I have a set of observations made of 10 features, each of these features being a real number in the interval (0,2). Say I wanted to train a simple neural network to classify whether the average of those features is above or below 1.0.
Unless I'm missing something, it should be enough with a two-layer network with one neuron on each layer. The activation functions would be a linear one (i.e. no activation function) on the first layer and a sigmoid on the output layer. An example of a NN with this architecture that would work is one that calculates the average on the first layer (i.e. all weights = 0.1 and bias=0) and asseses whether that is above or below 1.0 in the second layer (i.e. weight = 1.0 and bias = -1.0).
When I implement this using TensorFlow (see code below), I obviously get a very high accuracy quite quickly, but never get to 100% accuracy... I would like some help to understand conceptually why this is the case. I don't see why the backppropagation algorithm does not reach a set of optimal weights (may be this is related with the loss function I'm using, which has local minmums?). Also I would like to know whether a 100% accuracy is achievable if I use different activations and/or loss function.
import numpy as np
import tensorflow as tf
import matplotlib.pyplot as plt
X = [np.random.random(10)*2.0 for _ in range(10000)]
X = np.array(X)
y = X.mean(axis=1) >= 1.0
y = y.astype('int')
train_ratio = 0.8
train_len = int(X.shape[0]*0.8)
X_train, X_test = X[:train_len,:], X[train_len:,:]
y_train, y_test = y[:train_len], y[train_len:]
def create_classifier(lr = 0.001):
classifier = tf.keras.Sequential()
classifier.add(tf.keras.layers.Dense(units=1))
classifier.add(tf.keras.layers.Dense(units=1, activation='sigmoid'))#, input_shape=input_shape))
optimizer = tf.keras.optimizers.Adam(learning_rate=lr)
metrics=[tf.keras.metrics.BinaryAccuracy()],
classifier.compile(optimizer=optimizer, loss=tf.keras.losses.BinaryCrossentropy(from_logits=False), metrics=metrics)
return classifier
classifier = create_classifier(lr = 0.1)
history = classifier.fit(X_train, y_train, batch_size=1000, validation_split=0.1, epochs=2000)
Ignoring the fact that a neural network is an odd approach for this problem, and answering your specific question - it looks like your learning rate might be too high which could explain the fluctuations around the optimal point.
Related
As a learning exercise, I'm trying to use an LSTM model with the Keras framework to predict the stock market based on multiple data points. The size of my input array is roughly [5000, 100]. Based on other questions on this site and articles online, the approach seems fairly standard: put the data in a numpy array, scale it, reshape it to 3 dimensions for the LSTM, split it into train and test sections, and feed it through the model. Running only the training portion of the model, I am consistently getting loss scores around 400,000,000. This is not changed by altering the batch size, the number of epochs, the number of layers, replacing the normalization with dropout layers, changing the sizes of each layer, or using different optimizers and loss functions. Any idea why the loss is so high and what I can do to fix that? Attached is the code. All advice is greatly appreciated.
import pandas as pd
import numpy as np
import tensorflow as tf
from tensorflow.keras import layers, losses, optimizers, Model, preprocessing
from keras.utils import plot_model
from sklearn.preprocessing import MinMaxScaler
import matplotlib.pyplot as plt
scaler = MinMaxScaler(feature_range=(0, 1))
features_df = pd.read_csv("dataset.csv")
features_np = np.array(features_df)
features_np.astype(np.float64)
scaler.fit_transform(features_np)
num_features=features_np.shape[1]
features = np.reshape(features_np, (features_np.shape[0], 1, features_np.shape[1]))
labels_np = np.array(pd.read_csv("output.csv"))
scaler.fit_transform(labels_np)
test_in = features_np[int(features_np.shape[0] * 0.75):]
test_in = np.reshape(test_in, (test_in.shape[0], 1, test_in.shape[1]))
test_out = labels_np[int(labels_np.shape[0] * 0.75):]
test_out = np.reshape(test_out, (test_out.shape[0], 1, test_out.shape[1]))
inputs = layers.Input(shape=(1, features.shape[2]))
x = layers.LSTM(5000, return_sequences=True)(inputs)
lstm1 = layers.LSTM(1000, return_sequences=True)(x)
norm1 = layers.BatchNormalization()(lstm1)
lstm2 = layers.LSTM(1000, return_sequences=True)(norm1)
lstm3 = layers.LSTM(1000, return_sequences=True)(lstm2)
norm2 = layers.BatchNormalization()(lstm3)
lstm4 = layers.LSTM(1000, return_sequences=True)(norm2)
lstm5 = layers.LSTM(1000)(lstm4)
dense1 = layers.Dense(1000, activation='relu')(lstm5)
dense2 = layers.Dense(1000, activation='sigmoid')(dense1)
outputs = layers.Dense(2)(dense2)
model = Model(inputs=inputs, outputs=outputs)
model.compile(optimizer='adam', loss='mean_squared_error')
model.fit(features, labels_np, epochs=1, batch_size=4)
evaluate = model.evaluate(test_in, test_out, verbose=2)
While I have not solved the error, implementing the Sequential() model and using only two LSTM layers and a Dense layer changed the error: the training error is now very low while testing remains high. This now appears to be a (relatively) simple problem of overfitting rather than the more confusing error of high training loss. Hopefully, this helps anyone having a similar problem.
