SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Lemma 3. If  $\max_{\mathbf{x} \in \mathbb{X}} \operatorname{EI}(\mathbf{x}) > 0$ , then  $\arg \max_{\mathbf{x} \in \mathbb{X}} \operatorname{EI}(\mathbf{x}) = \arg \max_{\mathbf{x} \in \mathbb{X}, \operatorname{EI}(\mathbf{x}) > 0} \operatorname{LogEI}(\mathbf{x})$ .
Proof. Suppose  $\max_{\mathbf{x} \in \mathbb{X}} \operatorname{EI}(\mathbf{x}) > 0$ . Then  $\arg \max_{\mathbf{x} \in \mathbb{X}} \operatorname{EI}(\mathbf{x}) = \arg \max_{\mathbf{x} \in \mathbb{X}, \operatorname{EI}(\mathbf{x}) > 0} \operatorname{EI}(\mathbf{x})$ . For all  $\mathbf{x} \in \mathbb{X}$  such that  $\operatorname{EI}(\mathbf{x}) > 0$ ,  $\operatorname{LogEI}(\mathbf{x}) = \log (\operatorname{EI}(\mathbf{x}))$ . Since  $\log$  is monontonic, we have that  $\arg \max_{z \in \mathbb{R}_{>0}} z = \arg \max_{z \in \mathbb{R}_{>0}} \log (z)$ . Hence,  $\arg \max_{\mathbf{x} \in \mathbb{X}, \operatorname{EI}(\mathbf{x}) > 0} \operatorname{EI}(\mathbf{x}) = \arg \max_{\mathbf{x} \in \mathbb{X}, \operatorname{EI}(\mathbf{x}) > 0} \operatorname{LogEI}(\mathbf{x})$ .
Figure 9: Convergence behavior of the asymptotic Laurent expansion Eq. (16) of different orders.
A.2 Analytical LogEI's Asymptotics
As  $z$  grows negative and large, even the more robust second branch in Eq. (14), as well as the implementation of Hutter et al. [35] and Klein et al. [46] can suffer from numerical instabilities (Fig. 10, left). In our case, the computation of the last term of Eq. (12) is problematic for large negative  $z$ . For this reason, we propose an approximate asymptotic computation based on a Laurent expansion at  $-\infty$ . As a result of the full analysis in the following, we also attain a particularly simple formula with inverse quadratic convergence in  $z$ , which is the basis of the third branch of Eq. (14):
$$
\log \left(1 + \frac {z \Phi (z)}{\phi (z)}\right) = - 2 \log (| z |) + \mathcal {O} (| z ^ {- 2} |). \tag {15}
$$
In full generality, the asymptotic behavior of the last term can be characterized by the following result.
Lemma 4 (Asymptotic Expansion). Let  $z < -1$  and  $K \in \mathbb{N}$ , then
$$
\log \left(1 + \frac {z \Phi (z)}{\phi (z)}\right) = \log \left(\sum_ {k = 1} ^ {K} (- 1) ^ {k + 1} \left[ \prod_ {j = 0} ^ {k - 1} (2 j + 1) \right] z ^ {- 2 k}\right) + \mathcal {O} \left(| z ^ {- 2 (K - 1)} |\right). \tag {16}
$$
Proof. We first derived a Laurent expansion of the non-log-transformed  $z\Phi(z) / \phi(z)$ , a key quantity in the last term of Eq. (12), with the help of Wolfram Alpha [37]:
$$
\frac {z \Phi (z)}{\phi (z)} = - 1 - \sum_ {k = 1} ^ {K} (- 1) ^ {k} \left[ \prod_ {j = 0} ^ {k - 1} (2 j + 1) \right] z ^ {- 2 k} + \mathcal {O} (| z | ^ {- 2 K}). \tag {17}
$$
It remains to derive the asymptotic error bound through the log-transformation of the above expansion.
