Part 1¶
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using PyPlot
using Random
import Statistics.mean
import LinearAlgebra.dot
import LinearAlgebra.norm
import LinearAlgebra.I
import Printf.@printf
Random.seed!(4242);
Objective function is $$ f(w) = \frac{1}{2} w^2 $$ so the gradient is $$ f(w) = w $$
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x = randn(10); y = sqrt(0.95) * x + sqrt(0.05) * randn(10);
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scatter(x, y)
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# fit with linear regression
f = [x ones(10)];
w = y' / f';
xplot = collect(-1.5:0.1:1.5);
fplot = [xplot ones(length(xplot))]
plot(xplot, fplot * w'; color="red");
scatter(x, y);
xlim([-1.5, 1.5]);
ylim([-1.5, 1.5]);
training_error = round(mean((y - f * w').^2); digits=3)
println("training error: $(training_error)")
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# fit with degree 10 polynomial
f = hcat([x.^k for k = 0:10]...);
w = y' / f';
xplot = collect(-1.5:0.01:1.5);
fplot = hcat([xplot.^k for k = 0:10]...)
plot(xplot, fplot * w'; color="red");
scatter(x, y);
xlim([-1.5, 1.5]);
ylim([-1.5, 1.5]);
training_error = round(mean((y - f * w').^2); digits=3)
println("training error: $(training_error)")
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# make a test set and a validation set
test_x = randn(20); test_y = sqrt(0.9) * test_x + sqrt(0.1) * randn(20);
validation_x = randn(20); validation_y = sqrt(0.9) * validation_x + sqrt(0.1) * randn(20);
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# fit with linear regression
f = [x ones(10)];
w = y' / f';
xplot = collect(-1.5:0.1:1.5);
fplot = [xplot ones(length(xplot))]
plot(xplot, fplot * w'; color="red");
scatter(x, y);
scatter(test_x, test_y; color="orange");
xlim([-1.5, 1.5]);
ylim([-1.5, 1.5]);
training_error = round(mean((y - f * w').^2); digits=3)
println("training error: $(training_error)")
test_f = [test_x ones(length(test_x))];
test_error = round(mean((test_y - test_f * w').^2); digits=3)
println("test error: $(test_error)")
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# fit with degree 10 polynomial
f = hcat([x.^k / factorial(k) for k = 0:10]...);
w = y' / f';
xplot = collect(-1.5:0.01:1.5);
fplot = hcat([xplot.^k / factorial(k) for k = 0:10]...)
plot(xplot, fplot * w'; color="red");
scatter(x, y);
scatter(test_x, test_y; color="orange");
xlim([-1.5, 1.5]);
ylim([-1.5, 1.5]);
training_error = round(mean((y - f * w').^2); digits=3)
println("training error: $(training_error)")
test_f = hcat([test_x.^k / factorial(k) for k = 0:10]...)
test_error = round(mean((test_y - test_f * w').^2); digits=3)
@printf("test error: %f", test_error)
Part 2¶
Suppose that we use regularization such that for a polynomial of the form $$f_w(x) = \sum_{k = 0}^{10} \frac{w_k x^k}{k!}$$ the loss function is $$h(w) = \sum_{i=1}^N (f_w(x_i) - y_i)^2 + \sigma^2 \| w \|^2.$$ If $A$ is the matrix such that $$ A_{i,j} = \frac{x_i^j}{j!} $$ then equivalently $$h(w) = \left\| \left[\begin{array}{c} A \\ \sigma I \end{array} \right] w - \left[\begin{array}{c} y \\ 0 \end{array} \right] \right\|^2$$ so we can still solve this easily with linear regression.
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# fit with regularized degree 10 polynomial
sigma = 0.01
f = hcat([x.^k / factorial(k) for k = 0:10]...);
f_reg = vcat(f, sigma * Matrix{Float64}(I, 11, 11))
y_reg = vcat(y, zeros(11))
w = y_reg' / f_reg';
xplot = collect(-1.5:0.01:1.5);
fplot = hcat([xplot.^k / factorial(k) for k = 0:10]...)
plot(xplot, fplot * w'; color="red");
scatter(x, y);
scatter(test_x, test_y; color="orange");
xlim([-1.5, 1.5]);
ylim([-1.5, 1.5]);
training_error = round(mean((y - f * w').^2); digits=3)
println("training error: $(training_error)")
test_f = hcat([test_x.^k / factorial(k) for k = 0:10]...)
test_error = round(mean((test_y - test_f * w').^2); digits=3)
@printf("test error: %f", test_error)
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function regularized_fit_error(sigma)
test_x = randn(10000); test_y = sqrt(0.9) * test_x + sqrt(0.1) * randn(10000);
f = hcat([x.^k / factorial(k) for k = 0:10]...);
f_reg = vcat(f, sigma * Matrix{Float64}(I, 11, 11))
y_reg = vcat(y, zeros(11))
w = y_reg' / f_reg';
test_f = hcat([test_x.^k / factorial(k) for k = 0:10]...)
training_error = round(mean((y - f * w').^2); digits=3)
test_error = round(mean((test_y - test_f * w').^2); digits=3)
return (training_error, test_error)
end
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sigmas = 10 .^ (-3:0.1:2);
training_errors = zeros(length(sigmas));
test_errors = zeros(length(sigmas));
for i = 1:length(sigmas)
(training_errors[i], test_errors[i]) = regularized_fit_error(sigmas[i]);
end
loglog(sigmas, training_errors, label="training error");
loglog(sigmas, test_errors, label="test error");
xlabel("regularization coefficient");
ylabel("error");
legend();
Part 3¶
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f = hcat([x.^k / factorial(k) for k = 0:10]...);
validation_f = hcat([validation_x.^k / factorial(k) for k = 0:10]...);
test_f = hcat([test_x.^k / factorial(k) for k = 0:10]...);
w_sgd = zeros(11);
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models_at_epoch = []
for iepoch = 1:1000
for iters = 1:10
i = rand(1:10)
w_sgd -= 0.001 * (dot(f[i,:], w_sgd) - y[i]) * f[i,:];
end
val_loss = mean((validation_y - validation_f * w_sgd).^2);
push!(models_at_epoch, (val_loss, w_sgd));
end
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plot([val_loss for (val_loss, _) in models_at_epoch]);
xlabel("epochs");
ylabel("validation loss");
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(_, w_early_stopping) = minimum(models_at_epoch);
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xplot = collect(-1.5:0.01:1.5);
fplot = hcat([xplot.^k / factorial(k) for k = 0:10]...)
plot(xplot, fplot * w_early_stopping; color="red");
scatter(x, y);
scatter(test_x, test_y; color="orange");
xlim([-1.5, 1.5]);
ylim([-1.5, 1.5]);
training_error = round(mean((y - f * w_early_stopping).^2); digits=3)
println("training error: $(training_error)")
test_f = hcat([test_x.^k / factorial(k) for k = 0:10]...)
test_error = round(mean((test_y - test_f * w_early_stopping).^2); digits=3)
@printf("test error: %f", test_error)
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