There are two things i notice and dont understand why you use them. First one is , dense2 layer with sigmoid activation. I dont think sigmoid activation is benefical to when we are trying to solve a regression problem. Can you change that to relu and see what happens. Second one is you have two dense layers. You did not specify that but i think you are predicting two values with same inputs. If you are trying to predict just one value, you should you should change that to
outputs = layers.Dense(1)(dense2)
I am trying to find out, how exactly does BatchNormalization layer behave in TensorFlow. I came up with the following piece of code which to the best of my knowledge should be a perfectly valid keras model, however the mean and variance of BatchNormalization doesn't appear to be updated.
From docs https://www.tensorflow.org/api_docs/python/tf/keras/layers/BatchNormalization
in the case of the BatchNormalization layer, setting trainable = False on the layer means that the layer will be subsequently run in inference mode (meaning that it will use the moving mean and the moving variance to normalize the current batch, rather than using the mean and variance of the current batch).
I expect the model to return a different value with each subsequent predict call.
What I see, however, are the exact same values returned 10 times.
Can anyone explain to me why does the BatchNormalization layer not update its internal values?
import tensorflow as tf
import numpy as np
if __name__ == '__main__':
np.random.seed(1)
x = np.random.randn(3, 5) * 5 + 0.3
bn = tf.keras.layers.BatchNormalization(trainable=False, epsilon=1e-9)
z = input = tf.keras.layers.Input([5])
z = bn(z)
model = tf.keras.Model(inputs=input, outputs=z)
for i in range(10):
print(x)
print(model.predict(x))
print()
I use TensorFlow 2.1.0
Okay, I found the mistake in my assumptions. The moving average is being updated during training not during inference as I thought. This makes perfect sense, as updating the moving averages during inference would likely result in an unstable production model (for example a long sequence of highly pathological input samples [e.g. such that their generating distribution differs drastically from the one on which the network was trained] could potentially bias the network and result in worse performance on valid input samples).
The trainable parameter is useful when you're fine-tuning a pretrained model and want to freeze some of the layers of the network even during training. Because when you call model.predict(x) (or even model(x) or model(x, training=False)), the layer automatically uses the moving averages instead of batch averages.
