Develop Reference

LSTM Autoencoder problems

TLDR:
Autoencoder underfits timeseries reconstruction and just predicts average value.
Question Set-up:
Here is a summary of my attempt at a sequence-to-sequence autoencoder. This image was taken from this paper: https://arxiv.org/pdf/1607.00148.pdf
Encoder: Standard LSTM layer. Input sequence is encoded in the final hidden state.
Decoder: LSTM Cell (I think!). Reconstruct the sequence one element at a time, starting with the last element x[N].
Decoder algorithm is as follows for a sequence of length N:
Get Decoder initial hidden state hs[N]: Just use encoder final hidden state.
Reconstruct last element in the sequence: x[N]= w.dot(hs[N]) + b.
Same pattern for other elements: x[i]= w.dot(hs[i]) + b
use x[i] and hs[i] as inputs to LSTMCell to get x[i-1] and hs[i-1]
Minimum Working Example:
Here is my implementation, starting with the encoder:
class SeqEncoderLSTM(nn.Module):
def __init__(self, n_features, latent_size):
super(SeqEncoderLSTM, self).__init__()
self.lstm = nn.LSTM(
n_features,
latent_size,
batch_first=True)
def forward(self, x):
_, hs = self.lstm(x)
return hs
Decoder class:
class SeqDecoderLSTM(nn.Module):
def __init__(self, emb_size, n_features):
super(SeqDecoderLSTM, self).__init__()
self.cell = nn.LSTMCell(n_features, emb_size)
self.dense = nn.Linear(emb_size, n_features)
def forward(self, hs_0, seq_len):
x = torch.tensor([])
# Final hidden and cell state from encoder
hs_i, cs_i = hs_0
# reconstruct first element with encoder output
x_i = self.dense(hs_i)
x = torch.cat([x, x_i])
# reconstruct remaining elements
for i in range(1, seq_len):
hs_i, cs_i = self.cell(x_i, (hs_i, cs_i))
x_i = self.dense(hs_i)
x = torch.cat([x, x_i])
return x
Bringing the two together:
class LSTMEncoderDecoder(nn.Module):
def __init__(self, n_features, emb_size):
super(LSTMEncoderDecoder, self).__init__()
self.n_features = n_features
self.hidden_size = emb_size
self.encoder = SeqEncoderLSTM(n_features, emb_size)
self.decoder = SeqDecoderLSTM(emb_size, n_features)
def forward(self, x):
seq_len = x.shape[1]
hs = self.encoder(x)
hs = tuple([h.squeeze(0) for h in hs])
out = self.decoder(hs, seq_len)
return out.unsqueeze(0)
And here's my training function:
def train_encoder(model, epochs, trainload, testload=None, criterion=nn.MSELoss(), optimizer=optim.Adam, lr=1e-6, reverse=False):
device = 'cuda' if torch.cuda.is_available() else 'cpu'
print(f'Training model on {device}')
model = model.to(device)
opt = optimizer(model.parameters(), lr)
train_loss = []
valid_loss = []
for e in tqdm(range(epochs)):
running_tl = 0
running_vl = 0
for x in trainload:
x = x.to(device).float()
opt.zero_grad()
x_hat = model(x)
if reverse:
x = torch.flip(x, [1])
loss = criterion(x_hat, x)
loss.backward()
opt.step()
running_tl += loss.item()
if testload is not None:
model.eval()
with torch.no_grad():
for x in testload:
x = x.to(device).float()
loss = criterion(model(x), x)
running_vl += loss.item()
valid_loss.append(running_vl / len(testload))
model.train()
train_loss.append(running_tl / len(trainload))
return train_loss, valid_loss
Data:
