October 13, 2025
Learning-to-Defer (L2D) enables hybrid decision-making by routing inputs either to a predictor or to external experts. While promising, L2D is highly vulnerable to adversarial perturbations, which can not only flip predictions but also manipulate deferral decisions. Prior robustness analyses focus solely on two-stage settings, leaving open the end-to-end (one-stage) case where predictor and allocation are trained jointly. We introduce the first framework for adversarial robustness in one-stage L2D, covering both classification and regression. Our approach formalizes attacks, proposes cost-sensitive adversarial surrogate losses, and establishes theoretical guarantees including \(\mathcal{H}\), \((\mathcal{R }, \mathcal{F})\), and Bayes consistency. Experiments on benchmark datasets confirm that our methods improve robustness against untargeted and targeted attacks while preserving clean performance.
With the growing adoption of routing-based methods, Learning-to-Defer (L2D) has emerged as a principled framework for hybrid decision-making [1], [2]. An L2D system may either predict directly or defer to an expert, thereby trading off predictive performance against selective reliance on external decision-makers. This framework is particularly relevant in safety-critical applications [3]–[5] and also offers a unifying perspective on a broad class of routing problems [6], [7], where the central challenge is to design allocation policies that distribute decisions optimally across multiple agents.
Yet, despite its promise, L2D inherits the adversarial vulnerabilities of standard machine learning models [8]–[10] while introducing new ones unique to the routing setting. Recent work [11] shows that L2D systems can be even more fragile to adversarial perturbations than conventional systems. An adversary may not only induce misclassification but also manipulate whether, and to whom, the system defers—for example, bypassing a reliable expert or forcing deferral to one known to perform poorly. Such attacks threaten both predictive performance and the reliability of decision allocation, raising pressing concerns for deployment in safety-critical environments.
Existing robustness studies focus exclusively on the simplified two-stage L2D setting, where the predictor and experts are trained offline and only the allocation policy is learned [11]–[16]. This formulation is convenient but fails to capture the complexity of end-to-end (one-stage) L2D, in which the predictor and allocation policy must be optimized jointly. We address two problems: (i) characterizing attacks that target one-stage L2D systems, and (ii) designing one-stage L2D methods with guarantees for the proposed adversarial losses.
In this work, we develop a framework for adversarial robustness in one-stage L2D, encompassing both classification and regression. Our contributions are threefold:
We introduce both untargeted and targeted attacks that reveal how adversaries can manipulate prediction and deferral simultaneously;
We propose new outcome-wise adversarial surrogate deferral losses with tractable relaxations and establish theoretical guarantees for the proposed adversarial losses, including \(\mathcal{H}\)-consistency for classification and \((\mathcal{R},\mathcal{F})\)-consistency for regression;
We demonstrate empirically, on image classification and tabular regression benchmarks, that our methods substantially improve robustness against both attack types while maintaining competitive clean accuracy.
Together, these results provide a theoretically grounded and practically effective approach to adversarially robust one-stage L2D.
One-Stage L2D. [1] introduced the first formal framework for Learning-to-Defer (L2D), incorporating expert predictions via a defer-to-expert mechanism. A key advance was made by [2], who proposed a score-based formulation of L2D. In this approach, the classifier is augmented with a shared scoring function that jointly determines both prediction and allocation, thereby unifying two decisions that had previously been treated separately.
Subsequent work has pursued several directions: strengthening theoretical guarantees such as \(\mathcal{H}\)-consistency and realizability [17]–[21]; broadening the class of statistically sound surrogate losses [18], [22]–[24]; improving estimation methods [25]–[27]; addressing budgeted, imbalanced, or non-stationary deferral regimes [28]–[30]; and extending policies to top-\(k\) expert selection [16]. The score-based methodology has also been applied across diverse classification domains [31]–[38]. Regression-based variants have been proposed by [14], who employ a dedicated allocation policy alongside a trainable predictor. Robustness in L2D. Robustness has received comparatively little attention in the L2D literature. To the best of our knowledge, the only prior work addressing robustness is that of [11], who introduced an adversarially consistent formulation. Their analysis, however, is restricted to the two-stage setting, where experts are fixed and not jointly optimized with the allocation policy. This assumption simplifies the problem: robustness reduces to modifying the allocation strategy without accounting for the interaction between learning the predictor and adapting to the experts.
By contrast, the one-stage setting presents a qualitatively harder challenge as it requires jointly learning the predictor as well as the allocation policy. In this work, we extend the robustness framework of [11] to the technically more demanding one-stage L2D scenario, where adversarial perturbations may influence all components of the system simultaneously.
Let \(\mathcal{X}\subseteq\mathbb{R}^d\) be the input space and let \(\mathcal{Z}\) denote the target space, with \[\mathcal{Z} \;=\; \begin{cases} \mathcal{Y}=\{1,\dots,K\}, & \text{classification},\\ \mathcal{T}\subseteq\mathbb{R}^m, & \text{regression}. \end{cases}\] Each training label satisfies \(z \in \mathcal{Z}\). We assume access to i.i.d.samples \(\mathcal{S}_n = \{(x_i, z_i, \mathbf{m}_i)\}_{i=1}^n \sim \big(\mathcal{D} \times \mathcal{M}\big)^n\), where \((x,z) \sim \mathcal{D}\) is drawn from the underlying data distribution and \(\mathbf{m} \sim \mathcal{M}\!\mid(x,z)\) denotes the outputs of the \(J\) fixed experts conditioned on \((x,z)\) [1], [2], [25]. Here \(\mathbf{m} = (m_{1}, \dots, m_{J})\), with each \(m_{j}\) belonging to the appropriate output space: in classification, \(m_{j} \in \mathcal{Y}\) is a categorical prediction, while in regression, \(m_{j} \in \mathcal{T}\) is a real-valued prediction. All expert predictions can thus be regarded as elements of the common target space \(\mathcal{Z}\).
Classification L2D. We specialize to \(\mathcal{Z}=\mathcal{Y}=\{1,\dots,K\}\). Each training label \(y_i\in\mathcal{Y}\) and each expert output \(m_{ij}\in\mathcal{Y}\) is categorical. Alongside the \(J\) experts \(m_1,\dots,m_J\), we train a hypothesis \(h\in\mathcal{H}\) with \(h:\mathcal{X}\to\mathbb{R}^{K+J}\), where the first \(K\) coordinates correspond to class scores and the last \(J\) to expert scores [2], [16], [18], [27]. Let \(\mathcal{A}^{c} = \mathcal{Y} \cup \{K+1,\dots,K+J\}\) denote the augmented action space. For a vector \(\xi(x)\), we write \(\hat{\xi}(x)\) to denote its \(\arg\max\) index. The induced decision rule is then \(\hat{h}(x) = \arg\max_{k \in \mathcal{A}^{c}} h(x)_k\), with ties broken uniformly at random. If \(\hat{h}(x) \in \mathcal{Y}\), the system predicts the corresponding category; if \(\hat{h}(x) = K+j\) for some \(j \in \{1,\dots,J\}\), the system defers to expert \(m_j\).
The predictor \(h \in \mathcal{H}\) is trained to minimize the expected risk of the true deferral loss for classification \[\label{class} \begin{align} \mspace{-8mu}\ell_{\text{def}}^c(\hat{h}(x),y, \mathbf{m}) \mspace{-3mu} = \mspace{-3mu} \begin{cases} \mathbf{1}\{\hat{h}(x)\neq y\}, \mspace{-10mu} & \text{if } \hat{h}(x)\in\mathcal{Y},\\[0.3em] c^c_j(m_j,y), & \text{if } \hat{h}(x)=K+j. \end{cases} \end{align}\tag{1}\] where \(c^c_j(m_j,y) \in [0,1]\) is the cost of deferring to expert \(m_j\). We set \(c^c_j(m_j,y) \;=\; \alpha_j\,\mathbf{1}\{m_j\neq y\} + \beta_j\), with \(\alpha_j\ge 0\) a scaling coefficient and \(\beta_j\) the fixed consultation cost for querying expert \(j\).
A common way to minimize such a discontinuous loss is to replace it with a consistent surrogate [39]–[53], ensuring that the learned predictor approximates the Bayes classifier \(h^B \in \mathcal{H}\) that minimizes the expected true deferral loss. Given the augmented action space \(\mathcal{A}^c\), we define the surrogate deferral loss for classification as \[\label{eq:surr95classification} \begin{align} \Phi_{\text{def}}^{c, u}(h&(x),y, \mathbf{m}) \; = \; \Phi_{\text{cls}}^u(h(x),y) \\ & +\sum_{j=1}^J (1-c_j^c(m_j,y))\,\Phi_{\text{cls}}^u(h(x),K+j), \end{align}\tag{2}\] where \(\Phi_{\text{cls}}^u\) is a classification surrogate from the cross-entropy family [54], defined as \(\Phi_{\text{cls}}^u(h(x),y) =\Psi^u\!\Big(\sum_{y'\in\mathcal{Y}}\exp(h(x)_{y'}-h(x)_y) -1\Big)\) with outer transform \[\Psi^u(v)= \begin{cases} \log(1+v), & u=1,\\[0.3em] \tfrac{1}{1-u}\big[(1+v)^{1-u}-1\big], & u\neq 1. \end{cases}\] This family recovers well-known surrogates, including logistic loss [55], generalized cross-entropy [56], and mean absolute error [57]. The surrogate risk is \(\mathcal{E}_{\Phi_{\text{def}}^{c, u}}(h)=\mathbb{E}[\Phi_{\text{def}}^{c,u}(h(X), Y, M)]\) with optimal value \(\mathcal{E}_{\Phi_{\text{def}}^{c, u}}^\ast(\mathcal{H})=\inf_{h\in\mathcal{H}}\mathcal{E}_{\Phi_{\text{def}}^{c, u}}(h)\). A surrogate is called consistent if minimizing its excess risk also minimizes the true excess risk [39]–[41], [58].
