BILBO: BILevel Bayesian Optimization

Bryan Kian Hsiang Low; Quoc Phong Nguyen; Ruth Wan Theng Chew

REVIEW 2 major objections 3 minor 1 cited by

BILBO: BILevel Bayesian Optimization

T0 review · 2 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper proves that BILBO, a Bayesian optimization algorithm for bilevel problems with black-box, noisy, constrained, derivative-free functions, has sublinear cumulative regret while querying only one function per iteration.

desk verdict New simultaneous bilevel BO algorithm with a real but repairable proof gap; the regret guarantee as written doesn't follow. read the letter →

arxiv 2502.02121 v2 pith:XCAA6CVJ submitted 2025-02-04 cs.LG stat.ML

Ruth Wan Theng Chew , Quoc Phong Nguyen , Bryan Kian Hsiang Low This is my paper · ORCID

classification cs.LGstat.ML

keywords bileveloptimizationBayesianGaussianprocessesregretboundsblack-boxtrustedsetsconstraineddecoupledfunctionqueries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

BILBO tackles bilevel optimization problems in which the upper-level objective can only be evaluated at points that are optimal for a separate lower-level problem, with both functions black box, noisy, derivative free, and constrained. The paper's central claim is that both levels can be optimized simultaneously, with one function query per iteration, while still achieving sublinear cumulative regret: Theorem 4.9 bounds $R_T \leq \sqrt{4T|\mathcal{F}|\beta_T \max_{h\in\mathcal{F}} C_h \gamma_{h,T}}$ with high probability. The method works by building "trusted sets" from Gaussian process confidence bounds—points that are probably feasible and probably lower-level optimal—and by a conditional reassignment that queries the lower-level objective at its estimated optimum when that estimate is uncertain. If the claim holds, BILBO removes the repeated lower-level optimization that makes existing nested bilevel Bayesian optimization sample inefficient, and gives a no-regret algorithm for a class of problems where gradients and cheap lower-level solves are unavailable. The paper supports the claim with regret proofs and with experiments on synthetic and real-world energy and chemical process problems.

What carries the argument

The central object is the trusted set, defined for functions $h$ using Gaussian process upper and lower confidence bounds $u_{h,t}, l_{h,t}$. The feasible trusted set $S^+_t$ keeps points whose upper confidence bound on every constraint is nonnegative, so every point in it has constraint regret bounded by $2\beta_t^{1/2}\sigma_{c,t-1}(x,z)$. The lower-level trusted set $P^+_t$ keeps points $(x,z)$ for which the upper confidence bound of the lower-level objective at $z$ reaches the lower confidence bound at the estimated lower-level solution $\bar{z}_t(x)=\arg\max_{z\in S^+_{\mathrm{lo},t}(x)} u_{f,t}(x,z)$, making their lower-level regret bounded by $2\beta_t^{1/2}(\sigma_{f,t-1}(x,\bar{z}_t(x))+\sigma_{f,t-1}(x,z))$. The query point is the maximizer of $u_{F,t}$ over $S^+_t\cap P^+_t$; the function queried is the one with largest estimated regret; and if the lower-level function is queried while $\bar{z}_t(x_t)$ is more uncertain than $z_t$, the query is reassigned to $\bar{z}_t(x_t)$. That reassignment is what converts the estimated lower-level regret into the factor $4\beta_t^{1/2}\sigma_{f,t-1}(x_t,z_t)$ in Lemma 4.8 and makes the one-query-per-round scheme no regret.

What would settle it

One concrete check: on a domain with a single upper-level point $x_0$ and two lower-level points $z_1,z_2$, set lower-level constraints and initial data so that $z_2$ is actually infeasible but has the higher upper confidence bound, making $\bar{z}_t(x_0)=z_2$. If the algorithm's conditional reassignment ever queries $f$ or another function at $(x_0,z_2)$ and the observed value of $f$ there exceeds $f(x_0,z^*)$, then the trusted-set membership claim in Lemma C.2 fails at that round, and the finite-$T$ cumulative regret bound cannot hold for that instance.