Large dataset of events scraped from the news (ICEWS). Various categories exist that describe each event. I initially one-hot encoded these variables, expanding the data to 274 dimensions. However, in order to debug the model, I've cut it down to a single sequence that is 14 timesteps long and only contains 5 variables. Here is the sequence I'm trying to overfit:
tensor([[0.5122, 0.0360, 0.7027, 0.0721, 0.1892],
[0.5177, 0.0833, 0.6574, 0.1204, 0.1389],
[0.4643, 0.0364, 0.6242, 0.1576, 0.1818],
[0.4375, 0.0133, 0.5733, 0.1867, 0.2267],
[0.4838, 0.0625, 0.6042, 0.1771, 0.1562],
[0.4804, 0.0175, 0.6798, 0.1053, 0.1974],
[0.5030, 0.0445, 0.6712, 0.1438, 0.1404],
[0.4987, 0.0490, 0.6699, 0.1536, 0.1275],
[0.4898, 0.0388, 0.6704, 0.1330, 0.1579],
[0.4711, 0.0390, 0.5877, 0.1532, 0.2201],
[0.4627, 0.0484, 0.5269, 0.1882, 0.2366],
[0.5043, 0.0807, 0.6646, 0.1429, 0.1118],
[0.4852, 0.0606, 0.6364, 0.1515, 0.1515],
[0.5279, 0.0629, 0.6886, 0.1514, 0.0971]], dtype=torch.float64)
And here is the custom Dataset class:
class TimeseriesDataSet(Dataset):
def __init__(self, data, window, n_features, overlap=0):
super().__init__()
if isinstance(data, (np.ndarray)):
data = torch.tensor(data)
elif isinstance(data, (pd.Series, pd.DataFrame)):
data = torch.tensor(data.copy().to_numpy())
else:
raise TypeError(f"Data should be ndarray, series or dataframe. Found {type(data)}.")
self.n_features = n_features
self.seqs = torch.split(data, window)
def __len__(self):
return len(self.seqs)
def __getitem__(self, idx):
try:
return self.seqs[idx].view(-1, self.n_features)
except TypeError:
raise TypeError("Dataset only accepts integer index/slices, not lists/arrays.")
Problem:
The model only learns the average, no matter how complex I make the model or now long I train it.
Predicted/Reconstruction:
Actual:
My research:
This problem is identical to the one discussed in this question: LSTM autoencoder always returns the average of the input sequence
The problem in that case ended up being that the objective function was averaging the target timeseries before calculating loss. This was due to some broadcasting errors because the author didn't have the right sized inputs to the objective function.
In my case, I do not see this being the issue. I have checked and double checked that all of my dimensions/sizes line up. I am at a loss.
Other Things I've Tried
I've tried this with varied sequence lengths from 7 timesteps to 100 time steps.
I've tried with varied number of variables in the time series. I've tried with univariate all the way to all 274 variables that the data contains.
I've tried with various reduction parameters on the nn.MSELoss module. The paper calls for sum, but I've tried both sum and mean. No difference.
The paper calls for reconstructing the sequence in reverse order (see graphic above). I have tried this method using the flipud on the original input (after training but before calculating loss). This makes no difference.
I tried making the model more complex by adding an extra LSTM layer in the encoder.
I've tried playing with the latent space. I've tried from 50% of the input number of features to 150%.
I've tried overfitting a single sequence (provided in the Data section above).
Question:
What is causing my model to predict the average and how do I fix it?

