SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Ignacio Hounie
Luiz F. O. Chamon
When deploying machine learning solutions, they must satisfy multiple requirements beyond accuracy, such as fairness, robustness, or safety. These requirements are imposed during training either implicitly, using penalties, or explicitly, using constrained optimization methods based on Lagrangian duality. Either way, specifying requirements is hindered by the presence of compromises and limited prior knowledge about the data. Furthermore, their impact on performance can often only be evaluated by actually solving the learning problem. This paper presents a constrained learning approach that adapts the requirements while simultaneously solving the learning task. To do so, it relaxes the learning constraints in a way that contemplates how much they affect the task at hand by balancing the performance gains obtained from the relaxation against a user-defined cost of that relaxation. We call this approach resilient constrained learning after the term used to describe ecological systems that adapt to disruptions by modifying their operation. We show conditions under which this balance can be achieved and introduce a practical algorithm to compute it, for which we derive approximation and generalization guarantees. We showcase the advantages of this resilient learning method in image classification tasks involving multiple potential invariances and in heterogeneous federated learning.
Requirements are integral to engineering and of growing interest in machine learning (ML) [1]. This growing interest is evident in, e.g., the advancement towards designing ML systems that are fair [2], robust [3], and safe [4], as well as numerous applications in which we want to attain good performance with respect to more than one metric [5-24]. Two concrete applications that we will use as examples (Section 5) are heterogeneous federated learning, where each client realizes a different loss due to distribution shifts (as in, e.g., [18-20]) and invariant learning, where we seek to achieve good performance even after the data has undergone a variety of transformations (as in, e.g., [15, 21-23]).
The goal in these settings is to strike a compromise between some top-line objective metric and the requirements. To this end, an established approach is to combine the top-line and requirement violation metrics in a single training loss. This leads to penalty methods that are ubiquitous in ML, as attested by, e.g., fairness [25-28] and robustness [29, 30] applications. Another approach to balance objective and requirements is formulating and solving constrained learning problems. Though less typical in ML practice, they are not uncommon [5-18, 21, 24]. In particular, they have also been used in fair [31-34] and robust [12, 35] learning, to proceed with a common set of applications. It is worth noting that penalty and constrained methods are not unrelated. They are in fact equivalent in convex optimization, in the sense that every constrained problem has an equivalent penalty-based formulation. A similar result holds in non-convex ML settings for sufficiently expressive parametrizations [7, 8].
In either case, the compromise between objective and different requirements are specified by (hyper)parameters, be they penalty coefficients or constraint levels (Section 2). Finding penalties or constraints specifications that yield reasonable trade-offs is particularly difficult in ML, which often
involves statistical requirements, such as fairness and robustness, that have intricate dependencies with the model and unknown data distributions. Case in point, consider invariant learning for image classification [23]. While we know that invariance to rotations and translations is desirable, we do not know how much invariance to these transformations is beneficial. This depends on the level of invariance of the data distribution, its prevalence in the dataset, and the capability of the model to represent invariant functions. The standard solution to this problem involves time consuming and computationally expensive hyperparameter searches.
This paper addresses this issue by automating the specification of constraint levels during training. To do so, it begins by interpreting constraints as nominal specifications that can be relaxed to find a better compromise between objective and requirements (Section 3). We call this approach resilient constrained learning after the term used to describe ecological systems that adapt to disruptions by modifying their operation [36, 37]. Our first contribution is the following insight:
(C1) We relax constraints according to their relative difficulty, which we define as the sensitivity of the objective to perturbations of the constraint (Section 3.1).
That difficult constraints should be relaxed more is a natural choice. The value of (C1) is in defining what is a difficult constraint. We then seek constraint levels such that the objective loss is relatively insensitive to changes in those levels. This relative insensitivity incorporates a user-defined cost that establishes a price for relaxing nominal specifications.
The learning problem implied by (C1) seems challenging. Our next contribution is to show that it is not:
(C2) We use duality and perturbation theory to present reformulations of the resilient learning problem from (C1) (Section 3.2) that lead to a practical resilient learning algorithm (Section 4) for which we derive statistical approximation bounds (Thm. 1).
Our final contribution is the experimental evaluation of the resilient learning algorithm:
(C3) We evaluate resilient formulations of federated learning and invariant learning (Section 5).
Our experiments show that (C1)-(C2) effectively relaxes constraints according to their difficulty, leading to solutions that are less sensitive to the requirement specifications. It illustrates how resilient learning constitutes an interpretable and flexible approach to designing requirements while contemplating performance trade-offs.