Watch

Extended reading notes

Core claim

Stated on its own terms, the paper establishes that general black-box bilevel optimization—where $F$ and $f$ are modelled by Gaussian processes, observations are noisy, constraints appear at both levels, and the domains are finite—admits a simultaneous, decoupled-query algorithm with a sublinear cumulative regret bound. The key bound is Theorem 4.9: with probability at least $1-\delta$, Algorithm 1 satisfies $R_T \leq \sqrt{4T|\mathcal{F}|\beta_T \max_{h\in\mathcal{F}} C_h \gamma_{h,T}}$, and because $\gamma_{h,T}$ is sublinear for squared-exponential and Mat\'ern kernels, $R_T/T \to 0$. The derivation rests on Lemma 4.8, which bounds the instantaneous regret of the queried point by $4\beta_t^{1/2} \max_{h} \sigma_{h,t-1}(x_t,z_t)$, and on the construction of trusted sets in which every sampled point has bounded constraint regret and bounded lower-level objective regret. Empirically, BILBO is reported to converge faster than random trusted-set sampling and a nested Bayesian optimization baseline on multimodal synthetic problems and on two real-world problems.

Load-bearing premise

The load-bearing assumption is that the true optimal lower-level response lies inside the trusted set the algorithm builds; the proof needs the true optimum to dominate the confidence-bound maximizer $\bar{z}_t(x^*)$, and it also needs the conditional reassignment to keep the query inside the trusted set, so if either fails the instantaneous and cumulative regret bounds do not cover the point actually evaluated.

Editorial extensions

If this is right

If Theorem 4.9 is correct, BILBO is a no-regret algorithm for noisy, constrained, derivative-free bilevel problems: for squared-exponential and Mat\'ern kernels with $\nu>1/2$, $R_T/T\to 0$, so the average instantaneous bilevel regret vanishes.
The decoupled query rule means only one function—upper objective, lower objective, or one constraint—is evaluated per iteration, so the total query budget scales with the number of iterations rather than with repeated lower-level solves at each upper-level point.
The simple-regret estimator of Lemma 4.10 gives a point recommendation with regret $O(\sqrt{|\mathcal{F}|\beta_T \max_h C_h\gamma_{h,T}/T})$, so the same algorithm can be used for final-solution selection, not only for minimizing cumulative regret.
Because the trusted feasible set certifies every point in it, active constraints on the boundary of the feasible region do not require special handling, and upper-level points whose lower-level problem is infeasible are filtered out through $S^+_{\mathrm{lo},t}$.
Multiple lower-level optima for the same upper-level point are represented simultaneously inside $P^+_t$, which is a capability single-solution nested solvers lack.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same trusted-set construction could be adapted to continuous domains, where the paper notes $\beta_t$ would require modification; in that setting the bottleneck would likely shift from grid cardinality to the maximum information gain of the kernels, the same quantity that already controls the regret bound.
The conditional reassignment is essentially an explore step for the lower-level optimum; a natural and testable variant would trigger reassignment based on the estimated regret $\bar{r}_{f,t}$ rather than only on the raw variance comparison, which may improve exploration when the estimated lower-level solution is far from the queried point.
For applications where lower-level evaluations are cheap, the per-iteration overhead of updating trusted sets could outweigh the sample savings—the paper itself notes BILBO is about 26 times slower per iteration than its nested baseline in wall-clock time—so the practical win is largest when evaluations are expensive, noisy, or multimodal.
If the gap in Lemma C.2 is closed by using a conservative estimate of the true lower-level optimum instead of the confidence-bound maximizer, the same framework might extend to robust or adversarial lower-level problems where the lower level is not a rational optimizer.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a genuinely new algorithm for derivative-free bilevel Bayesian optimization, but the headline regret guarantee is not supported by the proof as written. I agree with the reader's REJECT, and the stress-test note correctly pinpoints the load-bearing break.

