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2_01_machine_learning.jl
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2_01_machine_learning.jl
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using DeepPumas
using CairoMakie
using Distributions
using Random
set_theme!(deep_light())
#
# TABLE OF CONTENTS
#
#
# 0. BASIC SAMPLE PARAMETERS
#
# 1. A SIMPLE MACHINE LEARNING (ML) MODEL
#
# 1.1. Sample subjects with an obvious `true_function`
# 1.2. Model `true_function` with a linear regression
#
# 2. CAPTURING COMPLEX RELATIONSHIPS
#
# 2.1. Sample subjects with a more complex `true_function`
# 2.2. Exercise: Reason about using a linear regression to model the current `true_function`
# 2.3. Use a neural network (NN) to model `true_function`
#
# 3. BASIC UNDERFITTING AND OVERFITTING
#
# 3.1. Exercise: Investigate the impact of the number of fitting iterations in NNs
# (Hint: Train again the NN for few and for many iterations.)
# 3.2. Exercise: Reason about Exercise 2.2 again (that is, using a linear regression
# to model a quadratic relationship). Is the number of iterations relevant there?
# 3.3. The impact of the NN size
#
# 4. INSPECTION OF THE VALIDATION LOSS AND REGULARIZATION
#
# 4.1. Validation loss as a proxy for generalization performance
# 4.2. Regularization to prevent overfitting
# 4.3. Hyperparameter tuning
#
#
# 0. BASIC SAMPLE PARAMETERS
#
num_samples = 100
uniform = Uniform(-1, 1)
normal = Normal(0, 1)
σ = 0.25
#
# 1. A SIMPLE MACHINE LEARNING (ML) MODEL
#
# 1.1. Sample from an obvious `true_function`
# 1.2. Model `true_function` with a DeepPumas linear regression
#
# 1.1. Sample from an obvious `true_function`
true_function = x -> x
x = rand(uniform, 1, num_samples) # samples stored columnwise
ϵ = rand(normal, 1, num_samples) # samples stored columnwise
y = true_function.(x) + σ * ϵ
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend(; position=:rb);
fig
# 1.2. Model `true_function` with a linear regression
target = preprocess(x, y) # DeepPumas `target`
linreg = MLPDomain(1, (1, identity); bias = true) # DeepPumas multilayer perceptron
# y = a * x + b
fitted_linreg = fit(linreg, target; optim_alg = DeepPumas.BFGS())
coef(fitted_linreg) # `true_function` is y = x + noise (that is, a = 1 b = 0)
ŷ = fitted_linreg(x)
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
scatter!(vec(x), vec(ŷ); label = "prediction");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend(; position=:rb);
fig
#
# 2. CAPTURING COMPLEX RELATIONSHIPS
#
# 2.1. Sample from a more complex `true_function`
# 2.2. Exercise: Reason about using a linear regression to model the current `true_function`
# 2.3. Use a neural network (NN) to model `true_function`
#
# 2.1. Sample from a more complex `true_function`
true_function = x -> x^2
x = rand(uniform, 1, num_samples)
ϵ = rand(normal, 1, num_samples)
y = true_function.(x) + σ * ϵ
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend();
fig
# 2.2. Exercise: Reason about using a linear regression to model `true_function`
target = preprocess(x, y)
fitted_linreg =
fit(linreg, target; optim_alg = DeepPumas.BFGS(), optim_options = (; iterations = 50))
coef(fitted_linreg)
ŷ_ex22_50iter = fitted_linreg(x)
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
scatter!(vec(x), vec(ŷ_ex22_50iter); label = "prediction");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend();
fig
# 2.3. Use a neural network (NN) to model `true_function`
nn = MLPDomain(1, (8, tanh), (1, identity); bias = true)
fitted_nn =
fit(nn, target; optim_alg = DeepPumas.BFGS(), optim_options = (; iterations = 50))
coef(fitted_nn) # try to make sense of the parameters in the NN
ŷ = fitted_nn(x)
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
scatter!(vec(x), vec(ŷ), label = "prediction");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend();
fig
#
# 3. BASIC UNDERFITTING AND OVERFITTING
#
# 3.1. Exercise: Investigate the impact of the number of fitting iterations in NNs
# (Hint: Train again the NN for few and for many iterations.)
# 3.2. Exercise: Reason about Exercise 2.2 again (that is, using a linear regression
# to model a quadratic relationship). Is the number of iterations relevant there?
# 3.3. The impact of the NN size
#
# 3.1. Exercise: Investigate the impact of the number of fitting iterations in NNs
# (Hint: Train again the NN for few and for many iterations.)
underfit_nn =
fit(nn, target; optim_alg = DeepPumas.BFGS(), optim_options = (; iterations = 2))
ŷ_underfit = underfit_nn(x)
overfit_nn =
fit(nn, target; optim_alg = DeepPumas.BFGS(), optim_options = (; iterations = 1_000))
ŷ_overfit = overfit_nn(x) # clarification on the term "overfitting"
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
scatter!(vec(x), vec(ŷ_underfit), label = "prediction (5 iterations)");
scatter!(vec(x), vec(ŷ), label = "prediction (50 iterations)");
scatter!(vec(x), vec(ŷ_overfit), label = "prediction (1000 iterations)");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend();
