Constrained Multi-objective Bayesian Optimization

Venue: NeurIPS_cc_2024_Workshop_BDU
Published: 2024-09-02 05:51:45

Abstract: Multi-objective Bayesian optimization has been widely adopted in scientific experiment design, including drug discovery and hyperparameter optimization. In practice, regulatory or safety concerns often impose additional thresholds on certain attributes of the experimental outcomes. Previous work has primarily focused on constrained single-objective optimization tasks or active search under constraints. We propose CMOBO, a sample-efficient constrained multi-objective Bayesian optimization algorithm that balances learning of the feasible region (defined on multiple unknowns) with multi-objective optimization within the feasible region in a principled manner. We provide both theoretical justification and empirical evidence, demonstrating the efficacy of our approach on various synthetic benchmarks and real-world applications.

Main paper: 
 introduction: 
 Multi-objective Bayesian optimization (MBO) is essential in scientific experiment design, such as drug discovery [Fromer and Coley, 2023] and hyper-parameter optimization [Gardner et al., 2019] , where efficient exploration of experimental space is crucial. However, real-world applications often require meeting additional safety or regulatory thresholds. For instance, drug discovery must balance therapeutic effectiveness with safety standards [Mellinghoff and Cloughesy, 2022] , and hyper-parameter optimization must avoid overfitting or violating constraints [Karl et al., 2023] .Previous studies have focused on single-objective constrained Bayesian optimization, unconstrained multi-objective Bayesian optimization, or constrained active search, leaving a gap in constrained multi-objective Bayesian optimization (CMOBO). Research in Constrained Bayesian Optimization (CBO) has primarily extended unconstrained problems with early work by Schonlau et al. [1998] and subsequent studies incorporating posterior sampling methods [Eriksson and Poloczek, 2021] and information-based approaches [Hernández-Lobato et al., 2014, Wang and Jegelka, 2017] to scale CBO and improve feasibility analysis [Hernández-Lobato et al., 2015 , Perrone et al., 2019 , Takeno et al., 2022] . The augmented Lagrangian framework further transformed constrained tasks into unconstrained ones [Gramacy et al., 2016 , Picheny et al., 2016 , Ariafar et al., 2019] , though it often lacked guarantees on feasibility and regret. Recent advancements [Zhou and Ji, 2022 , Lu and Paulson, 2022 , Xu et al., 2023 , Guo et al., 2023] focus on relaxed CBO objectives to ensure theoretical convergence. In Constrained Active Search, active learning for the level sets offers sample efficiency guarantees [Gotovos et al., 2013] but struggles with multiple unknown functions. Recent approaches [Malkomes et al., 2021 , Komiyama et al., 2022] emphasize diversity but fall short of balancing learning constraints with objective optimization. Multi-objective Bayesian Optimization [Golovin and Zhang, 2020 , Daulton et al., 2022 , Suzuki et al., 2020] typically relies on the scalarization of objectives combined and resort to generalized expected improvements. A principled integrated treatment of the constraints and multiple objectives in Bayesian optimization remains challenging.We propose a sample-efficient constrained multi-objective Bayesian optimization (CMOBO) algorithm that balances learning of level sets on multiple unknowns with multi-objective optimization within feasible regions. The insight is that we constrain the search space to areas with the potential of being feasible, while the random scalarization [Golovin and Zhang, 2020] allows an efficient and theoretically justified acquisition within the region. We offer theoretical justification, together with empirical evidence on both synthetic benchmarks and real-world applications, demonstrating the effectiveness and efficiency of the proposed method. 
problem statement: 
 Consider a constrained multi-objective optimization problem:max x∈X {f 1 (x), f 2 (x), ..., f m (x)} := F (x) s.t. g i (x) ≥ 0, ∀i ∈ [c]where f i and g j :X → R, ∀i ∈ [m], ∀j ∈ [c] are black-box functions, X ⊂ R d is the search space of F : R d → R m ,m is the number of objectives and c is the number of constraints. We denote [c] = {1, ..., c} and [m] = {1, ..., m}. The goal is to find the Pareto frontier P of F . For the main paper, we assume the search space X to be finite, and extend our discussion of the continuous and compact search space in Appendix D. Definition 2.1 (Pareto Front). Define the feasible region of this problem asF = {x|x ∈ X and g i (x) ≥ 0, ∀i ∈ [c]}. For points x 1 , x 2 , in the context of this problem, x 1 , x 2 ∈ F. x 1 is said to dominate x 2 if 1) f i (x 1 ) ≥ f i (x 2 ), ∀i ∈ [m] and 2) ∃j ∈ [m] s.t. f j (x 1 ) > f j (x 2 ).Denote X * = {x ∈ F|x is not dominated by any point ∈ F}, then the Pareto front is defined asP = {F (x)|x ∈ X * }.Assumption 1 (Gaussian Process). Following Srinivas et al. [2009] (shown in Definition B.5), we assume that objectives and constraint functions are drawn from their GP priors.Definition 2.2 (Simple Hypervolume Regret). Let simple hypervolume (HV) regret at step t, r t be the difference between the real hypervolume indicator of the Pareto front and the current approximation of the Pareto front. Define Y t to be a set of observed objectives, s.t.|Y t | = t r t = HV z (P ) -HV z (Y t ∩ F (F))where F (F) is the range of the objective in feasible area, and HV z (Y ) is defined as vol({y|y ≥ z, y is dominated by some point ∈ Y })(x ≥ y means term-by-term inequality). Maximizing HV z (Y t ∩ F (F)) reflects the exploration of the Pareto Front since it cannot be greater than H z (P ). Definition 2.3 (Simple Constraint Violation). We define the simple constraint violation of constraint g j at the t th observation x t as v j,t = [-g j (x t )] + where [•] + = max(0, •). Definition 2.4 (Constraint Regret). Based on the above definitions, we define a metric that considers both Constraint Violations and Hypervolume Regret as Constraint Regret at step t as cr t , the minimum of the sum of simple HV regret and simple constraint violation of all objectives among steps before t, which was originally proposed in Xu et al. [2023] cr t = min τ ∈{1,...,t}   r τ + c j=1 v j,τ   In the experiments, we rescale r τ and c j=1 v j,τ to the same range.3 Constrained Multi-Objective Bayesian Optimization Definition 3.1 (Acquisition Function). Our acquisition function in iteration t is defined as s θt (U t (x)), where s θ (y) = min i (max(0, yi θi )) m proposed in Golovin and Zhang [2020] . θ t is uniformly sampled form space S + m-1 = {y ∈ R m |∥y∥ = 1, y ≥ 0}. U t (x) = (u 1,t (x)z 1 , ..., u m,t (x) -z m ). z = (z 1 , ..., z m ) is a chosen sub-optimal reference point. u i,t (x), w j,t (x) are upper confidence bounds of f i , g j based on Srinivas et al. [2009] , and the detailed definition of them are in B.7.Algorithm 1 Constrained Multi-Objective Bayesian Optimization (CMOBO) Update GP posterior with the incoming observations. 9: end for In each step of the algorithm, we apply a random scalarization, s θt , to the upper confidence bound surrogates of m objectives. We substitute the c constraint functions g j with their upper confidence bound surrogates, following the approach of Xu et al. [2023] , and maximize the scalarized function subject to the new constraints. In line 2, we solve an auxiliary optimization problem, the solution of which helps determine whether to declare infeasibility. It can also be used in the subsequent optimization of the acquisition. See A for further discussion. Trade-off Between Simple HV Regret and Cumulative Violation. As shown in Figure 2 in Appendix E, while qNEHVI converges on simple HV regret at a rate comparable to CMOBO and achieves a higher final value, it incurs significantly more constraint violations. In more complex objectives, qNEHVI's simple HV performance is near-random, though its constraint violations remain sub-random. Scalarization-based qParEGO surpasses random search in Simple HV regret at around 30 steps, with near-random over-all constraint violation performance. In this case, the Constraint Violation performance is dominated by Simple HV regret. This trade-off, measured by constraint regret (Definition 2.4), reflects the ability to explore the Pareto Front within feasible regions. High constraint regret indicates failure to explore or maintain constraints. CMOBO outperforms all benchmarks, achieving the best balance with faster constraint regret reduction. 
theoretical results: 
 In this section, we provide the theoretical analysis of our algorithm CMOBO. We leverage the maximum mutual information gain γ i,T of the i th objective on GP after T iterations, and γ T is an upper bound for γ i,T , ∀i ∈ [m] and γ j,T , ∀j ∈ [c]. Detailed definitions are shown in B.6. The corresponding upper bounds for common kernels are previously studied by Srinivas et al. [2009] .Case 1: F ̸ = ∅. With assumptions stated in Section 2.1, given F ̸ = ∅ and the existence of the Pareto Front, we could bound the cumulative regret and corresponding violation in the following, with detailed proof deferred to Appendix C. Theorem 1 (Cumulative HV regret bound). After T iterations, under the conditions in Lemma 1, for δ ∈ (0, 1) our Algorithm CMOBO satisfies thatEQUATIONwith probability at least 1 -δ.Theorem 2 (Cumulative constraint violation bound). Let c denote the number of constraints, then ∀j ∈ [c] and under the conditions in Lemma 1, with probability at least 1 -δ, our Algorithm CMOBO satisfies thatEQUATIONCombining Theorem 2 and Theorem 1, we can bound Constraint Regret defined in Definition 2.4 by O(cm 2 [γ T ln T /T ] 1/2 ) by taking the sum of two cumulative terms and taking minimum with respect to step indexes.Case 2: F = ∅. Now we assume F = ∅. We conclude that we can declare infeasibility in (1) in Algorithm 1 within a certain number of steps with high probability.Theorem 3 (Declaration of infeasibility when the problem is infeasible). With conditions in Lemma 1, and that lim T →∞√ ln T γ T √ T = 0. If the problem is infeasible, i.e. ∃j ∈ [c], max x g j (x) = ϵ < 0 Then, given δ ∈ (0, 1), Algorithm 1 will declare infeasibility within number of steps equivalent to T = min T ∈N + {T | √ ln T γ T √ T≤ Cϵ} with probability at least 1 -δ. 
conclusions: 