Theorem 1 (\(\mathcal{H}\)-consistency bounds [42]). The surrogate \(\Phi_{\text{def}}^{c, u}\) is \(\mathcal{H}\)-consistent with respect to \(\ell_{\text{def}}^{c}\) if there exists a non-decreasing function \(\Gamma^u:\mathbb{R}^+\to\mathbb{R}^+\) such that for every distribution \(\mathcal{D}\), \[\begin{align} &\mathcal{E}_{\ell_{\mathrm{def}}^c}(h)-\mathcal{E}_{\ell_{\mathrm{def}}^c}^B(\mathcal{H})+\mathcal{U}_{\ell_{\mathrm{def}}^c}(\mathcal{H}) \leq \\ & \qquad \Gamma^u\!\Big(\mathcal{E}_{\Phi_{\mathrm{def}}^{c, u}}(h)-\mathcal{E}_{\Phi_{\mathrm{def}}^{c, u}}^\ast(\mathcal{H})+\mathcal{U}_{\Phi_{\mathrm{def}}^{c, u}}(\mathcal{H})\Big). \end{align}\]
Here \(\mathcal{U}_{\Phi_{\mathrm{def}}^{c, u}}(\mathcal{H})\) is the minimizability gap, quantifying the discrepancy between the best achievable excess risk within \(\mathcal{H}\) and the expected pointwise minimum. This gap vanishes when \(\mathcal{H}=\mathcal{H}_{\mathrm{all}}\), recovering Bayes-consistency in the asymptotic limit [39], [40]. Beyond the classification setting studied here, this hypothesis-dependent notion of \(\mathcal{H}\)-consistency has been developed into a broad theory [42], [59], with guarantees established for multi-class abstention and deferral [60]–[62], for ranking and cardinality-aware prediction [63]–[65], and for structured prediction [66], as well as in recent algorithms for generalized-metric optimization and robust generative modeling [67]–[69].
Regression L2D. Let \(\mathcal{Z}=\mathcal{T}\subseteq\mathbb{R}^m\). Each training label \(t_i\in\mathcal{T}\) and each expert output \(m_{ij}\in\mathcal{T}\) is a real-valued vector. The system consists of a predictor \(f\in\mathcal{F}\), with \(f:\mathcal{X}\to\mathcal{T}\), and a rejector \(r\in\mathcal{R}\), with \(r:\mathcal{X}\to\mathbb{R}^{J+1}\). Let \(\mathcal{A}^{r}=\{1,2,\dots,J+1\}\) denote this action space. The induced decision rule is \(\hat{r}(x)= \arg\max_{k\in\mathcal{A}^{r}} r(x)_k\), so that if \(\hat{r}(x)=1\), the system outputs \(f(x)\), while if \(\hat{r}(x)=j\) with \(j\ge 2\), the system defers to expert \(m_{j-1}\). The true deferral loss for regression is \[\label{regression} \ell_{\text{def}}^r(f(x),\hat{r}(x),t, \mathbf{m}) \mspace{-3mu}=\mspace{-3mu} \sum_{j=1}^{J+1} \mspace{-3mu}c_j^r(f(x), m_{j-1}, t)\mspace{-1mu}\mathbf{1}\{\hat{r}(x)=j\},\tag{3}\] where each \(c_j^r(f(x), m_{j-1}, t)\ge 0\) is the cost of action \(j\). For the predictor we define \(c_1^r(f(x),t)=\alpha_1\,L(f(x),t)+\beta_1\), and for expert \(j\ge 2\), \(c_j^r(m_{j-1},t)=\alpha_j\,L(m_{j-1},t)+\beta_j\), where \(L:\mathcal{T}\times\mathcal{T}\to\mathbb{R}_+\) is a regression loss (e.g.squared error). Similarly to the classification case, we approximate the discontinuous loss with a consistent surrogate, which we refer to as the surrogate deferral loss for regression: \[\begin{align} \Phi_{\text{def}}^{r,u}(f(x),r(x),t, \mathbf{m}) &= \sum_{j=1}^{J+1} \tau_j^r(f(x), \mathbf{m}, t)\,\Phi_{\text{cls}}^u(r(x),j) \\ &\qquad - (J-1)\,c_1^r(f(x),t), \end{align}\] where \(\Phi_{\text{cls}}^u\) is the same cross-entropy–type surrogate used in classification, and the weights \(\tau_j^r(f(x),\mathbf{m}, t)\) are defined as \(\tau_j^r(f(x),\mathbf{m}, t)=\sum_{i\not= j}c_i^r(f(x), m_{i-1}, t)\). This surrogate has been shown to be both Bayes consistent and \((\mathcal{R},\mathcal{F})\)-consistent [14].
Adversarially robust classification aims to train classifiers that remain reliable under small, often imperceptible, perturbations of the input [8], [9]. The goal is to minimize the true multiclass loss \(\ell_{01}\) evaluated on an adversarial input \(x' = x + \delta\) [42], [70]. A perturbation \(\delta\) is constrained by its magnitude, with the adversarial region around \(x\) defined as \(B_p(x,\gamma) = \{\, x' \in \mathbb{R}^d \mid \|x' - x\|_p \leq \gamma \,\}\), where \(\|\cdot\|_p\) denotes the \(p\)-norm and \(\gamma > 0\) bounds the perturbation size.
The adversarial \(0\)–\(1\) loss is \(\widetilde{\ell}_{01}(h,x,y) = \sup_{x' \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x') \neq y\}\), and its adversarial margin surrogate is \[\widetilde{\ell}_{01}(h,x,y)\leq \sup_{x' \in B_p(x,\gamma)} \Phi_{\text{cls}}^{\rho, u}(h(x'),y),\] with \(\Phi_{\text{cls}}^{\rho, u}(h(x'),y)=\Psi^u\!\Big(\sum_{k \neq y} \Psi_\rho\big(h(x')_k - h(x')_y\big)\Big)\). Here, \(\Psi^u\) and \(\Psi_\rho\) are transformations that characterize the surrogate family. In the analysis below, we use the exponential \(\rho\)-soft margin transform \(\Psi_\rho(v)=\exp(-v/\rho)\). Recent works show that algorithms based on smooth, regularized variants of these comp-sum \(\rho\)-margin losses achieve strong calibration and consistency guarantees [42], [54], [71].
As discussed in Section 2, robustness has so far been investigated only in the two-stage setting of Learning-to-Defer [11], while the one-stage formulation remains unexplored. Addressing robustness in this setting requires novel surrogate losses that simultaneously preserve consistency and provide robustness guarantees.
Learning-to-Defer seeks to route each query to the most reliable agent—either the predictor or one of the experts [1], [2]. In classification, the augmented classifier jointly handles both prediction and deferral. In regression, allocation and prediction are decoupled. Such formulations perform well on clean inputs [17], [26]. The natural question is: do they remain effective under noisy or adversarial perturbations? We answer this in the negative. Extending the conclusions of [11], we show that adversarial attacks not only compromise the allocation policy but also predictive performance in both the regression and classification settings of one-stage Learning-to-Defer.
In the untargeted setting, the adversary seeks a perturbation \(\delta \in B_p(0,\gamma)\) that maximally degrades the system’s performance, without targeting a specific outcome.
definitionuntargeted An untargeted adversarial attack in L2D is the problem of finding a perturbed input \(x' = x + \delta\), with \(\delta \in B_p(0,\gamma)\), that maximizes a differentiable attack objective aligned with the deferral loss in either classification or regression. In our experiments, we instantiate this objective with the corresponding surrogate loss. Formally, \[x' = \underset{x' \in B_p(x,\gamma)}{\arg\max}\; \Phi_{\mathrm{def}}^{\bullet,u}(x'),\] where \(\bullet \in \{c,r\}\) indicates whether the task is classification (\(c\)) or regression (\(r\)).
The attacker perturbs an input \(x\) so that, even if the clean input would be processed optimally, the perturbed version \(x'\) induces a worse outcome under the deferral policy. Such degradation can occur in several ways in the one-stage setting: (i) causing the predictor to output an incorrect class in classification or a high-error estimate in regression; (ii) obstructing deferral to a reliable expert when such deferral would reduce error; or (iii) inducing unnecessary deferral when the predictor is already sufficiently accurate. For instance, if the clean input \(x\) should be handled by the main predictor, an untargeted perturbation may instead redirect the query to an inappropriate expert, thereby increasing the overall system error.
In the targeted setting, the adversary specifies a desired outcome \(\nu\) and perturbs the input so that the system’s decision is steered toward this target. In classification, \(\nu\) may correspond to a class label, \(\nu \in \{1,\dots,K\}\), or to a deferral action, \(\nu \in \{K+1,\dots,K+J\}\). In regression, the target space is defined analogously: \(\nu = 1\) denotes trusting the predictor’s output, while \(\nu \in \{2,\dots,J+1\}\) corresponds to deferral to expert \(\nu\). Let \(\boldsymbol{\delta} = (\delta_1,\dots,\delta_{|\mathcal{A}|})\) denote the family of outcome-specific perturbations, with \(\delta_j \in \mathcal{X}\).
definitiontargeted A targeted adversarial attack in L2D is the problem of finding a perturbation \(\delta_\nu \in \mathcal{X}\) from the family \(\boldsymbol{\delta}\) such that the adversarial input \(x'_\nu = x + \delta_\nu\) lies within the \(p\)-norm ball \(B_p(x,\gamma)\) and drives the allocation policy toward the specified target \(\nu \in \mathcal{A}\). Formally, \[x'_\nu = \underset{x'_\nu \in B_p(x,\gamma)}{\arg\min}\; \Phi_{\mathrm{cls}}^u\big(\pi(x_\nu'), \nu\big),\] where \(\pi\) is \(h\) in classification or \(r\) in regression.
Unlike untargeted attacks, which aim only to degrade performance, targeted attacks explicitly redirect the decision to a chosen outcome \(\nu\). The surrogate \(\Phi_{\mathrm{cls}}^u\) acts as a differentiable proxy for the indicator \(\mathbf{1}\{\hat{\pi}(x_\nu') \neq \nu\}\), thereby encouraging \(\hat{\pi}(x'_\nu) = \nu\). Such attacks are particularly powerful: (i) in classification, an adversary may compel predictions into sensitive categories of \(\mathcal{Y}\); and (ii) the adversary may force deferral to a specific expert—possibly one known to perform poorly or to exhibit exploitable dependencies. For example, even if the clean input \(x\) should defer to expert 1, a targeted perturbation may redirect the system to predict an unrelated class or to defer to a less reliable expert.
In the two-stage case [11], both targeted and untargeted attacks focus on the allocation policy, whereas in our setting adversaries can attack both the predictive and allocation policies.
As previously discussed, current Learning-to-Defer approaches are highly sensitive to adversarial perturbations. This motivates the development of a formulation that explicitly incorporates adversarial robustness. Building on [11], we generalize and extend the analysis to the one-stage L2D setting and introduce outcome-wise adversarial deferral losses for both classification and regression.
