Meta-VBO: Utilizing Prior Tasks in Optimizing Risk Measures with Gaussian Processes
Authors: Quoc Phong Nguyen
Bryan Kian Hsiang Low
Patrick Jaillet
Venue: ICLR 2024 poster (2024)
Keywords: meta-learning
Bayesian optimization
risk measure
value-at-risk
conditional value-at-risk
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Research on optimizing the risk measure of a blackbox function using Gaussian processes, especially Bayesian optimization (BO) of risk measures, has become increasingly important due to the inevitable presence of uncontrollable variables in real-world applications. Nevertheless, existing works on BO of risk measures start the optimization from scratch for every new task without considering the results of prior tasks. In contrast, its vanilla BO counterpart has received a thorough investigation on utilizing prior tasks to speed up the current task through the body of works on meta-BO which, however, have not considered risk measures. To bridge this gap, this paper presents the first algorithm for meta-BO of risk measures (i.e., value-at-risk (VaR) and the conditional VaR), namely meta-VBO, by introducing a novel adjustment to the upper confidence bound acquisition function. Our proposed algorithm exhibits two desirable properties: (i) invariance to scaling and vertical shifting of the blackbox function and (ii) robustness to prior harmful tasks. We provide a theoretical performance guarantee for our algorithm and empirically demonstrate its performance using several synthetic function benchmarks and real-world objective functions.

Published as a conference paper at ICLR 2024
META-VBO: UTILIZING PRIOR TASKS IN OPTIMIZING RISK MEASURES WITH GAUSSIAN PROCESSES
Quoe Phong Nguyen', Bryan Kian Hsiang Low? & Patrick Jaillet' 'LIDS and EECS, Massachusetts Institute of Technology. USA Scchoo of Computing, National University Singapore, Singapore qphongmpe gma 1. com, lowkh@conme nus edu. sg, jaillalllli Weilletton @mit.
ABSTRACT
Research on optimizing the risk measure of a blackbox function using Gaussian processes, especially Bayesian optimization (BO) of risk measures, has become increasingly important due to the inevitable presence of uncontrollable variables in real-world applications. Nevertheless, existing works on BO of risk measures start the optimization from scratch for every new task without considering the results of prior tasks. contrast, its vanilla BÔ counterpart has received a thorough investigation on utilizing prior tasks to speed up the current task through the body of works on meta-BO -BO which, however, have not considered risk measures. To bridge this gap, this paper presents the first algorithm for mela-BO of risk measures (i.e., value-at-risk (VaR) and the conditional VaR), namely meta- meta-VBO, by introducing a novel adjustment the upper confidence bound acquisition function. Our proposed algorithm exhibits two desirable properties: (i) invariance to scaling and vertical shifting of the blackbox function and (ii) robustness to prior harmful tasks. We provide a theoretical performance guarantee for our algorithm and empirically demonstrate its performance using several synthetic function benchmarks and real-world objective functions.
INTRODUCTION
Real world optimization problems, such as maximizing the performance manufacturing line, the crop yield of a farm, and the performance of machine learning (ML) models, often face with the challenge of dealing with blackbox objective functions (i.e., without closed-form fooem expressions, convexity, or derivatives) and timeccosummig or expensive noisy zeroth-order evaluations (Snoek jet al. 20122 Nguyen et al., 2021b). These issues have attracted significant attention, particularly on the Bayesian optimization (BO) framework (Srinivas et al. 2010 Snoek al. 2012: Shahriari et al., 2013 Hernández- Lobato et al. 2014; Wang and Jegelka| 2017] Garnett 2022). Recently, there has been an increasing interest in B0 of risk measures, driven by the need to address practical aspects related to uncontrollable variables referred to as environmental random variables (Cakmak et al. 2020), e.g., random enrors in sensors of the manufacturing line, weather conditions in farms (Nguyen et al. 2021b(a), and random perturbation in training data of MI. models (Bogunovic et al.) 2018). BO of risk measure algorithms specifically search for the controllable decision variables (c.g., applied torque, input force, nutrient amounts, and ML model hyperparameters) that optimize risk measure of the blackbox function where the risk is induced by the environmental random variables. Risk measures investigated in existing works include the worst-cass case value (Bogunovic et al. 2018), value- at-risk (VaR) and conditional VaR (CVaR) (Cakmak et al.
2020 Nguyen et al. 2222 a Picheny
et al. 2022), probability threshold robustness (Iwazaki et al.
2021a) and the variance (Iwazaki et al.
