Verifiably Robust Conformal Prediction

Authors: linus  jeary
tom  kuipers
mehran  hosseini
nicola  paoletti
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-12 15:34:56

Abstract: Conformal Prediction (CP) is a popular uncertainty quantification method that provides distribution-free, statistically valid prediction sets, assuming that training and test data are exchangeable. In such a case, CP's prediction sets are guaranteed to cover the (unknown) true test output with a user-specified probability. Nevertheless, this guarantee is violated when the data is subjected to adversarial attacks, which often result in a significant loss of coverage. Recently, several approaches have been put forward to recover CP guarantees in this setting. These approaches leverage variations of randomised smoothing to produce conservative sets which account for the effect of the adversarial perturbations. They are, however, limited in that they only support $\ell_2$-bounded perturbations and classification tasks. This paper introduces VRCP (Verifiably Robust Conformal Prediction), a new framework that leverages recent neural network verification methods to recover coverage guarantees under adversarial attacks. Our VRCP method is the first to support perturbations bounded by arbitrary norms including $\ell_1$, $\ell_2$, and $\ell_\infty$, as well as regression tasks. We evaluate and compare our approach on image classification tasks (CIFAR10, CIFAR100, and TinyImageNet) and regression tasks for deep reinforcement learning environments. In every case, VRCP achieves above nominal coverage and yields significantly more efficient and informative prediction regions than the SotA.