Okay, after some debugging I think I know the reasons.
TLDR
You try to predict next timestep value instead of difference between current timestep and the previous one
Your hidden_features number is too small making the model unable to fit even a single sample
Analysis
Code used
Let's start with the code (model is the same):
import seaborn as sns
import matplotlib.pyplot as plt
def get_data(subtract: bool = False):
# (1, 14, 5)
input_tensor = torch.tensor(
[
[0.5122, 0.0360, 0.7027, 0.0721, 0.1892],
[0.5177, 0.0833, 0.6574, 0.1204, 0.1389],
[0.4643, 0.0364, 0.6242, 0.1576, 0.1818],
[0.4375, 0.0133, 0.5733, 0.1867, 0.2267],
[0.4838, 0.0625, 0.6042, 0.1771, 0.1562],
[0.4804, 0.0175, 0.6798, 0.1053, 0.1974],
[0.5030, 0.0445, 0.6712, 0.1438, 0.1404],
[0.4987, 0.0490, 0.6699, 0.1536, 0.1275],
[0.4898, 0.0388, 0.6704, 0.1330, 0.1579],
[0.4711, 0.0390, 0.5877, 0.1532, 0.2201],
[0.4627, 0.0484, 0.5269, 0.1882, 0.2366],
[0.5043, 0.0807, 0.6646, 0.1429, 0.1118],
[0.4852, 0.0606, 0.6364, 0.1515, 0.1515],
[0.5279, 0.0629, 0.6886, 0.1514, 0.0971],
]
).unsqueeze(0)
if subtract:
initial_values = input_tensor[:, 0, :]
input_tensor -= torch.roll(input_tensor, 1, 1)
input_tensor[:, 0, :] = initial_values
return input_tensor
if __name__ == "__main__":
torch.manual_seed(0)
HIDDEN_SIZE = 10
SUBTRACT = False
input_tensor = get_data(SUBTRACT)
model = LSTMEncoderDecoder(input_tensor.shape[-1], HIDDEN_SIZE)
optimizer = torch.optim.Adam(model.parameters())
criterion = torch.nn.MSELoss()
for i in range(1000):
outputs = model(input_tensor)
loss = criterion(outputs, input_tensor)
loss.backward()
optimizer.step()
optimizer.zero_grad()
print(f"{i}: {loss}")
if loss < 1e-4:
break
# Plotting
sns.lineplot(data=outputs.detach().numpy().squeeze())
sns.lineplot(data=input_tensor.detach().numpy().squeeze())
plt.show()
What it does:
get_data either works on the data your provided if subtract=False or (if subtract=True) it subtracts value of the previous timestep from the current timestep
Rest of the code optimizes the model until 1e-4 loss reached (so we can compare how model's capacity and it's increase helps and what happens when we use the difference of timesteps instead of timesteps)
We will only vary HIDDEN_SIZE and SUBTRACT parameters!
NO SUBTRACT, SMALL MODEL
HIDDEN_SIZE=5
SUBTRACT=False
In this case we get a straight line. Model is unable to fit and grasp the phenomena presented in the data (hence flat lines you mentioned).
1000 iterations limit reached
SUBTRACT, SMALL MODEL
HIDDEN_SIZE=5
SUBTRACT=True
Targets are now far from flat lines, but model is unable to fit due to too small capacity.
1000 iterations limit reached
NO SUBTRACT, LARGER MODEL
HIDDEN_SIZE=100
SUBTRACT=False
It got a lot better and our target was hit after 942 steps. No more flat lines, model capacity seems quite fine (for this single example!)
SUBTRACT, LARGER MODEL
HIDDEN_SIZE=100
SUBTRACT=True
Although the graph does not look that pretty, we got to desired loss after only 215 iterations.
Finally
Usually use difference of timesteps instead of timesteps (or some other transformation, see here for more info about that). In other cases, neural network will try to simply... copy output from the previous step (as that's the easiest thing to do). Some minima will be found this way and going out of it will require more capacity.
When you use the difference between timesteps there is no way to "extrapolate" the trend from previous timestep; neural network has to learn how the function actually varies
Use larger model (for the whole dataset you should try something like 300 I think), but you can simply tune that one.
Don't use flipud. Use bidirectional LSTMs, in this way you can get info from forward and backward pass of LSTM (not to confuse with backprop!). This also should boost your score
Questions
Okay, question 1: You are saying that for variable x in the time
series, I should train the model to learn x[i] - x[i-1] rather than
the value of x[i]? Am I correctly interpreting?
Yes, exactly. Difference removes the urge of the neural network to base it's predictions on the past timestep too much (by simply getting last value and maybe changing it a little)
Question 2: You said my calculations for zero bottleneck were
incorrect. But, for example, let's say I'm using a simple dense
network as an auto encoder. Getting the right bottleneck indeed
depends on the data. But if you make the bottleneck the same size as
the input, you get the identity function.
Yes, assuming that there is no non-linearity involved which makes the thing harder (see here for similar case). In case of LSTMs there are non-linearites, that's one point.
Another one is that we are accumulating timesteps into single encoder state. So essentially we would have to accumulate timesteps identities into a single hidden and cell states which is highly unlikely.
One last point, depending on the length of sequence, LSTMs are prone to forgetting some of the least relevant information (that's what they were designed to do, not only to remember everything), hence even more unlikely.
Is num_features * num_timesteps not a bottle neck of the same size as
the input, and therefore shouldn't it facilitate the model learning
the identity?
It is, but it assumes you have num_timesteps for each data point, which is rarely the case, might be here. About the identity and why it is hard to do with non-linearities for the network it was answered above.
One last point, about identity functions; if they were actually easy to learn, ResNets architectures would be unlikely to succeed. Network could converge to identity and make "small fixes" to the output without it, which is not the case.
I'm curious about the statement : "always use difference of timesteps
instead of timesteps" It seem to have some normalizing effect by
bringing all the features closer together but I don't understand why
this is key ? Having a larger model seemed to be the solution and the
substract is just helping.
Key here was, indeed, increasing model capacity. Subtraction trick depends on the data really. Let's imagine an extreme situation:
We have 100 timesteps, single feature
Initial timestep value is 10000
Other timestep values vary by 1 at most
What the neural network would do (what is the easiest here)? It would, probably, discard this 1 or smaller change as noise and just predict 1000 for all of them (especially if some regularization is in place), as being off by 1/1000 is not much.
What if we subtract? Whole neural network loss is in the [0, 1] margin for each timestep instead of [0, 1001], hence it is more severe to be wrong.
And yes, it is connected to normalization in some sense come to think about it.