We now move from the one-stage attack setting to optimization objectives that enable learning a robust hypothesis \(h \in \mathcal{H}\). To this end, we define an outcome-wise adversarial deferral loss by distinguishing perturbations according to their decision outcomes. This quantity upper-bounds the conventional single-perturbation adversarial deferral loss.
Given an input \(x\), let \(x'_j \in B_p(x,\gamma)\) denote the adversarial example associated with outcome \(j \in \mathcal{A}\), written as \(x'_j = x + \delta_j\) where \(\delta_j \in \mathcal{X}\) is the perturbation leading to outcome \(j\). Collecting these perturbations, we write \(\boldsymbol{\delta} = (\delta_1, \dots, \delta_{K+J})\) for the family of outcome-specific perturbations. For example, \(x'_1\) denotes the adversarial input aligned with outcome \(1 \in \mathcal{A}\).
lemmatlcrobustness For any hypothesis \(h \in \mathcal{H}\), input–label pair \((x,y)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the outcome-wise adversarial deferral loss for classification is \[\widetilde{\ell}_{\mathrm{def}}^c\big(h, x, y, \mathbf{m}\big) = \sum_{j=1}^{K+J} \mu_j(j, m_j, y)\mspace{-19mu} \sup_{x'_j \in B_p(x,\gamma)} \mspace{-19mu} \mathbf{1}\{\hat{h}(x'_j) = j\}.\]
Here, \(\mu_j\) denotes a shifted cost. For \(j \in \{1,\dots,K\}\), we set \(\mu_j(j,y) = \alpha_j \mathbf{1}\{j \neq y\} + \beta_j\), which encodes the cost of predicting class \(j\). For \(j \in \{K+1,\dots,K+J\}\), we define \(\mu_j(m_j,y) = c_{j-K}(m_{j-K},y)\), capturing the cost of deferring to expert \(j\).
We prove in Appendix 8.3 that this quantity upper-bounds the conventional single-perturbation adversarial deferral loss, and therefore also upper-bounds the clean deferral loss in classification (Equation 1 ). Since this formulation is NP-hard, we next introduce a tractable surrogate.
definitionscrobustness For any hypothesis \(h \in \mathcal{H}\), input–label pair \((x,y)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the adversarial surrogate deferral loss for classification is defined as \[\begin{align} \widetilde{\Phi}_{\mathrm{def}}^{c, u}(h&, x,y, \mathbf{m}) \mspace{-3mu}=\mspace{-8mu} \sum_{j=1}^{K+J}\mspace{-6mu} \sum_{i\not=j}\mspace{-3mu}\mu_i(i, m_i, y) \mspace{-24mu}\sup_{x'_j \in B_p(x,\gamma)}\mspace{-24mu} \Phi^{\rho, u}_{\mathrm{cls}}(h(x_j'),j). \end{align}\]
Here, \(\Phi^{\rho,u}_{\mathrm{cls}}(h(x'_j), j)\) denotes the margin-based surrogate loss evaluated on the adversarial input \(x'_j\) associated with outcome \(j \in \mathcal{A}\).
The surrogate in Definition [lem:scrobustness] is a tractable relaxation of the outcome-wise adversarial loss in Lemma [lem:tlcrobustness]. A key requirement is that it be both Bayes- and \(\mathcal{H}\)-consistent with respect to the defined outcome-wise adversarial loss, ensuring that minimization recovers an optimal allocation policy for this loss. We establish these guarantees in the sense of [39], [40], showing that optimization over the surrogate yields asymptotically optimal policies for the proposed adversarial loss.
theoremconsistency Let \(\mathcal{H}\) be symmetric and locally \(\rho\)-consistent. Then, for the set \(\mathcal{A}\), any hypothesis \(h \in \mathcal{H}\), and any distribution \(\mathcal{D}\), the following holds: \[\begin{align} & \mathcal{E}_{\widetilde{\ell}^c_{\text{def}}}(h) - \mathcal{E}_{\widetilde{\ell}^c_{\text{def}}}^B(\mathcal{H}) + \mathcal{U}_{\widetilde{\ell}^c_{\text{def}}}(\mathcal{H}) \\ & \quad \leq \Psi^u(1) \Big( \mathcal{E}_{\widetilde{\Phi}^{c,u}_{\text{def}}}(h) - \mathcal{E}_{\widetilde{\Phi}^{c,u}_{\text{def}}}^\ast(\mathcal{H}) + \mathcal{U}_{\widetilde{\Phi}^{c,u}_{\text{def}}}(\mathcal{H}) \Big), \end{align}\] with \(\Psi^u(1) = \begin{cases} \log(2), & u=1, \\[0.3em] \tfrac{1}{1-u}\big(2^{1-u}-1\big), & u \neq 1. \end{cases}\)
We prove this theorem using novel proof techniques, provided in Appendix 8.4. Intuitively, Theorem [theo:consistency] shows that the surrogate loss \(\widetilde{\Phi}^{c,u}_{\text{def}}\) is \(\mathcal{H}\)-consistent up to the multiplicative constant \(\Psi^u(1)\). If a sequence \((h_t)\subset\mathcal{H}\) satisfies \[\mathcal{E}_{\widetilde{\Phi}^{c,u}_{\text{def}}}(h_t)-\mathcal{E}_{\widetilde{\Phi}^{c,u}_{\text{def}}}^\ast(\mathcal{H})+\mathcal{U}_{\widetilde{\Phi}^{c,u}_{\text{def}}}(\mathcal{H})\to 0,\] then by Theorem [theo:consistency] we have \(\mathcal{E}_{\widetilde{\ell}^c_{\text{def}}}(h_t)-\mathcal{E}_{\widetilde{\ell}^c_{\text{def}}}^B(\mathcal{H})+\mathcal{U}_{\widetilde{\ell}^c_{\text{def}}}(\mathcal{H})\to 0\). Thus, minimizing the surrogate excess risk directly controls the excess risk of the defined outcome-wise adversarial deferral loss, ensuring that optimization with respect to \(\widetilde{\Phi}^{c,u}_{\text{def}}\) converges to the Bayes-optimal policy for this loss. Finally, note that the minimizability gap vanishes for realizable distributions or when \(\mathcal{H}\) is the set of all measurable functions \(\mathcal{H}_{\mathrm{all}}\) [39].
While we established the consistency of the surrogate introduced in Definition [lem:scrobustness], the formulation remains non-convex due to its reliance on the margin-based surrogate \(\widetilde{\Phi}_{\text{cls}}^{\rho,u}\). Non-convexity complicates optimization and limits the practical applicability of the guarantee. To address this, we construct a smooth upper bound by replacing the margin loss with a tractable relaxation, following the principled approaches of [11], [54], [71]. This relaxation preserves theoretical soundness while enabling tractable and stable optimization in practice.
lemmasurrogatemulti Let \(x \in \mathcal{X}\) be a clean input, and let \(\rho > 0\) and \(\kappa > 0\) be hyperparameters. The smooth adversarial surrogate losses are defined as \[\begin{align} \widetilde{\Phi}_{\mathrm{cls, s}}^{u}(h, x,j) = &\, \Phi_{\mathrm{cls}}^u (h(x)/\rho, j) \\ & + \kappa \mspace{-15mu}\sup_{x'_j \in B_p(x,\gamma)} \mspace{-15mu} \big\| \overline{\Delta}_h(x'_j, j) - \overline{\Delta}_h(x, j) \big\|_2, \end{align}\]
We defer the proof in Appendix 8.2. For any \(x \in \mathcal{X}\), we define the pairwise margin differences as \(\Delta_h(x, j, j') = h(x)_j - h(x)_{j'}\), and let \(\overline{\Delta}_h(x, j) \in \mathbb{R}^{|\mathcal{A}|-1}\) be the vector of all pairwise differences, i.e., \(\overline{\Delta}_h(x, j) = \big(\Delta_h(x, j, 1), \ldots, \Delta_h(x, j, j-1), \Delta_h(x, j, j+1), \ldots, \Delta_h(x, j, K+J)\big)\). The first term, \(\Phi_{\mathrm{cls}}^u(h(x)/\rho, j)\), is the standard multiclass surrogate loss scaled by \(\rho\), while the second term penalizes local instability by bounding the worst-case deviation of pairwise margins under adversarial perturbations. This yields a smooth tractable relaxation of the surrogate in Definition [lem:scrobustness].
definitionsmoothscrobustness For any hypothesis \(h \in \mathcal{H}\), input–label pair \((x,y)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the smooth adversarial surrogate deferral loss for classification is defined as \[\begin{align} \widetilde{\Phi}_{\mathrm{def,s}}^{c, u}(h&, x,y, \mathbf{m}) \mspace{-3mu}=\mspace{-8mu} \sum_{j=1}^{K+J}\mspace{-6mu} \sum_{i\not=j}\mspace{-3mu}\mu_i(i, m_i, y) \mspace{-24mu}\sup_{x'_j \in B_p(x,\gamma)}\mspace{-24mu} \Phi^{u}_{\mathrm{cls,s}}(h(x_j'),j). \end{align}\]
which yields the following pointwise upper bound.
corollarysmoothconsistency For any hypothesis \(h \in \mathcal{H}\), input–label pair \((x,y)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the following holds: \[\widetilde{\Phi}^{c,u}_{\mathrm{def}}(h,x,y,\mathbf{m}) \leq \widetilde{\Phi}^{c,u}_{\mathrm{def,s}}(h,x,y,\mathbf{m}).\]
Corollary [theo:smooth-consistency] shows that the smooth surrogate \(\widetilde{\Phi}^{c,u}_{\mathrm{def,s}}\) pointwise upper-bounds \(\widetilde{\Phi}^{c,u}_{\mathrm{def}}\). Combined with Theorem [theo:consistency], this motivates optimizing the smooth relaxation as a tractable proxy for the nonsmooth adversarial surrogate.