Main paper: 
 introduction: 
 Conformal Prediction (CP) (Vovk et al., 2005; Angelopoulos and Bates, 2021 ) is a popular uncertainty quantification method. In essence, it is a model-agnostic, distribution-free framework that allows one to construct prediction sets that are guaranteed to include the true (unknown) output with probability greater than 1 -α, where α ∈ (0, 1) is a user-specified miscoverage/error rate. In other words, for a test point (x n+1 , y n+1 ), CP seeks to construct a prediction set C(x n+1 ) such that the following coverage (a.k.a. validity) guarantee holds:P xn+1,yn+1 [y n+1 ∈ C(x n+1 )] ≥ 1 -α.(1) Importantly, the above guarantee holds when the calibration data, used to construct C(x n+1 ), and the test point are exchangeable (a special case is when calibration and test data are i.i.d.). 2), as well as for our proposed robust algorithms (VRCP-I and VRCP-C) along with the SotA (RSCP+ and RSCP+ (PTT), see Section 3) on the CIFAR10 dataset. As we observe, VRCP-I and VRCP-C closely resemble the spread of vanilla CP prediction set sizes, whilst the SotA falls short of achieving this. Here we use an adversarial perturbation of radius ϵ = 0.02, error rate α = 0.1, number of splits n splits = 50 and smoothing parameter (used in RSCP+ and RSCP+ (PTT)) σ = 2ϵ.When exchangeability is violated, e.g., in the presence of test-time distribution shifts, CP's coverage guarantee (1) ceases to hold, and we cannot rely on the prediction sets it produces. In this work, we address shifts induced by adversarial perturbations on the test inputs. In particular, we focus on perturbations in the form of additive ℓ p -bounded noise.To recover guarantees under adversarial inputs, the general mechanism is to inflate the prediction set to permit larger degrees of uncertainty. However, special care must be taken to avoid producing overly large or even trivial sets -i.e. those containing all possible outputs -as such sets do not provide any useful inference.Contributions We propose a CP framework that provides statistically valid prediction sets despite the presence of ℓ p -bounded adversarial perturbations at inference time. Formally, for any adversarially perturbed test point xn+1 = x n+1 + δ, our method produces adversarially robust sets C ϵ that enjoy the following guarantee:EQUATIONWhile CP uses an underlying predictor f , often a neural network (NN), to construct prediction regions, the novelty of our approach is to leverage NN verification algorithms to compute upper and lower output bounds of f (x ′ ) for any x ′ ∈ B ϵ (x) = {x ′ : ∥x ′ -x∥ p ≤ ϵ}. We use these bounds to inflate the CP regions, resulting in provably robust and efficient prediction sets. To the best of our knowledge, this is the first work that combines NN verification algorithms and CP to construct adversarially robust prediction sets. We call our method VRCP (Verifiably Robust Conformal Prediction).Recent work (discussed in Section 3) achieves adversarially robust coverage using probabilistic methods, specifically, randomised smoothing (Cohen et al., 2019) . Our approach overcomes some of the theoretical and empirical drawbacks of these prior methods, which are restricted to classification tasks with ℓ 2 -norm bounded guarantees and are overly conservative in practice.Thanks to our verification-based approach, VRCP is the first to extend adversarially robust conformal prediction to regression tasks and the first to go beyond ℓ 2 -norm bounded guarantees. In Section 4, we introduce two versions of VRCP that apply verification at calibration and inference time, respectively. Further, in Section 5, we empirically validate our theoretical guarantees and demonstrate a direct improvement over previous work in terms of prediction set efficiency (i.e., average set size) compared to prior work. Fig. 1 shows an extract of our results on CIFAR10, demonstrating that VRCP yields more informative (tighter) prediction regions, a trend that we observe experimentally across all our benchmarks. 
preliminaries: 
 We denote with R + the set of positive real numbers. Vectors x ∈ R d are shown in bold italic and scalars x ∈ R in italic typeface. We denote the norm used to make R d a normed vector space by ∥•∥. This could for instance be ℓ 1 , ℓ 2 , or ℓ ∞ -norm. Whenever a specific norm is intended, we indicate it using an index, e.g., ∥•∥ 2 indicates the ℓ 2 -norm. We denote the ϵ-ball around a point x ∈ R d with respect to the used norm by B ϵ (x).We provide a brief overview of the inductive (or split) vanilla CP approach. Suppose we have a dataset D containing pairs (x, y) sampled i.i.d. from an (unknown) data-generating distribution over a feature space X ⊆ R d and label space Y such that D = {(x 1 , y 1 ), . . . , (x m , y m )}.We partition the dataset into disjoint training and calibration sets D train and D cal , letting n = |D cal |.We fit a predictor f on D train and define a score function S : (X × Y ) → R as some notion of prediction error, such as S(x, y) = ∥f (x) -y∥ when f is a regressor, or S(x, y) = 1 -f (x) y when f is a classifier with f (•) y being y's predicted likelihood.After applying the score function to all calibration points, we construct the score distribution asF = δ∞ /(n+1) + n i=1δs i/n+1, where δ s is the Dirac distribution with parameter s, s i = S(x i , y i ) and δ ∞ represents the unknown score (potentially infinite) of the test point.Given a miscoverage/error rate α and a test point (x n+1 , y n+1 ), we define the prediction set C(x n+1 ) by including all labels that appear sufficiently likely w.r.t. the score distribution:C(x n+1 ) = {y ∈ Y : S(x n+1 , y) ≤ Q 1-α (F )}, where Q 1-α (F )is the 1 -α quantile of F . This set satisfies the marginal coverage guarantee in Eq. ( 1) if the test point and the calibration points are exchangeable.Neural networks have been shown to be vulnerable to adversarial attacks, i.e., small changes to their input that jeopardise the prediction (Szegedy et al., 2014; Biggio and Roli, 2018) . This notion can be formally defined as maximising an adversarial objective function (e.g., the loss of the true label) subject to ∥x -x∥ ≤ ϵ. Alternatively, it can be defined as finding an adversarial example x ∈ R m , such that ∥x -x∥ ≤ ϵ and ∥f ( x) -y∥ ≥ δ for a given neural network f : R d → R n .Various approaches have been proposed to verify the robustness of NNs against adversarial attacks. These approaches can be divided into complete and incomplete algorithms. Given a neural network f , a verifier is complete if it allows computing exact bounds f ⊥ and f ⊤ for the image f (B ϵ (x)) = {f (x ′ ) : x ′ ∈ B ϵ (x)}, i.e., such thatEQUATIONwhere min and max are computed coordinate-wise for vector-valued NNs. A verifier is incomplete, but sound, if it computes bounds that are valid but not exact, i.e., such that:EQUATIONOur results are verifier-agnostic, meaning that they are valid for any verifier that can produce exact bounds (as in Eq. ( 3)) or conservative bounds (as in Eq. ( 4)), depending on the completeness or incompleteness of the verifier used. The fastest and simplest way to compute the bounds in Eq. ( 4) is to propagate the bounds on the input B ϵ (x) through the network to compute the output bounds.Several methods based on this approach have been proposed (Gowal et al., 2018; Wang et al., 2018; Zhang et al., 2018a; Batten et al., 2024; Lopez et al., 2023) . At the expense of fast computation speed, these methods may result in loose bounds in Eq. ( 4). On the other hand, several complete methods (Pulina and Tacchella, 2010; Katz et al., 2017; Hosseini and Lomuscio, 2023) for NN verification have been put forward. Even though these methods compute exact bounds, their downside is their high computational cost. 
related work: 
 Adversarially Robust Conformal Prediction Gendler et al. ( 2021) introduced an algorithm called Randomly Smoothed Conformal Prediction (RSCP) that integrates randomised smoothing (Duchi et al., 2012; Cohen et al., 2019; Salman et al., 2019) with CP to provide robust coverage under adversarial attacks. RSCP replaces the CP score function S(x, y) with a smoothed score S(x, y) obtained by averaging the values of S(x + v, y) over n MC realisations of a Gaussian noise vector v ∼ N (0, σ2 I), for a given smoothing level σ. To correct for potential ℓ 2 -norm bounded ϵ perturbations at inference time, the critical value computed over the smoothed scores is inflated by ϵ/σ. This method produces empirically sound results, but the provided formal guarantees were found to be invalid in a later work (Yan et al., 2023) , as discussed below.Provably Robust Conformal Prediction Yan et al. (2023) address the issue with the robustness guarantee of Gendler et al. (2021) by correctly bounding the estimation error caused by the Monte-Carlo sampling used when generating the smoothed scores. The bound introduces an additional hyperparameter β such that they now find the Q 1-α+2β of smooth calibrated scores and inflate by Hoeffding's bound -ln(β) /2nMC before correcting by ϵ/σ. Furthermore, the smoothed scores of the test points are decreased by an empirical Bernstein bound. This further inflation of the critical value and deflation of smooth scores for each test point often cause their amended algorithm, so-called RSCP+, to generate trivial prediction sets.To address this issue, the authors introduce two methods to improve the efficiency of RSCP+. Firstly, they use robustly calibrated training (RCT), a training-time regularisation technique that penalises NN parameters that contribute to high scores for the true label. Our approach assumes that the underlying classifier is given; hence, we do not evaluate RCT in our experiments.Secondly, they implement a post-training transformation (PTT), which aims to decrease the values of the smoothed calibration scores that lie between Q 1-α (the critical value of the base scores) and Q 1-α (that of the smoothed scores). To this purpose, they transform the CDF of the smoothed scores S by composing learned ranking and sigmoid transformations with hyperparameters b and T using a holdout set D hold sampled i.i.d from D cal . PTT however is not theoretically guaranteed to improve the average set sizes computed by RSCP+ and, in many cases, its efficacy is largely dependent on how representative the sampled holdout set is of the calibration set. We demonstrate the effect of PTT's sample dependence empirically in Section 5.1.Probabilistically Robust Conformal Prediction Ghosh et al. ( 2023) also focus on the adversarial setting but maintain a relaxed form of robust coverage, where input perturbations δ are drawn from a specific distribution and only a proportion of such perturbations are sought to be covered. In contrast, we do not make assumptions about the noise distribution, and we account for any ϵ-bounded perturbation.All the works 2 discussed here rely on randomised smoothing Duchi et al. (2012) and as such are limited to the ℓ 2 -norm. In contrast, our VRCP approach relies on NN verifiers, can be used with any ℓ p -norms supported by the verification method, and does not require smoothing hyperparameters or holdout sets.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [Yes] Justification: Covered in Section 4. Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.• All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
verifiably robust conformal prediction (vrcp): 
 In this section, we formally introduce two variants of VRCP. Both methods allow us to construct adversarially robust prediction sets at inference time.The first variant, VRCP via Robust Inference (VRCP-I), employs NN verification at inference time to compute a lower bound of the score for the given test input (best-case score), thereby obtaining more conservative regions. The calibration procedure is computed as in standard CP. The second variant, VRCP via Robust Calibration (VRCP-C), instead uses NN verification at calibration time to derive upper bounds for the calibration scores (worst-case), thereby obtaining a more conservative calibration threshold (critical value). This allows us to use the regular scores at inference time, without requiring NN verification.Given a calibration set D cal , a test input x n+1 , and score function S(•, •), we compute the prediction set for x n+1 as follows.1. For each y ∈ Y we compute,s ⊥ (x n+1 , y) ≤ inf x ′ ∈Bϵ(xn+1)S(x ′ , y).