 We proposed CMOBO, a stochastic scalarization-based Bayesian optimization algorithm for constrained multi-objective problems. CMOBO outperforms the qParEGO in various tests and excels in metrics that account for hypervolume regret and constraint violations, even if it doesn't always match qNEHVI in hypervolume regret convergence speed. We also provide theoretical analysis of the metrics used. x t = arg max E F (x) [ HV z (Y t )|x ∈ F].qParEGO Daulton et al.[2020] applys an random augmented Chebychev scalarization s θt (y) = min i∈[m] θ i,t (y i -z i ) in each step to the objectives and uses GP to model the scalarized outcomes Knowles [2006] denoted as sθt (x). Then it applies conditional expected improvement (EI) to surrogate of scalarized objective.x t = arg max E F (x) [s θt (x) -s * θt ] + |x ∈ F = arg max E F (x) θ t [s θt (x) -s * θt ] + × I(x ∈ F)where I(x) is an indicator for x being feasible, also approximated by surrogates of g j , s * θt is the best observation of scalarized objective in current step.Inner Loop Optimization when Generalized to Continuous Search Space. Some of the test objectives has continuous search space, so we can leverage the information of the maxima of auxiliary function proposed in Line 2 of Algorithm 1. BoTorch uses L-BFGS algorithm to conduct the optimization with non-linear constraints in Line 6 if the search space is continuous, which requires sufficient initial candidates of feasible solutions. The default candidate sampler of BoTorch cannot always generate sufficient candidates due to the complexity of constraints. We show in the analysis(Theorem 4) that the solutions of the auxiliary problem can be used as the initial candidates for L-BFGS with high probability. We will provide a discussion in D to show the algorithm could be applied to general continuous objectives.To Match The Parallel Settings. Though the benchmark algorithms were designed for batchedoutput, we take number of queries q = 1 in each step to make them comparable to our approach, which is also seen in the experiments in Daulton et al. [2021] . We can still benefit from the parallel evaluation in optimizing the acquisition function.Penicillin Function. The objective was proposed in Liang and Lai [2021] . We add a Gaussian noise with standard deviation 0.05 to the observations. We define the constraint to make penicillin production ≥ 10, CO 2 production ≤ 60 and reaction time ≤ 350.We take β j,t = c log(t + 1), where c = 0.1 or c = 0.05. We used the RBF kernelk RBF (x 1 , x 2 ) = a • exp ∥x 1 -x 2 ∥ 2 bWe fit the GP models' parameters in each step in all experiments.Since the distribution of objective is highly skewed, we observed high over-fitting of GP model even with large training data. To fix this, we apply a filtering trick to the initial samples of the GP model of each benchmark. We apply Voxel Grid Sampling trick to 64 randomly sampled candidates to get around 20 samples uniformly distributed in the range of initial candidate set in each trial. That would make the performances of all benchmarks differentiable to random search.Toy Function. The objective is defined as:F (x 1 , x 2 ) = - 1 x 1 -x 2 , -x 1 -x 2 s.t. x 1 , x 2 ∈ [1, 1.5] - 1 x 1 -x 2 ≥ -1.9, -x 1 -x 2 2 ≥ -2.25β j,t = 0.4 log (4 • (1 + t)). We used Matérn kernel for the GP model with 0.05 standard deviation noise. We took 10 random initial candidates in each trial.k Matérn (x 1 , x 2 ) = 2 1-ν Γ(ν) √ 2νd ν K ν √ 2νd • d = (x 1 -x 2 ) ⊤ Θ -2 (x 1 -x 2 )is the distance between x 1 and x 2 , scaled by the lengthscale parameter Θ.• ν is a smoothness parameter that takes values 1 2 , 3 2 , or 5 2 . Smaller values correspond to less smoothness.• K ν is a modified Bessel function.Branin-Currin Function. A 2-D objective consisting of a Branin function and a Currin function.f 1 = 15x 2 -5.1 • (15x 1 -5) 2 4π 2 + 5(15x 1 -5) π -5 2 + 10 - 10 8π cos(15x 1 -5) f 2 = 1 -exp - 1 2x 2 • 2300x 3 1 + 1900x 2 1 + 2092x 1 + 60 100x 3 1 + 500x 2 1 + 4x 1 + 20 s.t. x 1 , x 2 ∈ [0, 1] f 1 ≥ -20, f 2 ≥ -6β j,t = 0.4 log (4 • (1 + t)). We use Matérn kernel with 0.01 standard deviation and 10 random initial candidates in each trial.Caco-2++. The original version Caco-2+(d = 2133, m = 3) was proposed in Park et al. [2024] , whose objective contains permeability, an experimentally tested value and 2 extra objectives, Crip-penClogP, TPSA. The search space contains 906 drug molecules. We modified this dataset to come up with Caco-2++(d = 2175, m = 3, c = 3). For the domain, we augmented a new feature, mqn feature, to the domain of Caco-2+. We changed the objectives to permeability, TPSA, drug-likeliness score(QED). The search space contains 909 molecules. We constrain the objectives so that QED ≥ 0.5, TPSA ≥ 80, permeability ≥ -5.β j,t = c log (1 + t), c = 0.1 or 0.05. We used Tanimoto kernel specialized for molecule representation. We take a 0.01 standard deviation observation noise and 64 initial candidates in each trial.k Tanimoto (x 1 , x 2 ) = a • x 1 • x 2 ∥x 1 ∥ 2 + ∥x 2 ∥ 2 -x 1 • x 2 ESOL+.The original ESOL (Delaney [2004] ) dataset contains 1,144 organic molecules and an experimentally measured metric called log(Solubility). We added three additional objectives-LogP, TPSA, and QED-to the original objective. For the domain, we use the same molecule representation, fragprint, as was used by Caco-2+ (d = 2133). We constrain the objectives so that LogP ≥ 2.5, QED ≥ 0.5, TPSA ≥ 55, and log(Solubility) ≥ -4. β j,t = c log (1 + t), c = 0.1 or 0.05. We used Tanimoto kernel and a 0.005 standard deviation observation noise and 64 initial candidates in each trial. 
 