2 Learning with Constraints
Let  $\mathcal{D}_0$  be a distribution over data pairs  $(\mathbf{x},y)$  composed of the feature vector  $\mathbf{x} \in \mathcal{X} \subset \mathbb{R}^d$  and the corresponding output  $y \in \mathcal{Y} \subset \mathbb{R}$ . Let  $f_{\theta}: \mathcal{X} \to \mathbb{R}^{k}$  be the function associated with parameters  $\theta \in \Theta \subset \mathbb{R}^{p}$  and  $\ell_0: \mathbb{R}^k \times \mathcal{Y} \to [-B,B]$  be the loss that evaluates the fitness of the estimate  $f_{\theta}(\mathbf{x})$  relative to  $y$ . Let  $\mathcal{F}_{\theta} = \{f_{\theta} \mid \theta \in \Theta\}$  be the hypothesis class induced by these functions. Different from traditional (unconstrained) learning, we do not seek  $f_{\theta}$  that simply minimizes  $\mathbb{E}[\ell_0(\phi(\mathbf{x}),y)]$ , but also account for its expected value with respect to additional losses  $\ell_i: \mathbb{R}^k \times \mathcal{Y} \to [-B,B]$  and distributions  $\mathcal{D}_i$ ,  $i = 1,\dots,m$ . These losses/distributions typically encode statistical requirements, such as robustness (where  $\mathcal{D}_i$  denote distribution shifts or adversarial perturbations) and fairness (where  $\mathcal{D}_i$  are conditional distributions of protected subgroups).
Explicitly, the constrained statistical learning (CSL) problem is defined as
$$
\mathsf {P} ^ {\star} = \min  _ {\theta \in \Theta} \quad \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {0}} \left[ \ell_ {0} \left(f _ {\theta} (\mathbf {x}), y\right) \right] \tag {P}
$$
$$
\text {s u b j e c t} \quad \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {i}} \left[ \ell_ {i} \left(f _ {\theta} (\mathbf {x}), y\right) \right] \leq 0, \quad i = 1, \dots , m.
$$
Without loss of generality, we stipulate the nominal constraint specification to be zero. Other values can be achieved by offsetting  $\ell_{i}$ , i.e.,  $\mathbb{E}\big[\tilde{\ell} (f_{\theta}(\mathbf{x}),y)\big]\leq c$  is obtained using  $\ell_i(\cdot) = \tilde{\ell} (\cdot) - c$  in (P). We also let  $\mathsf{P}^{\star}$  take values on the extended real line  $\mathbb{R}\cup \{\infty \}$  by defining  $\mathsf{P}^{\star} = \infty$  whenever (P) is infeasible, i.e., whenever for all  $\theta \in \Theta$  there exists  $i$  such that  $\mathbb{E}_{\mathcal{D}_i}\left[\ell_i(f_\theta (\mathbf{x}),y)\right] > 0$ .
A challenge in formulating meaningful CSL problems lies in specifying the constraints, i.e., the  $\ell_i$ . Indeed, while a solution  $f_{\theta^{\star}}$  always exists for unconstrained learning  $[m = 0$  in (P)], there may be no  $\theta$  that satisfies the constraints in (P). This issue is exacerbated when solving the problem using data: even arbitrarily good approximations of the expectations in (P) may introduce errors that hinder the estimation of  $\mathsf{P}^{\star}$  (see Appendix A for a concrete example). In practice, landing on feasible requirements may require some constraints to be relaxed relative to their initial specification. Then, in lieu of (P), we would use the relaxed problem
$$
\mathsf {P} ^ {\star} (\mathbf {u}) = \min  _ {\theta \in \Theta} \quad \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {0}} \left[ \ell_ {0} \left(f _ {\theta} (\mathbf {x}), y\right) \right] \tag {P_u}
$$
$$
\text {s u b j e c t} \quad \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {i}} \left[ \ell_ {i} \left(f _ {\theta} (\mathbf {x}), y\right) \right] \leq u _ {i}, \quad i = 1, \dots , m,
$$
where  $\mathbf{u} \in \mathbb{R}_+^m$  collects the relaxations  $u_i \geq 0$ . The value  $\mathsf{P}^\star(\mathbf{u})$  is known as the perturbation function of (P) since it describes the effect of the relaxation  $\mathbf{u}$  on the optimal value. Given  $\mathsf{P}^\star(\mathbf{0}) = \mathsf{P}^\star$  and  $(\mathbf{P_0})$  is equivalent to (P), this abuse of notation should not lead to confusion.