2021b Makarova et 202T
While there is an extensive research on BO of risk measures, using learning experiences from prior tasks to accelerate the optimization of the current task has not been explored. In contrast, this approach has been shown to effectively reduce the number of costly observations required by the vanilla BO counterpart through numerous works on meta-BO. These works encompass wide spectrum of methodologies, ranging from those that involve the transfer of the probabilistic models from prior tasks to the current one (Swersky et al. 2013). to more nuanced approaches centered on transferring the optimization result (Feurer et al." 2018 Wistuba et al., 2016 2018 Perrone et al.

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2019 Ramachandran al., 2019 Volpp al. 2020 Dai et al. 2022
Although the former category,
such as the multi-task Gaussian processes (GP) (Swersky et al. 2013 is well-defined, it often suffers from computational challenges discussed by Dai et al. (2222) and is not specifically tailored to the goal of optimization the blackbox function 's maximizer. In contrast, the latter category, which attracts the majority of research attention, presents greater challenges and lacks clearly defined framework. For example, [eurer et al. (2018) transfers the ranking of function evaluations. Perrone ct al. (2019) narrows down the search space. Ramachandran et al. (2019) considers the distribution of the optima. Dai et al. 2022) considers weighted average of the acquisition functions.
In this paper, we focus on the challenge of transferring the optimization outcome. Specifically, we pioneer the first solution to the meta-Bo BO of risk measure problem, namely meta- VBO. Our work accounts for the risk introduced by the environmental random variables a crucial aspect not addressed in any existing meta-BO works. We focus on two well-known risk measures: value-at-
risk (VaR) and conditional VaR (CVaR) (Cakmak et al., 2020 Torossian et al., 2020). Moreover, our solution also extends to the adversarially robust BO (Bogunovic et al. 2018) and applies to the meta-BO problem. We assume that the distribution of the environmental random variables is assume that
given, e.g., by estimating from historical data or domain knowledge. Additionally, during the optimization, noisy evaluations of the blackbox function can be acquired (queried) at any realizations of the environmental random variables. In the above examples, this can be achieved running simulations of manufacturing lines with controlled random errors or growing indoor plants in a laboratory environment where weather conditions are regulated. After the optimization is completed, we recommend implementing the best decision variable in the real-world environment where the environmental random variables are beyond our control. These assumptions are commonly made in several BO works with risk measures (Cakmak et al. 2020 Nguyen et al. 2021b at.
Under the above settings, there are several major challenges in addressing meta-BO of risk measures. First, the choice of the information to be transferred from prior tasks to the current task should be tailored to the optimization goal of determining the optima's 's locations. Since they are invariant under scaling and vertical-shifting (i.e., increasing or decreasing the function by a constant) of the blackbox function, it is crucial to ensure that the proposed meta-BO of risk measure solution maintains this invariance. Hence, it rules out the use of the function gap (i.c., the maximum difference between function evaluations) in the theoretical study of Dai et al. (2022) as a similarity measure between tasks. Second, as the number of prior tasks grows, the possibility of encountering task that hampers the current optimization task arises (Feurer et al. 2018 Dai et al. 2022). Thus, it is essential to be resilient against such prior harmful tasks.
Moreover, one cannot easily extend existing meta-BO solutions to address the BO problem with risk measures. While the posterior belief of the blackbox function follows a GP in the vanilla BO, many risk measures including VaR and CVaR do not follow any well-known stochastic process. Hence, it is challenging adopt approaches based on the Gaussian posterior distribution in the works of |Ramachandran et al. (019) and Dai et al. (2022). Furthermore, risk measures are not directly observed. They are only estimated from noisy evaluations of the blackbox function. As result, although FFurer et al. (2018) is able to discover and assign low weights to prior harmful tasks by proposing a ranking loss function, extending this loss function to tackle risk measures not immediately obvious. Last, there are not any existing meta-BO works that possess both theoretical performance guarantees and robustness against scaling and vertical-shi -hifial- of the blackbox function and prior harmful tasks.
To address the above problems, we introduce a new perspective in constructing our solution, namely meta- VBO. Specifically, while the classic Gaussian process- upper confidence bound (GP-UCB) algorithm (Srinivas et al., 2010) offers only one option of the input query at each BO iteration, we pursue different approach of construcing versatile query sel (V-set) in Sec. 3.. Interestingly, we can obtain a no-regret algorithm by selecting any input in this set as the input query (Theorem To the best of our knowledge, this characteristic has not been introduced in any BO algorithm. On one hand, it implies that by restricting the choice of the input query to V-set, the algorithm is no-regret regardless of prior harmful tasks. On the other hand, the flexibility in choosing the input query from a set (as opposed to one option in GP-UCB) allows us to transfer information from prior tasks to the current task by assigning different preferences/prioritiss to inputs in V-set (Sec. B.2pp. Essentially, inputs that are likely to be local (or global) maximizers of many prior tasks are assigned high priorities.
