Ranking

Rank a set of items. Each item has a hidden score, correlated with observable input features. The goal is to learn to sort items by their hidden scores, using observable features alone.

using DecisionFocusedLearningBenchmarks
using Plots

b = RankingBenchmark()

RankingBenchmark(instance_dim=10, nb_features=5)

Observable input

At inference time the decision-maker observes only the feature matrix x (rows = features, columns = items):

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: feature matrix (rows = features, columns = items; observable at train and test time)
• θ: true item costs (training supervision only, hidden at test time)
• y: ordinal ranks derived from θ (y[i] = 1 means item i has the lowest cost)

The full training triple (features, true costs, and derived ranking):

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 costs

Chain(
  Dense(5 => 1; bias=false),            # 5 parameters
  vec,
) 

and a maximizer ranking items by those scores:

maximizer = generate_maximizer(b)         # ordinal ranking via sortperm

ranking (generic function with 1 method)

A randomly initialized policy produces an arbitrary ranking:

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)

4.48195f0

Problem Description

In the Ranking benchmark, a feature matrix $x \in \mathbb{R}^{p \times n}$ is observed. A hidden linear encoder maps $x$ to a cost vector $\theta \in \mathbb{R}^n$. The task is to compute the ordinal ranking of the items by cost:

\[y_i = \mathrm{rank}(\theta_i \mid \theta_1, \ldots, \theta_n) = \mathop{\mathrm{argmax}}\limits_{y\in\sigma(n)} \theta^\top y\]

where $y_i = 1$ means item $i$ has the lowest cost.

Key Parameters

DFL Policy

\[\xrightarrow[\text{Features}]{x} \fbox{Linear model} \xrightarrow{\theta} \fbox{ranking} \xrightarrow{y}\]

Model: Chain(Dense(nb_features → 1; bias=false), vec): predicts one score per item.

Maximizer: ranking(θ): returns a vector of ordinal ranks via invperm(sortperm(θ)).

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