# Neural network only converges when data cloud is close to 0

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
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,
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 occurring are very much welcome!

Thanks, Joker

• The network isn't learning..... Lower your learning rate .5 is too high and correspondingly increase the epochs and report back what you found... Happy learning – Aditya May 11 '18 at 1:41
• Hey Aditya, thank you for your comment. I have tried different learning rates (5,0.5,0.05,0.005,...) and have gone up to several houndred thousand iterations but results remain unchanged. – Joker123 May 11 '18 at 10:51

## 2 Answers

Gradient descent (on which neural networks rely for learning) is sensitive to feature scaling; you should probably normalize the x-values to start with.

In your particular shift-by-500 case, I'd guess that the optimal weights lie in a small range of near-zero numbers, and so the gradient descent has a hard time finding those appropriate weights. So, perhaps not a local minimum so much as a plateau, i.e. the sigmoids are getting saturated?
https://datascience.stackexchange.com/a/13221/55122

the input is simply not enough to correctly predict the output, the model can not learn the output conditional distribution P(y|x).

Either you have to add more features to the Naive NN model, e.g concat the previous x to the current x to predict the current y or use RNN like models to model the problem as
p(y_t|x_t,x_t-1,x_t-2,....x_t0)