The notion of the best decision variable is discussed in Sec. 3.3
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The size of V-set can be tuned via two hyperparameters, which allows trade-off between the amount of information to be incorporated from prior tasks and the cumulative regret bound of the current ask. This reflects the intuition that without knowing the similarity between prior tasks and the current task, utilizing more information from prior tasks may not benefit the current task. We justify the selected hyperparameters in the experiments based on the incurred regret. Furthermore, V-set is adaptively updated as observations are accumulated from one iteration to the next. This naturally adjusts the effect of prior tasks on the current task as discussed in Remark [3.4] ensuring meta-VBO's robustness to prior harmful tasks. In Sec. 3.3] we propose a strategy to recommend an input that approximates the optimal solution at each iteration such that its instantaneous regret diminishes towards zero as the iteration progresses (Theorem (.7).. Last, we empirically evaluate the performance of meta- meta-VBO in optimizing VaR and CVaR of several synthetic and real-world objective functions Sec.
PRELIMINARY
2.1 BO OF RISK MEASURES
We are interested in maximizing blackbox objective function f (c.g.. the performance of a system)
that depends on both the decision variable x E X' and the environmental variable z E 2, where X and 2 are bounded subsets of Rd and R respectively. We assume that X' and Z are discrete for simplicity, though the results can be extended continuous domains (Srinivas et al. 2010 Chowdhury and Gopalan 2017). Furthermore, in the real-world environment, x is controllable while (e.g., weather conditions and random errors) is not. Hence, the environmental variable is represented as a random variable, denoted as Z. We assume that the distribution of Z is known, c.g.. by estimating from historical data or domain knowledge. These settings are adopted from the existing works of Cakmak et al. (2020): Nguyen et al. 2
Due to the randomness in Z, evaluating the blackbox function at a particular value of x can result in high value under one realization of Z but leading to low value under another realization of Z. In other words, the evaluation of f at x is a random variable, denoted as f(x, Z), whose randomness is induced by Z. Therefore, to determine the "optimal" design variable x,, one cannot directly compare the function evaluations at different values of x. An alternative approach is to consider a risk measure of the distribution of fx, Z) which accounts for the probability (or the risk) that f(x, Z) suffers from poor realizations. In this paper, we examine two risk measures for f(x Z): value-at-risk (VaR), denoted as vy (xx; a),, and conditional VaR (CVaR), denoted cx(x; a). Here, (0, represents a known risk level. VaR and CVaR are defined as follows:
( uy(x; a') da'.
(x; A inf(y: P(f(x, Z) S 7) a}
cf(x: a) &
(I)
While vy (x; ar) can be viewed as 100a 100a%--pereentil of f(x Z) (i.e., the event that f(x, Z) suffers from realizations lower than uy a) happens with probability of at most a), cy a) is the expected value of fx, Z) in the worst 100a% of cases. To avoid unnecessary repetition, we unify the notations of both VaR and CVaR as (x; The notations uy(x; a)) and ej (x; ()) are only used when VaR and CVaR require distinct treatments.
Given the risk measure pf(x; and the distribution of Z, the BO of risk measure problem aims to search for the maximizer x, A arg max (x a) (Cakmak et al., 2020). Since f is unknown, determining this maximizer typically involves iteratively observing noisy evalua- tions y xt, - f(xx, 2,) c(x,, 2,) for iteration 2 1 where c(x,, 2,) ~ AN(0, are i.i.d. Gaus- sian noises Furthermore, it often expensive to obtain these evaluations (e.g.. requiring running expensive simulations). Thus, the crux of the problem lies in devising a selection strategy of the input query (Xe, z.) at iteration f, considering observations in the previous t - 1 iterations, denoted as YD. e [y(xe, Zr.) ) where D, e ((x,, Zy ))+)) to quickly discover X..
We model f with a Gaussian process (GP) (Rasmussen and Williams 2006 Specifically, we use a constant GP prior mean function Po and a squared exponential kernel kxxx)),(xxxe)) covif(x, 2), f(x', z')]
-.5((2 0.5(2-z')) A- zz)) where Ag A
We adopt the settings outlined in previous works of Cakmak et al. 2020 Nguyen et al. (001a)) such that is controllable during the optimization (discussed Sec.
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diag(11.... ......;. (d), A. diag(la+1,.... ld+d'). ), and 03 are the lengthscales and the signal variance, respectively. Given YD., the posterior belief of f(x, 2) is a Gaussian distribution with the following mean and variance
[x(x)2) z) _ M40 K (x.)).D AD. (YD. - Ho)
ox z) A k(xxz) K (x.)),,, AD. K (D.,.x.))
where AD. A (KD..D, + t K is the covariance matrix, and is the identity matrix.