It is ready that  $\mathsf{P}^{\star}(\mathbf{u})$  is a componentwise non-increasing function, i.e., that for comparable arguments  $v_{i}\leq w_{i}$  for all  $i$  (denoted  $\mathbf{v}\preceq \mathbf{w}$ ), it holds that  $\mathsf{P}^{\star}(\mathbf{v})\geq \mathsf{P}^{\star}(\mathbf{w})$ . However, large relaxations  $\mathbf{u}$  drive  $(\mathrm{P_u})$  away from (P), the learning problem of interest. Thus, relaxing (P) too much can be as detrimental as relaxing it too little (see example in Appendix A). The goal of this paper is to exploit properties of  $\mathsf{P}^{\star}(\mathbf{u})$  together with a relaxation cost to strike a balance between these conflicting objectives. We call this balance a resilient version of (P).
3 Resilient Constrained Learning
We formulate resilient constrained learning using a functional form of  $(\mathrm{P_u})$ . Explicitly, consider a convex function class  $\mathcal{F} \supseteq \mathcal{F}_{\theta}$  and define the relaxed functional problem
$$
\tilde {\mathbf {P}} ^ {\star} (\mathbf {u}) = \min  _ {\phi \in \mathcal {F}} \quad \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {0}} \left[ \ell_ {0} (\phi (\mathbf {x}), y) \right] \tag {P_u}
$$
$$
\text {s u b j e c t} \quad \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {i}} \left[ \ell_ {i} (\phi (\mathbf {x}), y) \right] \leq u _ {i}, \quad i = 1, \dots , m.
$$
The difference between  $(\mathrm{P_u})$  and  $(\tilde{\mathrm{P}}_{\mathbf{u}})$  is that the latter does not rely on a parametric model. Instead, its solutions take values on the convex space of functions  $\mathcal{F}$ . Still, if  $\mathcal{F}_{\theta}$  is a sufficiently rich parameterization of  $\mathcal{F}$  (see Section 4 for details), then  $\mathsf{P}^{\star}(\mathbf{u})$  and  $\tilde{\mathsf{P}}^{\star}(\mathbf{u})$  are close [38, Sec. 3]. Throughout the rest of the paper, we use  $\sim$  to identify these functional problems.
The advantage of working with  $(\tilde{\mathbf{P}}_{\mathbf{u}})$  is that under mild conditions the perturbation function  $\tilde{\mathbb{P}}^{\star}$  is convex. This holds readily if the losses  $\ell_i$  are convex (e.g., quadratic or cross-entropy) [39, chap. 5]. However, the perturbation function is also convex for a variety of non-convex programs, e.g.,  $(\tilde{\mathbf{P}}_{\mathbf{u}})$  with non-convex losses, non-atomic  $\mathcal{D}_i$ , and decomposable  $\mathcal{F}$  (see Appendix B). For conciseness, we encapsulate this hypothesis in an assumption.
Assumption 1. The perturbation function  $\hat{\mathrm{P}}^{\star}(\mathbf{u})$  is a convex function of the relaxation  $\mathbf{u}\in \mathbb{R}_+^m$ .
A consequence of Assumption 1 is that the perturbation function  $\tilde{\mathbb{P}}^{\star}$  has a non-empty subdifferential at every point. Explicitly, let its subdifferential  $\partial \tilde{\mathbb{P}}^{\star}(\mathbf{u}^{o})$  at  $\mathbf{u}^{o} \in \mathbb{R}_{+}^{m}$  be defined as the set of vectors describing supporting hyperplanes at  $\mathbf{u}^{o}$  of the epigraph of  $\tilde{\mathbb{P}}^{\star}$ , i.e.,
$$
\partial \tilde {\mathbf {P}} ^ {\star} (\mathbf {u} ^ {o}) = \left\{\mathbf {p} \in \mathbb {R} _ {+} ^ {m} \mid \tilde {\mathbf {P}} ^ {\star} (\mathbf {v}) \geq \tilde {\mathbf {P}} ^ {\star} (\mathbf {u} _ {o}) + \mathbf {p} ^ {T} (\mathbf {v} - \mathbf {u} _ {o}), \text {f o r a l l} \mathbf {v} \in \mathbb {R} _ {+} ^ {m} \right\}. \tag {1}
$$
The elements of  $\partial \tilde{\mathsf{P}}^{\star}(\mathbf{u}^{o})$  are called subgradients of  $\tilde{\mathsf{P}}^{\star}$  at  $\mathbf{u}^o$ . If  $\tilde{\mathsf{P}}^{\star}$  is differentiable at  $\mathbf{u}^o$ , then it has a single subgradient that is equal to its gradient, i.e.,  $\partial \tilde{\mathsf{P}}^{\star}(\mathbf{u}^{o}) = \{\nabla \tilde{\mathsf{P}}^{\star}(\mathbf{u}^{o})\}$ . In general, however, the subdifferential is a non-singleton set [40]. Further notice that since  $\tilde{\mathsf{P}}^{\star}(\mathbf{u})$  is componentwise nonpositive the subgradients are componentwise negative,  $p_{\mathbf{u}} \preceq \mathbf{0}$ .