What the paper does well: it optimizes both levels simultaneously without nested lower-level loops, builds trusted sets from confidence bounds to bound lower-level suboptimality, and queries only one function per iteration with a conditional reassignment rule. That is a real departure from the existing nested frameworks, and the experimental setup is reasonable, with code provided. The writing is clear and the authors are honest about limitations.

The soft spot is in the proof of Lemma C.2. It asserts f(x*, z*(x*)) ≥ f(x*, \bar z_t(x*)), but \bar z_t is the UCB maximizer over S+_lo,t, which can contain points that are infeasible for the true lower-level constraints. An infeasible point can have larger true f than the feasible optimum, so the inequality does not hold. The example with f(x,z) = -z and constraint z ≥ 0.5 is enough to show the mechanism. Because (x*, z*) may not lie in S+_t ∩ P+_t, the argmax step in Lemma C.3 cannot bound the upper-level regret, and Theorem 4.9 falls apart. This is not a minor technicality; it is the central justification for no-regret convergence.

There is a second, smaller gap: the reassignment in Eq. 4.8 can move the query outside S+_t, so the constraint-regret bound and the argmax identity no longer apply to the actually queried point. That one is easier to patch.

Neither gap looks unfixable. You could restrict \bar z_t to high-probability feasible points, add a violation term to the trusted-set condition, or constrain the reassignment to stay in S+_t. But as it stands, the proof does not go through.

Who should read this: anyone working on bilevel BO or constrained BO. The algorithm is plausible and the experiments suggest it works, but the theory needs repair. I would send it to peer review rather than desk reject; the core idea deserves referee time, with the expectation of major revision.

Referee Report

2 major / 3 minor

Summary. BILBO proposes a Bayesian optimization algorithm for constrained bilevel problems in which both upper- and lower-level functions are black-box, noisy, and derivative-free. The algorithm builds two confidence-bound trusted sets, one for feasible points and one for approximately optimal lower-level solutions, selects a query by maximizing the upper-level UCB over their intersection, chooses a single function to observe in a decoupled manner, and may conditionally reassign the lower-level coordinate to the estimated lower-level maximizer. The paper claims an instantaneous regret bound, a sublinear cumulative regret bound (Theorem 4.9), and a simple regret bound (Lemma 4.10), and reports experiments on synthetic and real-world problems. The central theoretical claim is a no-regret guarantee with only one function query per round.

Significance. If the regret bound were valid, BILBO would be a meaningful contribution: it avoids repeated lower-level optimization, handles constraints and noise, works without gradients, and would provide sublinear cumulative regret for standard kernels. The empirical study covers varied problems, the code is released, and the paper identifies an interesting method for decoupled query selection. However, the proof of the central theorem contains a substantive gap in Lemma C.2, and a second gap arises from the conditional reassignment in Eq. (4.8). As written, the theoretical contribution is not established.

major comments (2)

[Appendix C.3, Lemma C.2] The proof asserts u_f,t(x*,z*) ≥ f(x*,z*) ≥ f(x*,\bar z_t(x*)) ≥ l_f,t(x*,\bar z_t(x*)), and uses this chain to conclude (x*,z*) ∈ P+t. The middle inequality is not justified. By Definition 4.5, \bar z_t(x*) is the maximizer of u_f,t over the optimistic feasible set S+_lo,t(x*), which only requires u_c,t(x*,z) ≥ 0 for lower-level constraints. A point in S+_lo,t can be genuinely infeasible, and for such a point the true value f(x*,\bar z_t(x*)) can exceed the true constrained optimum f(x*,z*(x*)). For example, with f(x,z) = -z and lower-level constraint z ≥ 0.5, an unobserved infeasible point z = 0 has f(x,0) = 0 > f(x,0.5) = -0.5 and can be the UCB maximizer. Consequently the inference (x*,z*) ∈ S+t ∩ P+t does not follow. This invalidates Lemma C.3's replacement of u_F,t(x*,z*) by max over S+t ∩ P+t, and therefore Theorem 4.9 and Lemma 4.10, both of which rely on Lemma C.3, are unsupported.
[Section 4.3, Eq. (4.8) and Lemma C.3] The conditional reassignment zt ← \bar z_t(xt) can move the query outside S+t ∩ P+t. Lemma 4.4 bounds constraint regret only for points in S+t, and Lemma C.3 bounds the upper-level objective regret using the equality u_F,t(xt,zt) - l_F,t(xt,zt), which is only valid when (xt,zt) is the argmax in Eq. (4.6). After reassignment, neither premise holds: \bar z_t(xt) need not satisfy the upper-level constraints, and the queried point is no longer the UCB maximizer over the trusted set. Thus even if Lemma C.2 were repaired, Lemma 4.8's instantaneous regret bound and Theorem 4.9 would still fail on rounds in which the reassignment occurs. The condition in Eq. (4.8) compares posterior standard deviations and gives no feasibility guarantee for the reassigned point.