The performance of BO algorithm often analyzed through the cumulative regret, denoted a) is the instantaneous regret at the input
as Rr _ Σ -/=r(xx, r(xx) where r(x,) _ px(xx:a) - -p ( query Xx. The algorithm is no-regret if its cumulative regret is sublinear, i.e., limy-+00 R Br/TT - O. In such a case, the simple regrel ST _ 1nin, rx,,) I R approaches O. This suggests that one should recommend the best observed imput so far as the outcome of the optimization, e.g.. in the vanilla BO problem with a small noise. Unfortunately, in BO of risk measures, P (Xx; a) is unobserved due to the unobserved f(x,, z) for (x,, 2) & D.- This prevents us from determining the best observed imput. Therefore, we will design recommendation strategy such that the instantaneous regret rxx) at the recommended input x² approaches zero & + in Sec. 3.3
2.2 META-BO OF RISK MEASURES
Unlike existing works on BO of risk measures, our approach in this paper considers a set T of prior tasks. Let f X' 2 + R denote the blackbox function of the current task and fr X' 2 RR denote the blackbox function in a prior task T E T. Furthermore, to facilitate later theoretical analysis, we follow the works of Chowdhury and Gopalan (2017) assume that all blackbox functions belong to the reproducing kernel Hilbert space (RKHS) associated with the kernel & and bounded RKHS norm, i.e., Il S and VT T, ll S B. The optimization result of prior task E 'T includes the GP posterior mean and variance of S., denoted as ūī, X' x 2 - IR and 82 X x + R respectively. Meta-BO of the risk measure p aims to utilize (ī+, to improve the optimization of ps. However, the similarity between f and fr (hence, between Ps and pf.) ) is unknown for all 5 E T. In particular, if ps. differs from py. naively utilizing Pr. o² to optimize P can deteriorate the optimization of pf-
Under the assumption that blackbox functions belong to the RKHS space with norm bounded by B, we first revisit the well-known confidence bound of fxx 2) from the work Chowdhury and Gopalan 2017p. Let " denote the maximum information gain about f that can be obtained from any set of ( -- observations. Choosing 8 E (0, 1). 8, (B on /2(7) + +1 log 1/ then the event that 2 1, Vx E X, Vz E Z, 4x(x 2) S f(x, 2) S un u. 2) holds with probability 2 1 - where the lower and upper confidence bounds are defined
4(x,2) z) A (x. z) - 3//3 (x, and ux(x, z) & (x + (,,/,, 2).
resp. (2)
Leveraging on the above confidence bounds and ut, Nguyen et al. (2021b a propose the fol- lowing confidence interval of the risk measure (y(x; a)) (applicable to both VaR and CVaR): with probability& 2 ò,
14 1, Vx X, (x(x))) (x; Pa, (x;o))
where 4, and ue are defined in equation ( and Pf. Pt. Pu, can be replaced by uy, UI,, and Va, for For a prior task r E T, we denote the confidence interval of f (x, 2) as [ (x, 2), 01 (x, 2)] which is defined in equation ② using the GP posterior belief (p,, o²) of fo and the index of the last optimization itcration of task T. Then, the confidence interval of the risk measure Pr. (x; a:) of prior task 7 is [pr, (x; a), Pa, ax).
VaR or by Cys Ct Cu, for CVaR in equation
3 OUR PROPOSED SOLUTION TO MEO OF RISK MEASURES
Typically, at each iteration of the GP-UCB algorithm (Srinivas et al. 2010 there is only one option to select the input query from (assuming a unique maximizer of the GCB acquisition function). Hence, in order to incorporate the information from prior tasks, the existing RM-GP-UCE (Dai et al., 2022) takes a weighted average of the GP-UCT acquisition functions of the current task and prior tasks. However, these weights depend on the maximum difference between f and fr evaluations,
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Pa. (..1x7000 ai)
max Pt. (x))))
mar ft, (xc)i))
Pi. (x)))
(x:a)
Pa, (M)
(x:00)
Pt. (2))0)
P-1V
PrIv
Xg
14
(a)
(b)
(c)
Figure 1: (a) Plot V. (xu, x x2) with X 0 and 1 = 1. The dashed line shows the confidence
interval of py px(x;; o.). While x and violate UC, X3 violates OC. (b) Plot of informative prior
task T where V2 Priv, # 0. (c) Plot of an uninformative prior task where P Priv. = Ve.
known as the function gap, at observed inputs. Defining such a gap notion for risk measures is not obvious. Moreover, the function gap is susceptible function transformations such scaling and vertical-shifting. Last, in order to achieve theoretical performance guarantee, RM-GP-UCB requires the weights of prior tasks to decay zero.