Next, we use this property to formalize resilient constrained learning as a compromise between reducing  $\tilde{\mathbf{P}}^{\star}(\mathbf{u})$  by increasing  $\mathbf{u}$  and staying close to the original problem  $(\tilde{\mathbf{P}}_0)$ .
(b)
Figure 1: Resilient equilibrium from Def. 1 for  $h(u) = u^2 / 2$ . The shaded area indicates infeasible specifications: (a) nominal specification ( $u = 0$ ) is feasible and easy to satisfy; (b) nominal specification is feasible but difficult to satisfy (close to infeasible); (c) nominal specification is infeasible.
3.1 Resilient Equilibrium
Consider the effect of increasing the value of a specific relaxation  $u_{i}$  in  $(\tilde{\mathbf{P}}_{\mathbf{u}})$  while keeping the rest unchanged. The solution of  $(\tilde{\mathbf{P}}_{\mathbf{u}^{\prime}})$  is allowed to suffer higher losses on the constraints  $(\ell_{i}$  for  $i \geq 1$ , which may lead to a smaller objective loss  $(\ell_0)$ . Hence, while larger relaxations are detrimental because they violate more the requirements, they are also beneficial because they reduce the objective loss. To balance these conflicting outcomes of constraint relaxations, we introduce a function  $h$  to capture their costs. Then, since relaxing both increases the costs and decreases the objective value, we conceptualize resilience as an equilibrium between these variations.
Definition 1 (Resilient Equilibrium). Let  $h: \mathbb{R}_+^m \to \mathbb{R}_+$  be a convex, differentiable, normalized (i.e.,  $h(0) = 0$ ), and componentwise increasing (i.e.,  $h(\mathbf{v}) < h(\mathbf{w})$  for  $\mathbf{v} \prec \mathbf{w}$ ) function. A resilient equilibrium of  $(\tilde{\mathrm{P}}_{\mathbf{u}})$  is a relaxation  $\mathbf{u}^\star$  satisfying
$$
\nabla h \left(\mathbf {u} ^ {\star}\right) \in - \partial \tilde {\mathrm {P}} ^ {\star} \left(\mathbf {u} ^ {\star}\right). \tag {2}
$$
The resilient constrained learning problem amounts to solving  $(\tilde{\mathbf{P}}_{\mathbf{u}^{\star}})$ , i.e., solving  $(\tilde{\mathbf{P}}_{\mathbf{u}})$  for a relaxation  $\mathbf{u}^{\star}$  that satisfies (2). We call this equilibrium resilient because it describes how far  $(\tilde{\mathbf{P}}_0)$  can be relaxed before we start seeing diminishing returns. Indeed,  $\mathbf{u}^{\star}$  from (2) is such that relaxing by an additional  $\epsilon \succ 0$  would incur in a relaxation cost at least  $\nabla h(\mathbf{u}^{\star})^{T}\epsilon$  larger, whereas tightening to  $\mathbf{u}^{\star} - \epsilon$  would incur in an optimal value increase of at least the same  $\nabla h(\mathbf{u}^{\star})^{T}\epsilon$ .
Notice that resilient constrained learning is defined in terms of sensitivity. Indeed, the resilient equilibrium in Def. (1) specifies a learning task that is as sensitive to changes in its requirements as it is sensitive to changes in the relaxation cost. This has the marked advantage of being invariant to constant translations of  $\ell_0$ , as is also the case for solutions of  $(\tilde{\mathbf{P}}_{\mathbf{u}})$ . Sensitivity also measures the difficulty of satisfying a constraint, since  $\partial \tilde{\mathbb{P}}^{\star}(\mathbf{u})$  quantifies the impact of each constraint specification on the objective loss. Hence, the equilibrium in (2) has the desirable characteristic of affecting stringent requirements more.