The code below demonstrates this clearly
import tensorflow as tf
import numpy as np
if __name__ == '__main__':
np.random.seed(1)
x = np.random.randn(10, 5) * 5 + 0.3
z = input = tf.keras.layers.Input([5])
z = tf.keras.layers.BatchNormalization(trainable=True, epsilon=1e-9, momentum=0.99)(z)
model = tf.keras.Model(inputs=input, outputs=z)
# a dummy loss function
model.compile(loss=lambda x, y: (x - y) ** 2)
# a dummy fit just to update the batchnorm moving averages
model.fit(x, x, batch_size=3, epochs=10)
# first predict uses the moving averages from training
pred = model(x).numpy()
print(pred.mean(axis=0))
print(pred.var(axis=0))
print()
# outputs the same thing as previous predict
pred = model(x).numpy()
print(pred.mean(axis=0))
print(pred.var(axis=0))
print()
# here calling the model with training=True results in update of moving averages
# furthermore, it uses the batch mean and variance as in training,
# so the result is very different
pred = model(x, training=True).numpy()
print(pred.mean(axis=0))
print(pred.var(axis=0))
print()
# here we see again that the moving averages are used but they differ slightly after
# the previous call, as expected
pred = model(x).numpy()
print(pred.mean(axis=0))
print(pred.var(axis=0))
print()
In the end, I found that the documentation (https://www.tensorflow.org/api_docs/python/tf/keras/layers/BatchNormalization) mentions this:
When performing inference using a model containing batch normalization, it is generally (though not always) desirable to use accumulated statistics rather than mini-batch statistics. This is accomplished by passing training=False when calling the model, or using model.predict.
Hopefully this will help someone with similar misunderstanding in the future.
I trained a simple fully connected network on CIFAR-10 dataset:
import torch
import torch.nn as nn
import torch.nn.functional as F
import torchvision
import torchvision.transforms as transforms
class Net(nn.Module):
def __init__(self):
super(Net, self).__init__()
self.fc1 = nn.Linear(3*32*32, 300, bias=False)
self.fc2 = nn.Linear(300, 10, bias=False)
def forward(self, x):
x = x.reshape(250, -1)
self.x2 = F.relu(self.fc1(x))
x = self.fc2(self.x2)
return x
def train():
# The output of torchvision datasets are PILImage images of range [0, 1].
transform = transforms.Compose([transforms.ToTensor()])
trainset = torchvision.datasets.CIFAR10(root='./data', train=True, download=True, transform=transform)
trainloader = torch.utils.data.DataLoader(trainset, batch_size=250, shuffle=True, num_workers=4)
testset = torchvision.datasets.CIFAR10(root='./data', train=False, download=True, transform=transform)
testloader = torch.utils.data.DataLoader(testset, batch_size=args.bs, shuffle=False, num_workers=4)
net = Net()
criterion = nn.CrossEntropyLoss()
optimizer = torch.optim.SGD(net.parameters(), lr=0.02, momentum=0.9, weight_decay=0.0001)
for epoch in range(20):
correct = 0
total = 0
for data in trainloader:
inputs, labels = data
outputs = net(inputs)
loss = criterion(outputs, labels)
optimizer.zero_grad()
loss.backward()
optimizer.step()
_, predicted = torch.max(outputs.data, 1)
total += labels.size(0)
correct += (predicted == labels).sum().item()
acc = 100. * correct / total
This network gets to ~50% test accuracy with the parameters specified, after 20 epochs.
Note that I didn't do any whitening of the inputs (no per channel mean subtraction)
Next I scaled up the model inputs by 255, by replacing outputs = net(inputs) with outputs = net(inputs*255). After this change, the network no longer converges. I looked at the gradients and they seem to grow explosively after just a few iterations, leading to all model outputs being zero. I'd like to understand why this is happening.
Also, I tried scaling down the learning rate by 255. This helps, but the network only gets to ~43% accuracy. Again, I don't understand why this helps, and more importantly why the accuracy is still degraded compared to the original settings.
EDIT: forgot to mention that I don't use biases in this network.
EDIT2: I can recover the original accuracy if I scale down the initial weights in both layers by 255 (in addition to scaling down the learning rate). I also tried to scale down the initial weights only in the first layer, but the network had trouble learning (even when I did scale down the learning rate in both layers). Then I tried scaling down the learning rate only in the first layer - this also didn't help. Finally I tried reducing learning rate in both layer even more (by 255*255) and this suddenly worked. This does not make sense to me - scaling down the initial weights by the same factor the inputs have been scaled up should have completely eliminated any difference from the original network, the input to the second layer is identical. At that point the learning rate should be scaled down in the first layer only, but in practice both layers need significantly lower learning rate...
Scaling up the inputs will lead to exploding gradients because of a few observations:
The learning rate is common to all the weights in a given update step.
Hence, the same scaling factor (ie: the learning rate) is applied to a given weight's cost derivative regardless of it's magnitude, so large and small weights get updated by the same scale.