minor comments (3)

[Definition 4.5] The displayed definition of P+t has an unbalanced parenthesis: 'lf,t(x, ¯zt(x)}' should close with ')'.
[Definition 4.5] The estimated lower-level solution \bar z_t(x) is defined as an argmax over S+_lo,t(x); when S+_lo,t(x) is empty, this argmax is undefined. The paper should state how such upper-level points are handled, even though they cannot belong to P+t.
[Section 5] The comparison against TrustedRand is on a per-query basis even though BILBO queries one function per round while TrustedRand queries all functions. This is a reasonable sample-efficiency comparison, but it should be stated more explicitly, since wall-clock efficiency is only discussed in Appendix E.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the regret analysis is a self-contained GP-UCB-style derivation; the Lemma C.2 concern is a proof gap, not an input-output equivalence.

full rationale

I walked the derivation chain from Definitions 4.1, 4.3, and 4.5 through Lemmas 4.4, 4.6, 4.8, and Theorem 4.9. The argument is a standard confidence-bound GP-UCB construction: Corollary 4.2 supplies GP concentration from an external result (Srinivas et al. 2010), the trusted sets S+t and P+t are defined directly from the confidence bounds, and the regret bounds are proven from those definitions rather than assumed. No parameter is fitted to the target regret value, and no prediction is renamed from a fitted input. The citations to Nguyen et al. (2023), which share authors with this paper, are contextual comparisons of constrained-BO techniques and are not load-bearing: the paper proves its own trusted-set and regret bounds instead of importing that paper's theorems. The only notable issue is a soundness gap in Lemma C.2: to show (x*, z*) is in P+t, the proof uses the inequality f(x*, z*) >= f(x*, bar-z_t(x*)), but bar-z_t(x*) is only required to satisfy the lower-level UCB constraint and may be infeasible, so the true lower-level optimum need not dominate f at bar-z_t. That is a correctness concern in the proof, not a circular reduction: P+t is not defined in terms of the true optimum, and Theorem 4.9 does not reduce to Lemma C.2 by construction. Similarly, the conditional reassignment in Eq. 4.8 can move the query outside S+t, but this is a proof gap about the queried point's constraint regret, not a circular equation. I therefore find no significant circularity and set the score to 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central theoretical claim rests on the standard GP-UCB confidence-bound machinery and on two unstated assumptions about the estimated lower-level solution \bar{z}_t: that it is no worse than the true optimum and that reassigning the query to it preserves the regret bound. These assumptions are not guaranteed by the construction and constitute the main burden. No new physical entities are introduced.

free parameters (3)

GP kernel hyperparameters (length scale, prior mean) = Tuned via maximum likelihood at each iteration; initial values in Table 1
The regret bound assumes a known kernel, but experiments re-fit hyperparameters. The tuned values are free choices that affect the empirical results and the validity of the GP confidence bounds.
Discretization grid size per dimension = 100 (Branin), 25 (SMD2/SMD6), 16 (SMD12), 15 (Energy), 10 (Chemical)
Chosen by hand. The regret bound scales with |X||Z| through beta_t, so the grid resolution is a free parameter that shapes the theoretical guarantee.
Noise level sigma_n = 0.01
Set by hand for all experiments; the theory requires the noise level to be known.