In contrast, our meta-V VBO approach tackles the problem from a different perspective that not only exhibits robustness against scaling and vertical-shifting transformations but also eliminates the explicit
need for diminishing the effect of prior tasks via weight decay. Specifically, in the next section, we
offer multiple choices of the input query in the form of versatile query set (V-set), denoted as V
(depending on iteration f), such that any input query sequence in the Cartesian product [ V incurs sublinear cumulative regret,
((x))_
V. II,
lim 1=1 T-000
1 T T T
Rr
lim
1
rx,,) =0 Σ t=1
4
The flexibility of choosing any input query x, E V grants us the opportunity to incorporate informa- tion from prior tasks into the current optimization task elaborated in Sec. 3.2
3.1 VERSATILE QUERY SET
Intuitively, the versatile query set (V-set) V₁ should (C1) contain those inputs that are likely be the maximizer of py. Furthermore, we would like to avoid over-exploitation at inputs with well-estimated risk measures where gathering observations leads to minimal progress towards discovering x.. This is done by introducing another criterion (C2) to exclude inputs with well-estimated risk measures from V. These two key criteria (C1) and (C2) are formulated follows.
(C1) To determine whether an input is likely to be the maximizer of Py. we obtain the following confidence interval of p (x.; a) in App. A With probability
6,
Y1 2 1, (x.; o) E [pr, (x7 ; :0), Pu. (xF :x)]
(5)
where x, A arg x Pt. (x; input query at iteration V-UCB (Nguyen et al.) (02Tb) and CV-UCB (Nguyen et al. 2021a). For an input x' to be the maximizer x. of Ps (i.e., the above condition (C1)), the confidence interval of its risk measure py (xx;a)) should overlap with that of the maximum risk measure py(x;; a). i... Pr, a)) 2 pr, ax) from equation (3) and equation 5). Hence, we propose to formulate (C1) as the following overlapping condition (OC): For an input x' to be in V4,
a) and x A arg max x Pr. (x: a). It is noted that x, is the
Pu, (x); 2 Pt. (x;: + > (xt: a) Pt, (x); for A E [0,
As A + 1, Pu, (x); + Pa, (x+: a), i.e., the overlap between the confidence interval of P. (x'; a) and that of (x, a) at itcration t increases. When À = 1, the set of inputs satisfying OC reduce to only the input query x of V-UCB (Nguyen et al. 2021b or CV-UCB (Nguyen et al. 2021
(C2) We would like to exclude inputs with well-estimated risk measures by ensuring that the uncertainty of their risk measures is relatively high compared to that of the maximum risk mea- sure pf (x.3 Specifically, propose formulate (C2) the following uncertainty condition
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(UC): For input x' to be in Vi,
Pug (x (x'; ;0) (
Pu, (ut (xj (xT;aa
for 1/ E [1,1/A]
(7)
uncertainty of Py (x)
uncertainty of (xx;;
If > 1/X, OC implies UC since pt, (x;;a) 2 Pt, (x; Vx E X (from the definition of x Hence, we are only interested in UC when 1) S 1/A as specified in equation 7
To sum up, we define V-set V2 as the set of inputs x E X' that satisfy both OC in equation 66 and UC in equation (7) (illustrated in Fig. [T]. The conditions of non-empty which follows from the following Lemma 3.1 (proved in App. B.
and 1 ensure V; always
Lemma 3.1. For all 2 1, Vj contains x""
We note that the size of V. is non-decreasing when À decreases or increases. Specifically, À = 1 implies 17 (since 17 E [1, 1/ and OC implies UC. In this case, Vi {xx") (assuming Pv., has a unique maximizer), i.e., V/ reduces to the UCB-ttppd input query such as those in V-UCB (Nguyen jet al. b021b) and CV-UCB (Nguyen et al., 2021a). There not any flexibility in choosing X from V. [x]), ). which prevents us from incorporating the information of prior tasks into the current task. However, when À < 1 and 17 E [1. 1/ Vi contains other inputs than the popular input query x of UCB-typed algorithms. This a trade-off between prior tasks (existing knowledge) and the current task (new knowledge). When À = 1 (V, = {x; H). we utilize only observations from the current task. Conversely, decreasing À (increasing [v:] ) promotes the exploitation//uilizaiion of prior tasks to select an input query in V/. This lask-specific trade-off is distinct from the conventional (fanction- evaluation-spedt specific) ipploration-epplitatttti
trade-off associated with estimating the maximizer in V-UCB (guuyen et al., 2021b) and CV-UCB (Nguyen et al. 2021a) (pertaining to particular task).