fig
# 3.2. Exercise: Reason about Exercise 2.2 again (that is, using a linear regression
# to model a quadratic relationship). Is the number of iterations relevant there?
# Investigate the effect of `max_iterations`.
max_iterations = 2
fitted_linreg = fit(
linreg,
target;
optim_alg = DeepPumas.BFGS(),
optim_options = (; iterations = max_iterations),
)
ŷ_linreg = fitted_linreg(x)
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
scatter!(vec(x), vec(ŷ_linreg), label = "$max_iterations iterations");
scatter!(vec(x), vec(ŷ_ex22_50iter), label = "50 iterations");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend();
fig
# 3.3. The impact of the NN size
nn = MLPDomain(1, (32, tanh), (32, tanh), (1, identity); bias = true)
fitted_nn = fit(
nn,
target;
optim_alg = DeepPumas.BFGS(),
optim_options = (; iterations = 1_000)
)
ŷ = fitted_nn(x)
fig = scatter(vec(x), vec(y); axis = (xlabel = "x", ylabel = "y"), label = "data");
scatter!(vec(x), vec(ŷ), label = "prediction MLP(1, 32, 32, 1)");
lines!(-1:0.1:1, true_function.(-1:0.1:1); color = :gray, label = "true");
axislegend();
fig
#
# 4. INSPECTION OF THE VALIDATION LOSS, REGULARIZATION AND HYPERPARAMETER TUNING
#
# 4.1. Validation loss as a proxy for generalization performance
# 4.2. Regularization to prevent overfitting
# 4.3. Hyperparameter tuning
#
# 4.1. Validation loss as a proxy for generalization performance
x_train, y_train = x, y
target_train = target
ϵ = rand(normal, 1, num_samples)
x_valid = rand(uniform, 1, num_samples)
y_valid = true_function.(x_valid) + σ * ϵ
target_valid = preprocess(x_valid, y_valid)
fig = scatter(vec(x_train), vec(y_train); axis = (xlabel = "x", ylabel = "y"), label = "training data");
scatter!(vec(x_valid), vec(y_valid); label = "validation data");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend();
fig
loss_train_l, loss_valid_l = [], []
fitted_nn = fit(
nn,
target_train;
optim_alg = DeepPumas.BFGS(),
optim_options = (; iterations = 10),
)
push!(loss_train_l, sum((fitted_nn(x_train) .- y_train) .^ 2))
push!(loss_valid_l, sum((fitted_nn(x_valid) .- y_valid) .^ 2))
iteration_blocks = 100
for _ = 2:iteration_blocks
global fitted_nn = fit(
nn,
target_train,
coef(fitted_nn);
optim_alg = DeepPumas.BFGS(),
optim_options = (; iterations = 10),
)
push!(loss_train_l, sum((fitted_nn(x_train) .- y_train) .^ 2))
push!(loss_valid_l, sum((fitted_nn(x_valid) .- y_valid) .^ 2))
end
iteration = 10 .* (1:iteration_blocks)
fig, ax = scatterlines(
iteration,
Float32.(loss_train_l);
label = "training",
axis = (; xlabel = "Iteration", ylabel = "Mean squared loss"),
);
scatterlines!(iteration, Float32.(loss_valid_l); label = "validation");
axislegend();
fig
# 4.2. Regularization to prevent overfitting
reg_nn = MLPDomain(1, (32, tanh), (32, tanh), (1, identity); bias = true, reg = L2(0.1))
reg_loss_train_l, reg_loss_valid_l = [], []
fitted_reg_nn = fit(
reg_nn,
target_train;
optim_alg = DeepPumas.BFGS(),
optim_options = (; iterations = 10),
)
push!(reg_loss_train_l, sum((fitted_reg_nn(x_train) .- y_train) .^ 2))
push!(reg_loss_valid_l, sum((fitted_reg_nn(x_valid) .- y_valid) .^ 2))
iteration_blocks = 100
for _ = 2:iteration_blocks
global fitted_reg_nn = fit(
reg_nn,
target_train,
coef(fitted_reg_nn);
optim_alg = DeepPumas.BFGS(),
optim_options = (; iterations = 10),
)
push!(reg_loss_train_l, sum((fitted_reg_nn(x_train) .- y_train) .^ 2))
push!(reg_loss_valid_l, sum((fitted_reg_nn(x_valid) .- y_valid) .^ 2))
end
iteration = 10 .* (1:iteration_blocks)
fig, ax = scatterlines(
iteration,
Float32.(loss_train_l);
label = "training",
axis = (; xlabel = "Blocks of 10 iterations", ylabel = "Mean squared loss"),
);
scatterlines!(iteration, Float32.(loss_valid_l); label = "validation");
scatterlines!(iteration, Float32.(reg_loss_train_l); label = "training (L2)");
scatterlines!(iteration, Float32.(reg_loss_valid_l); label = "validation (L2)");
axislegend();
fig
# 4.3. Programatic hyperparameter tuning
nn_ho = hyperopt(reg_nn, target_train)
nn_ho.best_hyperparameters
ŷ_ho = nn_ho(x_valid)
fig = scatter(vec(x_valid), vec(y_valid); label = "validation data");
scatter!(vec(x_valid), vec(ŷ_ho), label = "prediction (hyperparam opt.)");
lines!(-1..1, true_function; color = :gray, label = "true");
axislegend(; position=:ct);
fig