 Appendix: Definition B.1 (Scalarized Regret).r θ (X t ) = max x∈F s θ (F (x) -z) -max x∈Xt s θ (F (x) -z) (3) Definition B.2 (Bayes Regret). is the expectation of scalarized regret over all possible θ ∼ S + k-1 : R(X t ) = E θ [r θ (X t )](4) Definition B.3 (Instantaneous Regret). is defined as:r(x t , θ t ) = max x∈F s θt (F (x) -z) -s θt (F (x t ) -z) (5)where θ t and x t are the parameter and corresponding observation in the t th iteration of our algorithm. Definition B.4 (Cumulative Regret). is the cumulative sum of instantaneous regret: ∼ Gaussian(0, σ 2 ), ∀i ∈ [m] models the noise of observation with fixed variance σ 2 . Assumef i ∼ GP(0, k i (•, •)), where k i (x, x ′ ) is a kernel function that models the covariance of f i (.) at any two observations x, x ′ . A predictive distribution of f i (x) given t observations is the posterior distribution: Srinivas et al. [2009] , we further assume k(•, •) ≤ 1.EQUATIONf i (x)|(y 1 i , ..., y t i ) ∼ Gaussian(µ i,t (x), σ 2 i,t (x)). Like• µ i,t (x) = k i (x 1:t , x) T (K i + λI) -1 y 1:t i • σ 2 i,t (x) = k i (x, x) -k i (x 1...t , x) T (K i + λI) -1 k i (x 1...t , x) • K i = [k i (x, x ′ ) x,x ′ ∈{x 1 ,...,x t } ] ∈ R t×t • k i (x 1:t , x) = [k i (x 1 , x), ..., k i (x t , x)] T • λ = σ 2Similarly, we define the posterior mean and variance function for constraint functions g j , ∀j ∈ [c] as µ ′ j,t (x) and σ 2 j,t (x). Definition B.6 (Maximum Mutual Information). The information gain of of a set of newly sampled points A t whose corresponding set of objective values is Y t , is the mutual information of the distribution of f i and the distribution of Y t . Both f i and Y t follow the assumption of noise observation and Gaussian process with kernel k(•, •) in B.5.EQUATIONwhere H(•) is the Shannon entropy function. The closed form of I(Y t ; f i ) is given by Srinivas et al.[2009]:1 2 log det(I +λ -1 K i,t ) where K i,t = [k i (x, x ′ ) x,x ′ ∈At ].Accordingly, the maximum mutual information for objective f i given t observations is defined as:EQUATIONSimilarly, we define γ j,t for constraint functionsEQUATIONEQUATIONEQUATIONwhere β i,t , β j,t are defined in Lemma 1.We assume the search space is finite as stated in Section 2.1.Proof of Theorem 1. From Lemma 5 in Golovin and Zhang [2020] :EQUATIONFurthermore, from basic arithmetic relationship:EQUATIONLemma 6 in Golovin and Zhang [2020] shows c m L ≤ m for s θ . Combining with Lemma 2, we obtainEQUATION≤ O(m 2 [γ T T ln(T )] 1/2 )with probability at least 1 -δ.Proof of Theorem 2.EQUATIONEQUATIONEQUATIONThe inequality in (17) directly comes from Lemma 3. The last inequality is the conclusion of Lemma 4.Proof of Theorem 3. Given iteration T and δ ∈ (0, 1), suppose infeasibility has not yet been declared. It follows that minEQUATIONwhere x ′ t = arg max x min j∈[c] w j,t (x). From the infeasibility of the problem, assume j * satisfies the condition for j in (3), and assume Combining ( 19) and ( 21) to ( 23) and the fact that ϵ < 0, we haveEQUATIONEQUATIONwith probability at least 1 -δ. Furthermore,EQUATIONBy picking J = arg minEQUATIONwe haveEQUATIONWhat follows is similar to the proof of Theorem 5. 