Definition 1 (RERM-C: Regularized ERM for \(\widetilde{\Phi}^{c,u}_{\mathrm{def,s}}\)). Assume \(\mathcal{H}\) is symmetric and locally \(\rho\)-consistent. Let \(\Omega: \mathcal{H} \to \mathbb{R}^+\) be a regularizer and let \(\eta > 0\) be a hyperparameter. We define the regularized empirical risk minimization (ERM) objective as \[\min_{h \in \mathcal{H}} \Bigg[ \frac{1}{n} \sum_{k=1}^n \widetilde{\Phi}^{c,u}_{\mathrm{def,s}}\big(h,x_k, y_k, \mathbf{m}_k\big) + \eta \, \Omega(h) \Bigg],\]
Unlike classification, regression requires jointly learning both the main predictor \(f \in \mathcal{F}\) and the allocation policy defined by the rejector \(r \in \mathcal{R}\) (see Equation 3 ). This interdependence introduces additional complexity, making a direct extension of the classification analysis infeasible. To address this, we formalize an outcome-wise adversarial deferral loss for regression, which characterizes a worst-case upper bound under perturbations. This formulation extends the deferral framework to settings where both prediction and deferral are adversarially sensitive. The result is summarized in Lemma [lem:tlcrobustness-reg], with the proof deferred to Appendix 8.5.
lemmatlcrobustnessreg For any hypothesis \(r \in \mathcal{R}\), predictor \(f \in \mathcal{F}\), input–label pair \((x,t)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the outcome-wise adversarial deferral loss for regression is defined as \[\widetilde{\ell}_{\mathrm{def}}^r\big(r, f, t, x, \mathbf{m}\big) = \sum_{j=1}^{J+1} \tilde{c}_j^r \mspace{-10mu} \sup_{x'_j \in B_p(x,\gamma)} \mspace{-10mu} \mathbf{1}\{\hat{r}(x'_j) = j\},\] Here, \(\tilde{c}_j^r = \begin{cases} \alpha_1\sup_{x'_1 \in B_p(x,\gamma)} L(f(x'_1), t)+\beta_1, & j = 1, \\[0.4em] c_j^r(m_{j-1}, t), & j >1. \end{cases}\)
To make the regression setting amenable to optimization, we construct a surrogate that corrects for the adaptive penalization introduced by adversarial perturbations.
definitionsrrobustness For any hypothesis \(r \in \mathcal{R}\), predictor \(f \in \mathcal{F}\), input–label pair \((x,t)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the adversarial surrogate deferral loss for regression is defined as \[\begin{align} \widetilde{\Phi}_{\mathrm{def}}^{r, u}(r&,f,x, t, \mathbf{m}) = -(J-1)\tilde{c}_1^r(f,x,t) \\ & + \sum_{j=1}^{J+1}\mspace{-6mu} \sum_{i\not=j}\tilde{c}_i^r(f,x,m_{i-1},t) \mspace{-24mu}\sup_{x'_j \in B_p(x,\gamma)}\mspace{-24mu} \Phi^{\rho, u}_{\mathrm{cls}}(r(x_j'),j). \end{align}\]
This surrogate loss provides a tractable relaxation of the outcome-wise adversarial deferral loss, while explicitly incorporating adaptive penalization through \(\tilde{c}_1^r\).
This surrogate differs fundamentally from both its classification counterpart and the two-stage setting [11], owing to the explicit involvement of the learnable predictor \(f \in \mathcal{F}\).
Similarly to the classification setting, we introduced both an adversarial surrogate deferral loss and its outcome-wise counterpart. It is therefore necessary to establish that the surrogate in Definition [lem:rcrobustness] is Bayes- and \((\mathcal{R}, \mathcal{F})\)-consistent. We prove the following theorem in Appendix 8.6.
theoremrconsistency Let \(\mathcal{R}\) be symmetric and locally \(\rho\)-consistent. Then, for the set \(\mathcal{A}^{r}\), any hypothesis \(r \in \mathcal{R}\) and \(f \in \mathcal{F}\), and any distribution \(\mathcal{D}\), the following holds: \[\begin{align} & \mathcal{E}_{\widetilde{\ell}^r_{\text{def}}}(r,f) - \mathcal{E}_{\widetilde{\ell}^r_{\text{def}}}^B(\mathcal{R}, \mathcal{F}) + \mathcal{U}_{\widetilde{\ell}^r_{\text{def}}}(\mathcal{R}, \mathcal{F}) \\ & \leq \overline{\Gamma}^u(1) \Big( \mathcal{E}_{\widetilde{\Phi}^{r,u}_{\text{def}}}(r,f) - \mathcal{E}_{\widetilde{\Phi}^{r,u}_{\text{def}}}^\ast(\mathcal{R}, \mathcal{F}) + \mathcal{U}_{\widetilde{\Phi}^{r,u}_{\text{def}}}(\mathcal{R}, \mathcal{F}) \Big), \end{align}\] with \(\overline{\Gamma}^u(1) = \max\{1, \Psi^u(1)\}\).
Theorem [theo:r95consistency] shows that the adversarial surrogate loss for regression is \((\mathcal{R}, \mathcal{F})\)-consistent up to the factor \(\max\{1, \Psi^u(1)\}\) with respect to the defined outcome-wise adversarial loss. Importantly, this factor differs from the classification case (see Theorem [theo:consistency]) precisely because regression involves the learnable predictor \(f \in \mathcal{F}\) in addition to the rejector \(r \in \mathcal{R}\). As a result, the analysis must bound a joint function of \((r,f)\) rather than a single function, which makes the regression case fundamentally more challenging. Indeed, if two sequences \((f_t) \subset \mathcal{F}\) and \((r_t) \subset \mathcal{R}\) satisfy \(\mathcal{E}_{\widetilde{\Phi}^{r,u}_{\text{def}}}(r_t,f_t) - \mathcal{E}_{\widetilde{\Phi}^{r,u}_{\text{def}}}^\ast(\mathcal{R}, \mathcal{F}) + \mathcal{U}_{\widetilde{\Phi}^{r,u}_{\text{def}}}(\mathcal{R}, \mathcal{F}) \rightarrow 0\), then by Theorem [theo:r95consistency] we have \(\mathcal{E}_{\widetilde{\ell}^r_{\text{def}}}(r_t,f_t) - \mathcal{E}_{\widetilde{\ell}^r_{\text{def}}}^B(\mathcal{R}, \mathcal{F}) + \mathcal{U}_{\widetilde{\ell}^r_{\text{def}}}(\mathcal{R}, \mathcal{F}) \rightarrow 0\). Therefore, minimizing the surrogate recovers an optimal predictor-rejector pair for the defined adversarial loss. In this case, the minimizability gap vanishes under realizability with respect to the pair \((r,f)\), provided that \(\mathcal{F}=\mathcal{F}_{\text{all}}\) and \(\mathcal{R}=\mathcal{R}_{\text{all}}\).
Analogous to the classification case, we apply Lemma [lemma:surrogate95class] to upper-bound the surrogate defined in Definition [lem:rcrobustness] by a smooth relaxation.
corollarysmoothsrrobustness For any hypothesis \(r \in \mathcal{R}\), predictor \(f \in \mathcal{F}\), input–label pair \((x,t)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the smooth adversarial surrogate deferral loss for regression is defined as \[\begin{align} \widetilde{\Phi}^{r,u}_{\mathrm{def,s}}(r, &f, x,t,\mathbf{m}) = -(J-1)\tilde{c}_1^r(f(x),t) \\ &+ \sum_{j=1}^{J+1} \sum_{i\not=j}\tilde{c}_i^r(f,x,m_{i-1},t) \widetilde{\Phi}^{u}_{\mathrm{cls, s}}(r(x),j). \end{align}\]
This yields a novel surrogate with guarantees while ensuring tractable optimization.
corollaryrconsistencycor For any hypothesis \(r \in \mathcal{R}\), predictor \(f \in \mathcal{F}\), input–label pair \((x,t)\), and expert outputs \(\mathbf{m} = (m_1,\dots,m_J)\), the following holds: \[\widetilde{\Phi}^{r,u}_{\mathrm{def}}(r,f,x,t,\mathbf{m}) \leq \widetilde{\Phi}^{r,u}_{\mathrm{def,s}}(r,f,x,t,\mathbf{m}).\]
Corollary [corr:r95consistency] shows that the smooth surrogate \(\widetilde{\Phi}^{r,u}_{\mathrm{def,s}}\) pointwise upper-bounds \(\widetilde{\Phi}^{r,u}_{\mathrm{def}}\). This provides a tractable optimization objective aligned with the adversarial surrogate analyzed in Theorem [theo:r95consistency].
Hence, it motivates the introduction of a regularized ERM algorithm for regression (RERM-R).
Definition 2 (RERM-R: Regularized ERM for \(\widetilde{\Phi}^{r,u}_{\mathrm{def,s}}\)). Assume \(\mathcal{R}\) is symmetric and locally \(\rho\)-consistent. Let \(\Omega: \mathcal{R} \times \mathcal{F} \to \mathbb{R}_{+}\) be a convex regularizer, and let \(\eta > 0\) be a hyperparameter. The regularized ERM objective is \[\min_{\,r \in \mathcal{R},\, f \in \mathcal{F}} \mspace{-5mu}\Bigg[ \frac{1}{n} \sum_{k=1}^n \widetilde{\Phi}^{r,u}_{\mathrm{def,s}}\!\big(r, f,x_k, t_k, \mathbf{m}_k\big) + \eta \,\Omega(r,f) \Bigg]\]
We will use this algorithm in Subsection 6.2. Its per-epoch computational cost in the \(h\)-score setting is summarized below.
Consider \(n\) training examples processed in mini-batches of size \(B\), and let \(\mathcal{A}\) denote the action space (for instance, \(\mathcal{A}=\{1,\dots,K+J\}\)). Suppose that each inner maximization is carried out by \(\mathrm{PGD}(T)\), that is, by \(T\) projected-gradient steps. Let \(C_{\mathrm{fwd}}\) and \(C_{\mathrm{bwd}}\) denote the costs of one forward and one backward pass through the score network \(h\), respectively. Then one epoch of RERM-C has computational cost \[\label{eq:epoch-cost95label} n\bigl(1+|\mathcal{A}|T\bigr)\bigl(C_{\mathrm{fwd}}+C_{\mathrm{bwd}}\bigr).\tag{4}\] Moreover, the peak memory requirement is the same as that of a standard forward-backward pass, up to the additional storage needed for one adversarial copy of each input currently being optimized.
We evaluate the robustness of one-stage L2D under adversarial perturbations, focusing on both classification and regression tasks. Our experiments show that standard one-stage L2D baselines can degrade sharply under adversarial perturbations, with attacks affecting both predictive accuracy and deferral decisions. By contrast, our algorithms (RERM-C and RERM-R) improve attacked performance while maintaining competitive clean performance. All results are averaged over three independent runs.
[11] proposes a defense mechanism tailored to a more restrictive two-stage setting, in which only the router is learned while the experts are fixed. As a result, a direct comparison is not meaningful, since the learning setup and objectives differ fundamentally from ours. We therefore focus on standard one-stage L2D baselines; adapting generic adversarial training methods developed for pure classification problems [72] to the augmented L2D action space is beyond the scope of this work.
