Constrained Max-Value Entropy Search

• The paper replaces improvement-based heuristics with a mutual information framework, using Gaussian processes to model both objectives and constraints for precise optimization.
• Constrained Max-Value Entropy Search is a set of Bayesian methods that target expected information gain about feasible optimum values or Pareto fronts under complex constraints.
• The method demonstrates practical advantages by accommodating both real-valued and binary feedback through joint GP sampling and variational approximations, ensuring robust performance.

Constrained Max-value Entropy Search denotes a family of information-theoretic Bayesian optimization methods that select evaluations by maximizing expected information gain about an optimum under unknown constraints. In the scalar setting, the target is the constrained optimum value itself; in multi-objective extensions, the target is typically the constrained Pareto frontier in function space. The defining move is to replace improvement-based heuristics with mutual information or an analytically tractable surrogate of it, while modeling the objective and constraints with Gaussian processes and updating acquisition values as feasibility uncertainty changes (Perrone et al., 2019, Takeno et al., 2021, Belakaria et al., 2020).

1. Information-theoretic basis

Max-value Entropy Search (MES) was introduced for unconstrained Bayesian optimization as an acquisition function that maximizes the mutual information between a prospective observation and the unknown maximum value $y_*$, rather than the argmax location $x_*$. In its standard form,

$$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$

and the conditional term is approximated through an upper-truncated Gaussian predictive distribution. This yields a cheap scalar-information objective and leads to the familiar closed-form dependence on $\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$ (Wang et al., 2017).

A key structural fact is that one-sample MES is equivalent to EST with $m=y_*$, and can be mapped to GP-UCB and PI through appropriate parameterizations. That equivalence clarifies why MES often behaves like a randomized thresholding rule, but with the threshold drawn from an approximation to the posterior over the optimum value (Wang et al., 2017).

Constrained variants inherit this logic but replace the unconstrained optimum with a feasible optimum. For a scalar minimization problem with latent objective $y(\mathbf{x})$ and latent constraint $c(\mathbf{x})$, cMES writes the constrained optimum as

$$y_\star=\min_{\mathbf{x}\in\mathcal X}\left\{y(\mathbf{x})\;\big\|\; c(\mathbf{x})\le \delta\right\},$$

or, under binary feasibility feedback,

$$y_\star=\min_{\mathbf{x}\in\mathcal X}\left\{y(\mathbf{x})\;\big\|\; \sigma(c(\mathbf{x}))\le \sigma(\delta)\right\}.$$

The acquisition is then the mutual information between the future objective/constraint observation and this constrained optimum value (Perrone et al., 2019).

2. Scalar constrained MES

In cMES, the objective and constraint are modeled by independent GPs over a continuous domain $\mathcal{X}\subset\mathbb{R}^p$. The framework treats two feedback regimes. In the real-valued setting, both objective and constraint are observed through Gaussian noise. In the binary setting, the objective is real-valued but the constraint feedback is a Bernoulli indicator $x_*$0, with logistic link $x_*$1. The binary formulation is intended for cases such as out-of-memory failures, where only feasibility is returned and the objective may be unavailable on infeasible trials (Perrone et al., 2019).

The corresponding mutual-information objectives are

$x_*$2

for real-valued constraints, and

$x_*$3

for binary constraints. In both cases, cMES uses a local truncation argument: if the point is feasible, then consistency with $x_*$4 requires the objective value at that point to be no better than the constrained optimum; if the point is infeasible, the objective value does not constrain $x_*$5. For real-valued constraints this produces a masked Gaussian density $x_*$6, while for binary feedback the non-Gaussian term $x_*$7 is replaced by a Gaussian approximation $x_*$8, yielding closed-form entropy differences after integrating over the feasibility event (Perrone et al., 2019).

A practical contribution of cMES is its critique of the factorized approximation used in MES for sampling the optimum value. The paper shows empirically that sampling the constrained minimum from independent marginal posteriors biases the minimum downward and does not stabilize as the discretization grows. The recommended alternative is joint GP sampling on a discretized set $x_*$9, followed by direct computation of the constrained minimum on each sampled path. In the reported experiments, a Sobol discretization with $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$0 and only $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$1 to $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$2 samples of $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$3 was already sufficient for strong performance, and joint sampling consistently outperformed the marginal approximation (Perrone et al., 2019).

The method also explicitly covers the “unobserved-objective” scenario in which $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$4 is only available when the constraint is satisfied. That extension is one reason cMES is usually discussed as more general than feasibility-weighted EI heuristics in realistic hyperparameter optimization settings (Perrone et al., 2019).

3. CMES-IBO and the information lower bound

CMES-IBO reformulates constrained MES around a different random variable and a different approximation principle. Its starting point is that the feasible set can be empty with positive posterior probability, so the constrained optimum must be defined on that event. The paper therefore sets

$$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$5

which makes $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$6 a proper random variable and incorporates uncertainty about feasibility directly into the acquisition target (Takeno et al., 2021).