Letting  $L(z, K) = \sum_{k=1}^{K} (-1)^{k+1} \left[ \prod_{j=0}^{k-1} (2j + 1) \right] z^{-2k}$ , we get
$$
\begin{array}{l} \log \left(1 + \frac {z \Phi (z)}{\phi (z)}\right) = \log \left(L (z, K) + \mathcal {O} (| z | ^ {- 2 K})\right) \\ = \log L (z, K) + \log \left(1 + \mathcal {O} \left(\left| z \right| ^ {- 2 K}\right) / L (z, K)\right) \tag {18} \\ = \log L (z, K) + \mathcal {O} (\mathcal {O} (| z | ^ {- 2 K}) / L (z, K)) \\ = \log L (z, K) + \mathcal {O} (| z | ^ {- 2 (K - 1)}). \\ \end{array}
$$
The penultimate equality is due to  $\log (1 + x) = x + \mathcal{O}(x^2)$ , the last due to  $L(z,K) = \Theta (|z|^{-2})$ .
Figure 10: Left: Comparison of LogEI values in single-precision floating point arithmetic, as a function of  $z = (\mu(x) - f^*) / \sigma(x)$  to RoBO's LogEI implementation [46]. Notably, RoBO's LogEI improves greatly on the naive implementation (Fig. 8), but still exhibits failure points (red) well above floating point underflow. The implementation of Eq. (14) continues to be stable in this regime. Right: Comparison of LogEI values to HEBO's approximate LogEI implementation [12]. At the time of writing, HEBO's implementation exhibits a discontinuity and error of  $> 1.02$  at the threshold  $z = -6$ , below which the approximation takes effect. The discontinuity could be ameliorated, though not removed, by correcting the log normalization constant (turquoise). The figure also shows that the naive implementation used by HEBO for  $z > -6$  starts to become unstable well before  $z = -6$ .
SMAC 1.0 and RoBO's Analytic LogEI To our knowledge, SMAC 1.0's implementation of the logarithm of analytic EI due to Hutter et al. [35], later translated to RoBO [46], was the first to improve the numerical stability of analytic EI through careful numerics. The associated implementation is mathematically identical to  $\log \circ \mathrm{EI}$ , and greatly improves the numerical stability of the computation. For large negative  $z$  however, the implementation still exhibits instabilities that gives rise to floating point infinities through which useful gradients cannot be propagated (Fig. 10, left). The implementation proposed herein remedies this problem by switching to the asymptotic approximation of Eq. (15) once it is accurate to machine precision  $\epsilon$ . This is similar to the use of asymptotic expansions for the computation of  $\alpha$ -stable densities proposed by Ament and O'Neil [2].
HEBO's Approximate Analytic LogEI HEBO [12] contains an approximation to the logarithm of analytical EI as part of its implementation of the MACE acquisition function [55], which – at the time of writing – is missing the log normalization constant of the Gaussian density, leading to a large discontinuity at the chosen cut-off point of  $z = -6$  below which the approximation takes effect, see here for the HEBO implementation. Notably, HEBO does not implement an exact stable computation of LogEI like the one of Hutter et al. [35], Klein et al. [46] and the non-asymptotic branches of the current work. Instead, it applies the approximation for all  $z < -6$ , where the implementation exhibits a maximum error of  $> 1.02$ , or if the implementation's normalization constant were corrected, a maximum error of  $> 0.1$ . By comparison, the implementation put forth herein is mathematically exact for the non-asymptotic regime  $z > -1/\sqrt{\epsilon}$  and accurate to numerical precision in the asymptotic regime due to the design of the threshold value.
A.3 Monte-Carlo Expected Improvement
For Monte-Carlo, we cannot directly apply similar numerical improvements as for the analytical version, because the utility values, the integrand of Eq. (4), on the sample level are likely to be mathematically zero. For this reason, we first smoothly approximate the acquisition utility and subsequently apply log transformations to the approximate acquisition function.
To this end, a natural choice is  $\mathrm{softplus}_{\tau_0}(x) = \tau_0\log (1 + \exp (x / \tau_0))$  for smoothing the  $\max(0,x)$ , where  $\tau_0$  is a temperature parameter governing the approximation error. Further, we approximate the  $\max_i$  over the  $q$  candidates by the norm  $\| \cdot \|_{1 / \tau_{\max}}$  and note that the approximation error introduced by both smooth approximations can be bound tightly as a function of two "temperature" parameters  $\tau_0$  and  $\tau_{\mathrm{max}}$ , see Lemma 2.