1 in Xu et al. [2023] , by taking the sum for each t, and Lemma 4 shows:EQUATIONHere, we leverage the fact β J,T ∼ O(ln T ). Then, there exists a C > 0 such that EQUATIONg j (x) ≥ 0 (31)then, in each iteration, x ′ t = arg max x∈X min j∈[c] w j (x) in Algorithm 1 is feasible for set {x|w i,t (x) ≥ 0, ∀i ∈ [c]}, which is equivalent to infeasibility not being declared, with probability ≥ 1 -δ.Proof of Theorem 4. Until the T th iteration, given δ ∈ (0, 1) from Lemma 1, we conclude that the following holds with probability at least 1 -δ w j,t (x) ≥ g j (x) (32) ∀x ∈ X, ∀t ∈ {1, ..., T }, ∀i ∈ [c] then, it follows that maxx∈X min j∈[c] w j,t (x) ≥ max x∈X min j∈[c] g j,t (x) ≥ 0 (33) with probability at least 1 -δ. Then, let x ′ t = arg max x∈X min j∈[c] w j,t (x), we have w j,t (x ′ t ) ≥ 0, ∀j ∈ [c].Lemma 1 (Lemma 5. 1, Srinivas et al. [2009] ). Let cardinality of X, |X| < ∞. Then, ∀δ ∈ (0, 1)|µ i,t-1 (x) -f i (x)| ≤ β 1/2 i,t σ i,t-1 (x) (34) |µ ′ j,t-1 (x) -g j (x)| ≤ β 1/2 j,t σ j,t-1 (x) (35) ∀x ∈ F, ∀i ∈ [m], ∀j ∈ [c], ∀t ∈ {1, ..., T }with probability ≥ 1-δ. µ i,t-1 , µ ′ j,t , σ i,t-1 , σ j,t-1 and σ, are defined in B.5. µ 0,0 and σ 0,0 are prior mean and standard deviation. β i,t , β j,t are defined as 2 log ((m + c)|X|π t /δ) , ∀i ∈ [m], j ∈ [c]. And π t = π 2 t 2 6 .Lemma 2 (Modified version of Theorem 7 in Golovin and Zhang [2020] ). In our algorithm, suppose s θ (y) is L-Lipschitz for all possible θ. With conditions in Lemma 1, the expected cumulative regret(6) is bounded E[R C (T )] = O(Lm[γ T T ln(T )] 1/2 ) (36) with probability at least 1 -δ for any δ ∈ (0, 1). where γ T is an upper bound for γ k,T .Proof of Lemma 2. The unconstrained and definite version of this lemma was directly used in Golovin and Zhang [2020] and was proved by Paria et al. [2019] . Define F t = {x | w j,t (x) ≥ 0, ∀j ∈ [c]}. Let's pick x * t = arg max x∈Ft s θt (F (x)) and x t = arg max x∈Ft s θt (U t (x)) in our problem. Then it follows thatE[R C (T )] = E T t=1 max x∈F s θt (F (x)) -s θt (F (x t )) (37) ≤ E T t=1 s θt (U t (x t )) -s θt (F (x t )) B1 + (38) E T t=1 s θt (F (x * t )) -s θt (U t (x * t )) B2 (38) holds if s θt (U t (x t )) -s θt (U t (x * t )) > 0 and s θt (F (x * t )) -max x∈F s θt (F (x)) > 0.The first condition is always true by the definition of x t . From Lemma 1, we know w j,t (x) ≥ g j (x), ∀j ∈ [c], t ∈ {1, ..., T }, x ∈ X with probability at least 1 -δ, then F t ⊂ F with probability at least 1 -δ, which means the second condition holds with the same probability. Given (38), for finite X: Lemma 3 in Paria et al. [2019] showsEQUATION))where β T is an upper bound for β i,T , ∀i ∈ [m]. Lemma 2 in Paria et al. [2019] showsEQUATION))The final bound comes from the fact that β i,t ∼ O(ln t) and exp(EQUATIONLemma 4 (A corollary of Lemma 5.4 in Srinivas et al. [2009] ). ∀j ∈ [c], with x 1 , ..., x T selected by our algorithm, T t=1 2β 1/2 j,t σ j,t-1 (x t ) ≤ 2 T M β j,T γ j,Twhere M = We now consider X to be continuous and compact. WLOG, X := [0, 1] d . We keep the assumptions for GPs in Assumption By result in Ghosal and Roy [2006] , if kernel k is stationary and 4 th -differentiable, f i , g j ∼ GP i , GP j respectively. ∃a i , b j > 0 s.t. ∀k ∈ {1, ..., d}, ∀Q > 0 EQUATIONWith Lemma 6, we can follow the idea of proof of Theorem 1 to bound the cumulative HV regret with continuous and compact search space by O(m 2 d 1/2 [γ T T ln T ] 1/2 ). Note that the cumulative HV regret bound for continuous and compact search space additionally considers the size of instance vector d, which arises from a dynamic discretization density 1/τ t defined in Definition D.1.  