We report the following evaluation metrics: Clean Accuracy (C.Acc, %) — the overall accuracy of the L2D system, accounting for both model predictions and expert deferrals on clean inputs; Untargeted Accuracy (U.Acc, %) — system accuracy under untargeted adversarial attack; Targeted Accuracy (T.Acc, %) — system accuracy under a targeted attack toward a randomly selected target action; Adversarial Deferral Loss (Def.Loss) — the empirical outcome-wise adversarial deferral loss.
We evaluate on CIFAR-10 [73] and DermaMNIST from the MedMNIST benchmark suite [74]–[77]. As a baseline, we compare against the consistent deferral framework of [2], [18], which represent the standard approach for one-stage L2D. In particular, we instantiate our method with the logistic loss (\(u=1\)) for RERM-C, as introduced in Definition 1. The main paper reports results on CIFAR-10, while additional results on DermaMNIST are deferred to the Appendix.
The augmented classifier is implemented using ResNet-4 [78] and trained with the AdamW optimizer [79] for 400 epochs. As experts, we employ three ResNet-16 models, each trained on a subset of the dataset; their accuracies are reported in Appendix 8.8.2. The consultation costs are set as follows: \(\beta_{j \leq K}=0\) for predictions, and \(\beta_{K+1}=0.05\), \(\beta_{K+2}=0.075\), and \(\beta_{K+3}=0.1\) for the experts. The baseline method uses a learning rate of \(0.005\), while our approach employs a learning rate of \(0.01\). For the PGD attack, we set \(\epsilon=0.03137\), and in our method we additionally use the hyperparameters \(\rho=1.0\) and \(\nu=0.002\).
| C.Acc | U.Acc | T.Acc | Def.Loss | |||
|---|---|---|---|---|---|---|
| [18] | \(82.67 \pm 2.06\) | \(27.47 \pm 0.63\) | \(19.80 \pm 0.48\) | \(0.75 \pm 0.02\) | ||
| Ours | \(75.60 \pm 1.87\) | \(52.00 \pm 1.31\) | \(68.67 \pm 1.70\) | \(0.52 \pm 0.01\) |
Table 1 reports performance on the CIFAR-10 dataset, comparing our approach with [18]. The baseline achieves slightly higher clean accuracy (\(82.67\) vs.\(75.60\)), but this advantage vanishes under adversarial perturbations: its accuracy collapses to \(27.47\) (untargeted) and \(19.80\) (targeted). In contrast, our method preserves robustness, improving untargeted accuracy by more than \(24\%\) and targeted accuracy by nearly \(49\%\). Furthermore, our approach attains a substantially lower empirical adversarial deferral loss under the outcome-wise evaluation metric of Lemma [lem:tlcrobustness]. These results indicate that our surrogate-based methods improve attacked performance while retaining competitive clean performance.
DermaMNIST is a subset of the MedMNIST dataset consisting of biomedical images for 7-class classification. The augmented classifier is implemented using ResNet-18 and trained for 100 epochs. As experts, we construct three specialized classifiers, each responsible for a randomly assigned subset of three classes (with overlap), predicting correctly with probability \(p=0.85\) on their assigned classes and uniformly at random otherwise; their accuracies are reported in the Appendix. The consultation costs are set as \(\beta_{j \leq K}=0\) for predictions, and \(\beta_{K+1}=0.05\), \(\beta_{K+2}=0.075\), and \(\beta_{K+3}=0.125\) for the experts. Both the baseline method and our approach use a learning rate of \(0.005\). For the PGD attack, we set \(\epsilon=0.03137\), while our approach additionally uses the hyperparameters \(\rho=1.75\) and \(\nu=0.001\).
| C.Acc | U.Acc | T.Acc | Def.Loss | |||
|---|---|---|---|---|---|---|
| [18] | \(83.39\) | \(30.82\) | \(27.08\) | \(69.60\) | ||
| Ours | \(81.80\) | \(71.12\) | \(80.65\) | \(31.66\) |
On the DermaMNIST dataset, Table 2 shows that our approach remains close to the baseline under the clean setting. Under adversarial perturbations, our approach preserves substantially higher targeted and untargeted accuracy, while also achieving a lower empirical adversarial deferral loss.
We evaluate our approach on the Communities and Crime dataset [80] and the Insurance Company Benchmark (COIL 2000) [81]. As a baseline, we compare against [14], which represents the standard one-stage L2D approach. For our method, we instantiate the logistic loss (\(u=1\)) within RERM-R, as introduced in Definition 2. The main paper reports results on Communities, while additional results on COIL 2000 are deferred to the Appendix.
The rejector is implemented as an MLP and trained using the AdamW optimizer [79] for 500 epochs. The main predictor is a linear layer. As experts, we employ three MLPs specialized in different socio-economic factors (e.g., demographics, economics, housing), and report their accuracies in the Appendix. The consultation costs are set as \(\beta_{1}=0\) for the main predictor, and \(\beta_{2}=0.04\), \(\beta_{3}=0.05\), and \(\beta_{4}=0.07\) for experts \(m_1, m_2, m_3\). Both the baseline and our method are trained with AdamW; the baseline uses a learning rate of \(0.005\), while our approach employs \(0.01\). For the PGD attack, we set \(\epsilon=0.5\); our method additionally uses the hyperparameters \(\rho=2.5\) and \(\nu=0.005\).
| C.Acc | U.Acc | T.Acc | Def.Loss | |||
|---|---|---|---|---|---|---|
| [14] | \(9.96 \pm 0.21\) | \(18.08 \pm 0.44\) | \(56.74 \pm 1.57\) | \(18.93 \pm 0.40\) | ||
| Ours | \(12.09 \pm 0.39\) | \(12.13 \pm 0.31\) | \(19.67 \pm 0.12\) | \(12.94 \pm 0.26\) |
Table 3 compares our method with [14] on the Communities & Crime dataset. All accuracies are reported as RMSE, where lower values indicate higher accuracy. On clean inputs, our approach performs comparably to the baseline. Under adversarial perturbations, however, our method provides substantial gains: in the targeted setting, RMSE is reduced by nearly \(37\%\), and in the untargeted setting, it remains consistently lower than the baseline. Equally important, our approach achieves a markedly lower deferral loss. This gap suggests that our method attains a more favorable trade-off between predictive error, robustness, and consultation cost under the reported evaluation pipeline.
The rejector is implemented using an MLP and trained with the AdamW optimizer [79] for 25 epochs. The main predictor is a linear layer. As experts, we employ four regression MLPs, each focusing on different customer segments (demographics, product ownership, high-value customers) and generating predictions using rules and noise; their accuracies are reported in the Appendix. The consultation costs are set as follows: \(\beta_{1}=0\) for the main predictor, and \(\beta_2=0.035\), \(\beta_3=0.04\), \(\beta_4=0.045\) and \(\beta_5=0.05\) for the experts. The baseline method uses a learning rate of \(0.005\), while our approach employs a learning rate of \(0.01\). For the PGD attack, we set \(\epsilon=2\), and in our method we additionally use the hyperparameters \(\rho=2.75\) and \(\nu=0.01\).
| C.Acc | U.Acc | T.Acc | Def.Loss | |||
|---|---|---|---|---|---|---|
| [14] | \(7.02\) | \(11.61\) | \(8.31\) | \(11.98\) | ||
| Ours | \(7.39\) | \(7.41\) | \(7.40\) | \(7.81\) |
On the COIL-2000 dataset, our approach remains close to the baseline under the clean setting while improving both attacked RMSE and adversarial deferral loss. This suggests that the approach transfers reasonably well across the reported regression benchmarks.
We presented a framework for adversarial robustness in one-stage Learning-to-Defer (L2D), addressing both classification and regression tasks. Our work makes three key advances: we formalized untargeted and targeted attacks that reveal how adversaries can jointly exploit prediction and deferral; we proposed outcome-wise adversarial surrogate losses with tractable relaxations and established consistency guarantees for the proposed adversarial losses; and we demonstrated empirically, across diverse benchmarks, that our methods improve attacked performance while maintaining competitive clean accuracy.
This research was supported by the National Research Foundation, Singapore, under its AI Singapore Programme (AISG Award No.AISG2-PhD-2023-01-041-J) and by A*STAR, and forms part of the DesCartes programme, which is supported by the National Research Foundation, Prime Minister’s Office, Singapore, under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.
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Definition 3 (Symmetric Hypothesis Class). Let \(\mathcal{A}\) denote the set of possible actions (predictions and deferrals), and let \(Q\) be a class of score-valued hypotheses \(q:\mathcal{X}\to\mathbb{R}^{|\mathcal{A}|}\). We say that \(Q\) is symmetric* if it is closed under permutations of the coordinates indexed by \(\mathcal{A}\), i.e., for any \(q \in Q\) and any permutation \(\Pi:\mathcal{A}\to\mathcal{A}\), the permuted hypothesis \(q^\Pi:\mathcal{X}\to\mathbb{R}^{|\mathcal{A}|}\) defined by \[q^\Pi(x)_j = q(x)_{\Pi^{-1}(j)}, \qquad \forall x \in \mathcal{X}, \;\forall j \in \mathcal{A},\] also belongs to \(Q\).*
definitionrhoconsistency A hypothesis set \(\mathcal{Q}\) is locally \(\rho\)-consistent if, for any \(x \in \mathcal{X}\), there exists a hypothesis \(q \in \mathcal{Q}\) such that: \[\inf_{x' \in B_p(x, \gamma)} |q(x')_i - q(x')_j| \geq \rho,\] where \(\rho > 0\), \(i \neq j \in \mathcal{A}\), and \(x' \in B_p(x, \gamma)\). Moreover, the ordering of the values \(\{q(x')_j\}\) is preserved with respect to \(\{q(x)_j\}\) for all \(x' \in B_p(x, \gamma)\).