Importantly, the smoothing alone only solves the problem of having mathematically zero gradients, not that of having numerically vanishing gradients, as we have shown for the analytical case above. For this reason, we transform all smoothed computations to log space and thus need the following special implementation of  $\log \circ$  softplus that can be evaluated stably for a very large range of inputs:
$$
\operatorname {l o g s o f t p l u s} _ {\tau} (x) = \left\{ \begin{array}{l l} [ \log \circ \operatorname {s o f t p l u s} _ {\tau} ] (x) & x / \tau > l \\ x / \tau + \log (\tau) & x / \tau \leq l \end{array} \right.
$$
where  $\tau$  is a temperature parameter and  $l$  depends on the floating point precision used, around  $-35$  for double precision in our implementation.
Note that the lower branch of  $\log \text{softplus}$  is approximate. Using a Taylor expansion of  $\log (1 + z) = z - z^2 / 2 + \mathcal{O}(z^3)$  around  $z = 0$ , we can see that the approximation error is  $\mathcal{O}(z^2)$ , and therefore,  $\log (\log (1 + \exp (x))) = x + \mathcal{O}(\exp (x)^2)$ , which converges to  $x$  exponentially quickly as  $x \to -\infty$ . In our implementation,  $l$  is chosen so that no significant digit is lost in dropping the second order term from the lower branch.
Having defined logsoftplus, we further note that
$$
\begin{array}{l} \log \| \mathbf {x} \| _ {1 / \tau_ {\max}} = \log \left(\sum_ {i} x _ {i} ^ {1 / \tau_ {\max}}\right) ^ {\tau_ {\max}} \\ = \tau_ {\max } \log \left(\sum_ {i} \exp \left(\log \left(x _ {i}\right) / \tau_ {\max }\right)\right) \\ = \tau_ {\max } \operatorname {l o g s u m e x p} _ {i} \left(\log \left(x _ {i}\right) / \tau_ {\max }\right) \\ \end{array}
$$
Therefore, we express the logarithm of the smoothed acquisition utility for  $q$  candidates as
$$
\tau_ {\max } \operatorname {l o g s u m e x p} _ {j} ^ {q} \left(\operatorname {l o g s o f t p l u s} _ {\tau_ {0}} \left(\left(\xi^ {i} (x _ {j}) - y ^ {*}\right) / \tau_ {\max }\right). \right.
$$
Applying another logsumexp to compute the logarithm of the mean of acquisition utilities over a set of Monte Carlo samples  $\{\xi_i\}_i$  gives rise to the expression in Eq. (10).
In particular for large batches (large  $q$ ), this expression can still give rise to vanishing gradients for some candidates, which is due to the large dynamic range of the outputs of the logsoftplus when  $x < 0$ . To solve this problem, we propose a new class of smooth approximations to the "hard" non-linearities that decay as  $\mathcal{O}(1 / x^2)$  as  $x \to -\infty$  in the next section.
A.4 A Class of Smooth Approximations with Fat Tails for Larger Batches
A regular  $\mathrm{softplus}(x) = \log (1 + \exp (x))$  function smoothly approximates the ReLU non-linearity and - in conjunction with the log transformations - is sufficient to achieve good numerical behavior for small batches of the Monte Carlo acquisition functions. However, as more candidates are added,  $\log \mathrm{softplus}(x) = \log (\log (1 + \exp (x)))$  is increasingly likely to have a high dynamic range as for  $x\ll 0$ ,  $\log \mathrm{softplus}_{\tau}(x)\sim -x / \tau$ . If  $\tau >0$  is chosen to be small,  $(-x / \tau)$  can vary orders of magnitude within a single batch. This becomes problematic when we approximate the maximum utility over the batch of candidates, since  $\mathrm{logsumexp}$  only propagates numerically non-zero gradients to inputs that are no smaller than approximately  $(\max_j x_j - 700)$  in double precision, another source of vanishing gradients.