Reviews: review: title: title: The paper introduces a Constrained Multi-Objective Bayesian Optimization (CMOBO) algorithm that efficiently balances learning the level sets of multiple unknowns with multi-objective optimization within feasible regions, using random scalarization. This approach addresses gaps in the literature by focusing on constrained optimization in a multi-objective context, where previous work primarily focused on single-objective constrained optimization or unconstrained multi-objective tasks. The proposed algorithm demonstrates its effectiveness through both theoretical analysis and empirical evidence across various benchmarks, including synthetic and real-world applications.  Key Contributions:  Novel Algorithm: CMOBO integrates constraint management with multi-objective optimization by leveraging random scalarization, ensuring a principled search within feasible regions. This innovation addresses the challenge of balancing constraint learning and optimization in high-dimensional, uncertain spaces. Empirical Validation: The algorithm shows superior performance, particularly in constraint handling and hypervolume regret reduction, compared to existing benchmarks like qNEHVI and qParEGO.  Theoretical Guarantees: The paper provides bounds on cumulative regret and constraint violation, enhancing the credibility of the approach.  Advantages:  Efficiency: CMOBO is highly sample-efficient, which is critical for real-world applications like drug discovery and hyperparameter optimization, where constraints like safety or regulatory limits are pivotal. Balanced Exploration and Optimization: The integration of learning feasible regions while optimizing multiple objectives fills a significant gap in constrained optimization literature.  Drawbacks:  Model Misspecification: The paper acknowledges challenges related to model misspecification, which could affect performance. This is particularly relevant when dealing with noisy, real-world datasets.  Scalarization Dependence: The reliance on random scalarization, while effective, may be limited in certain scenarios where adaptive approaches could improve results.  Overall, the paper makes a meaningful contribution to the field of Bayesian optimization, particularly by addressing the complexities of constrained multi-objective tasks. It stands out in comparison to existing methods by providing a well-rounded solution, though future work might focus on improving model robustness and exploring adaptive scalarization techniques.
main_review: main_review: -----

