This script provides an illustration of constrained deep learning for 1-dimensional convex networks, along with the boundedness guarantees discussed for 1-dimensional convex functions. This example shows these steps:
- Generate a dataset with some sinusoidal additive noise contaminating a convex function signal.
- Prepare the dataset for custom training loop.
- Create a fully input convex neural network (FICNN) architecture.
- Train the FICNN using a custom training loop and apply projected gradient descent to guarantee convexity.
- Train a comparable MLP network without architectural or weight constraints.
- Compare the two networks and show convexity is violated in the unconstrained network.
- Compute guaranteed bounds of the FICNN over the interval and test the validity of these bounds with network inference on this domain.
- Show these bounds are violated for the unconstrained MLP.
First, take the convex function y=x^4 and uniformly randomly sample this over the interval [-1,1]. Add sinusoidal noise to create a dataset. You can change the number of random samples if you want to experiment.
numSamples = 512;
rng(0);
xTrain = -2*rand(numSamples,1)+1; % [-1 1]
xTrain = sort(xTrain);
tTrain = xTrain.^4 + 0.2*sin(20*xTrain);Visualize the data.
figure
plot(xTrain,tTrain,"k.")
grid on
xlabel("x")
To prepare the data for custom training loops, add the input and response to a minibatchqueue. You can do this by creating arrayDatastore objects and combining these into a single datastore using the combine function. Form the minibatchqueue with this combined datastore object.
xds = arrayDatastore(xTrain);
tds = arrayDatastore(tTrain);
cds = combine(xds,tds);
mbqTrain = minibatchqueue(cds,2, ...
"MiniBatchSize",numSamples, ...
"OutputAsDlarray",[1 1], ...
"MiniBatchFormat",["BC","BC"]);As discussed in AI Verification: Convex, fully input convex neural networks adhere to a specific class of neural network architectures with constraints applied to weights. In this proof of concept example, you build a simple FICNN using fully connected layers and ReLU activation functions. For more information on the architectural construction, see AI Verification: Convex.
inputSize = 1;
numHiddenUnits = [16 8 4 1];
ficnnet = buildConstrainedNetwork("fully-convex",inputSize,numHiddenUnits, ...
PositiveNonDecreasingActivation="relu")ficnnet =
dlnetwork with properties:
Layers: [14x1 nnet.cnn.layer.Layer]
Connections: [16x2 table]
Learnables: [14x3 table]
State: [0x3 table]
InputNames: {'input'}
OutputNames: {'add_4'}
Initialized: 1
View summary with summary.
You can view the network architecture in deepNetworkDesigner by setting the viewNetworkDND flag to true. Otherwise, plot the network graph.
viewNetworkDND = false;
if viewNetworkDND
deepNetworkDesigner(ficnnet) %#ok<UNRCH>
else
figure;
plot(ficnnet)
end
First, create a custom training options struct. For the trainConvexNetwork function, you can specify 4 hyperparameters: maxEpochs, initialLearnRate, decay, and lossMetric.
maxEpochs = 1200;
initialLearnRate = 0.05;
decay = 0.01;
lossMetric = "mae";Train the network with these options.
trained_ficnnet = trainConstrainedNetwork("fully-convex",ficnnet,mbqTrain, ...
MaxEpochs=maxEpochs, ...
InitialLearnRate=initialLearnRate, ...
Decay=decay, ...
LossMetric=lossMetric);
Evaluate the accuracy on the true underlying convex function from an independent random sampling from the interval [-1,1].
rng(1);
xTest = -2*rand(numSamples,1)+1; % [-1 1]
tTest = xTest.^4;
lossAgainstUnderlyingSignal = computeLoss(trained_ficnnet,xTest,tTest,lossMetric)lossAgainstUnderlyingSignal =
gpuArray single
0.0338
Create a multi-layer perception as a comparison, with the same depth as the FICNN defined previously. Increase the number of activations to give the network capacity to fit and roughly compensate for the lack of skip connections in terms of learnable parameter count.
mlpHiddenUnits = numHiddenUnits;
layers = featureInputLayer(inputSize,Normalization="none");
for ii=1:numel(mlpHiddenUnits)-1
layers = [layers ...
fullyConnectedLayer(mlpHiddenUnits(ii)*2) ...
tanhLayer];
end
% Add a final fully connected layer
layers = [layers ...
fullyConnectedLayer(mlpHiddenUnits(end))];Initialize the network.
rng(0);
mlpnet = dlnetwork(layers);Again, you can view the network architecture in deepNetworkDesigner by setting the viewNetworkDND flag to true. Otherwise, plot the network graph.
viewNetworkDND = false;
if viewNetworkDND
deepNetworkDesigner(mlpnet) %#ok<UNRCH>
else
figure;
plot(mlpnet)
end
Specify the training options and then train the network using the trainnet function.
options = trainingOptions("adam", ...
Plots="training-progress", ...
MaxEpochs=maxEpochs, ...
InitialLearnRate=initialLearnRate, ...
MiniBatchSize=numSamples, ...
LearnRateSchedule="piecewise", ...
LearnRateDropFactor=0.9, ...
LearnRateDropPeriod=30, ...