The sequential acquisition is defined as

$$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$7

but the paper does not approximate this MI directly. Instead, it introduces an auxiliary density $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$8 and applies a KL-based lower-bound argument. Choosing $$\alpha_t^{\mathrm{MES}}(x)=I(\{x,y\};y_* \mid D_t) =H\big(p(y\mid D_t,x)\big)-\mathbb{E}_{p(y_*\mid D_t)}\big[H(p(y\mid D_t,x,y_*))\big],$$9 to be the predictive density truncated to the complement of the “improving feasible hyperrectangle”

$\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$0

yields the information lower bound

$\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$1

where $\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$2 is the probability that the local outcome does not lie in $\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$3. Under independent GPs for objective and constraints,

$\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$4

so the final sequential acquisition is

$\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$5

with $\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$6 obtained by Monte Carlo sampling of constrained-optimum values from random Fourier feature GP path samples (Takeno et al., 2021).

This lower-bound construction is significant for two reasons. First, it guarantees non-negativity, whereas the paper proves that a direct constrained analogue of conventional MES can become negative when $\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$7. Second, CMES-IBO provides an explicit variance analysis: if $\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$8 is the Monte Carlo estimator of the lower bound, then $\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}$9, and the paper gives a concentration bound of Bernstein type,

$m=y_*$0

with

$m=y_*$1

The authors emphasize that such low-variability guarantees had not been investigated for prior information-theoretic BO acquisitions (Takeno et al., 2021).

The same logic extends to parallel selection by replacing MI with conditional MI,

$m=y_*$2

and greedily adding points to a batch. The resulting parallel CMES-IBO preserves the lower-bound structure and reuses joint samples of $m=y_*$3 and pending fantasy outcomes $m=y_*$4 (Takeno et al., 2021).

4. Multi-objective constrained max-value entropy search

In multi-objective constrained BO, the target is no longer a scalar feasible optimum but the constrained Pareto solution. PESMOC, MESMOC, and MESMOC+ differ mainly in whether they reduce entropy over the Pareto set $m=y_*$5 or over the Pareto front $m=y_*$6, and in how they approximate the conditional predictive distribution induced by that global object (Garrido-Merchán et al., 2016, Belakaria et al., 2020, Fernández-Sánchez et al., 2020).

Method	Target random object	Core approximation
PESMOC	Constrained Pareto set $m=y_*$7	Monte Carlo over Pareto sets + EP
MESMOC	Constrained Pareto front $m=y_*$8	Output-space entropy + truncated Gaussians
MESMOC+	Constrained Pareto front $m=y_*$9	Monte Carlo over fronts + ADF

PESMOC is the constrained multi-objective extension of predictive entropy search. It defines

$y(\mathbf{x})$0

and then uses the symmetry of mutual information to work with

$y(\mathbf{x})$1

Its conditional predictive depends on non-Gaussian feasibility and dominance factors involving Heaviside step functions, which are approximated with EP. The method is conceptually close to constrained entropy search over the solution set, but computationally demanding (Garrido-Merchán et al., 2016).

MESMOC moves from input space to output space. It targets the entropy of the constrained Pareto front $y(\mathbf{x})$2 and defines

$y(\mathbf{x})$3

To approximate the second term, it samples constrained Pareto fronts by drawing objective and constraint functions from independent GP posteriors with random Fourier features, solves a cheap constrained multi-objective problem on those samples with constrained NSGA-II, and then conditions each output on a one-sided upper bound induced by the sampled front. This yields truncated Gaussian entropies for each objective and constraint and produces the MESMOC acquisition

$y(\mathbf{x})$4

The acquisition is optimized only over points satisfying the mean-feasibility filter $y(\mathbf{x})$5 for all constraints (Belakaria et al., 2020).

MESMOC+ keeps the output-space focus on $y(\mathbf{x})$6 but replaces MESMOC’s approximate dominance condition with an explicit compatibility factor

$y(\mathbf{x})$7

where $y(\mathbf{x})$8 encodes that a candidate point is either infeasible or dominated by the front. The conditional predictive

$y(\mathbf{x})$9

is approximated by Assumed Density Filtering, producing factorized Gaussians and a final acquisition expressed as variance reduction rather than log-variance reduction: $c(\mathbf{x})$0 Because the acquisition decomposes into one term per objective or constraint, MESMOC+ supports decoupled evaluation. The paper states that its cost is linear in the number of objectives and constraints and often significantly smaller than methods based on minimizing the entropy of the Pareto set (Fernández-Sánchez et al., 2020).