lemmaqconsistency Assume \(\mathcal{Q}\) is symmetric and locally \(\rho\)-consistent. Then, for the set \(\mathcal{A}\), any hypothesis \(q \in \mathcal{Q}\), and any distribution \(\mathcal{P}\) with probabilities \(p = (p_1, \ldots, p_{|\mathcal{A}|}) \in \Delta^{|\mathcal{A}|}\), the following inequality holds: \[\begin{align} \sum_{j \in \mathcal{A}} p_j &\sup_{x_j'\in B_p(x,\gamma)} \mathbf{1}\{\hat{q}(x_j')\not=j\} - \inf_{q \in \mathcal{Q}} \sum_{j \in \mathcal{A}} p_j \sup_{x_j'\in B_p(x,\gamma)}\mathbf{1}\{\hat{q}(x_j')\not=j\} \leq \\ & \Psi^u(1) \Big( \sum_{j \in \mathcal{A}} p_j \sup_{x_j'\in B_p(x,\gamma)}\Phi^{\rho,u}_{\text{cls}}(q(x_j'), j) - \inf_{q \in \mathcal{Q}} \sum_{j \in \mathcal{A}} p_j \sup_{x_j'\in B_p(x,\gamma)}\Phi^{\rho,u}_{\text{cls}}(q(x_j'), j) \Big). \end{align}\] with \(\Psi^u(1) = \begin{cases} \log(2), & u=1, \\[0.3em] \tfrac{1}{1-u}\big(2^{1-u}-1\big), & u \neq 1. \end{cases}\)
Proof. Fix a target class \(j\in\mathcal{A}\). Define \[\Phi^{\rho,u}_{\text{cls}}(h(x),j) =\Psi^u\left(\sum_{\substack{j'\in\mathcal{A}\\ j'\neq j}} \Psi_\rho\big(h(x)_{j'}-h(x)_j\big)\right), \qquad \widetilde{\Phi}^{\rho,u}_{\text{cls}}(h,x,j) =\sup_{x'_j\in B_p(x,\gamma)} \Psi^u\left(\sum_{\substack{j'\in\mathcal{A}\\ j'\neq j}} \Psi_\rho\big(h_{j'}(x'_j)-h_j(x'_j)\big)\right).\] We take the exponential link \(\Psi_{\mathrm e}(v)=e^{-v}\) and the \(\rho\)-softening \[\Psi_\rho(v)=\Psi_{\mathrm e}\Big(\frac{v}{\rho}\Big)=\exp\Big(-\frac{v}{\rho}\Big),\qquad \rho>0.\] For \(u>0\) define \[\Psi^{u}(v)= \begin{cases} \log(1+v), & u=1,\\[2pt] \dfrac{(1+v)^{1-u}-1}{1-u}, & u\neq 1, \end{cases} \qquad v\ge 0.\] Then \(\Psi^{u}\) is nondecreasing and \(1\)-Lipschitz on \(\mathbb{R}_+\) since \(\big|\frac{\partial}{\partial v}\Psi^u(v)\big|=\frac{1}{(1+v)^u}\le 1\) for \(v\ge 0\) and \(u>0\). Moreover, \(\Psi_\rho\) is \(\frac{1}{\rho}\)-Lipschitz on \(\mathbb{R}\).
For \(j'\neq j\) define the pairwise margin \[\Delta_h(x,j,j')\;=\;h(x)_j-h(x)_{j'}.\] Collect these \(|\mathcal{A}|-1\) margins into \[\overline{\Delta}_h(x,j) =\big(\Delta_h(x,j,1),\ldots,\Delta_h(x,j,j-1),\Delta_h(x,j,j+1),\ldots,\Delta_h(x,j,|\mathcal{A}|)\big)\in\mathbb{R}^{|\mathcal{A}|-1}.\] Note that \(h(x)_{j'}-h(x)_j=-\Delta_h(x,j,j')\), hence \(\Psi_\rho\big(h_{j'}(\cdot)-h_j(\cdot)\big)=\Psi_\rho\big(-\Delta_h(\cdot,j,j')\big)\).
Using monotonicity and \(1\)-Lipschitzness of \(\Psi^u\) on \(\mathbb{R}_+\), \[\begin{align} \widetilde{\Phi}^{\rho,u}_{\text{cls}}(h,x,j) &=\sup_{x'_j\in B_p(x,\gamma)}\;\Psi^u\left(\sum_{j'\neq j}\Psi_\rho\big(-\Delta_h(x'_j,j,j')\big)\right)\nonumber\\ &\le \Psi^u\left(\sum_{j'\neq j}\Psi_\rho\big(-\Delta_h(x,j,j')\big)\right) +\sup_{x'_j\in B_p(x,\gamma)} \left|\sum_{j'\neq j}\Big(\Psi_\rho\big(-\Delta_h(x'_j,j,j')\big)-\Psi_\rho\big(-\Delta_h(x,j,j')\big)\Big)\right|\nonumber\\ &=\Phi^{\rho,u}_{\text{cls}}(h(x),j)\;+\;\sup_{x'_j\in B_p(x,\gamma)} \sum_{j'\neq j}\left|\Psi_\rho\big(-\Delta_h(x'_j,j,j')\big)-\Psi_\rho\big(-\Delta_h(x,j,j')\big)\right|. \label{eq:step1-h} \end{align}\tag{5}\]
Since \(\Psi_\rho\) is \(\frac{1}{\rho}\)-Lipschitz and by Cauchy–Schwarz, \[\begin{align} \sum_{j'\neq j}\left|\Psi_\rho\big(-\Delta_h(x'_j,j,j')\big)-\Psi_\rho\big(-\Delta_h(x,j,j')\big)\right| &\le \frac{1}{\rho}\sum_{j'\neq j}\big|\Delta_h(x'_j,j,j')-\Delta_h(x,j,j')\big|\nonumber\\ &\le \frac{\sqrt{|\mathcal{A}|-1}}{\rho}\,\big\|\overline{\Delta}_h(x'_j,j)-\overline{\Delta}_h(x,j)\big\|_2. \label{eq:step2-h} \end{align}\tag{6}\] Plugging 6 into 5 gives, for all \(j\), \[\label{eq:key-bound-h} \widetilde{\Phi}^{\rho,u}_{\text{cls}}(h,x,j) \;\le\;\Phi^{\rho,u}_{\text{cls}}(h(x),j)\;+\;\kappa\;\sup_{x'_j\in B_p(x,\gamma)} \big\|\overline{\Delta}_h(x'_j,j)-\overline{\Delta}_h(x,j)\big\|_2, \qquad \kappa\geq\frac{\sqrt{|\mathcal{A}|-1}}{\rho}.\tag{7}\]
Because \(\Psi_\rho(v)=\exp(-v/\rho)=\Psi_{\mathrm e}(v/\rho)\) and \(\Psi^u\) is nondecreasing, \[\Phi^{\rho,u}_{\text{cls}}(h(x),j) =\Psi^u\left(\sum_{j'\neq j}\Psi_{\mathrm e}\left(\frac{h(x)_{j'}-h(x)_j}{\rho}\right)\right) =\Psi^u\left(\sum_{j'\neq j}\Psi_{\mathrm e}\left(\frac{h(x)_{j'}}{\rho}-\frac{h(x)_j}{\rho}\right)\right) =\;\Phi^{u}_{\text{cls}}\left(\frac{h}{\rho},x,j\right),\] where \(\frac{h}{\rho}\) denotes the score map \(x\mapsto h(x)/\rho\) (componentwise). Hence 7 becomes \[\label{eq:final-upper-h} \widetilde{\Phi}^{\rho,u}_{\text{cls}}(h,x,j) \;\le\;\Phi^{u}_{\text{cls}}\left(\frac{h}{\rho},x,j\right)\;+\;\kappa\;\sup_{x'_j\in B_p(x,\gamma)} \big\|\overline{\Delta}_h(x'_j,j)-\overline{\Delta}_h(x,j)\big\|_2,\tag{8}\]
Define the smooth upper-bounding adversarial surrogate \[\widetilde{\Phi}^{\mathrm{smth},u}_{\text{cls}}(h,x,j) =\;\Phi^{u}_{\text{cls}}\left(\frac{h}{\rho},x,j\right)\;+\;\kappa\;\sup_{x'_j\in B_p(x,\gamma)} \big\|\overline{\Delta}_h(x'_j,j)-\overline{\Delta}_h(x,j)\big\|_2,\] Then 8 states, pointwise in \((x,j)\), \[\widetilde{\Phi}^{\rho,u}_{\text{cls}}(h,x,j)\;\le\;\widetilde{\Phi}^{\mathrm{smth},u}_{\text{cls}}(h,x,j).\] ◻
Proof. We first show that we can rewrite the usual score-based loss [2], [18] introduced in Equation 1 with a cost-sensitive formulation depending on a shifted cost \(\mu_j\). Let the following change of variable: \[\label{eq:1} \mu_j(j,m_j,y) = \begin{cases} \alpha_j\mathbf{1}\{j\not=y\} + \beta_j & \text{if } j\leq K \\ \alpha_j\mathbf{1}\{m_{j-K}\not=y\} + \beta_j & \text{if } j> K, \\ \end{cases}\tag{9}\] leading to \[\ell_{\text{def}}^c(\hat{h}(x),y,\mathbf{m}) = \sum_{j=1}^{K+J}\mu_j(j,m_j,y)\mathbf{1}\{\hat{h}(x)=j\}.\]
It is immediate that the formulation reduces to the standard score-based loss when setting \(\alpha_j=1\) and \(\beta_j=0\) for \(j \leq K\). For instance, if \(\mathcal{Y}=\{1,2\}\) and the prediction is \(\hat{h}(x)=1\), then by Equation 9 we obtain \(\ell_{\mathrm{def}}^c(\hat{h}(x),y,\mathbf{m})=\mathbf{1}\{1\neq y\}\), which coincides with the usual score-based loss in Equation 1 .