Shuffle="never");
trained_mlpnet = trainnet(mbqTrain,mlpnet,lossMetric,options); Iteration Epoch TimeElapsed LearnRate TrainingLoss
_________ _____ ___________ _________ ____________
1 1 00:00:00 0.05 0.26302
50 50 00:00:07 0.045 0.12781
100 100 00:00:11 0.03645 0.12262
150 150 00:00:14 0.032805 0.10849
200 200 00:00:18 0.026572 0.1102
250 250 00:00:22 0.021523 0.11806
300 300 00:00:25 0.019371 0.10301
350 350 00:00:29 0.015691 0.096023
400 400 00:00:32 0.012709 0.10675
450 450 00:00:36 0.011438 0.097555
500 500 00:00:39 0.0092651 0.094147
550 550 00:00:43 0.0075047 0.090284
600 600 00:00:46 0.0067543 0.088997
650 650 00:00:50 0.0054709 0.086944
700 700 00:00:53 0.0044315 0.085979
750 750 00:00:57 0.0039883 0.085362
800 800 00:01:00 0.0032305 0.08497
850 850 00:01:04 0.0026167 0.08464
900 900 00:01:07 0.0023551 0.084311
950 950 00:01:11 0.0019076 0.084135
1000 1000 00:01:15 0.0015452 0.083962
1050 1050 00:01:19 0.0013906 0.083793
1100 1100 00:01:22 0.0011264 0.08367
1150 1150 00:01:26 0.0009124 0.08356
1200 1200 00:01:29 0.00082116 0.083461
Training stopped: Max epochs completed
Evaluate the accuracy on an independent random sampling from the interval [-1,1]. Observe that the loss against the underlying monotonic signal here is higher as the network has fitted to the sinusoidal contamination.
lossAgainstUnderlyingSignal = computeLoss(trained_mlpnet,xTest,tTest,lossMetric)lossAgainstUnderlyingSignal = 0.0699
Compare the shape of the solution by sampling the training data and plotting this for both networks.
ficnnPred = predict(trained_ficnnet,xTrain);
mlpPred = predict(trained_mlpnet,xTrain);
figure;
plot(xTrain,ficnnPred,"r-.",LineWidth=2)
hold on
plot(xTrain,mlpPred,"b-.",LineWidth=2)
plot(xTrain,tTrain,"k.",MarkerSize=0.1)
xlabel("x")
legend("FICNN","MLP","Training Data")
It is visually evident that the MLP solution is not convex over the interval but the FICNN is convex, owing to its convex construction and constrained learning.
As discussed in AI Verification: Convex, fully input convex neural networks allows you to efficiently compute bounds over intervals by only evaluating the function, and its derivative, at the boundary. These boundedness guarantees play an important role in guaranteeing behavior on continuous regions of input space. Moreover, you only need to sample a small, finite number of test points. You can use more test points to tighten the bounds in different regions as needed.
numIntervals = 12;
intervalSet = linspace(-1,1,numIntervals+1);Collect the set of vertices into a cell array for use in the computeICNNBoundsOverNDGrid function. As this is 1-D, pass the interval as a cell into the function.
V = {intervalSet};Compute the network value at the vertices and also guaranteed output bounds over the subintervals specified by the set of vertices.
To refine the lower bound, set the name-value argument RefineLowerBounds=true. As described in AI Verification: Convex, this performs convex optimization over the intervals for which a minimum cannot be determined using the derivative conditions. If this option is set to false, NaN values are returned in the grid locations where the conditions on the derivatives of the network at the grid vertices are not sufficient to determine the minimum, as discussed in AI Verification: Convex.
Note that RefineLowerBounds=true requires Optimization Toolbox. If you do not have this toolbox, then the netMin variable will have NaN values in the positions of the intervals the minimum cannot be determined and the plots below will not show a lower bound over this interval.
[netMin,netMax] = convexNetworkOutputBounds(trained_ficnnet,V,RefineLowerBounds=true);Plot the bounds.
v = reshape([V{1}(1:end-1); V{1}(2:end)], 1, []);
yMax = reshape([netMax{1}'; netMax{1}'], 1, []);
yMin = reshape([netMin{1}'; netMin{1}'], 1, []);
figure
grid on
hold on
plot(v, yMin, "b-", "LineWidth", 1);
plot(v, yMax, "r-", "LineWidth", 1);
plot(xTrain,ficnnPred,"k-.")
hold off
xlabel("x")
legend("Network Lower Bound","Network Upper Bound",Location="northwest")
title("Guarantees of upper and lower bounds for FICNN network");
Repeat the analysis above for the MLP. Since this has not been trained with enforced convexity, the bounding guarantees do not hold and you observe violations of the bounds over the same interval.
[netMin,netMax,netPred] = convexNetworkOutputBounds(trained_mlpnet,V,RefineLowerBounds=true);Plot the bounds.
v = reshape([V{1}(1:end-1); V{1}(2:end)], 1, []);
yMax = reshape([netMax{1}'; netMax{1}'], 1, []);
yMin = reshape([netMin{1}'; netMin{1}'], 1, []);
figure; % Create a new figure
hold on;
plot(v, yMax, "r-", "LineWidth", 1);
plot(v, yMin, "b-", "LineWidth", 1);
plot(xTrain,mlpPred,"k-.")
hold off
xlabel("x");
legend("Network Lower Bound","Network Upper Bound",Location="northwest")
title("Violation of upper and lower bounds for MLP network");
grid on;
function loss = computeLoss(net,X,T,lossMetric)
Y = predict(net,X);
switch lossMetric
case "mse"
loss = mse(Y,T);
case "mae"
loss = mean(abs(Y-T));
end
endCopyright 2024 The MathWorks, Inc.