To solve this problem, we propose a new smooth approximation to the ReLU, maximum, and indicator functions that decay only polynomially as  $x \to -\infty$ , instead of exponentially, like the canonical softplus. The high level idea is to use  $(1 + x^2)^{-1}$ , which is proportional to the Cauchy density function (and is also known as a Lorentzian), in ways that maintain key properties of existing smooth approximations – convexity, positivity, etc – while changing the asymptotic behavior of the functions from exponential to  $\mathcal{O}(1 / x^2)$  as  $x \to -\infty$ , also known as a “fat tail”. Further, we will show that the proposed smooth approximations satisfy similar maximum error bounds as their exponentially decaying counterparts, thereby permitting a similar approximation guarantee as Lemma 2 with minor adjustments to the involved constants. While the derivations herein are based on the Cauchy density with inverse quadratic decay, it is possible to generalize the derivations to e.g.  $\alpha$ -stable distribution whose symmetric variants permit accurate and efficient numerical computation [2].
Fat Softplus We define
$$
\varphi_ {+} (x) = \alpha \left(1 + x ^ {2}\right) ^ {- 1} + \log \left(1 + \exp (x)\right), \tag {19}
$$
for a positive scalar  $\alpha$ . The following result shows that we can ensure the monotonicity and convexity - both important properties of the ReLU that we would like to maintain in our approximation - of  $g$  by carefully choosing  $\alpha$ .
Figure 11: The fat softplus approximates Figure 12: The fat softplus has an  $\mathcal{O}(1 / x^2)$  asymptotically tightly as the regular softplus is a monotonic decay, versus the  $\mathcal{O}(\exp (x))$  decay as  $x\to \infty$  and is also monotonic, convex, and positive. The  $-\infty$ , moderating the dynamic range of the quantitatively used a temperature of  $\tau_0 = 0.01$ .
Lemma 5 (Monotonicity and Convexity).  $\varphi_{+}(x)$  is positive, monotonically increasing, and strictly convex for  $\alpha$  satisfying
$$
0 \leq \alpha <   \frac {e ^ {1 / \sqrt {3}}}{2 \left(1 + e ^ {1 / \sqrt {3}}\right)}.
$$
Proof. Positivity follows due to  $\alpha \geq 0$ , and both sumands being positive. Monotonicity and convexity can be shown via canonical differential calculus and bounding relevant quantities.
In particular, regarding monotonicity, we want to select  $\alpha$  so that the first derivative is bounded below by zero:
$$
\partial_ {x} \varphi_ {+} (x) = \frac {e ^ {x}}{1 + e ^ {x}} - \alpha \frac {2 x}{(1 + x ^ {2}) ^ {2}}
$$
First, we note that  $\partial_x\varphi_+(x)$  is positive for  $x < 0$  and any  $\alpha$ , since both terms are positive in this regime. For  $x \geq 0$ ,  $\frac{e^x}{1 + e^x} = (1 + e^{-x})^{-1} \geq 1/2$ , and  $-1/(1 + x^2)^2 \geq -1/(1 + x^2)$ , so that
$$
\partial_ {x} \varphi_ {+} (x) \geq \frac {1}{2} - \alpha \frac {2 x}{(1 + x ^ {2})}
$$
Forcing  $\frac{1}{2} -\alpha \frac{2x}{(1 + x^2)} >0$ , and multiplying by  $(1 + x^{2})$  gives rise to a quadratic equation whose roots are  $x = 2\alpha \pm \sqrt{4\alpha^2 - 1}$ . Thus, there are no real roots for  $\alpha < 1 / 2$ . Since the derivative is certainly positive for the negative reals and the guaranteed non-existence of roots implies that the derivative cannot cross zero elsewhere,  $0\leq \alpha < 1 / 2$  is a sufficient condition for monotonicity of  $\varphi_{+}$ .
