Optimal Robust Recourse with $L^p$-Bounded Model Change

Recourse provides individuals who received undesirable labels (e.g., denied a loan) from algorithmic decision-making systems with a minimum-cost improvement suggestion to achieve the desired outcome. However, in practice, models often get updated to reflect changes in the data distribution or environment, invalidating the recourse recommendations (i.e., following the recourse will not lead to the desirable outcome). The robust recourse literature addresses this issue by providing a framework for computing recourses whose validity is resilient to slight changes in the model. However, since the optimization problem of computing robust recourse is non-convex (even for linear models), most of the current approaches do not have any theoretical guarantee on the optimality of the recourse. Recent work by Kayastha et. al. provides the first provably optimal algorithm for robust recourse with respect to generalized linear models when the model changes are measured using the $L^{\infty}$ norm. However, using the $L^{\infty}$ norm can lead to recourse solutions with a high price. To address this shortcoming, we consider more constrained model changes defined by the $L^p$ norm, where $p\geq 1$ but $p\neq \infty$, and provide a new algorithm that provably computes the optimal robust recourse for generalized linear models. Empirically, for both linear and non-linear models, we demonstrate that our algorithm achieves a significantly lower price of recourse (up to several orders of magnitude) compared to prior work and also exhibits a better trade-off between the implementation cost of recourse and its validity. Our empirical analysis also illustrates that our approach provides more sparse recourses compared to prior work and remains resilient to post-processing approaches that guarantee feasibility.

💡 Research Summary

Algorithmic recourse aims to give individuals who receive an unfavorable decision (such as a loan denial) a low‑cost plan to change their features so that a desired outcome is achieved. Traditional recourse methods assume a static predictive model, but in real‑world deployments models are frequently retrained or updated, which can invalidate previously generated recourse suggestions. Robust recourse addresses this by seeking feature changes that remain effective even under modest model shifts. However, computing robust recourse is a non‑convex problem, and existing approaches either lack optimality guarantees or become prohibitively expensive.

The first provably optimal algorithm for robust recourse was introduced by Kayastha et al., which handles model changes bounded in the L∞ norm. While theoretically sound, the L∞ bound is often overly conservative: it forces every model parameter to be allowed to change by the full budget α simultaneously, leading to recourse solutions with very high implementation cost. This paper proposes a more realistic perturbation model by bounding model changes with an Lp norm (p ≥ 1, p ≠ ∞). An Lp bound limits the overall magnitude of the parameter shift while permitting only a few parameters to move substantially, better reflecting typical model updates.

The authors formalize the robust recourse problem as follows. Given an initial model fθ₀, an unfavourable instance x₀ (so that fθ₀(x₀)=0), a convex loss ℓ(·,·) penalising deviation from the target label, and a cost function c (chosen as the L₁ distance between feature vectors), the price of a candidate recourse x under a model θ is J(x,θ)=ℓ(fθ(x),1)+λc(x,x₀). The robust objective is to minimise the worst‑case price over all models θ lying within an Lp‑ball of radius α around θ₀:
 x* = arg minₓ max_{‖θ−θ₀‖ₚ≤α} J(x,θ).

The core technical contribution is an algorithm that solves this problem optimally for any generalized linear model (GLM) when p is finite (i.e., p ≥ 1, p ≠ ∞). A GLM has the form fθ(x)=g(θᵀx) where g is a monotone link function (e.g., sigmoid for logistic regression). The algorithm proceeds in two stages. First, it approximates the original model locally around x₀ with a linear surrogate (using tools such as LIME). Let θ denote the parameters of this linear approximation. Second, instead of searching over the entire Lp‑ball, the authors prove that the worst‑case adversarial model for any candidate recourse must lie in a much smaller set Θ±(θ) consisting of 2d models obtained by adding or subtracting α in a single coordinate i (i = 1,…,d). This reduction follows from the structure of GLMs: the adversary’s optimal strategy is to tilt the decision boundary in the direction that most aligns with the sign and magnitude of the candidate’s feature changes. Consequently, the original non‑convex problem decomposes into 2d convex sub‑problems, each of the form
 minₓ ℓ(fθ′(x),1)+λc(x,x₀) subject to linear constraints that enforce θ′ being the worst‑case model for that x.

Each sub‑problem is convex (ℓ is convex, the GLM composition preserves convexity, and c is convex) but may be non‑differentiable because of the L₁ cost. The authors solve them efficiently using projected sub‑gradient descent, which converges to the global optimum for convex objectives with linear constraints. Since there are only O(d) sub‑problems and each can be solved in polynomial time, the overall algorithm runs in polynomial time in the feature dimension d, providing a provably optimal robust recourse for any finite‑p norm bound.

For completeness, the paper also revisits the L∞ case. In that setting the candidate adversarial models can differ in all coordinates simultaneously, leading to an exponential (2ᵈ) candidate set. Kayastha et al. previously proposed a greedy algorithm that iteratively adjusts the most influential coordinate and updates the adversarial model accordingly. The authors restate this algorithm (Algorithm 2) and confirm its optimality for GLMs, but note that it remains computationally more demanding than the finite‑p approach.

Empirical evaluation spans both linear and non‑linear classifiers on several real‑world datasets (e.g., German Credit). The experiments compare the proposed Lp‑algorithm (Algorithm 1) against the L∞ optimal algorithm, ROAR (a gradient‑based robust recourse method), and RBR (a Bayesian robust recourse method). Key findings include:

• Price Reduction: When model changes are measured with an L₁ or L₂ norm, the proposed method achieves recourse prices that are often 10‑1000× lower than those obtained under an L∞ bound, and substantially lower than ROAR/RBR, which suffer from the conservatism of the L∞ assumption.
• Sparsity: Recourse vectors produced by the Lp algorithm are markedly sparser (fewer features need to be altered), which improves interpretability and practical feasibility for end‑users.
• Robustness to Post‑Processing: The recourses remain valid after applying feasibility‑ensuring post‑processing steps (e.g., clipping to allowable feature ranges), demonstrating resilience to downstream adjustments.
• Scalability: Runtime scales linearly with the number of features, confirming the theoretical polynomial‑time claim. The method is competitive with, and often faster than, ROAR and RBR, especially in higher dimensions where the L∞ greedy search becomes costly.

In summary, this work introduces a more realistic model‑change constraint (Lp norm) for robust recourse, derives a provably optimal algorithm for generalized linear models, and validates its practical advantages through extensive experiments. By reducing the price of recourse, increasing sparsity, and maintaining robustness under realistic model updates, the proposed approach bridges a critical gap between theoretical guarantees and deployable recourse solutions. Future directions include extending the Lp‑bounded framework to richer model families (e.g., tree ensembles, deep neural networks) and integrating data‑driven predictions of future model shifts to further tighten the robustness‑cost trade‑off.