(5)2. The robust prediction set is then defined asEQUATIONBelow, we show that we are able to maintain the marginal coverage guarantee from Eq. ( 2) for any ℓ p -norm bounded adversarial attack. Theorem 1. Let xn+1 = x n+1 + δ for a clean test sample x n+1 and ∥δ∥ p ≤ ϵ. The prediction set C ϵ ( xn+1 ) defined in Eq. (6) satisfies P [y n+1 ∈ C ϵ ( xn+1 )] ≥ 1 -α.Proof. We obtain thatP [y n+1 ∈ C ϵ ( xn+1 )] = P s ⊥ ( xn+1 , y n+1 ) ≤ Q 1-α (F ) ≥ P inf x ′ ∈Bϵ( xn+1) S(x ′ , y n+1 ) ≤ Q 1-α (F )by Eq. ( 5)≥ P [S(x n+1 , y n+1 ) ≤ Q 1-α (F )] ≥ 1 -α.Given a calibration set D cal , a test input x n+1 , and score function S(•, •), we compute the robustly calibrated prediction set for x n+1 as follows.1. We compute the upper-bound calibration distribution as:EQUATION2. The robust post-calibration prediction set is then defined asC ϵ (x n+1 ) = {y : S(x n+1 , y) ≤ Q 1-α (F ⊤ )}. (8) Theorem 2. Let xn+1 = x n+1 + δ for a clean test sample x n+1 and ∥δ∥ p ≤ ϵ. The prediction set C ϵ ( xn+1 ) defined in Eq. (8) satisfies P [y n+1 ∈ C ϵ ( xn+1 )] ≥ 1 -α.Proof. We have thatP [y n+1 ∈ C ϵ ( xn+1 )] = P S( xn+1 , y n+1 ) ≤ Q 1-α F ⊤ ≥ P S( xn+1 , y n+1 ) ≤ Q 1-α sup x ′ ∈Bϵ(xi) S(x ′ , y i ) n i=1 ∪ {∞} ≥ P sup x ′ ∈Bϵ(xn+1) S(x ′ , y n+1 ) ≤ Q 1-α sup x ′ ∈Bϵ(xi) S(x ′ , y i ) n i=1 ∪ {∞} ≥ 1 -αLet P ⊤ denote the distribution of (x ⊤ , y) where x ⊤ = argsup x ′ ∈Bϵ(x) S(x ′ , y) and (x, y) ∼ P . The final inequality above holds since it is equivalent to constructing a CP prediction set using n i.i.d realisations of P ⊤ and evaluating it on x n+1 ∼ P ⊤ . The resulting set will include the true test output y n+1 with probability at least 1 -α.Classification In the classification setting, we use the score function proposed in (Lei et al., 2013; Gendler et al., 2021) :EQUATIONwhere f (x) y ∈ (0, 1) is the model-predicted likelihood for label y. In this setting, to compute s ⊥ and s ⊤ (required by VRCP-I and VRCP-C, respectively), it suffices to use NN verification algorithms to derive the output bounds for f (x). Specifically, in VRCP-I, for a test input x n+1 and for each y ∈ Y we derive s ⊥ (x n+1 , y) asEQUATIONwhere f (x n+1 ) ⊤ y denotes the upper bound computed by the neural network verifier for the modelpredicted likelihood of label y ∈ Y and input x n+1 .In VRCP-C, for each calibration point (x i , y i ) we compute s ⊤ (x i , y i ) asEQUATIONwhere f (x i )⊥ yi denotes the lower bound of the model output for label y i given input x i .Regression In the regression tasks, we follow the conformalized quantile regression (CQR) methodology proposed by (Romano et al., 2019) . We train quantile regressors f αlow and f αhigh to estimate the α low = α/2 and α high = 1 -α/2 quantiles of y | x. In CQR, we use the following score function:EQUATION)Unlike classification, where the label space is discrete, we cannot construct the region explicitly by enumerating all possible outputs y. Instead, the prediction region for a given test point C(x n+1 ) is constructed implicitly, by adjusting the quantile predictions by the critical value of the calibration distribution Q 1-α (F ), as follows:EQUATIONIn both VRCP-C and VRCP-I, the score function leverages an NN verifier to derive the bounds over the upper and lower quantiles of the model. In VRCP-C, we compute the worst-case calibration scores as:EQUATIONIn VRCP-I for classification, for each output we check inclusion in C ϵ by using the best-case score s ⊥ . As explained above, explicit enumeration is infeasible for regression, and so we construct our robust region by replacing predicted quantiles in Eq. ( 13) with their conservative approximations, as follows:EQUATIONThe above-defined region is equivalent to enumerating all possible outputs y, and for each, considering the best-case score s⊥ (x n+1 , y) = max {f ⊥ αlow (x n+1 ) -y, y -f ⊤ αhigh (x n+1 )}. A proof is available in Appendix A.A nice property of both VRCP-I and VRCP-C is that they guarantee that they can only increase the size of the prediction set for any input x compared to vanilla CP, thus will always attain at least as much coverage as the vanilla CP procedure. Moreover, as we show in Section 5, both algorithms do not trivially inflate the size of the prediction sets and maintain a similar distribution of set sizes. This is formalised in the Proposition 1, which is proved in Appendix A. Proposition 1. Let C(x) and C ϵ (x) be the prediction sets obtained using vanilla CP and VRCP (using VRCP-I or VRCP-C), respectively. For any input x, we have that C(x) ⊆ C ϵ (x).Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: All relevant details provided in Section 5, Appendix B and in our GitHub repository.Guidelines:• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example (a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. (b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. (c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 
evaluation: 
 We evaluate VRCP-I and VRCP-C on classification (image) and regression (RL) benchmarks, and compare them against the SotA approaches on each benchmark. For all the networks used, we did not perform adversarial training as we assume that the attack budget ϵ is unknown at training time. Nonetheless, both our approaches can benefit from adversarial training, as it results in models that are more verifiable and have tighter bounds for the same attack budget.3 We evaluate each method using a nominal coverage of 1 -α = 0.9 and report the 95% confidence intervals for coverage and average set sizes computed over 50 splits (n splits = 50) of the calibration, holdout and test set.Bounds We use the verification library auto_LiRPA (Xu et al., 2020a) to compute the output bounds for f (x) required in Eq. ( 10) and Eq. ( 11) for VRCP-I and VRCP-C respectively. In particular, we use two SotA GPU-parallelised incomplete NN verification algorithms, CROWN Zhang et al. (2018b) and α- CROWN Xu et al. (2020b) . In brief, CROWN performs linear bound propagation and α-CROWN employs a branch-and-bound algorithm to tighten the CROWN bounds at the expense of slower verification times. Therefore, we use CROWN to compute the output bounds for the TinyImageNet model and α-CROWN for the smaller CIFAR10 and CIFAR100 models.Our CIFAR10 model with α-CROWN takes ≈ 0.5s per image to compute bounds with ϵ = 0.03, whereas our larger CIFAR100 model takes ≈ 7.2s with ϵ = 