---
title: Constrained Max-Value Entropy Search
url: https://www.emergentmind.com/topics/constrained-max-value-entropy-search
type: topic
---

# Constrained Max-Value Entropy Search

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 [1910.07003] [2102.09788] [2009.01721].

## 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 \(\gamma_{y_*}(x)=\frac{y_*-\mu_t(x)}{\sigma_t(x)}\) [1703.01968].

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 [1703.01968].

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 [1910.07003].

## 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 \(z_c\in\{-1,+1\}\), with logistic link \(z_c\sim \sigma(z_c\,c(\mathbf{x}))\). 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 [1910.07003].

The corresponding mutual-information objectives are
\[
\mathcal I\big((y,c);y_\star\big)
=H[P(y,c)]-\mathbb E_{y_\star}\big[H(P(y,c\mid y_\star))\big]
\]
for real-valued constraints, and
\[
\mathcal I\big((y,z_c);y_\star\big)
=H[P(y,z_c)]-\mathbb E_{y_\star}\big[H(P(y,z_c\mid y_\star))\big]
\]
for binary constraints. In both cases, cMES uses a local truncation argument: if the point is feasible, then consistency with \(y_\star\) 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 \(y_\star\). For real-valued constraints this produces a masked Gaussian density \(P(y,c\mid y_\star)\propto P(y)P(c)\kappa(y,c)\), while for binary feedback the non-Gaussian term \(P(z_c\mid c)P(c)\) is replaced by a Gaussian approximation \(Q(z_c)Q(c\mid z_c)\), yielding closed-form entropy differences after integrating over the feasibility event [1910.07003].

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 \(\hat{\mathcal X}\), followed by direct computation of the constrained minimum on each sampled path. In the reported experiments, a Sobol discretization with \(|\hat{\mathcal X}|=2000\) and only \(2\) to \(10\) samples of \(y_\star\) was already sufficient for strong performance, and joint sampling consistently outperformed the marginal approximation [1910.07003].

The method also explicitly covers the “unobserved-objective” scenario in which \(z_y\) 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 [1910.07003].

## 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
\[
f_* \coloneqq
\begin{cases}
\max_{x\in\mathcal X_{\mathrm{feasible}}} f(x), & \text{if } \mathcal X_{\mathrm{feasible}}\neq \emptyset,\\
-\infty, & \text{if } \mathcal X_{\mathrm{feasible}}=\emptyset,
\end{cases}
\]
which makes \(f_*\in\mathbb R\cup\{-\infty\}\) a proper random variable and incorporates uncertainty about feasibility directly into the acquisition target [2102.09788].

The sequential acquisition is defined as
\[
\alpha(x)=I(\mathbf h_x;f_* \mid \mathcal D_{t-1}),
\qquad
\mathbf h_x=(f(x),g_1(x),\dots,g_C(x))^\top,
\]
but the paper does not approximate this MI directly. Instead, it introduces an auxiliary density \(q(\mathbf h_x\mid f_*)\) and applies a KL-based lower-bound argument. Choosing \(q\) to be the predictive density truncated to the complement of the “improving feasible hyperrectangle”
\[
R_{f_*}=(f_*,\infty)\times(z_1,\infty)\times\dots\times(z_C,\infty)
\]
yields the information lower bound
\[
L(x)=\mathbb E_{f_*}\big[-\log \overline Z_x(f_*)\big],
\]
where \(\overline Z_x(f_*)\) is the probability that the local outcome does not lie in \(R_{f_*}\). Under independent GPs for objective and constraints,
\[
\overline Z_x(f_*)
=
1-
\Bigl(1-\Phi(\gamma_x^{(f)}(f_*))\Bigr)
\prod_{c=1}^C
\Bigl(1-\Phi(\gamma_x^{(g_c)}(z_c))\Bigr),
\]
so the final sequential acquisition is
\[
\alpha^{\mathrm{IBO}}(x)\approx
-\frac{1}{K}\sum_{\tilde f_*\in\mathcal F_*}
\log \overline Z_x(\tilde f_*),
\]
with \(\mathcal F_*\) obtained by Monte Carlo sampling of constrained-optimum values from random Fourier feature GP path samples [2102.09788].

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 \(C>5\). Second, CMES-IBO provides an explicit variance analysis: if \(\hat L_K(x)\) is the Monte Carlo estimator of the lower bound, then \(\mathrm{Var}[\hat L_1(x)]\le 2\), and the paper gives a concentration bound of Bernstein type,
\[
\Pr\bigl(|\hat L_K(x)-L(x)|\ge \xi\bigr)\le \min\{U_1,U_2\},
\]
with
\[
U_1=\frac{2}{K\xi^2},\qquad
U_2=2\exp\left[-AK\min\left(\frac{\xi^2}{B^2},\frac{\xi}{B}\right)\right].
\]
The authors emphasize that such low-variability guarantees had not been investigated for prior information-theoretic BO acquisitions [2102.09788].

The same logic extends to parallel selection by replacing MI with conditional MI,
\[
I(\mathbf h_x;f_* \mid \mathbf H_q),
\]
and greedily adding points to a batch. The resulting parallel CMES-IBO preserves the lower-bound structure and reuses joint samples of \(f_*\) and pending fantasy outcomes \(\mathbf H_q\) [2102.09788].