Now, we upper-bound this loss with the outcome-wise worst-case scenario under the newly cost-sensitive formulation: \[\begin{align} \ell_{\text{def}}^c(\hat{h}(x),y,\mathbf{m}) & \leq \sup_{x' \in B_p(x,\gamma)} \sum_{j=1}^{K+J}\mu_j(j,m_j,y)\mathbf{1}\{\hat{h}(x')=j\} \\ & \leq \sum_{j=1}^{K+J}\mu_j(j,m_j,y)\sup_{x'_j \in B_p(x,\gamma)}\mathbf{1}\{\hat{h}(x'_j)=j\} \\ & = \widetilde{\ell}_{\text{def}}^c(h,x,y,\mathbf{m}) \end{align}\] ◻
Proof. Let \(\mathcal{A}=\{1,\dots,K{+}J\}\). For \(x\in\mathcal{X}\) and radius \(\gamma>0\), write \(B_p(x,\gamma)=\{x'\in\mathcal{X}:\|x'-x\|_p\le\gamma\}\). Given a score vector \(h(x)\in\mathbb{R}^{K+J}\), let the induced decision be \(\hat{h}(x)\in\mathcal{A}\) (e.g., argmax with fixed tie-breaking). We begin by recalling the outcome-wise adversarial deferral loss \[\widetilde{\ell}_{\mathrm{def}}^c(h,x,y,\mathbf{m}) = \sum_{j=1}^{K+J} \mu_j(j,m_j,y)\, \sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)=j\},\] together with its surrogate counterpart introduced in Definition [lem:scrobustness]: \[\widetilde{\Phi}_{\mathrm{def}}^{c,u}(h,x,y,\mathbf{m}) = \sum_{j=1}^{K+J} \sum_{i\neq j} \mu_i(i,m_i,y)\, \sup_{x'_j \in B_p(x,\gamma)} \Phi_{\mathrm{cls}}^{\rho,u}\big(h(x'_j),j\big).\]
Let define the conditional risk associated with the adversarial true loss: \[\begin{align} \mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x) & = \mathbb{E}_{Y\mid X=x}\,\mathbb{E}_{M\mid X=x,Y}\big[\widetilde{\ell}_{\mathrm{def}}^c(h,x,Y,M)\big] \\ & = \sum_{j=1}^{K+J} \mathbb{E}_{Y\mid X=x}\,\mathbb{E}_{M_j\mid X=x,Y}\Big[\mu_j(j,M_j,Y)\Big]\sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)=j\} \\ & = \sum_{j=1}^{K+J} \overline{\mu}_j(x)\sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)=j\} \end{align}\] with \(\overline{\mu}_j(x) = \mathbb{E}_{Y\mid X=x}\,\mathbb{E}_{M_j\mid X=x,Y}\Big[\mu_j(j,M_j,Y)\Big]\). Next, we assume there exist a function \(h \in \mathcal{H}\) that is local-\(\rho\)-consistent (see Lemma [rho95consistent]). For any \(h\) and \(x\), define the reachability set \[\mathcal{H}_\gamma(h,x)\;=\;\Big\{j\in\mathcal{A}:\;\exists\,x'_j\in B_p(x,\gamma)\;\text{s.t.}\;\hat{h}(x'_j)=j\Big\}.\] Then, for each \(j\in\mathcal{A}\), \[\sup_{x'_j\in B_p(x,\gamma)}\mathbf{1}\{\hat{h}(x'_j)=j\} =\mathbf{1}\{\,j\in\mathcal{H}_\gamma(h,x)\,\}.\] By definition, the supremum over \(x'_j\in B_p(x,\gamma)\) of the indicator equals \(1\) iff there exists at least one \(x'_j\) in the ball with \(\hat{h}(x'_j)=j\), i.e., iff \(j\in\mathcal{H}_\gamma(h,x)\); otherwise it is \(0\). Consequently, \[\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x) =\sum_{j=1}^{K+J}\overline{\mu}_j(x)\,\mathbf{1}\{\,j\in\mathcal{H}_\gamma(h,x)\,\}.\] For any \(h\), \(\mathcal{H}_\gamma(h,x)\neq\varnothing\) because \(x\in B_p(x,\gamma)\) and \(\hat{h}(x)\in\mathcal{A}\), hence at least one label is realized in the ball. \[\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x) =\sum_{j\in\mathcal{A}}\overline{\mu}_j(x)\,\mathbf{1}\{j\in\mathcal{H}_\gamma(h,x)\} \;\ge\;\min_{j\in\mathcal{A}}\overline{\mu}_j(x) \mathbf{1}\{\mathcal{H}_\gamma(h,x)\neq\varnothing\} =\min_{j\in\mathcal{A}}\overline{\mu}_j(x).\] Let \(j^\star\in\arg\min_{j\in\mathcal{A}}\overline{\mu}_j(x)\). By local-\(\rho\)-consistency, there exists \(h^{j^\star}\) with \(\hat{h}^{j^\star}(x')=j^\star\) for all \(x'\in B_p(x,\gamma)\). Therefore \(\mathcal{H}_\gamma(h^{j^\star},x)=\{j^\star\}\). \[\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h^{j^\star},x) =\sum_{j\in\mathcal{A}}\overline{\mu}_j(x)\,\mathbf{1}\{j=j^\star\} =\overline{\mu}_{j^\star}(x) =\min_{j\in\mathcal{A}}\overline{\mu}_j(x).\] Combining the lower bound with achievability yields \[\inf_{h \in \mathcal{H}}\;\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x) \;=\mathcal{C}^B_{\widetilde{\ell}_{\mathrm{def}}^c}(\mathcal{H},x) \;= \;\min_{j\in\mathcal{A}}\overline{\mu}_j(x).\] Next, let’s define the calibration gap: \[\begin{align} \Delta\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x) & = \mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x) - \mathcal{C}^B_{\widetilde{\ell}_{\mathrm{def}}^c}(\mathcal{H},x) \\ & = \mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x) - \min_{j\in\mathcal{A}}\overline{\mu}_j(x) \\ & = \sum_{j=1}^{K+J}\Big( \overline{\mu}_j(x)\sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)=j\}\Big) - \min_{j\in\mathcal{A}}\overline{\mu}_j(x) \end{align}\]
Now, for the surrogate, we have: \[\begin{align} \mathcal{C}_{\widetilde{\Phi}_{\mathrm{def}}^{c,u}}(h,x) = \sum_{j=1}^{K+J} \sum_{i\not=j}\overline{\mu}_i(x)\sup_{x'_j \in B_p(x,\gamma)} \Phi_{\mathrm{cls}}^{\rho,u}\big(h(x'_j),j\big) \end{align}\] Let’s define the probability distribution \(p_j(x) = \sum_{i\not=j}\overline{\mu}_i(x)/S(x)\) with \(S(x) =\sum_{j=1}^{K+J}\sum_{k\not=j}\overline{\mu}_k(x)\). It follows the calibration gap and by Lemma [lemma:qconsistency]: \[\begin{align} \Delta\mathcal{C}_{\widetilde{\Phi}_{\mathrm{def}}^{c,u}}(h,x) & = S(x) \Bigg( \sum_{j=1}^{K+J} p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \Phi_{\mathrm{cls}}^{\rho,u}\big(h(x'_j),j\big) - \inf_{h \in \mathcal{H}} \sum_{j=1}^{K+J}p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \Phi_{\mathrm{cls}}^{\rho,u}\big(h(x'_j),j\big)\Bigg) \\ & \geq S(x) [\Psi^u(1)]^{-1} \Bigg( \sum_{j=1}^{K+J} p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)\not=j\}\big) - \inf_{h \in \mathcal{H}} \sum_{j=1}^{K+J}p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)\not=j\}\big)\Bigg) \end{align}\]
We fix \(h\in\mathcal{H}\) and analyze its excess risks at a given \(x\). Define \[\mathcal{E}_p(h,x)=\sum_{j=1}^{K+J}p_j(x)\,\sup_{x'_j\in B_p(x,\gamma)}\mathbf{1}\{\hat{h}(x'_j)\neq j\}.\] Write for conciseness \[\mu_\ast(x)=\min_{j\in\mathcal{A}}\overline{\mu}_j(x),\] and use \[M(x)=\sum_{i=1}^{K+J}\overline{\mu}_i(x), \qquad S(x)=\sum_{j=1}^{K+J}\sum_{k\neq j}\overline{\mu}_k(x)=(K{+}J{-}1)\,M(x),\] so that \(p_j(x)=\dfrac{M(x)-\overline{\mu}_j(x)}{S(x)}\). For a fixed \(h\), \(\sup_{x'_j}\mathbf{1}\{\hat{h}(x'_j)\neq j\}=0\) iff \(j\in \mathcal{H}_\gamma(h,x)\) and \(\hat{h}(x')=j\) for all \(x'\in B_p(x,\gamma)\), and equals \(1\) otherwise. Hence \[\mathcal{E}_p(h,x) =1-\sum_{j\in\mathcal{A}}p_j(x)\,\mathbf{1}\big\{j\in \mathcal{H}_\gamma(h,x)\;\text{and}\;\hat{h} \text{ is constant }j\text{ on }B_p(x,\gamma)\big\}.\] Allowing a predictor that is constant on the ball with the most likely \(p\)-class yields the Bayes value \[\mathcal{E}_p^\star(x)=1-\max_{j\in\mathcal{A}}p_j(x)=1-\frac{M(x)-\mu_\ast(x)}{S(x)}.\] Therefore, \[\label{eq:robust-excess-identity-star} \mathcal{E}_p(h,x)-\mathcal{E}_p^\star(x) =\max_{j}p_j(x)\;-\;\sum_{j\in\mathcal{A}}p_j(x)\,\mathbf{1}\big\{j\in \mathcal{H}_\gamma(h,x)\;\text{and}\;\hat{h} \text{ is constant }j\text{ on }B_p(x,\gamma)\big\}.\tag{10}\]
We therefore have two different cases to inspect.
Case (i): \(h\) is constant on \(B_p(x,\gamma)\) with label \(j_0\).
Then the RHS of 10 equals \(\max_j p_j(x)-p_{j_0}(x)\). Using \(p_j(x)=\frac{M(x)-\overline{\mu}_j(x)}{S(x)}\) and that \(\max_j p_j\) occurs at any minimizer of \(\overline{\mu}_j\), \[\mathcal{E}_p(h,x)-\mathcal{E}_p^\star(x)=\frac{\overline{\mu}_{j_0}(x)-\mu_\ast(x)}{S(x)}.\] Since here \(\mathcal{H}_\gamma(h,x)=\{j_0\}\), the true excess is \(\Delta\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x)=\overline{\mu}_{j_0}(x)-\mu_\ast(x)\), hence \[\label{eq:key-bound-case1-star} \mathcal{E}_p(h,x)-\mathcal{E}_p^\star(x)=\frac{1}{S(x)}\,\Delta\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x).\tag{11}\]
Case (ii): \(h\) is not constant on \(B_p(x,\gamma)\).