0.02. Comparatively, computing the smoothed scores takes ≈ 0.09s per image to compute on both models under the same respective ϵ values. The largest model for the TinyImageNet dataset uses CROWN to compute bounds at a rate of ≈ 0.2s per image whereas the smoothed scores take ≈ 0.24s. All measurements are made with respect to the hardware details listed in Appendix B.Attacks We use the PGD attack algorithm (Madry et al., 2017) , which is a popular white-box attack algorithm to generate adversarial inputs with respect to either the ℓ 2 or ℓ ∞ -norm. In Table 1 , we benchmark both our methods against the baseline vanilla CP (which is agnostic of the attack), RSCP+ and RSCP+ with PTT. At inference time, images are attacked using PGD to generate ℓ 2 -norm bounded attacks with ϵ = 0.02 for CIFAR100 and TinyImageNet, and ϵ = 0.03 for CIFAR10.In all domains, the vanilla CP method fails to construct valid prediction sets with nominal marginal coverage, as expected. RSCP+ maintains robust marginal coverage but produces trivial prediction sets in all settings due to the highly conservative inflation of the threshold with respect to the calibration scores. Using PTT improves RSCP+'s performance but introduces significant variance in the set sizes: in many cases, PTT still produces trivial prediction sets and is heavily dependent on the sampled holdout set for RSCP+ to generate useful predictions.Both of our methods have minimal sample dependence, as demonstrated by a very small variability in coverage and size over the 50 splits. We obtain prediction sets with substantially smaller average sizes than the other robust approaches, and hence, they provide more informative uncertainty estimates. VRCP-I provides slightly more efficient regions than VRCP-C. Still, it implies additional computational overhead at inference time because it requires computing bounds via NN verification for each test sample. In contrast, in VRCP-C, bounds are computed only once at calibration time. On the other hand, in an environment where we may want to change ϵ for different test points at inference time, VRCP-I would be a sound choice, while VRCP-C would require re-calibration.Effect of increasing adversarial noise Fig. 2b shows the impact of increasing ϵ across all evaluated robust methods. Our methods consistently produce smaller average set sizes with minor sample dependence, and simultaneously provide a more conservative marginal coverage than RSCP+ (PTT). We remark that, unlike RSCP+, we do not require a holdout set or any score function transformations. improves the performances of randomised smoothing approaches, we incur a large computational overhead when computing the smoothed scores. In our experiments in Table 1 we fix n MC = 1024 which is four times larger than the value for n MC used in previous work (Gendler et al., 2021; Yan et al., 2023) as a trade-off between prediction quality and computation. 2 demonstrates that both of our methods generalise to other ℓ pbounded perturbations other than for when p = 2 which RSCP+ is limited to. In particular, we examine the ℓ ∞ , where even a small ϵ can cause misclassification. We experiment using CIFAR10 and use ϵ = 0.001. PGD is used to generate ℓ ∞ -bounded adversarial examples.Set size distribution From Fig. 1 we can visually examine the sample dependency issue that the PTT introduces. In the splits where the holdout set allows the PTT to make an informative transformation, RSCP+ is able to make quite reasonable predictions, otherwise, RSCP+ just returns trivial sets. This is clearly an undesirable property and adds significant variance to the predictions.Both of our methods increase the spread of the average set sizes to account for the presence of adversarial examples whilst still maintaining a consistent distribution.We evaluate our VRCP framework on regression tasks from the PettingZoo Multi-Particle Environment (MPE) library Terry et al. (2021) for deep reinforcement learning. In these environments, the world is a 2D space containing n agents (of which some may be adversarial) and m landmarks, which are defined as circles of fixed radii. The position of the landmarks is fixed, and agents traverse the space according to second-order motion laws. We evaluate our method on three tasks:• Adversary The good agents must try to reach a specific goal landmark whilst avoiding the adversaries. We use 2 good agents, 1 adversary and 2 landmarks.• Spread All agents collaborate and minimise the distance to each landmark. We set the number of agents and landmarks equal to 3.• Push In this task, there is a single good agent, adversary and landmark. The task is for the adversary to maximise the distance between the landmark and the good agent.In our experiments, for data-generation we select 5,000 random initial world configurations and, for each, simulate 25 Monte-Carlo trajectories of length k = 5. The regression task for all environments is to predict the upper and lower quantiles of the total cumulative reward over the k steps, given as input the initial world state. As in the classification experiments, we partition the dataset into the following partitions:|D train | = 1,000, |D cal | = 2,000 and |D test | = 2,000.For computing the bounds, we use CROWN Zhang et al. (2018b) with ℓ ∞ -bounded perturbations. To generate the adversarially perturbed test points, we use the Fast Gradient Sign Method as given in (Goodfellow et al., 2015) .As seen in Table 3 , both VRCP methods recover the marginal coverage guarantees in the presence of adversarial perturbations, whereas vanilla CP fails drastically after ϵ = 0.02. We note that the performance of VRCP-C and VRCP-I are similar, although VRCP-I tends to produce more conservative intervals (without sacrificing efficiency).Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [Yes] Justification: All code used for experiments is available in our GitHub repository.Guidelines:• The answer NA means that paper does not include experiments requiring code. • The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
limitations: 