Two important properties of the equilibrium in Def. 1 are summarized next (proofs are provided in appendices D.1 and D.2).
Proposition 1. Under Ass. 1, the resilient equilibrium (2) exists. If  $h$  is strictly convex, it is unique.
Proposition 2. Let  $\mathbf{v},\mathbf{w}\in \mathbb{R}_+^m$  be such that  $[\mathbf{v}]_i = [\mathbf{w}]_i,$  for  $i\neq j$  , and  $[\mathbf{v}]_j <   [\mathbf{w}]_j$  . Under Ass. 1, (i)  $[\nabla h(\mathbf{v})]_j\leq [\nabla h(\mathbf{w})]_j$  and (ii)  $[-{\bf p}_v]_j\geq [-{\bf p}_w]_j$  for all  ${\bf p}_v\in \partial \tilde{\mathrm{P}}^\star (\mathrm{v})$  and  $\mathbf{p}_w\in \partial \tilde{\mathrm{P}}^\star (\mathbf{w})$
Prop. 1 shows that the equilibrium in (2) is well-posed. Prop. 2 states that, all things being equal, relaxing the  $j$ -th constraint increases the sensitivity of the cost  $h$  to it, while simultaneously decreasing its effect on the objective value  $\tilde{\mathbb{P}}^{\star}$ . To illustrate these points better, Fig. 1 considers prototypical learning problems with a single constraint  $[m = 1$  in  $(\tilde{\mathbb{P}}_{\mathbf{u}})]$ , differentiable  $\tilde{\mathbb{P}}^{\star}(u)$ , and relaxation cost  $h(u) = u^{2} / 2$ . According to (2), the resilient relaxations are obtained at  $h'(u^{\star}) = u^{\star} = \tilde{\mathbb{P}}^{\star \prime}(u^{\star})$  where we let  $g'(u) = dg(u) / du$  denote the derivative of the function  $g$ . As per Prop. 2,  $h'$  is increasing and  $-\tilde{\mathbb{P}}^{\star \prime}$  is decreasing. Further observe that the sensitivity  $-\tilde{\mathbb{P}}^{\star \prime}(u)$  diverges as  $u$  approaches the value that makes the problem infeasible and vanishes as the constraint is relaxed. These two curves must therefore intersect, claimed by Prop. 1.
The illustrations in Fig. 1 represent progressively more sensitive/difficult problems. In Fig. 1a, the nominal problem  $(u = 0)$  is easy to solve (small  $\tilde{\mathbb{P}}^{\star \prime}(0)$ ), making the resilient equilibrium  $u_{a}^{*}\approx 0$ . The original problem and the relaxed problem are essentially equivalent. In Fig. 1b, the nominal problem is difficult to solve (large  $\tilde{\mathbb{P}}^{\star \prime}(0)$ ), inducing a significant change in the constraint and objective losses. In Fig. 1c, the nominal problem is unsolvable, but the resilient relaxation recovers a feasible problem.
Having motivated the resilient compromise in Def. 1 and proven that it is well-posed, we proceed to obtain equivalent formulations that show it is also computationally tractable. These formulations are used to show traditional learning tasks that can be seen as resilient learning problems (Sec. 3.3), before deriving a practical algorithm to tackle the resilient learning problem  $(\tilde{\mathrm{P}}_{\mathbf{u}^{\star}})$  (Sec. 4).
3.2 Equivalent Formulations
While we have shown that the resilient relaxation from Def. 1 exists (Prop. 1), the equilibrium in (2) does not provide a straightforward way to compute it. In this section, we show two more computationally amenable reformulations of Def. 1 by relating  $\mathbf{u}^{\star}$  to the Lagrange multipliers of  $(\tilde{\mathrm{P}}_{\mathbf{u}})$  and to the solution of a related optimization problem.