Regarding convexity, our goal is to prove a similar condition on  $\alpha$  that guarantees the positivity of the second derivative:
$$
\partial_ {x} ^ {2} \varphi_ {+} (x) = \alpha \frac {6 x ^ {2} - 2}{(1 + x ^ {2}) ^ {3}} + \frac {e ^ {- x}}{(1 + e ^ {- x}) ^ {2}}
$$
Note that  $\frac{6x^2 - 2}{(1 + x^2)^3}$  is symmetric around  $0$ , is negative in  $(- \sqrt{1/3}, \sqrt{1/3})$  and has a minimum of  $-2$  at  $0$ .  $\frac{e^{-x}}{(1 + e^{-x})^2}$  is symmetric around zero and decreasing away from zero. Since the rational polynomial is only negative in  $(- \sqrt{1/3}, \sqrt{1/3})$ , we can lower bound  $\frac{e^{-x}}{(1 + e^{-x})^2} > \frac{e^{-\sqrt{1/3}}}{(1 + e^{-\sqrt{1/3}})^2}$  in
$(-\sqrt{1 / 3},\sqrt{1 / 3})$  . Therefore,
$$
\partial_ {x} ^ {2} \varphi_ {+} (x) \geq \frac {e ^ {- x}}{(1 + e ^ {- x}) ^ {2}} - 2 \alpha
$$
Forcing  $\frac{e^{-\sqrt{1/3}}}{(1 + e^{-\sqrt{1/3}})^2} - 2\alpha > 0$  and rearranging yields the result. Since  $\frac{e^{-\sqrt{1/3}}}{(1 + e^{-\sqrt{1/3}})^2} / 2 \sim 0.115135$ , the convexity condition is stronger than the monotonicity condition and therefore subsumes it.
Importantly  $\varphi$  decays only polynomially for increasingly negative inputs, and therefore  $\log \varphi$  only logarithmically, which keeps the range of  $\varphi$  constrained to values that are more manageable numerically. Similar to Lemma 7, one can show that
$$
\left| \tau \varphi_ {+} (x / \tau) - \operatorname {R e L U} (x) \right| \leq (\alpha + \log (2)) \tau . \tag {20}
$$
There are a large number of approximations or variants of the ReLU that have been proposed as activation functions of artificial neural networks, but to our knowledge, none satisfy the properties that we seek here: (1) smoothness, (2) positivity, (3) monotonicity, (4) convexity, and (5) polynomial decay. For example, the leaky ReLU does not satisfy (1) and (2), and the ELU does not satisfy (5).
Fat Maximum The canonical logsumexp approximation to  $\max_i x_i$  suffers from numerically vanishing gradients if  $\max_i x_i - \min_j x_j$  is larger a moderate threshold, around 760 in double precision, depending on the floating point implementation. In particular, while elements close to the maximum receive numerically non-zero gradients, elements far away are increasingly likely to have a numerically zero gradient. To fix this behavior for the smooth maximum approximation, we propose
$$
\varphi_ {\max } (\mathbf {x}) = \max  _ {j} x _ {j} + \tau \log \sum_ {i} \left[ 1 + \left(\frac {x _ {i} - \max  _ {j} x _ {j}}{\tau}\right) ^ {2} \right] ^ {- 1}. \tag {21}
$$
This approximation to the maximum has the same error bound to the true maximum as the logsumexp approximation:
Lemma 6. Given  $\tau >0$
$$
\max  _ {i} x _ {i} \leq \tau \phi_ {\max } (x / \tau) \leq \max  _ {i} x _ {i} + \tau \log (d). \tag {22}
$$
Proof. Regarding the lower bound, let  $i = \arg \max_{j} x_{j}$ . For this index, the associated summand in (21) is 1. Since all sumands are positive, the entire sum is lower bounded by 1, hence
$$
\tau \log \sum_ {i} \left[ 1 + \left(\frac {x _ {i} - \operatorname* {m a x} _ {j} x _ {j}}{\tau}\right) ^ {2} \right] ^ {- 1} > \tau \log (1) = 0
$$
Adding  $\max_j x_j$  to the inequality finishes the proof for the lower bound.