Attention is all you need, keeping only the encoding part for video classification

I am trying to modify a code that could find in the following link in such a way that the proposed Transformer model that is related to the paper: all you need is attention would keep only the Encoder part of the whole Transformer model. Furthermore, I would like to modify the input of the Network, instead of being a sequence of text to be a sequence of images (or better-extracted features of images) coming from a video. In a sense, I would like to figure out which frames are related to each other from my input and encode that info in an output embedding in the same way that is happening to the Transformers model.
The project as it is in the link provided is mainly performing sequence-sequence transformation. The input is text from one language and the output is text in another language. The main formation of the model is happening in the lines 386-463. Where the model is initialized and the compile of the Model is happening. For me I would like to do something like:
#414-416
self.encoder = SelfAttention(d_model, d_inner_hid, n_head, layers, dropout)
#self.decoder = Decoder(d_model, d_inner_hid, n_head, layers, dropout)
#self.target_layer = TimeDistributed(Dense(o_tokens.num(), use_bias=False))
#434-436
enc_output = self.encoder(src_emb, src_seq, active_layers=active_layers)
#dec_output = self.decoder(tgt_emb, tgt_seq, src_seq, enc_output, active_layers=active_layers)
#final_output = self.target_layer(dec_output)
Furthermore, since I would like to combine the output of the Encoder which is the output of MultiHeadAttention and PositionwiseFeedForward using an LSTM and a Dense layer which will tune the whole Encoding procedure using classification optimization. Therefore, I add when I define my model the following layers:
self.lstm = LSTM(units = 256, input_shape = (None, 256), return_sequences = False, dropout = 0.5)
self.fc1 = Dense(64, activation='relu', name = "dense_one")
self.fc2 = Dense(6, activation='sigmoid', name = "dense_two")
and then pass the output of the encoder, in line 434 using the following code:
enc_output = self.lstm(enc_output)
enc_output = self.fc1(enc_output)
enc_output = self.fc2(enc_output)
Now the video data that I would like to replace the text data provided with the Github code, have the following dimensionality: Nx10x256 where N is the number of samples, 10 is the number of frames and 256 the number of features for each frame. I have some difficulties to understand some parts of the code, in order to successfully, modified it to my needs. I guess, that now the Embedding layer is not necessary for me anymore since it is related to text classification and NLP.
Furthermore, I need to modify the input to 419-420 to be sth like:
src_seq_input = Input(shape=(None, 256,), dtype='float32') # source input related to video
tgt_seq_input = Input(shape=(6,), dtype='int32') # the target classification size (since I have 6 classes)
What other parts of the code do I need to skip or modify? What is the usefulness of the PosEncodingLayer that is used in the following line:
self.pos_emb = PosEncodingLayer(len_limit, d_emb) if self.src_loc_info else None
Is it needed in my case? Can I skip it?
After my modification in the code I noticed that when I run the code, I can check the loss function from the def get_loss(y_pred, y_true), however, in my case it is crucial to define a loss for the classification task that returns also the accuracy. How can I do so, with the provided code?
Edit:
I have to add that I treat my input as the output of the Embedding layer from the initial NLP code. Therefore, for me (in the version of code that functioned for me):
src_seq_input = Input(shape=(None, 256,), dtype='float32')
tgt_seq_input = Input(shape=(6,), dtype='int32')
src_seq = src_seq_input
#src_emb_ = self.i_word_emb(src_seq)
src_emb = src_seq
enc_output = self.encoder(src_emb, src_emb, active_layers=active_layers)
I treat src_emb as my input and completely ignore src_seq.
Edit:
The way that the loss is calculated is using the following code:
def get_loss(y_pred, y_true):
y_true = tf.cast(y_true, 'int32')
loss = tf.nn.sparse_softmax_cross_entropy_with_logits(labels=y_true, logits=y_pred)
mask = tf.cast(tf.not_equal(y_true, 0), 'float32')
loss = tf.reduce_sum(loss * mask, -1) / tf.reduce_sum(mask, -1)
loss = K.mean(loss)
return loss
loss = get_loss(enc_output, tgt_seq_input)
self.ppl = K.exp(loss)
Edit:
As it is the loss function (sparse_softmax_cross_entropy_with_logits) returns a loss score. Even if the whole procedure is about classification. How, can I further, tune my system to return also the accuracy?