Let  $\lambda \in \mathbb{R}_+^m$  collect multipliers  $\lambda_{i}$  associated to the  $i$ -th constraint of  $(\tilde{\mathrm{P}}_{\mathbf{u}})$  and define the Lagrangian
$$
\mathcal {L} (\phi , \boldsymbol {\lambda}; \mathbf {u}) = \mathbb {E} _ {\mathcal {D} _ {0}} [ \ell_ {0} (\phi (\mathbf {x}), y) ] + \sum_ {i = 1} ^ {m} \lambda_ {i} [ \mathbb {E} _ {\mathcal {D} _ {i}} [ \ell_ {i} (\phi (\mathbf {x}), y) ] - u _ {i} ]. \tag {3}
$$
Based on (3), define dual functions  $g(\lambda; \mathbf{u})$  and dual problems  $\tilde{\mathrm{D}}^*(\mathbf{u})$  for given constraint level  $\mathbf{u}$ ,
$$
\tilde {\mathbf {D}} ^ {\star} (\mathbf {u}) = \max  _ {\boldsymbol {\lambda} \succeq \mathbf {0}} g (\boldsymbol {\lambda}; \mathbf {u}) = \max  _ {\boldsymbol {\lambda} \succeq \mathbf {0}} \min  _ {\phi \in \mathcal {F}} \mathcal {L} (\phi , \boldsymbol {\lambda}; \mathbf {u}). \quad \left(\tilde {\mathbf {D}} _ {\mathbf {u}}\right)
$$
While  $\tilde{\mathsf{D}}^{\star} \leq \tilde{\mathsf{P}}^{\star}$  (weak duality) in general, there are cases in which  $\tilde{\mathsf{D}}^{\star} = \tilde{\mathsf{P}}^{\star}$  (strong duality), e.g., in convex optimization. The constrained  $(\tilde{\mathbf{P}}_{\mathbf{u}})$  is then essentially equivalent to  $(\tilde{\mathbf{D}}_{\mathbf{u}})$  that can be tackled by solving an unconstrained, penalized problem (minimizing (3) with respect to  $\phi$  and  $\mathbf{u}$ ), while adapting the weights  $\lambda_{i}$  of the penalties (maximizing (3) with respect to  $\lambda$ ). This is, in fact, the basis of operation of primal-dual constrained optimization algorithms [39, Chapter 5]. The penalties  $\lambda^{\star}(\mathbf{u})$  that achieve this equivalence, known as Lagrange multipliers, are solutions of  $(\tilde{\mathbf{D}}_{\mathbf{u}})$  and subgradients of the perturbation function. Strong duality also holds under the milder Assumption 1 and a constraint qualification requirement stated next.
Assumption 2. There exist a finite relaxation  $\mathbf{u} \preceq \infty$  and a function  $\phi \in \mathcal{F}$  such that all constraints are met with margin  $c > 0$ , i.e.,  $\mathbb{E}_{\mathcal{D}_i}[\ell_i(\phi(\mathbf{x}), y)] \leq u_i - c$ , for all  $i = 1, \dots, m$ .
Note that since  $\mathbf{u}$  can be large, this requirement is mild. We can thus leverage strong duality to provide an alternative definition of the resilient relaxation in (2).
Proposition 3. Let  $\lambda^{\star}(\mathbf{u})$  be a solution of  $(\tilde{\mathrm{D}}_{\mathbf{u}})$  for the relaxation  $\mathbf{u}$ . Under Assumptions 1 and 2,  $\tilde{\mathrm{P}}^{\star}(\mathbf{u}^{*}) = \tilde{\mathrm{D}}^{\star}(\mathbf{u}^{*})$  and any  $\mathbf{u}^{\star} \in \mathbb{R}_{+}^{m}$  such that  $\nabla h(\mathbf{u}^{\star}) = \lambda^{\star}(\mathbf{u}^{\star})$  is a resilient relaxation of  $(\tilde{\mathrm{P}}_{\mathbf{u}})$ .
See appendix D.3 for proof. Whereas Def. 1 specifies the resilience relaxation in terms of the marginal performance gains  $\partial \tilde{\mathbb{P}}^{\star}$ , Prop. 3 formulates it in terms of the penalties  $\lambda$ . Indeed, it establishes that the penalty  $\lambda^{\star}(\mathbf{u}^{\star})$  that achieves the resilient relaxation of  $(\tilde{\mathrm{P}}_{\mathbf{u}})$  is encoded in the relaxation cost as  $\nabla h(\mathbf{u}^{\star})$ . That is not to say that  $h$  directly weighs the requirements as in fixed penalty formulations, since the equilibrium in Prop. 3 also depends on the relative difficulty of satisfying those requirements. Though Prop. 3 does not claim that all resilient relaxations satisfy the Lagrange multiplier relation, this holds under additional continuity conditions on  $(\tilde{\mathrm{P}}_{\mathbf{u}})$  [40].