Portfolio Optimization

Allocate wealth across assets to maximize expected return subject to a risk constraint: asset returns are unknown and must be predicted from contextual features.

using DecisionFocusedLearningBenchmarks
using Plots

b = PortfolioOptimizationBenchmark()

PortfolioOptimizationBenchmark(d=50, p=5, deg=1, ν=1.0)

Observable input

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

dataset = generate_dataset(b, 20; seed=0)
sample = first(dataset)
plot_context(b, sample)

A training sample

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

• x: contextual feature vector (observable at train and test time)
• θ: true expected asset returns (training supervision only, hidden at test time)
• y: optimal portfolio weights solving the Markowitz QP given θ

Top: feature vector x. Bottom left: true returns θ. Bottom right: optimal weights y:

plot_sample(b, sample)

Untrained policy

A DFL policy chains two components: a statistical model predicting expected asset returns:

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

Dense(5 => 50)      # 300 parameters

and a maximizer allocating the optimal portfolio given those returns:

maximizer = generate_maximizer(b)         # Markowitz QP solver (Ipopt via JuMP)

(::DecisionFocusedLearningBenchmarks.PortfolioOptimization.var"#portfolio_maximizer#portfolio_maximizer##0"{Float32, Matrix{Float32}, Int64}) (generic function with 1 method)

A randomly initialized policy predicts arbitrary returns, leading to a suboptimal allocation:

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)

-0.30309997827918106

Problem Description

A Markowitz portfolio optimization problem where asset expected returns are unknown. Given contextual features $x \in \mathbb{R}^p$, the learner predicts returns $\theta \in \mathbb{R}^d$ and solves:

\[\begin{aligned} \max_{y} \quad & \theta^\top y \\ \text{s.t.} \quad & y^\top \Sigma y \leq \gamma \\ & \mathbf{1}^\top y \leq 1 \\ & y \geq 0 \end{aligned}\]

where $\Sigma$ is the asset covariance matrix and $\gamma$ is the risk budget. The solver uses Ipopt.jl via JuMP.

Key Parameters

Data is generated following the process in Mandi et al., 2023.

DFL Policy

\[\xrightarrow[\text{Features}]{x \in \mathbb{R}^p} \fbox{Linear model} \xrightarrow[\text{Predicted returns}]{\theta \in \mathbb{R}^d} \fbox{QP solver (Ipopt)} \xrightarrow[\text{Portfolio}]{y \in \mathbb{R}^d}\]

Model: Dense(p → d), predicts one expected return per asset.

Maximizer: Ipopt QP solver enforcing the variance and budget constraints.

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