When the loss landscape is highly erratic, this leads to exploding gradients.(like a snowball effect, one overshot update - in say, the axis of one particular weight - causes another in the opposite direction in the next update which overshoots again and so on..)
The range of values of the pixels are 0 to 255, hence scaling the data by 255 will ensure all inputs are between 0 and 1 and hence more smooth convergence as all the gradients will be uniform with respect to the learning rate. But here you scaled the learning rate which adjusts some of the problems mentioned above but is not as effective as scaling the data itself. This reduces the learning rate hence making convergence time longer, that might be the reason why it reaches 43% at 20 epochs, maybe it needs more epochs..
Also:
CIFAR-10 is a significant step up from something like the MNIST dataset, hence, fully connected neural networks do not have the representation power needed to accurately predict these images. CNNs are the way to go for any image classification task beyond MNIST. ~50% accuracy is the max you can get with a fully connected neural network unfortunately.
Maybe decrease the learning rate by 1/255 ... just a guess
I'm currently working on data with rare binary outcome, i.e. the response vector contains mostly 0 and only a few 1 (approximately 1.5% ones). I've got about 20 continuous explanatory variables. I tried to train models using GBM, Random Forests, TensorFlow with Keras backend.
I observed a special behavior of the models, regardless which method I used:
The accuracy is high (~98%) but the model predicts probabilities for class "0" for all outcomes as ~98.5% and for class "1" ~1,5%.
How can I prevent this behavior?
I'm using RStudio. For Example a TF model with Keras would be:
model <- keras_model_sequential()
model %>%
layer_dense(units = 256, activation = "relu", input_shape = c(20)) %>%
layer_dense(units = 256, activation = "relu") %>%
layer_dense(units = 2, activation = "sigmoid")
parallel_model <- multi_gpu_model(model, gpus=2)
parallel_model %>% compile(
optimizer = "adam",
loss = "binary_crossentropy",
metrics = "binary_accuracy")
histroy <- parallel_model %>% fit(
x_train, y_train,
batch_size = 64,
epochs = 100,
class_weight = list("0"=1,"1"=70),
verbose = 1,
validation_split = 0.2
)
But my observation is not limited to TF. This makes my question more general. I'm not asking for specific adjustments for the model above, rather I'd like to discuss at what point all outcomes are assigned the same probability.
I can guess, the issue is connected to the loss-function.
I know there is no way to use AUC as loss functions, since it's not differentiable. If I test models with AUC with unknown data, the result is not better than random guessing.
I don't mind answers with code in Python, since this isn't a problem about coding rather than general behavior and algorithms.
When your problem has unbalanced classes, I suggest using SMOTE (on the training data only!!! never use smote on your testing data!!!) before training the model.
For example:
from imblearn.over_sampling import SMOTE
X_trn_balanced, Y_trn_balanced = SMOTE(random_state=1, ratio=1).fit_sample(X_trn, Y_trn)
#next fit the model with the balanced data
model.fit(X_trn_balanced, Y_trn_balanced )
In my (not so big) experience with AUC problems and rare positives, I see models with one class (not two). It's either "result is positive (1)" or "result is negative (0)".
Metrics like accuracy are useless for these problems, you should use AUC based metrics with big batch sizes.
For these problems, it doesn't matter whether the outcome probabilities are too little, as long as there is a difference between them. (Forests, GBM, etc. will indeed output these little values, but this is not a problem)
For neural networks, you can try to use class weights to increase the output probabilities. But notice that if you split the result in two separate classes (considering only one class should be positive), it doesn't matter if you use weights, because:
For the first class, low weights: predict all ones is good
For the second class, high weights: predict all zeros is good (weighted to very good)
So, as an initial solution, you can:
Use a 'softmax' activation (to guarantee your model will have only one correct output) and a 'categorical_crossentropy' loss.
(Or, preferrably) Use a model with only one class and keep 'sigmoid' with 'binary_crossentropy'.
I always work with the preferrable option above. In this case, if you use batch sizes that are big enough to contain one or two positive examples (batch size around 100 for you), weights may even be discarded. If the batch sizes are too little and many batches don't contain positive results, you may have too many weight updates towards plain zeros, which is bad.
