First and foremost, we need synthetic data to work with. The data should exhibit some non-linear dependency. Let’s define it like this:
In python it will have the following shape:
np.random.seed(42)
X = np.random.normal(1, 4.5, 10000)
y = np.piecewise(X, [X < -2,(X >= -2) & (X < 2), X >= 2], [lambda X: 2*X + 5, lambda X: 7.3*np.sin(X), lambda X: -0.03*X**3 + 2]) + np.random.normal(0, 1, X.shape)
After visualization:
Since we are visualizing a 3D space, our neural network will only have 2 weights. This means the ANN will consist of a single hidden neuron. Implementing this in PyTorch is quite intuitive:
class ANN(nn.Module):
def __init__(self, input_size, N, output_size):
super().__init__()
self.net = nn.Sequential()
self.net.add_module(name='Layer_1', module=nn.Linear(input_size, N, bias=False))
self.net.add_module(name='Tanh',module=nn.Tanh())
self.net.add_module(name='Layer_2',module=nn.Linear(N, output_size, bias=False))
def forward(self, x):
return self.net(x)
Important! Don’t forget to turn off the biases in your layers, otherwise you’ll end up having x2 more parameters.
To build the error surface, we first need to create a grid of possible values for W1 and W2. Then, for each weight combination, we will update the parameters of the network and calculate the error:
W1, W2 = np.arange(-2, 2, 0.05), np.arange(-2, 2, 0.05)
LOSS = np.zeros((len(W1), len(W2)))
for i, w1 in enumerate(W1):
model.net._modules['Layer_1'].weight.data = torch.tensor([[w1]], dtype=torch.float32)
for j, w2 in enumerate(W2):
model.net._modules['Layer_2'].weight.data = torch.tensor([[w2]], dtype=torch.float32)
model.eval()
total_loss = 0
with torch.no_grad():
for x, y in test_loader:
preds = model(x.reshape(-1, 1))
total_loss += loss(preds, y).item()
LOSS[i, j] = total_loss / len(test_loader)
It may take some time. If you make the resolution of this grid too coarse (i.e., the step size between possible weight values), you might miss local minima and maxima. Remember how the learning rate is often schedule to decrease over time? When we do this, the absolute change in weight values can be as small as 1e-3 or less. A grid with a 0.5 step simply won’t capture these fine details of the error surface!
At this point, we don’t care at all about the quality of the trained model. However, we do want to pay attention to the learning rate, so let’s keep it between 1e-1 and 1e-2. We’ll simply collect the weight values and errors during the training process and store them in separate lists:
model = ANN(1,1,1)
epochs = 25
lr = 1e-2
optimizer = optim.SGD(model.parameters(),lr =lr)
model.net._modules['Layer_1'].weight.data = torch.tensor([[-1]], dtype=torch.float32)
model.net._modules['Layer_2'].weight.data = torch.tensor([[-1]], dtype=torch.float32)
errors, weights_1, weights_2 = [], [], []
model.eval()
with torch.no_grad():
total_loss = 0
for x, y in test_loader:
preds = model(x.reshape(-1,1))
error = loss(preds, y)
total_loss += error.item()
weights_1.append(model.net._modules['Layer_1'].weight.data.item())
weights_2.append(model.net._modules['Layer_2'].weight.data.item())
errors.append(total_loss / len(test_loader))
for epoch in tqdm(range(epochs)):
model.train()
for x, y in train_loader:
pred = model(x.reshape(-1,1))
error = loss(pred, y)
optimizer.zero_grad()
error.backward()
optimizer.step()
model.eval()
test_preds, true = [], []
with torch.no_grad():
total_loss = 0
for x, y in test_loader:
preds = model(x.reshape(-1,1))
error = loss(preds, y)
test_preds.append(preds)
true.append(y)
total_loss += error.item()
weights_1.append(model.net._modules['Layer_1'].weight.data.item())
weights_2.append(model.net._modules['Layer_2'].weight.data.item())
errors.append(total_loss / len(test_loader))
Finally, we can visualize the data we have collected using plotly. The plot will have two scenes: surface and SGD trajectory. One of the ways to do the first part is to create a figure with a plotly surface. After that we will style it a little by updating a layout.
The second part is as simple as it is — just use Scatter3d function and specify all three axes.
import plotly.graph_objects as go
import plotly.io as pio
plotly_template = pio.templates["plotly_dark"]
fig = go.Figure(data=[go.Surface(z=LOSS, x=W1, y=W2)])
fig.update_layout(
title='Loss Surface',
scene=dict(
xaxis_title='w1',
yaxis_title='w2',
zaxis_title='Loss',
aspectmode='manual',
aspectratio=dict(x=1, y=1, z=0.5),
xaxis=dict(showgrid=False),
yaxis=dict(showgrid=False),
zaxis=dict(showgrid=False),
),
width=800,
height=800
)
fig.add_trace(go.Scatter3d(x=weights_2, y=weights_1, z=errors,
mode='lines+markers',
line=dict(color='red', width=2),
marker=dict(size=4, color='yellow') ))
fig.show()
Running it in Google Colab or locally in Jupyter Notebook will allow you to investigate the error surface more closely. Honestly, I spent a buch of time just looking at this figure:)
I’d love to see you surfaces, so please feel free to share it in comments. I strongly believe that the more imperfect the surface is the more interesting it is to investigate it!
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