Then the second term in 10 vanishes and \[\mathcal{E}_p(h,x)-\mathcal{E}_p^\star(x)=\max_j p_j(x)=\frac{M(x)-\mu_\ast(x)}{S(x)}.\] Since \(\sum_{j\in \mathcal{H}_\gamma(h,x)}\overline{\mu}_j(x)\le M(x)\), we obtain \[\label{eq:key-bound-case2-star} \mathcal{E}_p(h,x)-\mathcal{E}_p^\star(x)\;\ge\;\frac{\sum_{j\in \mathcal{H}_\gamma(h,x)}\overline{\mu}_j(x)-\mu_\ast(x)}{S(x)} =\frac{1}{S(x)}\,\Delta\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x).\tag{12}\]
Combining 11 –12 yields, for all \(h\in\mathcal{H}\), \[\label{eq:robust01-vs-true-star} \mathcal{E}_p(h,x)-\mathcal{E}_p^\star(x)\;\ge\;\frac{1}{S(x)}\,\Delta\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x).\tag{13}\]
Using (13 ), we obtain: \[\begin{align} \Delta\mathcal{C}_{\widetilde{\Phi}_{\mathrm{def}}^{c,u}}(h,x) & = S(x) \Bigg( \sum_{j=1}^{K+J} p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \Phi_{\mathrm{cls}}^{\rho,u}\big(h(x'_j),j\big) - \inf_{h \in \mathcal{H}} \sum_{j=1}^{K+J}p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \Phi_{\mathrm{cls}}^{\rho,u}\big(h(x'_j),j\big)\Bigg) \\ & \geq S(x) [\Psi^u(1)]^{-1} \Bigg( \sum_{j=1}^{K+J} p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)\not=j\}\big) - \inf_{h \in \mathcal{H}} \sum_{j=1}^{K+J}p_j(x)\sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{h}(x'_j)\not=j\}\big)\Bigg) \\ & \geq S(x) [\Psi^u(1)]^{-1} \Bigg(\frac{1}{S(x)}\,\Delta\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x)\Bigg) \\ & = [\Psi^u(1)]^{-1} \Bigg(\Delta\mathcal{C}_{\widetilde{\ell}_{\mathrm{def}}^c}(h,x)\Bigg) \end{align}\]
Applying the expectation \(\mathbb{E}_X[\Delta\mathcal{C}_{\ell}(h,X)] = \mathcal{E}_{\ell}(h) - \mathcal{E}_{\ell}^B(\mathcal{H}) + \mathcal{U}_{\ell}(\mathcal{H})\), leads to the desired inequality: \[\begin{align} & \mathcal{E}_{\widetilde{\ell}^c_{\text{def}}}(h) - \mathcal{E}_{\widetilde{\ell}^c_{\text{def}}}^B(\mathcal{H}) + \mathcal{U}_{\widetilde{\ell}^c_{\text{def}}}(\mathcal{H}) \leq \Psi^u(1) \Big( \mathcal{E}_{\widetilde{\Phi}^{c,u}_{\text{def}}}(h) - \mathcal{E}_{\widetilde{\Phi}^{c,u}_{\text{def}}}^\ast(\mathcal{H}) + \mathcal{U}_{\widetilde{\Phi}^{c,u}_{\text{def}}}(\mathcal{H}) \Big), \end{align}\] with \(\Psi^u(1) = \begin{cases} \log(2), & u=1, \\[0.3em] \tfrac{1}{1-u}\big(2^{1-u}-1\big), & u \neq 1. \end{cases}\) ◻
Proof. We begin by recalling the standard true deferral loss for regression from [14]: \[\ell_{\mathrm{def}}^r\big(\hat{r}(x), f(x), t, \mathbf{m}\big) = \sum_{j=1}^{J+1} c_j^r\big(f(x), m_{j-1}, t\big)\, \mathbf{1}\{\hat{r}(x) = j\}.\] To capture the worst-case scenario, we account for adversarial perturbations of the input while noting that the cost \(c_1^r(f(x),t)\) depends explicitly on \(x\) and that the predictor \(f \in \mathcal{F}\) is trainable: \[\begin{align} \ell_{\mathrm{def}}^r\big(\hat{r}(x), f(x), t, \mathbf{m}\big) &\leq \sup_{x' \in B_p(x,\gamma)} \sum_{j=1}^{J+1} c_j^r\big(f,x, m_{j-1}, t\big)\, \mathbf{1}\{\hat{r}(x') = j\} \\[0.5em] &\leq \sum_{j=1}^{J+1} \tilde{c}_j^r\big(f,x, m_{j-1}, t\big)\, \sup_{x'_j \in B_p(x,\gamma)} \mathbf{1}\{\hat{r}(x'_j) = j\} \\[0.5em] &= \tilde{\ell}_{\mathrm{def}}^r\big(r, f, x, t, \mathbf{m}\big). \end{align}\] ◻
Proof. The outcome-wise adversarial deferral loss for regression is defined as \[\begin{align} \tilde{\ell}^r_{\mathrm{def}}(f,r,x,t,\mathbf{m}) = \sum_{j=1}^{J+1} \tilde{c}_j^r(f,x,m_{j-1},t)\, \sup_{x_j' \in B_p(x,\gamma)} \mathbf{1}\{\hat{r}(x_j') = j\}, \end{align}\] ◻
Process \(n\) training examples in mini-batches of size \(B\). Let \(\mathcal{A}\) be action space ( e.g.\(\mathcal{A}=\{1,\dots,K{+}J\}\) ) and let \(T\in\mathbb{N}\) be the number of projected-gradient steps used in each inner maximization (PGD(\(T\))). Write \(C_{\mathrm{fwd}}\) and \(C_{\mathrm{bwd}}\) for the costs of one forward and one backward pass through the score network \(h\). Then one epoch of RERM-C minimization incurs \[\label{eq:epoch-cost} n\,\bigl(1+|\mathcal{A}|\,T\bigr)\,\bigl(C_{\mathrm{fwd}}+C_{\mathrm{bwd}}\bigr)\tag{14}\] network traversals, while the peak memory equals that of a single forward–backward pass plus the storage of one adversarial copy of each input currently being optimized.
Proof. Consider one mini-batch. RERM-C performs:
one clean forward pass of \(h\) to compute scores \(h(x)\) and the loss terms;
for each \(j\in\mathcal{A}\) and each of the \(T\) PGD steps that update the adversarial proxy \(x'_j\in B_p(x,\gamma)\), one forward and one backward pass of \(h\) (to obtain gradients w.r.t.the input);
one backward pass to update \(\theta\) (the parameter gradient of the batch loss).
Thus, per mini-batch, the total number of network traversals equals \[\underbrace{1}_{\text{clean forward}} +\underbrace{|\mathcal{A}|\,T}_{\text{\mathrm{\small PGD} forwards}} +\underbrace{|\mathcal{A}|\,T}_{\text{\mathrm{\small PGD} backwards}} +\underbrace{1}_{\text{parameter backward}} \;=\;2\bigl(1+|\mathcal{A}|\,T\bigr).\] Multiplying by the number of mini-batches \(\lceil n/B\rceil\) gives the total cost \[\lceil n/B\rceil \cdot 2\bigl(1+|\mathcal{A}|\,T\bigr) \;\approx\; \frac{n}{B}\cdot 2\bigl(1+|\mathcal{A}|\,T\bigr),\] which, when expressed in units of per-example forward/backward costs, yields 14 : each example induces \(\bigl(1+|\mathcal{A}|\,T\bigr)\) forwards and the same number of backwards, for a total of \(n\bigl(1+|\mathcal{A}|\,T\bigr)\bigl(C_{\mathrm{fwd}}+C_{\mathrm{bwd}}\bigr)\). ◻
All experiments were conducted on an internal cluster using an NVIDIA A100 GPU with 80 GB of VRAM.
| Expert | 1 | 2 | 3 |
|---|---|---|---|
| Accuracy | 37.55 | 35.92 | 38.54 |
Analysis in the main paper.
| Expert | 1 | 2 | 3 |
|---|---|---|---|
| Accuracy | 28.48 | 30.52 | 71.72 |
DermaMNIST is a subset of the MedMNIST dataset consisting of biomedical images for 7-class classification. The augmented classifier is implemented using ResNet-18 and trained for 100 epochs. As experts, we construct three specialized classifiers, each responsible for a randomly assigned subset of three classes (with overlap), predicting correctly with probability \(p=0.85\) on their assigned classes and uniformly at random otherwise; their accuracies are reported above. The consultation costs are set as \(\beta_{j \leq K}=0\) for predictions, and \(\beta_{K+1}=0.05\), \(\beta_{K+2}=0.075\), and \(\beta_{K+3}=0.125\) for the experts. Both the baseline method and our approach use a learning rate of \(0.005\). For the PGD attack, we set \(\epsilon=0.03137\), while our approach additionally uses the hyperparameters \(\rho=1.75\) and \(\nu=0.001\).
| C.Acc | U.Acc | T.Acc | Def.Loss | |||
|---|---|---|---|---|---|---|
| [18] | \(83.39\) | \(30.82\) | \(27.08\) | \(69.60\) | ||
| Ours | \(81.80\) | \(71.12\) | \(80.65\) | \(31.66\) |
On the DermaMNIST dataset, Table 7 shows that our approach remains close to the baseline under the clean setting while substantially improving targeted and untargeted attacked accuracy, together with a lower empirical adversarial deferral loss.
| Expert | 1 | 2 | 3 |
|---|---|---|---|
| RMSE | 0.5442 | 1.1373 | 1.5613 |
Analysis in the main paper.
| Expert | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| RMSE | 0.0744 | 0.0747 | 0.0741 | 0.831 |
The rejector is implemented using an MLP and trained with the AdamW optimizer [79] for 25 epochs, while the main predictor is a linear layer. As experts, we employ four regression MLPs, each focusing on different customer segments (demographics, product ownership, high-value customers) and generating predictions using rules and noise; their accuracies are reported above. The consultation costs are set as follows: \(\beta_{1}=0\) for the main predictor, and \(\beta_2=0.035\), \(\beta_3=0.04\), \(\beta_4=0.045\) and \(\beta_5=0.05\) for the experts. The baseline method uses a learning rate of \(0.005\), while our approach employs a learning rate of \(0.01\). For the PGD attack, we set \(\epsilon=2\), and in our method we additionally use the hyperparameters \(\rho=2.75\) and \(\nu=0.01\).
| C.Acc | U.Acc | T.Acc | Def.Loss | |||
|---|---|---|---|---|---|---|
| [14] | \(7.02\) | \(11.61\) | \(8.31\) | \(11.98\) | ||
| Ours | \(7.39\) | \(7.41\) | \(7.40\) | \(7.81\) |
On the COIL-2000 dataset, our approach remains close to the baseline under the clean setting while improving both attacked RMSE and adversarial deferral loss, suggesting that the method transfers reasonably well across the reported regression benchmarks.