y = K.conv1d(merged, 'same')
model = Model([x,ker], y)
EDIT:
#user2179331 thanks for clarifying your intention. Now you are using Lambda Class incorrectly, that is why the error message is showing.
But what you are trying to do can be achieved using keras.backend layers.
Though be noted that when using lower level layers you will lose some higher level abstraction. E.g when using keras.backend.conv1d you need to have input shape of (BATCH_SIZE,width, channels) and kernel with shape of (kernel_size,input_channels,output_channels). So in your case let as assume the x has channels of 1(input channels ==1) and y also have the same number of channels(output channels == 1).
So your code now can be refactored as follows
from keras import backend as K
def convolve1d(x,kernel):
y = K.conv1d(x,kernel, padding='valid', strides=1,data_format="channels_last")
return y
input_channels = 1
output_channels = 1
kernel_width = 5
input_width = 100
ker = K.variable(K.random_uniform([kernel_width,input_channels,output_channels]),K.floatx())
x = Input(shape=(input_width,input_channels)
y = convolve1d(x,ker)
I guess I have understood what you mean. Given the wrong example code below:
input_signal = Input(shape=(L), name='input_signal')
input_h = Input(shape=(N), name='input_h')
faded= Lambda(lambda x: tf.nn.conv1d(input, x))(input_h)
You want to convolute each signal vector with different fading coefficients vector.
The 'conv' operation in TensorFlow, etc. tf.nn.conv1d, only support a fixed value kernel. Therefore, the code above can not run as you want.
I have no idea, too. The code you given can run normally, however, it is too complex and not efficient. In my idea, another feasible but also inefficient way is to multiply with the Toeplitz matrix whose row vector is the shifted fading coefficients vector. When the signal vector is too long, the matrix will be extremely large.
I have a Scipy sparse CSR matrix created from sparse TF-IDF feature matrix in SVM-Light format. The number of features is huge and it is sparse so I have to use a SparseTensor or else it is too slow.
For example, number of features is 5, and a sample file can look like this:
0 4:1
1 1:3 3:4
0 5:1
0 2:1
After parsing, the training set looks like this:
trainX = <scipy CSR matrix>
trainY = np.array( [0,1,00] )
I have two important questions:
1) How I do convert this to a SparseTensor (sp_ids, sp_weights) efficiently so that I perform fast multiplication (W.X) using lookup: https://www.tensorflow.org/versions/master/api_docs/python/nn.html#embedding_lookup_sparse
2) How do I randomize the dataset at each epoch and recalculate sp_ids, sp_weights to so that I can feed (feed_dict) for the mini-batch gradient descent.
Example code on a simple model like logistic regression will be very appreciated. The graph will be like this:
# GRAPH
mul = tf.nn.embedding_lookup_sparse(W, X_sp_ids, X_sp_weights, combiner = "sum") # W.X
z = tf.add(mul, b) # W.X + b
cost_op = tf.reduce_sum(tf.nn.sigmoid_cross_entropy_with_logits(z, y_true)) # this already has built in sigmoid apply
train_op = tf.train.GradientDescentOptimizer(0.05).minimize(cost_op) # construct optimizer
predict_op = tf.nn.sigmoid(z) # sig(W.X + b)
I can answer the first part of your question.
def convert_sparse_matrix_to_sparse_tensor(X):
coo = X.tocoo()
indices = np.mat([coo.row, coo.col]).transpose()
return tf.SparseTensor(indices, coo.data, coo.shape)
First you convert the matrix to COO format. Then you extract the indices, values, and shape and pass those directly to the SparseTensor constructor.