I'm afraid this approach is not going to work.
Video data has massive dependence between adjacent frames, with each frame very similar to the last. There is also a weaker dependence on prior frames, because objects tend to continue to move relative to other objects in similar ways. Modern video formats use this redundancy to achieve high compression rates by modelling the motions.
This means that your network will have an extremely strong attention on the previous image. As you suggest, you could subsample frames several seconds apart to destroy much of the dependence on the previous frame, but if you did so I really wonder whether you would find structure at all in the result? Even if you feed it hand-coded features optimised for the purpose, there are are few general rules about which features will be in motion and which will not, so what structure can your attention network learn?
The problem of handling video is just radically different from handling sentences. Video has very complex elements (pictures) that are largely static over time and have locally predictable motions over a few frames in very simple ways. Text has simple elements (words) in a complex sentence structure with complex dependence extending over many words. These differences mean they require fundamentally different approaches.

Neural network versus random forest performance discrepancy

I want to run some experiments with neural networks using PyTorch, so I tried a simple one as a warm-up exercise, and I cannot quite make sense of the results.
The exercise attempts to predict the rating of 1000 TPTP problems from various statistics about the problems such as number of variables, maximum clause length etc. Data file https://github.com/russellw/ml/blob/master/test.csv is quite straightforward, 1000 rows, the final column is the rating, started off with some tens of input columns, with all the numbers scaled to the range 0-1, I progressively deleted features to see if the result still held, and it does, all the way down to one input column; the others are in previous versions in Git history.
I started off using separate training and test sets, but have set aside the test set for the moment, because the question about whether training performance generalizes to testing, doesn't arise until training performance has been obtained in the first place.
Simple linear regression on this data set has a mean squared error of about 0.14.
I implemented a simple feedforward neural network, code in https://github.com/russellw/ml/blob/master/test_nn.py and copied below, that after a couple hundred training epochs, also has an mean squared error of 0.14.
So I tried changing the number of hidden layers from 1 to 2 to 3, using a few different optimizers, tweaking the learning rate, switching the activation functions from relu to tanh to a mixture of both, increasing the number of epochs to 5000, increasing the number of hidden units to 1000. At this point, it should easily have had the ability to just memorize the entire data set. (At this point I'm not concerned about overfitting. I'm just trying to get the mean squared error on training data to be something other than 0.14.) Nothing made any difference. Still 0.14. I would say it must be stuck in a local optimum, but that's not supposed to happen when you've got a couple million weights; it's supposed to be practically impossible to be in a local optimum for all parameters simultaneously. And I do get slightly different sequences of numbers on each run. But it always converges to 0.14.
Now the obvious conclusion would be that 0.14 is as good as it gets for this problem, except that it stays the same even when the network has enough memory to just memorize all the data. But the clincher is that I also tried a random forest, https://github.com/russellw/ml/blob/master/test_rf.py
... and the random forest has a mean squared error of 0.01 on the original data set, degrading gracefully as features are deleted, still 0.05 on the data with just one feature.
Nowhere in the lore of machine learning is it said 'random forests vastly outperform neural nets', so I'm presumably doing something wrong, but I can't see what it is. Maybe it's something as simple as just missing a flag or something you need to set in PyTorch. I would appreciate it if someone could take a look.
import numpy as np
import pandas as pd
import torch
import torch.nn as nn
# data
df = pd.read_csv("test.csv")
print(df)
print()
# separate the output column
y_name = df.columns[-1]
y_df = df[y_name]
X_df = df.drop(y_name, axis=1)
# numpy arrays
X_ar = np.array(X_df, dtype=np.float32)
y_ar = np.array(y_df, dtype=np.float32)
# torch tensors
X_tensor = torch.from_numpy(X_ar)
y_tensor = torch.from_numpy(y_ar)
# hyperparameters
in_features = X_ar.shape[1]
hidden_size = 100
out_features = 1
epochs = 500
# model
class Net(nn.Module):
def __init__(self, hidden_size):
super(Net, self).__init__()
self.L0 = nn.Linear(in_features, hidden_size)
self.N0 = nn.ReLU()
self.L1 = nn.Linear(hidden_size, hidden_size)
self.N1 = nn.Tanh()
self.L2 = nn.Linear(hidden_size, hidden_size)
self.N2 = nn.ReLU()
self.L3 = nn.Linear(hidden_size, 1)
def forward(self, x):
x = self.L0(x)
x = self.N0(x)
x = self.L1(x)
x = self.N1(x)
x = self.L2(x)
x = self.N2(x)
x = self.L3(x)
return x
model = Net(hidden_size)
criterion = nn.MSELoss()
optimizer = torch.optim.Adam(model.parameters(), lr=0.1)
# train
print("training")
for epoch in range(1, epochs + 1):
# forward
output = model(X_tensor)
cost = criterion(output, y_tensor)
# backward
optimizer.zero_grad()
cost.backward()
optimizer.step()
# print progress
if epoch % (epochs // 10) == 0:
print(f"{epoch:6d} {cost.item():10f}")
print()
output = model(X_tensor)
cost = criterion(output, y_tensor)
print("mean squared error:", cost.item())