5. Later reinterpretations and conceptual analogues

Several later works clarify or generalize the MES viewpoint in ways that bear directly on constrained variants. Joint Entropy Search (JES) replaces the scalar optimum value $c(\mathbf{x})$1 with the joint optimum pair $c(\mathbf{x})$2 and defines

$c(\mathbf{x})$3

Its implementation conditions the GP on fantasized noiseless optimal input/output pairs and then imposes a local truncation $c(\mathbf{x})$4. The paper argues that this is more noise-aware than standard MES, which conflates noisy observations and noiseless optima in the truncated-Gaussian approximation. Although JES is unconstrained, it provides a template in which the optimum can be expanded into a richer latent object than a scalar max-value (Hvarfner et al., 2022).

Variational Entropy Search (VES) develops a different reinterpretation. It applies the Barber–Agakov lower bound to MES and defines an entropy-search lower bound

$c(\mathbf{x})$5

With an exponential variational family, the method collapses to EI in the noise-free scalar setting; with a Gamma family, it yields an EI term plus a log-gap regularizer. The same paper explicitly suggests that a constrained extension would redefine the latent variable as a feasible max-value $c(\mathbf{x})$6 and then apply the same variational machinery to joint objective/constraint observations (Cheng et al., 2024).

A conceptually distinct but structurally related line appears in automatic search synthesis. There, a discrete search problem is represented by a program, symbolic execution extracts logical outcome constraints $c(\mathbf{x})$7, model counting computes exact outcome probabilities

$c(\mathbf{x})$8

and the next query is chosen by maximizing Shannon entropy

$c(\mathbf{x})$9

The paper explicitly describes this as a “discrete, constraint-based MES,” with the worthwhile-query predicate $$y_\star=\min_{\mathbf{x}\in\mathcal X}\left\{y(\mathbf{x})\;\big\|\; c(\mathbf{x})\le \delta\right\},$$0 playing the role of a feasibility condition on the search space (Downing et al., 2020).

6. Interpretation, practical use, and limitations

A persistent misconception is that constrained max-value entropy search is merely feasibility-weighted EI. The literature does not support that identification. cMES and CMES-IBO are formulated around mutual information with a constrained optimum value, not improvement over the current incumbent, and cMES was designed to work even when no feasible point has yet been observed. CMES-IBO goes further by encoding the event of an empty feasible region directly through the atom at $$y_\star=\min_{\mathbf{x}\in\mathcal X}\left\{y(\mathbf{x})\;\big\|\; c(\mathbf{x})\le \delta\right\},$$1 in $$y_\star=\min_{\mathbf{x}\in\mathcal X}\left\{y(\mathbf{x})\;\big\|\; c(\mathbf{x})\le \delta\right\},$$2, so the acquisition can explicitly trade off learning feasibility against learning objective values (Perrone et al., 2019, Takeno et al., 2021).

A second distinction concerns what “max-value” means in the multi-objective case. MESMOC and MESMOC+ do not optimize a scalar feasible max-value. They minimize entropy over the constrained Pareto frontier, which is the output-space analogue of the optimum in multi-objective optimization. This is why their acquisitions decompose over objectives and constraints but still depend on sampled fronts rather than only on scalar thresholds (Belakaria et al., 2020, Fernández-Sánchez et al., 2020).

Across the family, the main approximations recur. Scalar cMES uses local truncation and, in the binary case, Gaussian approximations to the feasibility likelihood; it also depends on discretization plus joint GP sampling to approximate the constrained optimum value. CMES-IBO relies on a lower bound rather than exact MI, and its performance depends on the quality of random Fourier feature path samples and the constrained optimization used to obtain $$y_\star=\min_{\mathbf{x}\in\mathcal X}\left\{y(\mathbf{x})\;\big\|\; c(\mathbf{x})\le \delta\right\},$$3. MESMOC and MESMOC+ require Monte Carlo sampling of Pareto fronts and either truncated-Gaussian or ADF approximations to the induced conditional predictive distributions. These are principled but still approximate constructions (Perrone et al., 2019, Takeno et al., 2021, Fernández-Sánchez et al., 2020).

The practical trade-off is therefore consistent across the literature. Information-theoretic constrained acquisitions are usually more complex than EI-based baselines, but they aim directly at uncertainty reduction about the constrained solution itself. In scalar constrained BO, this has yielded methods that handle continuous and binary constraints, including cases where the objective is unobserved on infeasible trials. In multi-objective constrained BO, output-space max-value formulations have provided competitive performance with substantially lower computational cost than entropy-search methods over the Pareto set. This suggests that constrained max-value entropy search is best understood not as a single algorithm but as a design pattern: represent the constrained optimum as a random object, approximate how a candidate query would reduce uncertainty about that object, and optimize the resulting information criterion under the current surrogate model (Perrone et al., 2019, Takeno et al., 2021, Fernández-Sánchez et al., 2020).