 VRCP's scalability depends on that of the underlying neural network verifier. We evaluated VRCP on small to medium-sized neural networks. For large networks, existing complete verification methods would become computationally infeasible, while incomplete methods would produce bounds that are too loose to be useful. However, it is important to note that since VRCP is agnostic of the specific verification tool used, it would directly benefit from any future advances in neural network verification. Thus, as neural network verification tools continue to evolve and improve, so does VRCP. 
conclusion: 
 We introduced Verifiably Robust Conformal Prediction (VRCP), a novel framework that leverages conformal prediction and neural network verification to produce prediction sets that maintain marginal coverage under adversarial perturbations. We presented two variants: VRCP-C, which applies verification at calibration time, and VRCP-I, which applies verification at inference time.Extensive experiments on classification and regression tasks demonstrated that VRCP recovers valid marginal coverage in the presence of ℓ 1 , ℓ 2 , and ℓ ∞ -norm bounded adversarial attacks while producing more accurate prediction sets than existing methods. VRCP is the first adversarially robust CP framework supporting regression tasks and perturbations beyond the ℓ 2 -norm, achieving strong results without relying on probabilistic smoothing or posthoc corrections. VRCP's theoretical guarantees and empirical performance showcase the potential of leveraging verification tools for uncertainty quantification of machine learning models under attack.Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] Justification: All experiments in Section 5 provide a 95% confidence interval taken over 50 splits. Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
experiments compute resources: 
 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes] Justification: Compute resources discussed in Section 5 and Appendix B Guidelines:• The answer NA means that the paper does not include experiments. Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
broader impacts: 
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safeguards: 
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 Appendix: Here we prove Proposition 1 regarding the prediction sets obtained from VRCP-I and VRCP-C.Proof of Proposition 1. To prove C(x) ⊆ C ϵ (x) for VRCP-I it suffices to observe thatC ϵ (x) = {y ∈ Y : s ⊥ (x, y) ≤ Q 1-α (F )} ⊇ x ′ ∈Bϵ(xn+1) {y ∈ Y : S(x ′ , y) ≤ Q 1-α (F )} = x ′ ∈Bϵ(xn+1) C(x ′ ) ⊇ C(x).To prove the same for VRCP-C, we observe that sinceQ 1-α (F ⊤ ) ≥ Q 1-α (F ), we have that C ϵ (x) = {y ∈ Y : S(x, y) ≤ Q 1-α (F ⊤ )} ⊇ {y ∈ Y : S(x, y) ≤ Q 1-α (F )} = C(x).Next, we prove the validity of the VRCP-I region for the regression case, defined in Eq. ( 15).Proof. It suffices to show that all y ∈ C ϵ (x n+1 ) = [f ⊥ -q, f ⊤ + q] satisfy s ⊥ (x n+1 , y) = max {f ⊥ -y, y -f ⊤ } ≤ q and all y ̸ ∈ C ϵ (x n+1 ) do not. For simplicity of notation, we abbreviatedf ⊤ αhigh (x n+1 ) with f ⊤ , f ⊥ αlow (x n+1 ) with f ⊥ and Q 1-α (F ) with q. Assume y ∈ C ϵ (x n+1). We divide the proof into two cases:1. s ⊥ (x n+1 , y) = f ⊥ -y, which implies that y ∈ [f ⊥ -q, f ⊤ +f ⊥ 2 ]. It suffices to show that f ⊥ -y ≤ q for y = f ⊥ -q, which is clearly satisfied. 2. s ⊥ (x n+1 , y) = y -f ⊤ , which implies that y ∈ [ f ⊤ +f ⊥ 2 , f ⊤ + q]. It suffices to show that y -f ⊤ ≤ q for y = f ⊤ + q, which is clearly satisfied.Finally, we show that y ̸ ∈ C ϵ (x n+1 ) implies s ⊥ (x n+1 , y) > q: if y < f ⊥ -q, we have that s ⊥ (x n+1 , y) = f ⊥ -y > q. Similarly, if y > f ⊥ + q, we have that s ⊥ (x n+1 , y) = y -f ⊤ > q.All experimental results were obtained from running the code provided in our GitHub repository on a server with 2x Intel Xeon Platinum 8360Y (36 cores, 72 threads, 2.4GHz), 512GB of RAM and an NVIDIA A40 48GB GPU. All pre-trained models as well as the training scripts are also provided in the GitHub repository. In summary, the models' train and test performances are provided in Tables 4 and 5 . CIFAR100 We use 1 convolution layer with average pooling, 2 further convolution layers with average pooling and dropout followed by 2 linear layers. ReLU activations across all layers.TinyImageNet We use 4 convolution layers with dropout followed by 2 linear layers with dropout. Leaky ReLU activation function with a = 0.1For all models we train using images augmented with random crop with 4 pixels of padding and random horizontal flip. We standardise the TinyImageNet models with µ = 0.5 and σ = 0.5 overall 3 RGB channels.As previously mentioned, we do not make any assumptions during training about the perturbations we expect to see at inference time. As such, unlike the existing SotA methods, we do not train on smoothed or adversarially attacked images.All models are trained for 200 epochs with a batch size of 128 using the stochastic gradient descent optimiser with momentum set to 0.9. We also employ a weight decay of 5 × 10 -4 and a cosine annealing learning rate scheduler.For the MPE datasets, we train Deep Q-Net policies for the RL tasks for the sole purposes of generating the appropriate datasets and provide these policies in the GitHub repository.The model used for the quantile regressors is a simple linear architecture consisting of 3 layers, separated with ReLU activation functions and dropout. We trained the model to estimate the α/2 and 1 -α/2 quantiles, where α = 0.1, as in the other experiments.The exact parameters for the RL policies can be found in the config files within the GitHub repository, however have little bearing on the efficiency of our results, being used only for the data-generating process. The quantile regressors are each trained for 400 epochs, with a learning rate of 10 -5 , dropout of 0.1 and a decay of 10 -5 . Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.Justification: There is no significant societal impact of the work provided outside of the fact that we provide a robust approach to defend against adversarial attacks.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.• If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).We are aware of related contemporaneous work byZargarbashi et al. (2024). However, at the time of submission, neither the manuscript nor the code were available.Code for the experiments is available at: https://github.com/ddv-lab/Verifiably_Robust_CP 