can you please print the shape of your input ?
I would say check those things first:
that your target y have the shape (-1, 1) I don't know if pytorch throws an Error in this case. you can use y.reshape(-1, 1) if it isn't 2 dim
your learning rate is high. usually when using Adam the default value is good enough or try simply to lower your learning rate. 0.1 is a high value for a learning rate to start with
place the optimizer.zero_grad at the first line inside the for loop
normalize/standardize your data ( this is usually good for NNs )
remove outliers in your data (my opinion: I think this can't affect Random forest so much but it can affect NNs badly)
use cross validation (maybe skorch can help you here. It's a scikit learn wrapper for pytorch and easy to use if you know keras)
Notice that Random forest regressor or any other regressor can outperform neural nets in some cases. There is some fields where neural nets are the heros like Image Classification or NLP but you need to be aware that a simple regression algorithm can outperform them. Usually when your data is not big enough.

PyTorch Binary Classification - same network structure, 'simpler' data, but worse performance?

To get to grips with PyTorch (and deep learning in general) I started by working through some basic classification examples. One such example was classifying a non-linear dataset created using sklearn (full code available as notebook here)
n_pts = 500
X, y = datasets.make_circles(n_samples=n_pts, random_state=123, noise=0.1, factor=0.2)
x_data = torch.FloatTensor(X)
y_data = torch.FloatTensor(y.reshape(500, 1))
This is then accurately classified using a pretty basic neural net
class Model(nn.Module):
def __init__(self, input_size, H1, output_size):
super().__init__()
self.linear = nn.Linear(input_size, H1)
self.linear2 = nn.Linear(H1, output_size)
def forward(self, x):
x = torch.sigmoid(self.linear(x))
x = torch.sigmoid(self.linear2(x))
return x
def predict(self, x):
pred = self.forward(x)
if pred >= 0.5:
return 1
else:
return 0
As I have an interest in health data I then decided to try and use the same network structure to classify some a basic real-world dataset. I took heart rate data for one patient from here, and altered it so all values > 91 would be labelled as anomalies (e.g. a 1 and everything <= 91 labelled a 0). This is completely arbitrary, but I just wanted to see how the classification would work. The complete notebook for this example is here.
What is not intuitive to me is why the first example reaches a loss of 0.0016 after 1,000 epochs, whereas the second example only reaches a loss of 0.4296 after 10,000 epochs
Perhaps I am being naive in thinking that the heart rate example would be much easier to classify. Any insights to help me understand why this is not what I am seeing would be great!