Given the selected x, E V,, we adopt the choice of z; in the existing works of Nguyen et al. (2021bla). In particular, if the risk measure is VaR, then 2, is selected as lacing value (LV) w.r.t. Xt, 4,, Ut. and a If the risk measure is CVaR, we first find the risk level a₁ a arg max,'e(0.o) Bu, (xx;a') m, (xe:a') Then, 2, is selected as LV w.r.t. X2, 4, ut, and or-
Interestingly, given the above choice of 2, any choice x; in our proposed Vi results in a sublinear cumulative regret shown in the following Theorem 3.2 (proved in App. D) In other words, we achieve the goal of constructing V, that satisfies equation (4)
Theorem 3.2. Given the selection of xt as any input in V. and the selection of z, from the works of Nguyen et al. (2021b a), the algorithm achieves a sublinear cumulative regret:
VT 2 1, RT S (17(1 -) 1)
(8)
holds with probability 2 1 5 where C1 A 8/ log(1 + )); yr, 3r, 5 are defined in equation (2 and 1 are defined in equation (6) and equation (7). respectively.
Given the flexibility of choosing x, in V2, tasks to the current task by assigning different priorities to inputs in V2.
now able to transfer the information from prior
3.2 ASSIGNING PRIORITY TO INPUTS IN VERSATILE QUERY SET V.
Since selecting any input query x, Ve results in a no-regret algorithm (Theorem 3.2p. would like to exploit the results of prior tasks to assign varying degrees of importance (i.e., prioritie) to inputs in Vr. To be robust scaling and vertical-shifting of the blackbox function, the priority is based on the location of the maximizer rather than the estimated risk measure value. Besides, to provide more opportunities for transferring information between tasks, we consider the case that the global maximizer of the current task is the local maximizer of prior task. Hence, we assign higher priorities to inputs that have a higher chance of being a local maximizer (restricted to V,) of prior tasks. For this reason, we propose the notion of the probuble local maximizer set, denoted as P,, V. which contains inputs that are probably a local maximizer restricted to Vn of ps.
Prpy fx E Val Pa, (x) 2 max Pi, (x': a))
X'EV.
P Lacing values Buy w.r.t. 46, ue, and a satisfy 4, z S m. (x; a) S rue (x; S ue Zx) Nguyen 2021b).

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An alternative assumption is that the global maximizer of the current task is close that of prior task, then we can consider the probable global maximizer set, defined as P+ix _ {x X Pa, (x: a) 2 ex Pt, a))) Since using Prix does not substantially change the implementation of meta- VBO in comparison to using Priye we only focus on the probable local maximizer set Priv, in this paper. Figs. D and I show the set Priv, of inputs x E V/ such that Pa. (x; o)) is above the dash-dotted line representing maX Př, a)). Essentially, we transfer the information about the local maximizer restricted to Vi of prior tasks to the current task. This is an unexplored direction in the current meta-BO literature.
Given the probable local maximizer set Prim,, the priority of an input x E Vr, denoted as Ty (x), is measured by the number of the probable local maximizer sets that x belongs to, i.e.,
Tee (x) A Σ IxEP-IV
(10)
TET
The higher (x(xx) is, the more likely that x is in the probable local maximizer set of a prior task. Hence, we select the input query X, as
X
max Pu, (x; a) where Ve.max
fx Vol (x(x) = max (x(x')}
(11)
XEV, max
x'EV.
Remark 3.3 (Prior harmful tasks). The local maximizers of Ps. in prior harmful task restricted to Vr are different from that of p of the current task. Contrary to the intuition suggesting that prior harmful tasks may bias meta- VBO away from the desirable maximizer x., Theorem 3.2 asserts that our algorithm maintains a sublinear cumulative regret by constraining the choice of x, to the V, However, one could reasonably anticipate potential reduction in the convergence rate, as empirically shown in Sec. 4 due to the influence of prior harmful tasks.
Remark 3.4 (Informativeness of prior task). If ne (x) is constant for all x E V,, it results in Vr.max Vi and x, x This happens when there are not any prior tasks when the BO results of prior tasks are uninformative in the local region V4, i.e., Prny = V Vr E T. As x; = x, in these cases, our meta- meta-VBO VBO algorithm reduces to V-UCB (Nguyen et al., 202Tb) or CV-UCB (Nguyen al. 102 Fig. [ illustrates an iteration t where prior task is uninformative as V = P-py.. while Fig. Lp shows The case that T is an informative task. is noted that the informativeness of a prior task w.r.t. the current task depends Ve, hence, the iteration A prior task may be informative when have not gathered many observations on the current task (V: is large such that Prpy, is a strict subset of V.), but it may become uninformative when many observations are acquired (V; is small such that PAIv. = V.). When a prior task is uninformative, it does not influence the optimization of the current task in our meta- meta-VBO algorithm (from the definition of Fy (x) in equation (39)). Hence, the effect of prior task on the current task is adaptively controlled by V as illustrated in Appendix [.I] This approach contrasts with that of Dai et al. (2022) that explicitly requires a schedule of the weight decay based the function gap.