assumptions (4)

standard math The functions F, f, and constraints c are drawn from GPs with known kernels, and the confidence bounds hold uniformly with probability 1-delta (Corollary 4.2).
Inherited from GP-UCB analysis (Srinivas et al. 2010) via a union bound over the finite function set and finite grid. This is the foundation of the trusted sets.
domain assumption The true optimal lower-level solution z*(x) is contained in the optimistic trusted set S+_lo,t(x) for all x and t.
Follows from Corollary 4.2 via S_lo subset of S+_lo,t; used in Lemma C.1. This is valid under the GP confidence event.
ad hoc to paper The UCB maximizer \bar{z}_t(x) achieves at least the true optimal lower-level value, i.e., f(x,z*(x)) >= f(x,\bar{z}_t(x)).
Invoked in the proof of Lemma C.2 without proof. It is false in general because \bar{z}_t(x) maximizes the upper confidence bound over the optimistic superset S+_lo,t(x), which can include infeasible points with larger true f than the feasible optimum.
ad hoc to paper The conditional reassignment of z_t to \bar{z}_t(x_t) does not invalidate the instantaneous regret bound.
Equation 4.8 can move the query outside S+_t (the full trusted set including upper-level constraints). The proof of Lemma 4.8 does not bound upper-level constraint regret at the reassigned point.

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Cite this review

Pith. "Pith review of BILBO: BILevel Bayesian Optimization." pith.science (2026). https://pith.science/paper/XCAA6CVJ

@misc{pith2026250202121,
  author       = {Pith},
  title        = {Pith review of: BILBO: BILevel Bayesian Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XCAA6CVJ}},
  note         = {Machine review of arXiv:2502.02121}
}

read the original abstract

Bilevel optimization is characterized by a two-level optimization structure, where the upper-level problem is constrained by optimal lower-level solutions, and such structures are prevalent in real-world problems. The constraint by optimal lower-level solutions poses significant challenges, especially in noisy, constrained, and derivative-free settings, as repeating lower-level optimizations is sample inefficient and predicted lower-level solutions may be suboptimal. We present BILevel Bayesian Optimization (BILBO), a novel Bayesian optimization algorithm for general bilevel problems with blackbox functions, which optimizes both upper- and lower-level problems simultaneously, without the repeated lower-level optimization required by existing methods. BILBO samples from confidence-bounds based trusted sets, which bounds the suboptimality on the lower level. Moreover, BILBO selects only one function query per iteration, where the function query selection strategy incorporates the uncertainty of estimated lower-level solutions and includes a conditional reassignment of the query to encourage exploration of the lower-level objective. The performance of BILBO is theoretically guaranteed with a sublinear regret bound for commonly used kernels and is empirically evaluated on several synthetic and real-world problems.

Figures

Figures reproduced from arXiv: 2502.02121 by the authors.

**Figure 2.** Key components of BILBO, where pink shaded ar [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** BraninHoo+GoldsteinPrice experiment details. LL refers to lower-level. (a) Upper-level objective, Branin-Hoo. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Instantaneous regrets (log-scale) over number of queries, averaged over 5 runs, for synthetic experiments. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Real-world experiments. (a-b) Functions from energy experiment and (c-d) BILBO outputs, with optimal solution [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Flowsheet of chemical process. R-101 is the reactor, and C101 is the distillation column. [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: Regret against wall-clock time for the BraninHoo+GoldsteinPrice experiment [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

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Forward citations

Cited by 1 Pith paper

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Bilevel Bayesian optimization that optimizes only black-box variables and solves the known white-box subproblem exactly reports 11x to 10^8x lower regret than full-space BO on 13 separable grey-box benchmarks.

Reference graph

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