Regarding the upper bound, (21) can be maximized when  $x_{i} = \max_{j}x_{j}$  for all  $i$ , in which case each  $(x_{i} - \max_{j}x_{j})^{2}$  is minimized, and hence each summand is maximized. In this case,
$$
\tau \log \sum_ {i} \left[ 1 + \left(\frac {x _ {i} - \max  _ {j} x _ {j}}{\tau}\right) ^ {2} \right] ^ {- 1} \leq \tau \log \left(\sum_ {i} 1\right) = \tau \log (d).
$$
Adding  $\max_j x_j$  to the inequality finishes the proof for the upper bound.
Fat Sigmoid Notably, we encountered a similar problem using regular (log)-sigmoids to smooth the constraint indicators for EI with black-box constraints. In principle the Cauchy cumulative distribution function would satisfy these conditions, but requires the computation of  $\arctan$ , a special function that requires more floating point operations to compute numerically than the following function. Here, we want the smooth approximation  $\iota$  to satisfy 1) positivity, 2) monotonicity, 3) polynomial decay, and 4)  $\iota(x) = 1/2 - \iota(-x)$ . Let  $\gamma = \sqrt{1/3}$ , then we define
$$
\iota (x) = \left\{ \begin{array}{l l} \frac {2}{3} \left(1 + (x - \gamma) ^ {2}\right) ^ {- 1} & x <   0, \\ 1 - \frac {2}{3} \left(1 + (x + \gamma) ^ {2}\right) ^ {- 1} & x \geq 0. \end{array} \right.
$$
Figure 13: We construct the fat sigmoid approximation (purple) by splicing together two Lorentzians (blue and teal) at one of their inflection points.
Figure 14: The fat sigmoid approximation  $(\tau = 0.01)$  decays as  $\mathcal{O}(1 / x^2)$  instead of  $\mathcal{O}(\exp (x))$  as  $x\to -\infty$  , minimizing the dynamic range of the numerical quantities in constrained qLogEI.
$\iota$  is monotonically increasing, satisfies  $\iota(x) \to 1$  as  $x \to \infty$ ,  $\iota(0) = 1/2$ , and  $\iota(x) = \mathcal{O}(1/x^2)$  as  $x \to -\infty$ . Further, we note that the asymptotics are primarily important here, but that we can also make the approximation tighter by introducing a temperature parameter  $\tau$ , and letting  $\iota_{\tau}(x) = \iota(x/\tau)$ . The approximation error of  $\iota_{\tau}(x)$  to the Heaviside step function becomes tighter point-wise as  $\tau \to 0+$ , except for at the origin where  $\iota_{\tau}(x) = 1/2$ , similar to the canonical sigmoid.
A.5 Constrained Expected Improvement
For the analytical case, many computational frameworks already provide a numerically stable implementation of the logarithm of the Gaussian cumulative distribution function, in the case of PyTorch, torch specialized.log_mdtr, which can be readily used in conjunction with our implementation of LogEI, as described in Sec. 4.3.
For the case of Monte-Carlo parallel EI, we implemented the fat-tailed  $\iota$  function from Sec. A.4 to approximate the constraint indicator and compute the per-candidate, per-sample acquisition utility using
$$
\left(\operatorname {l o g s o f t p l u s} _ {\tau_ {0}} \left(\xi_ {i} (\mathbf {x} _ {j}) - y ^ {*}\right) + \sum_ {k} \log \circ \iota \left(- \frac {\xi_ {i} ^ {(k)} (\mathbf {x} _ {j})}{\tau_ {\mathrm {c o n s}}}\right), \right.
$$
where  $\xi_{i}^{(k)}$  is the  $i$ th sample of the  $k$ th constraint model, and  $\tau_{\mathrm{cons}}$  is the temperature parameter controlling the approximation to the constraint indicator. While this functionality is in our implementation, our benchmark results use the analytical version.