Remark 3.5 (Scaling and vertical-shifting of the blackbox functions). Our proposed meta- VBO depends on V. and pr,v. These sets rely on comparing risk measures at different inputs which remains unchanged under scaling or vertical-shifting of the blackbox function. Hence, our algorithm is robust to these transformations.
Remark 3.6 (Meta-VBB for vanilla BO and adversarially robust BO). As proposed VBO depends on the upper and lower confidence bounds of function evaluations, it is sufficiently general to address the vanilla meta-BO problem (see Appendix F). Its potential applicability the meta- learning of other UCB-typed algorithms remains an area for future exploration. Furthermore, since adversarially robust BO is special case of BO of VaR - 0+ (Nguyen et al. 2021b). our proposed meta-VBO directly applies to adversarially robust BO (see App.
3.3 DECISION VARIABLE RECOMMENDATION STRATEGY
At iteration f, we propose to recommend x; c(t) where ((t) A arg max.e(l,...... Pt. (x7; o). We prove that its instantancous regret approaches zero in App. E
Theorem 3.7. Let (()) A arg max.re Pt, (x);)), then, by recommending x = X the instantaneous regret at x; is bounded: r(x;) S // C2 Br/tt with probability 2 1-6 where n, C₁ Sn. and 7/t are defined in Theorem 3.2
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As sublinear bounds of Y are established for common kernels such as the SE kernel in the work of Srinivas et al. (2010), the instantaneous regret at x
x ((t) approaches zero as I. + co. Further- more, we should choose 1 = to minimize the upper confidence bound of r(x;) in Theorem 3.7 Given 17 = 1, we choose À E [0. 1] to trade off between the flexibility in choosing x; (i.e., the size of V-set) and the upper confidence bound of the cumulative regret Rr at the input queries in Theorem 3.2 In the experiments, as we want to exploit the solutions of prior tasks, we maximize the size of set by setting À = O. In this case, the upper confidence bound of RT is two times that of V-UCB Nguyen et al. 2021b) and CV-UCB (Nguyen et al. 2021a): Rr S 3/T,T
EXPERIMENTS
Given the absence of prior research on meta- BO of risk measures, we compare the performance of our proposed algorithm under different sets of prior tasks to that of V-UCB (Nguyen et al.) 2021b) which is equivalent to our proposed algorithm when there are not any prior tasks, f.e., 7 = 0. Recall that our emphasis in this paper is transferring the optimization results, rather than transferring the surrogate model, as discussed in Sec. [ Therefore, we exclusively focus on independent GPs. They are also more practical due their lower computational overhead, while still enabling us to provide fair comparisons to empirically demonstrate our advantage over V-UCB Our goal is to empirically verify whether our proposed algorithm (i) able exploit the information from prior informative tasks to improve its converge rate in comparison to V-UCB even if harmful tasks exist in T and (ii) robust to prior harmful tasks even if there are not any useful tasks in T. To achieve this, we design different sets of prior tasks: Tu Juseful pos scale (i.e., positive scaling of the blackbox function of the current task: fr (x. z) = af z) for a > 0), Tuseful vshift (i.c., vertical-shifing f, (x, z) = f(x, z) + b for b E I), Tnazmful neg scale (i.c., negative scaling fo (x, 2) -x(x, z)), and Tha:mful hehift (i.e., horizontal-shittnng f=(x,z) z) = f(x + {,z) for E E ) Additionally, Tai1 is the union of the contains the most harmful task as the function evalua-
above sets. We note that Tnarmtul-neg-scss tions negated. On the other hand, porizontal-shiittin moves the location of the maximizers away from that of the current task. As explained in Sec. [33 set À 0 and 1 1. We present additional experimental results of meta- BO for CVaR in App.
along the x dimensions in Tharmful hharmful-hshit- -hsifft
The experiments are performed on 6 synthetic functions: a Gaussian curve, the Branin-Hoo, the Goldstein-Prriee, the six-hump camel, the Hartmann-3D,, and the Hartmann-6D (obtained from htttps://wwy s fu. ca/ -ssur" ) and using 2 real-world datasets: the yacht hydrodynam- ics dataset (Dua and Graff] 2017) and portfolio optimization dataset (obtained from the existing works Cakmak et al. of x and z are set 2 et al. (2011a). In the first 4 experiments, both the dimensions experiment,
I. In the Hartmann- the Hartmann- 6D experiment, d = 5 and d' = 1. In the yacht hydrodynamics experiment, the optimization problem is to minimize VaR of the residuary resistance per unit weight of displacement of yacht by controlling the its 5-dimensional hull geometry coefficients. The environmental random variable is the I-dimensional Froude number. The blackbox function is generated by fitting a GP to the yacht hydrodynamics dataset (Dua and Graff (017). In the portfolio optimization experiment, we follow the work of (Cakmak et al.) 2020) to optimize a portfolio by controlling d = 3 decision variables subject to the randomness in the bid-ask spread and the borrow cost (i.e., d' = 2). The blackbox function is generated by fitting a GP to the simulated portfolio dataset. The risk level is set 0.1 for all experiments.