Decision: This paper considers the problem of providing valid prediction sets using conformal prediction in the presence of adversarial perturbations. This is an active research area and the paper makes a good contribution by approaching this problem through the lens of neural network verification tools and is applicable to both classification and regression tasks (demonstrated by experiments).

The reviewers' appreciated the new ideas in the paper but also raised some good questions. Authors' responses were satisfactory to address those questions.

Therefore, I recommend accepting the paper and strongly encourage the authors' to incorporate all the discussion in the camera copy to further improve the paper.

Reviews: review: paper_summary: paper_summary: This submission proposes VRCP, a framework for verifiably robust conformal prediction under Lp-bounded adversarial attacks. The framework integrates existing bound propagation tools for verification of conformal predictions. Two variants, VRCP-I that triggers bound computation at inference time and VRCP-C that triggers bound computation at calibration time, are proposed. VRCP-I has inference overhead but the set size is smaller. Experiments on CIFAR10, CIFAR100, and a regression task demonstrate the effectiveness.
strength_weakness: strength_weakness: 1. The framework is relatively straightforward. The foundation is some probability relaxations that can be intuitively derived. I would not view this intuitiveness as a weakness. However, it would be great if the authors could discuss some extensions and optimizations, e.g., proof sharing for VRCP-C, alternative but more verification-friendly score function design, etc.

