Subset Selection

Select the k most valuable items from a set of n: items with unknown values must be identified from observable features alone.

using DecisionFocusedLearningBenchmarks
using Plots

b = SubsetSelectionBenchmark(; identity_mapping=false)

SubsetSelectionBenchmark(n=25, k=5)

Observable input

At inference time the decision-maker observes only the feature vector x:

dataset = generate_dataset(b, 50; seed=0)
sample = first(dataset)
plot_context(b, sample)

A training sample

Each sample is a labeled triple (x, θ, y):

• x: item feature vector (observable at train and test time)
• θ: true item values, derived from x via a hidden encoder (training supervision only)
• y: selection indicator (y[i] = 1 for the k highest-value items, 0 otherwise)

The full training triple (features, hidden values, and selection):

plot_sample(b, sample)

Untrained policy

A DFL policy chains two components: a statistical model predicting item scores:

model = generate_statistical_model(b)     # linear map: features → predicted item scores

Dense(25 => 25; bias=false)  # 625 parameters

and a maximizer selecting the top-k items by those scores:

maximizer = generate_maximizer(b)         # top-k selection

(::Base.Fix2{typeof(DecisionFocusedLearningBenchmarks.SubsetSelection.top_k), Int64}) (generic function with 2 methods)

A randomly initialized policy selects items with no relation to their true values:

θ_pred = model(sample.x)
y_pred = maximizer(θ_pred)
plot_sample(b, DataSample(sample; θ=θ_pred, y=y_pred))

Optimality gap on the dataset (lower is better):

compute_gap(b, dataset, model, maximizer)

1.0452117f0

Problem Description

In the Subset Selection benchmark, $n$ items have unknown values $\theta_i$. A feature vector $x \in \mathbb{R}^n$ is observed (identity mapping by default). The task is to select the $k$ items with the highest values:

\[\begin{aligned} y = \mathrm{top}_k(\theta) = & \mathop{\mathrm{argmax}}\limits_{y \in \{0,1\}^n} \; \theta^\top y \\ & \quad\text{s.t.} \quad \sum_{i=1}^n y_i = k \end{aligned}\]

where $y \in \{0,1\}^n$ with exactly $k$ ones.

Key Parameters

When identity_mapping=true, features equal item values directly (x = θ). When false, a random linear layer is used as the hidden mapping.

DFL Policy

\[\xrightarrow[\text{Features}]{x} \fbox{Linear model} \xrightarrow{\theta} \fbox{top-k} \xrightarrow{y}\]

Model: Dense(n → n; bias=false): predicts a score per item.

Maximizer: top_k(θ, k): returns a boolean vector with true at the k highest-scoring positions.

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