## 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 \(\mathcal X^\star\) or over the Pareto front \(\mathcal Y^\star\), and in how they approximate the conditional predictive distribution induced by that global object [1609.01051] [2009.01721] [2011.01150].

| Method | Target random object | Core approximation |
|---|---|---|
| PESMOC | Constrained Pareto set \(\mathcal X^\star\) | Monte Carlo over Pareto sets + EP |
| MESMOC | Constrained Pareto front \(\mathcal Y^\star\) | Output-space entropy + truncated Gaussians |
| MESMOC+ | Constrained Pareto front \(\mathcal Y^\star\) | Monte Carlo over fronts + ADF |

PESMOC is the constrained multi-objective extension of predictive entropy search. It defines
\[
\alpha(\mathbf x)
=
H(\mathcal X^\star\mid\mathcal D)
-
\mathbb E_{\mathbf y}\big[H(\mathcal X^\star\mid\mathcal D\cup\{(\mathbf x,\mathbf y)\})\big],
\]
and then uses the symmetry of mutual information to work with
\[
H(\mathbf y\mid\mathcal D,\mathbf x)-
\mathbb E_{\mathcal X^\star}\big[H(\mathbf y\mid\mathcal D,\mathbf x,\mathcal X^\star)\big].
\]
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 [1609.01051].

MESMOC moves from input space to output space. It targets the entropy of the constrained Pareto front \(\mathcal Y^\star\) and defines
\[
\alpha(\mathbf x)=I(\{\mathbf x,\mathbf y\},\mathcal Y^\star\mid\mathcal D)
=
H(\mathbf y\mid\mathcal D,\mathbf x)
-
\mathbb E_{\mathcal Y^\star}\big[H(\mathbf y\mid\mathcal D,\mathbf x,\mathcal Y^\star)\big].
\]
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
\[
\alpha(\mathbf x)\simeq \frac{1}{S}\sum_{s=1}^S
\left[
\sum_{j=1}^K
\frac{\gamma_s^{f_j}(\mathbf x)\phi(\gamma_s^{f_j}(\mathbf x))}{2\Phi(\gamma_s^{f_j}(\mathbf x))}
-\ln\Phi(\gamma_s^{f_j}(\mathbf x))
+
\sum_{i=1}^L
\frac{\gamma_s^{c_i}(\mathbf x)\phi(\gamma_s^{c_i}(\mathbf x))}{2\Phi(\gamma_s^{c_i}(\mathbf x))}
-\ln\Phi(\gamma_s^{c_i}(\mathbf x))
\right].
\]
The acquisition is optimized only over points satisfying the mean-feasibility filter \(\mu_{c_i}(\mathbf x)\ge 0\) for all constraints [2009.01721].

MESMOC+ keeps the output-space focus on \(\mathcal Y^\star\) but replaces MESMOC’s approximate dominance condition with an explicit compatibility factor
\[
p(\mathcal Y^\star\mid \mathbf f,\mathbf c)
\propto
\prod_{\mathbf f^\star\in\mathcal Y^\star}
\Omega(\mathbf f^\star,\mathbf f,\mathbf c),
\]
where \(\Omega\) encodes that a candidate point is either infeasible or dominated by the front. The conditional predictive
\[
p(\mathbf f,\mathbf c\mid \mathcal D,\mathbf x,\mathcal Y^\star)
\propto
p(\mathbf f,\mathbf c\mid \mathcal D,\mathbf x)\,
p(\mathcal Y^\star\mid \mathbf f,\mathbf c)
\]
is approximated by Assumed Density Filtering, producing factorized Gaussians and a final acquisition expressed as variance reduction rather than log-variance reduction:
\[
\alpha(\mathbf x)\approx
\sum_g \big(v_g(\mathbf x)+\sigma_g^2\big)
-
\frac{1}{M}\sum_{m=1}^M \sum_g
\big(\tilde v_{g,(m)}(\mathbf x)+\sigma_g^2\big).
\]
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 [2011.01150].

## 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 \(f^*\) with the joint optimum pair \((x^*,f^*)\) and defines
\[
\alpha_{\mathrm{JES}}(x)=I\big((x,y);(x^*,f^*)\mid\mathcal D_n\big).
\]
Its implementation conditions the GP on fantasized noiseless optimal input/output pairs and then imposes a local truncation \(f(x)\le f^*\). 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 [2206.04771].

Variational Entropy Search (VES) develops a different reinterpretation. It applies the Barber–Agakov lower bound to MES and defines an entropy-search lower bound
\[
ESLB(q,\mathbf x)
=
\mathbb E_{p(y^*,y_{\mathbf x}\mid\mathcal D_t)}
\big[\log q(y^*\mid\mathcal D_t,y_{\mathbf x})\big].
\]
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 \(y^*_{\mathrm{feas}}\) and then apply the same variational machinery to joint objective/constraint observations [2402.11345].

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 \(\phi_o(t,q)\), model counting computes exact outcome probabilities
\[
p(o\mid Q=q)=\frac{\#[\phi_o\land \kappa(t)](q)}{\#\kappa(t)},
\]
and the next query is chosen by maximizing Shannon entropy
\[
\mathcal H(q)= -\sum_{o_i\in O} p(o_i\mid Q=q)\log_2 p(o_i\mid Q=q).
\]
The paper explicitly describes this as a “discrete, constraint-based MES,” with the worthwhile-query predicate \(W(q)\) playing the role of a feasibility condition on the search space [2009.10877].

## 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 \(-\infty\) in \(f_*\), so the acquisition can explicitly trade off learning feasibility against learning objective values [1910.07003] [2102.09788].

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 [2009.01721] [2011.01150].

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 \(\tilde f_*\). 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 [1910.07003] [2102.09788] [2011.01150].

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 [1910.07003] [2102.09788] [2011.01150].

Source: https://www.emergentmind.com/topics/constrained-max-value-entropy-search