2. Some in-depth discussion could benefit the submission. Concretely, why does VRCP work with different Lp norms? Does the benefit come from the existing verifier's flexibility? Does the bound tightness differ with different Lp norms? Why does the method surpass RSCP?

3. The experimental setup is not very clear: What is the model's normal accuracy? Could you provide more background on the setup of regression experiments especially the physical meaning of the reward bound and nominal performance? The setting is not very common in the literature.

Minor:
Line 129-130: the lengthy sentence seems to be lacking a period or comma.
questions: questions: See weaknesses.
limitations: limitations: Yes, authors discuss the limitations in the last paragraph of the main text.
paper_summary: paper_summary: The paper provides a verifiably robust conformal prediction method via neural network verification. It considers two paradigms for considering the perturbation either at the calibration stage or the inference stage. They also consider the validity of method for both classification and regression.
strength_weakness: strength_weakness: 1. Novelty: I understand that it is a new angle that combines NN verification and conformal prediction to provide a certifiably robust conformal prediction framework, but I wonder technically, can we do more about the combination. For example, can we have new conformity score adapted to NN verification that can provide tighter bound? Or at least empirically, we can analyze what types of conformity scores are more suitable for randomized certification? What are more suitable for deterministic certification (NN verification).

2. What is the intuition that the method can outperform RSCP+ in Table 1? Since usually randomized smoothing provides better certification than NN verification, then coming to the conformal prediction context, what makes the difference?

3. Missing related work: Kang, Mintong, et al. COLEP: Certifiably Robust Learning-Reasoning Conformal Prediction via Probabilistic Circuits. ICLR 2024.
questions: questions: Please refer to the weakness part. Overall, I do not have many concerns about the paper's soundness. If the authors can provide more insights into comparisons between randomized certification and deterministic certification in robust conformal prediction, it would be beneficial to the community. Basically, the authors need more support of why NN verification-based conformal prediction is worthy besides it can provide a general $\ell_p$ norm guarantee.
limitations: limitations: discussed in Sec 6.1
paper_summary: paper_summary: The authors propose a novel method to recover coverage guarantees for conformal predictions in the presence of adversarial attacks. Unlike previously proposed approaches, the authors directly leverage verifiable methods for NN to compute prediction scores. Through empirical tests, the authors prove the benefit of the proposed approach against attacks bounded by different norms.
strength_weakness: strength_weakness: 1. In the paper, the authors focus on PDG. However, the literature offers more advanced and sophisticated attacks. It would be beneficial to assess the robustness of the proposed approach against other type of attacks. 

2. Since it is not specified in the paper, I assume the ML model under attack has not been trained using adversarial training. The reader might wonder whether the benefit of the proposed approach remains valid in the presence of an adversarially trained model. Concerning the latter case, will vanilla CP still violate the coverage guarantees?

3. Figure 2 is hard to read. Which curve represents the proposed approach, and which one represents the benchmarks?
Concerning Figure 2a: Why does the coverage go to one as the magnitude of the adversarial perturbation goes to one? Perhaps it is because all the methods converge to a trivial prediction set. However, this should be clearly explained in the text.
Concerning Figure 2b: Why does the coverage decrease as a function of the number of samples?
questions: questions: See Weaknesses
limitations: limitations: Limitations are discussed

Comments: title: Thanks for the rebuttal
comment: Thank you for your response.

> Sorry the question may not be clear enough. I was asking whether the gap between verifier's score bound and actual score minimum / maximum for perturbed inputs can become larger/smaller among different Lp norms.

Yes, it differs between different $\ell_p$ norms. For example, $\ell_\infty$ results in larger gaps for the same attack budget $\epsilon$. This is because the $\ell_\infty$ ball contains other $\ell_p$ balls. However, it should be noted that verifiers tend to over-approximate non-linear norms (e.g., $\ell_2, \ell_3, \dots$) more than linear norms (e.g., $\ell_1$ and $\ell_\infty$). This means for non-linear norms, we may get larger gaps than linear norms.

We hope this answers your questions.
Thanks for the response! Most of my concerns are resolved. So I maintain my score.

>  Does the bound tightness differ with different Lp norms?

Sorry the question may not be clear enough. I was asking whether the gap between verifier's score bound and actual score minimum / maximum for perturbed inputs can become larger/smaller among different Lp norms.
I would like to thank the authors to take the time to write the rebuttal to address my concerns. 

Although I appreciate the paper motivation, I think that the contribution is limited. I will keep my previous score.
Thank the authors for the rebuttal. Overall, I think this paper is technically sound and the first combination work of NN verification and conformal certification. I will maintain a borderline because the motivation for using it instead of RSCP is not very strong besides the arbitrary $\ell_p$ norm. Although, RSCP requires many samples, this work is also not efficient due to NN verification. I also do not think RSCP can not be tailored for regression tasks.