Papers › Near-Optimal Procedures for Model Discrimination with Non-Disclosure Properties
Near-Optimal Procedures for Model Discrimination with Non-Disclosure Properties
Dmitrii M. Ostrovskii, Mohamed Ndaoud, Adel Javanmard, Meisam Razaviyayn
Let θ₀,θ₁ ∈ℝᵈ be the population risk minimizers associated to some loss ℓ:ℝᵈ×𝒵→ℝ and two distributions ℙ₀,ℙ₁ on 𝒵. The models θ₀,θ₁ are unknown, and ℙ₀,ℙ₁ can be accessed by drawing i.i.d samples from them. Our work is motivated by the following model discrimination question: "What sizes of the samples from ℙ₀ and ℙ₁ allow to distinguish between the two hypotheses θ^*=θ₀ and θ^*=θ₁ for given θ^*∈{θ₀,θ₁}?" Making the first steps towards answering it in full generality, we first consider the case of a well-specified linear model with squared loss. Here we provide matching upper and lower bounds on the sample complexity as given by min{1/Δ²,√(r)/Δ} up to a constant factor; here Δ is a measure of separation between ℙ₀ and ℙ₁ and r is the rank of the design covariance matrix. We then extend this result in two directions: (i) for general parametric models in asymptotic regime; (ii) for generalized linear models in small samples (n≤r) under weak moment assumptions. In both cases we derive sample complexity bounds of a similar form while allowing for model misspecification. In fact, our testing procedures only access θ^* via a certain functional of empirical risk. In addition, the number of observations that allows us to reach statistical confidence does not allow to "resolve" the two models - that is, recover θ₀,θ₁ up to O(Δ) prediction accuracy. These two properties allow to use our framework in applied tasks where one would like to identify a prediction model, which can be proprietary, while guaranteeing that the model cannot be actually inferred by the identifying agent.
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