TL;DR
Your input data is not normalized.
use x_data = (x_data - x_data.mean()) / x_data.std()
increase the learning rate optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
You'll get
convergence in only 1000 iterations.
More details
The key difference between the two examples you have is that the data x in the first example is centered around (0, 0) and has very low variance.
On the other hand, the data in the second example is centered around 92 and has relatively large variance.
This initial bias in the data is not taken into account when you randomly initialize the weights which is done based on the assumption that the inputs are roughly normally distributed around zero.
It is almost impossible for the optimization process to compensate for this gross deviation - thus the model gets stuck in a sub-optimal solution.
Once you normalize the inputs, by subtracting the mean and dividing by the std, the optimization process becomes stable again and rapidly converges to a good solution.
For more details about input normalization and weights initialization, you can read section 2.2 in He et al Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification (ICCV 2015).
What if I cannot normalize the data?
If, for some reason, you cannot compute mean and std data in advance, you can still use nn.BatchNorm1d to estimate and normalize the data as part of the training process. For example
class Model(nn.Module):
def __init__(self, input_size, H1, output_size):
super().__init__()
self.bn = nn.BatchNorm1d(input_size) # adding batchnorm
self.linear = nn.Linear(input_size, H1)
self.linear2 = nn.Linear(H1, output_size)
def forward(self, x):
x = torch.sigmoid(self.linear(self.bn(x))) # batchnorm the input x
x = torch.sigmoid(self.linear2(x))
return x
This modification without any change to the input data, yields similar convergance after only 1000 epochs:
A minor comment
For numerical stability, it is better to use nn.BCEWithLogitsLoss instead of nn.BCELoss. For this end, you need to remove the torch.sigmoid from the forward() output, the sigmoid will be computed inside the loss.
See, for example, this thread regarding the related sigmoid + cross entropy loss for binary predictions.

Let's start first by understanding how neural networks work, neural networks observe patterns, hence the necessity for large datasets. In the case of the example, two what pattern you intend to find is when if HR < 91: label = 0, this if-condition can be represented by the formula, sigmoid((HR-91) * 1) , if you plug various values into the formula you can see you that all values < 91, label 0 and others label 1. I have inferred this formula and it could be anything as long as it gives the correct values.
Basically, we apply the formula wx+b, where x in our input data and we learn the values for w and b. Now initially the values are all random, so getting the b value from 1030131190 (a random value), to maybe 98 is fast, since the loss is great, the learning rate allows the values to jump fast. But once you reach 98, your loss is decreasing, and when you apply the learning rate, it takes it more time to reach closer to 91, hence the slow decrease in loss. As the values get closer, the steps taken are even slower.
This can be confirmed via the loss values, they are constantly decreasing, initially, the deceleration is higher, but then it becomes smaller. Your network is still learning but slowly.
Hence in deep learning, you use this method called stepped learning rate, wherewith the increase in epochs you decrease your learning rate so that your learning is faster

Why my one-filter convolutional neural network is unable to learn a simple gaussian kernel?

