A two-dimensional strategic regression problem in the spirit of Figure 4 of Perdomo, Zrnic, Mendler-Dünner, and Hardt (2020). The base population has features x ∼ N(0, Σ) and outcomes y = ⟨β⋆, x⟩ + noise. When the model θ ∈ R2 is deployed, individuals shift their features to x − εθ to lower their predicted value, and the Learner pays the squared loss (y − ⟨θ, x − εθ⟩)2. The distribution map is ε-sensitive. Everything below is computed in closed form from Σ, β⋆, and ε, except the no-regret algorithm, which sees samples.
Both procedures start from the risk minimizer on the base data (•). Retraining is the population version, θt+1 = G(θt). The no-regret algorithm deploys θt, collects a batch of shifted samples, and refits least squares on the entire history (follow the leader). The stable point (×) is the fixed point of G; the performative optimum (★) minimizes the risk. The plot is colored by the value of the performative risk PR(θ), light where it is low and dark where it is high, on a logarithmic color scale. Use Step or Play to trace the trajectories.
Left: ‖θt − θPS‖ per round, as in Figure 2 of the paper. Right: the stability gap of the uniform mixture over the first t models of each procedure, computed exactly from the deployed models. A negative gap means the mixture beats every single model on the data it induces. The main theorem bounds the no-regret mixture's gap by RegretT/T for every ε.