Evaluation:

1. Quality

The quality of this paper is credible, both in its technical execution and its depth of research. The authors present a well-defined and structured approach to Constrained Multi-Objective Bayesian Optimization (CMOBO). The mathematical rigor is evident by providing theoretical proofs that guarantee regret bounds for cumulative hypervolume and constraint violations.

Strengths in Quality:
Mathematical Rigor: The use of Gaussian Process (GP) priors, scalarization techniques, and the clear formulation of the Pareto front provide a solid foundation for the algorithm. The theoretical results are well-justified through proofs (e.g., the cumulative regret bound) and align with established literature in Bayesian optimization.
Empirical Validation: The algorithm is tested on synthetic functions and real-world problems (e.g., Penicillin production, molecular datasets), demonstrating its versatility and robustness.

Areas for Improvement in Quality:
Model Misspecification Handling: While the algorithm is robust in various scenarios, it relies heavily on the Gaussian Process assumption, which might be limiting in practical applications where GPs may not accurately model the data. Introducing methods to handle model misspecification would improve the reliability of the results.
Scalability to Higher Dimensions: While the method is tested on relatively high-dimensional problems, the scalability in extremely high-dimensional spaces, which are typical in real-world tasks, could be further explored and validated.


2. Clarity

The clarity of the paper is generally convincing, with clear explanations of the problem setup, the proposed algorithm, and the accompanying theoretical analysis. However, the clarity could be improved in some sections to better engage readers unfamiliar with the nuanced details of multi-objective optimization.

Strengths in Clarity:
Definitions and Problem Statement: The paper starts by clearly defining the constrained multi-objective problem with precise mathematical formulations. Definitions such as the Pareto frontier, hypervolume regret, and constraint violations are explained adequately.
Figures and Visualizations: The performance of CMOBO is well-illustrated through plots, comparing hypervolume regret and constraint violation against baselines (e.g., qNEHVI, qParEGO). These help in understanding the trade-offs involved in the optimization process.

Areas for Improvement in Clarity:
Technical Jargon: Some sections, particularly those involving detailed proofs or complex algorithms, could benefit from additional explanations or footnotes. Not all readers may have a deep background in GP-based optimization, and providing simpler overviews or intuitive explanations would increase accessibility.
Detailed Algorithm Explanation: While the algorithm is outlined, more focus on explaining its intuition and real-world applicability would benefit readers. A clearer breakdown of how the random scalarization interacts with the constraints in different settings could enhance comprehension.


3. Originality

The originality of the paper lies in its novel integration of random scalarization within the context of constrained multi-objective Bayesian optimization. This approach adds a unique angle to the existing literature, particularly in the way it tackles feasibility learning and optimization trade-offs simultaneously.

Strengths in Originality:
Unique Problem Addressing: Constrained multi-objective optimization is a relatively under-explored area compared to its unconstrained counterpart. The authors have contributed by addressing a significant gap in the literature: how to balance learning constraints with optimizing multiple objectives.
Scalarization Approach: The use of random scalarization to explore the feasible region is an innovative method. It allows the algorithm to adapt dynamically to the constraints while optimizing multiple objectives, which has not been widely explored in previous works.

Areas for Improvement in Originality:
Further Extension of Techniques: While the scalarization technique is a novel contribution, it is still somewhat based on established methods of GP optimization and scalarization in multi-objective contexts. Further innovation could come from adaptive or learning-based approaches to scalarization, where the parameters are adjusted based on real-time feedback from the problem space.


4. Significance

The significance of this work is adequate, particularly for practitioners and researchers working on constrained optimization in real-world applications such as drug discovery, material science, and hyperparameter optimization.