You may also resample your data and, for instance, multiply by 10 the number of positive examples, so your batches contain more positives and training becomes easier.
Example of AUC metric to determine when training should end:
#in python - considering outputs with only one class
def aucMetric(true, pred):
true= K.flatten(true)
pred = K.flatten(pred)
totalCount = K.shape(true)[0]
values, indices = tf.nn.top_k(pred, k = totalCount)
sortedTrue = K.gather(true, indices)
tpCurve = K.cumsum(sortedTrue)
negatives = 1 - sortedTrue
auc = K.sum(tpCurve * negatives)
totalCount = K.cast(totalCount, K.floatx())
positiveCount = K.sum(true)
negativeCount = totalCount - positiveCount
totalArea = positiveCount * negativeCount
return auc / totalArea
I am new to tensorflow and am learning the basics at the moment so please bear with me.
My problem concerns strange non-convergent behaviour of neural networks when presented with the supposedly simple task of finding a regression function for a small training set consisting only of m = 100 data points {(x_1, y_1), (x_2, y_2),...,(x_100, y_100)}, where x_i and y_i are real numbers.
I first constructed a function that automatically generates a computational graph corresponding to a classical fully connected feedforward neural network:
import numpy as np
import tensorflow as tf
import matplotlib.pyplot as plt
import math
def neural_network_constructor(arch_list = [1,3,3,1],
act_func = tf.nn.sigmoid,
w_initializer = tf.contrib.layers.xavier_initializer(),
b_initializer = tf.zeros_initializer(),
loss_function = tf.losses.mean_squared_error,
training_method = tf.train.GradientDescentOptimizer(0.5)):
n_input = arch_list[0]
n_output = arch_list[-1]
X = tf.placeholder(dtype = tf.float32, shape = [None, n_input])
layer = tf.contrib.layers.fully_connected(
inputs = X,
num_outputs = arch_list[1],
activation_fn = act_func,
weights_initializer = w_initializer,
biases_initializer = b_initializer)
for N in arch_list[2:-1]:
layer = tf.contrib.layers.fully_connected(
inputs = layer,
num_outputs = N,
activation_fn = act_func,
weights_initializer = w_initializer,
biases_initializer = b_initializer)
Phi = tf.contrib.layers.fully_connected(
inputs = layer,
num_outputs = n_output,
activation_fn = tf.identity,
weights_initializer = w_initializer,
biases_initializer = b_initializer)
Y = tf.placeholder(tf.float32, [None, n_output])
loss = loss_function(Y, Phi)
train_step = training_method.minimize(loss)
return [X, Phi, Y, train_step]
With the above default values for the arguments, this function would construct a computational graph corresponding to a neural network with 1 input neuron, 2 hidden layers with 3 neurons each and 1 output neuron. The activation function is per default the sigmoid function. X corresponds to the input tensor, Y to the labels of the training data and Phi to the feedforward output of the neural network. The operation train_step performs one gradient-descent step when executed in the session environment.
So far, so good. If I now test a particular neural network (constructed with this function and the exact default values for the arguments given above) by making it learn a simple regression function for artificial data extracted from a sinewave, strange things happen:
Before training, the network seems to be a flat line. After 100.000 training iterations, it manages to partially learn the function, but only the part which is closer to 0. After this, it becomes flat again. Further training does not decrease the loss function anymore.
This get even stranger, when I take the exact same data set, but shift all x-values by adding 500:
Here, the network completely refuses to learn. I cannot understand why this is happening. I have tried changing the architecture of the network and its learning rate, but have observed similar effects: the closer the x-values of the data cloud are to the origin, the easier the network can learn. After a certain distance to the origin, learning stops completely. Changing the activation function from sigmoid to ReLu has only made things worse; here, the network tends to just converge to the average, no matter what position the data cloud is in.
Is there something wrong with my implementation of the neural-network-constructor? Or does this have something do do with initialization values? I have tried to get a deeper understanding of this problem now for quite a while and would greatly appreciate some advice. What could be the cause of this? All thoughts on why this behaviour is occuring are very much welcome!
Thanks,
Joker