A.6 Parallel Expected Hypervolume Improvement
The hypervolume improvement can be computed via the inclusion-exclusion principle, see [13] for details, we focus on the numerical issues concerning qEHVI here. To this end, we define
$$
z _ {k, i _ {1}, \dots , i _ {j}} ^ {(m)} = \min  \left[ \mathbf {u} _ {k}, \mathbf {f} (\mathbf {x} _ {i _ {1}}), \dots , \mathbf {f} (\mathbf {x} _ {i _ {j}}) \right],
$$
where  $\mathbf{f}$  is the vector-valued objective function, and  $\mathbf{u}_k$  is the vector of upper bounds of one of  $K$  hyper-rectangles that partition the non-Pareto-dominated space, see [13] for details on the partitioning. Letting  $\mathbf{l}_k$  be the corresponding lower bounds of the hyper-rectangles, the hypervolume improvement can then be computed as
$$
\operatorname {H V I} \left(\left\{\mathbf {f} \left(\mathbf {x} _ {i}\right) \right\} _ {i = 1} ^ {q} = \sum_ {k = 1} ^ {K} \sum_ {j = 1} ^ {q} \sum_ {X _ {j} \in \mathcal {X} _ {j}} (- 1) ^ {j + 1} \prod_ {m = 1} ^ {M} \left[ z _ {k, X _ {j}} ^ {(m)} - l _ {k} ^ {(m)} \right] _ {+}, \right. \tag {23}
$$
where  $\mathcal{X}_j = \{X_j\subset \mathcal{X}_{\mathrm{cand}}:|X_j| = j\}$  is the superset of all subsets of  $\mathcal{X}_{\mathrm{cand}}$  of size  $j$  and  $z_{k,X_j}^{(m)} = z_{k,i_1,\ldots ,i_j}^{(m)}$  for  $X_{j} = \{\mathbf{x}_{i_{1}},\dots ,\mathbf{x}_{i_{j}}\}$ .
To find a numerically stable formulation of the logarithm of this expression, we first re-purpose the  $\varphi_{\mathrm{max}}$  function to compute the minimum in the expression of  $z_{k,i_1,\ldots ,i_j}^{(m)}$ , like so  $\varphi_{\min}(x) = -\varphi_{\max}(-x)$ . Further, we use the  $\varphi_{+}$  function of Sec. A.4 as for the single objective case to approximate  $[z_{k,X_j}^{(m)} - l_k^{(m)}]_+$ . We then have
$$
\log \prod_ {m = 1} ^ {M} \varphi_ {+} \left[ z _ {k, X _ {j}} ^ {(m)} - l _ {k} ^ {(m)} \right] = \sum_ {m = 1} ^ {M} \log \varphi_ {+} \left[ z _ {k, X _ {j}} ^ {(m)} - l _ {k} ^ {(m)} \right] \tag {24}
$$
Since we can only transform positive quantities to log space, we split the sum in Eq. (23) into positive and negative components, depending on the sign of  $(-1)^{j+1}$ , and compute the result using a numerically stable implementation of  $\log (\exp (\log$  of positive terms) -  $\exp (\log$  of negative terms). The remaining sums over  $k$  and  $q$  can be carried out by applying  $\mathrm{logsumexp}$  to the resulting quantity. Finally, applying  $\mathrm{logsumexp}$  to reduce over an additional Monte-Carlo sample dimension yields the formulation of qLogEHVI that we use in our multi-objective benchmarks.
A.7 Probability of Improvement
Numerical improvements for the probability of improvement acquisition which is defined as  $\alpha(x) = \Phi\left(\frac{\mu(x) - y^*}{\sigma(x)}\right)$ , where  $\Phi$  is the standard Normal CDF, can be obtained simply by taking the logarithm using a numerically stable implementation of  $\log(\Phi(z)) = \operatorname{logerfc}\left(-\frac{1}{\sqrt{2}}z\right) - \log(2)$ , where  $\operatorname{logerfc}$  is computed as
$$
\operatorname {l o g e r f c} (x) = \left\{ \begin{array}{l l} \log (\operatorname {e r f c} (x)) & x \leq 0 \\ \log (\operatorname {e r f c x} (x)) - x ^ {2} & x > 0. \end{array} \right.
$$
A.8 q-Noisy Expected Improvement