Strengths in Significance:
Practical Applications: The constrained optimization framework presented is particularly useful for fields where safety and regulatory constraints are critical. The case studies on penicillin production and molecular datasets highlight the practical applicability of the method.
Contribution to Multi-Objective Optimization Literature: This work addresses an important gap in constrained multi-objective optimization, offering a method that performs well in both theoretical and practical scenarios. The proposed algorithm could serve as a building block for future research and real-world optimization tasks.

Areas for Improvement in Significance:
Broader Applicability: While the paper tests the algorithm on a variety of datasets, demonstrating its application to larger-scale industrial problems or broader fields could significantly enhance its impact. Future work could focus on applying CMOBO to other complex domains. 


List of Pros and Cons:

Pros:

- Novel Algorithm: Introduces a sample-efficient approach for constrained multi-objective Bayesian optimization, integrating random
  scalarization with constraint handling.
- Theoretical Rigor: The paper provides theoretical guarantees for cumulative hypervolume regret and constraint violations, making the
   algorithm both practical and mathematically sound.
- Real-World Applications: Demonstrates effectiveness on synthetic and real-world datasets, particularly in drug discovery and
   hyperparameter optimization.
- Balanced Exploration-Exploitation: The algorithm manages a good trade-off between exploring the feasible region and optimizing
  objectives, which is crucial for constrained problems.

Cons:

- Model Misspecification: The reliance on GP models may limit the applicability of the algorithm in scenarios where GPs do not accurately
  reflect the underlying data structure.
- Technical Complexity: The detailed mathematical derivations, while necessary, could make the paper less accessible to non-specialists or
   those without a strong background in Bayesian optimization.
- Scalability: Although the algorithm is shown to perform well on relatively high-dimensional problems, its performance on extremely high
  -dimensional spaces could be further validated.
- Potential Over-reliance on Scalarization: The random scalarization approach, while innovative, may have limitations in certain
  applications. More adaptive methods could offer better performance in dynamically changing problem spaces.


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title: title: Reads well, but contributions should be toned-down and set-up clarified
main_review: main_review: # Summary

This paper proposes a method for *Constrained Multi-objective Bayesian Optimization* (CMOBO), based on previous work by Golovin & Zhang, 2020 and Deng & Zhang, 2019.

In a nutshell, the proposed method optimizes a random scalarization of the multivariate objective using the upper confidence bound. It is known that, by following this process, we end up optimizing the hypervolume improvement. The authors incorporate constraints by modelling them with GPs and choosing points such that the upper confidence bound of the constraints is positive.

The authors compare their CMOBO method against other well-known unconstrained alternatives, showing they can optimize multi-variate problems while violating the constraints less often.

# Review

The paper reads well and is well-motivated (indeed, practitioners in drug discovery and other applications need to optimize multiple objectives while satisfying certain constraints). However, the presentation of some of the content could be improved, some of the contributions should be toned down, and the experimental set-up and algorithm should be clarified further.

## On the contributions and other related work

Although not in our community, constrained multiobjective Bayesian Optimization was explored in the context of aerospace vehicle design:  https://hal.science/hal-02912982/document.

They take a similar approach to CEI (namely, multiplying by the probability that a constraint is satisfied). It might be interesting to benchmark against such a method in future versions of this paper.

## On the budget and experimental set-up

The choice of budget for most experiments feels unmotivated. Why stop at 70 for e.g. the Penicillin example?

Low budgets should be justifiable by talking to practitioners (who indeed sometimes rely on expensive experiments, for which they can afford ~100 evaluations). Still, it would be great to justify the choice of budget further.

## What are those 0.05 and 0.1?

Algorithm 1 has no inputs to it. We can see in the results, however, that there are two presentations of the algorithm: CMOBO-0.05 and CMOBO-0.1. What are the constants next to the name? Are they the UCB beta parameters? Could you clarify that further?

## On presentation

Minor details:

- The scale for cumulative HV regret could be presented in log-scale to make the differences clearer.
- The random scalarization is studied in Golovin and Zhang 2020, but it was originally proposed by Deng and Zhang in 2019. This might warrant citing Deng and Zhang as well.
- In the conclusions, I would change “we’ve” to “we have”.
- The name of the algorithm corresponds to a *meta name*. It is *not* the only algorithm for constrained multi-objective BO out there, so I would like to suggest a name change.