I was surprised that the deep learning algorithms I had implemented did not work, and I decided to create a very simple example, to understand the functioning of CNN better. Here is my attempt of constructing a small CNN for a very simple task, which provides unexpected results.
I have implemented a simple CNN with only one layer of one filter. I have created a dataset of 5000 samples, the inputs x being 256x256 simulated images, and the outputs y being the corresponding blurred images (y = signal.convolvded2d(x,gaussian_kernel,boundary='fill',mode='same')).
Thus, I would like my CNN to learn the convolutional filter which would transform the original image into its blurred version. In other words, I would like my CNN to recover the gaussian filter I used to create the blurred images. Note: As I want to 'imitate' the convolution process such as it is described in the mathematical framework, I am using a gaussian filter which has the same size as my images: 256x256.
It seems to me quite an easy task, and nonetheless, the CNN is unable to provide the results I would expect. Please find below the code of my training function and the results.
# Parameters
size_image = 256
normalization = 1
sigma = 7
n_train = 4900
ind_samples_training =np.linspace(1, n_train, n_train).astype(int)
nb_epochs = 5
minibatch_size = 5
learning_rate = np.logspace(-3,-5,nb_epochs)
tf.reset_default_graph()
tf.set_random_seed(1)
seed = 3
n_train = len(ind_samples_training)
costs = []
# Create Placeholders of the correct shape
X = tf.placeholder(tf.float64, shape=(None, size_image, size_image, 1), name = 'X')
Y_blur_true = tf.placeholder(tf.float64, shape=(None, size_image, size_image, 1), name = 'Y_true')
learning_rate_placeholder = tf.placeholder(tf.float32, shape=[])
# parameters to learn --should be an approximation of the gaussian filter
filter_to_learn = tf.get_variable('filter_to_learn',\
shape = [size_image,size_image,1,1],\
dtype = tf.float64,\
initializer = tf.contrib.layers.xavier_initializer(seed = 0),\
trainable = True)
# Forward propagation: Build the forward propagation in the tensorflow graph
Y_blur_hat = tf.nn.conv2d(X, filter_to_learn, strides = [1,1,1,1], padding = 'SAME')
# Cost function: Add cost function to tensorflow graph
cost = tf.losses.mean_squared_error(Y_blur_true,Y_blur_hat,weights=1.0)
# Backpropagation: Define the tensorflow optimizer. Use an AdamOptimizer that minimizes the cost.
opt_adam = tf.train.AdamOptimizer(learning_rate=learning_rate_placeholder)
update_ops = tf.get_collection(tf.GraphKeys.UPDATE_OPS)
with tf.control_dependencies(update_ops):
optimizer = opt_adam.minimize(cost)
# Initialize all the variables globally
init = tf.global_variables_initializer()
lr = learning_rate[0]
# Start the session to compute the tensorflow graph
with tf.Session() as sess:
# Run the initialization
sess.run(init)
# Do the training loop
for epoch in range(nb_epochs):
minibatch_cost = 0.
seed = seed + 1
permutation = list(np.random.permutation(n_train))
shuffled_ind_samples = np.array(ind_samples_training)[permutation]
# Learning rate update
if learning_rate.shape[0]>1:
lr = learning_rate[epoch]
nb_minibatches = int(np.ceil(n_train/minibatch_size))
for num_minibatch in range(nb_minibatches):
# Minibatch indices
ind_minibatch = shuffled_ind_samples[num_minibatch*minibatch_size:(num_minibatch+1)*minibatch_size]
# Loading of the original image (X) and the blurred image (Y)
minibatch_X, minibatch_Y = load_dataset_blur(ind_minibatch,size_image, normalization, sigma)
_ , temp_cost, filter_learnt = sess.run([optimizer,cost,filter_to_learn],\
feed_dict = {X:minibatch_X, Y_blur_true:minibatch_Y, learning_rate_placeholder: lr})
I have run the training on 5 epochs of 4900 samples, with a batch size equal to 5. The gaussian kernel has a variance of 7^2=49.
I have tried to initialize the filter to be learnt both with the xavier initiliazer method provided by tensorflow, and with the true values of the gaussian kernel we actually would like to learn. In both cases, the filter that is learnt results too different from the true gaussian one as it can be seen on the two images available at https://github.com/megalinier/Helsinki-project.

By examining the photos it seems like the network is learning OK, as the predicted image is not so far off the true label - for better results you can tweak some hyperparams but that is not the case.
I think what you are missing is the fact that different kernels can get quite similar results since it is a convolution.
Think about it, you are multiplying some matrix with another, and then summing all the results to create a new pixel. Now if the true label sum is 10, it could be a results of 2.5 + 2.5 + 2.5 + 2.5 and -10 + 10 + 10 + 0.
What I am trying to say, is that your network could be learning just fine, but you will get a different values in the conv kernel than the filter.

I think this would better serve as a comment as it's somewhat speculative, but it's too long...
Hard to say what exactly is wrong but there could be multiple culprits here. For one, squared error provides a weak signal in the case that target and prediction are already quite similar -- and while the xavier-initalized filter looks quite bad, the predicted (filtered) image isn't too far off the target. You could experiment with other metrics such as absolute error (e.g. 1-norm instead of 2-norm).
Second, adding regularization should help, i.e. add a weight penalty to the loss function to encourage the filter values to become small where they are not needed. As it is, what I suppose happens is: The random values in the filter average out to about 0, leading to a similar "filtering" effect as if they were actually all 0. As such, the learning algorithm doesn't have much incentive to actually pull them to 0. By adding a weight penalty, you provide this incentive.
Third, it could just be Adam messing up. It is known to provide "strange" non-optimal solutions in some very simple (e.g. convex) problems. Maybe try default Gradient Descent with learning rate decay (and possibly momentum).