2 and d' 1, while in
The average and the standard error of r(x;) (in Sec. 3.3 over 30 random repetitions are shown in Fig. [ It is noted that the curves of use
useful-vshi ft, and a all 1 overlap each other in Figs. 2 D. and 2g. In these figures, we observe that when T contains only prior useful tasks in Tuse houl-poa-scas and Tusetul-vshitt our proposed meta-VBe outperforms V-UCB which does not utilize any prior tasks. Notably, even when T contains both harmful and useful tasks in Ta11, meta- meta-VBO still outperforms V-UCB. In the extreme case when T contains only harmful tasks in Tharmful worrmful--g--aooo le and Tharmful -hehif meta- VBO does not outperform V-UCB, but it still converges to a low-regret solution. In Fig. 2 meta-VBO given the set Jharmful-neg-sealee performs poorly probably due to the large input space and the function surface of the Hartmann-611. We can
'When computation cost are affordable, one may combine a method that transfers the surrogate model with our algorithm by simply replacing independent GPs with multitask GP hecause our algorithm only relies on upper and lower confidence bounds that are readily obtainable from multitask GP model.
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Poefu--po--sae pos-scale
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Figure 2: Plots of the regret at the recommended input against the BO iteration in experiments with (a) Gaussian curve, (b) the Branin-Hoo function, (c) the Goldstein- Price function, (d) the six-hump camel function, (e) the Hartmann-3D function, (f) the Hartmann- -6 function, (g) the yacht hydrodynamics dataset, and (h) the portfolio optimization dataset.
43.3%
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Figure 3: Percentage of prior tasks whose probable local maximizer contain the input query (grouped by the transformation) at different BO iterations in the experiment with the Gaussian curve and the Tau of prior tasks. The percentage is computed over 30 repeated experiments.
also observe the percentage of prior tasks whose probable maximizer sets Priy, containing x, in Fig. 3 It shows that as we gather more observations, the percentage of harmful tasks whose Prpy. contains X, reduces.
5 CONCLUSION
In this paper, propose meta-VBO which is the first algorithm for meta-BO of risk measures by proposing new concept of V-set and leveraging prior tasks to prioritize inputs in this set. Our algorithm offers theoretical performance guarantec
when prior tasks are harmful. Furthermore, we propose a recommendation strategy that suggests an input as the result of the optimization, with proof of its instantaneous regret approaching zero. The empirical performance of the proposed solution is demonstrated through several synthetic function benchmarks and real-world objective functions. In the future, we would like study our meta- VBO approach perform meta-learning for other BO variants such as federated BO (Dai et al. 2020). preferential BO (González et al. 20177] Nguyen et (2021c). BO for finding Nash equilibria (Ta al. 2023). and BÔ with unknown constraints (guuyen et al. 2023).
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of both the theoretical and experimental results in our paper, have clearly elaborated on our assumptions, the proofs of the theoretical results, and provided the implementation of the experiments. In particular,
main assumptions stated in the third paragraph in Sec. and the first paragraph in Sec. The details proofs of our theoretical results are shown in Appendices A B and G

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Published as a conference paper at ICLR 2024
ACKNOWLEDGEMENT
This research supported by Al Singapore, under grant AISG2-RPP-0020-01..
This research is supported by the National Research Foundation (NRF), Prime Minister's Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme. The Mens, Manus, and Machina (M3S) is interdisciplinary research group (IRG) of the Singapore MIT Alliance for Research and Technology (SMART) centre.
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Reviews:
[{"review_id": "zUqeWBmkvp", "reviewer": "Reviewer_25FF", "summary": "This work considers how to leverage prior knowledge when performing Bayesian optimization of risk measures (VaR, CVaR). Prior works in the literature either consider the availability of prior knowledge to warm-start BO for non-risk measures, or optimization of risk measures without using prior knowledge. This work tackles both of these issues simultaneously. The key novelty of the work is the development of a \"no-regret query set\" that defines a set of input query sequences that would result in sub-linear cumulative regret. This set contains those inputs that are likely to be maximizers of the risk measure given the posterior belief about the underlying function and also provides additional information toward the estimation of risk measures. Once such a set is established, prior knowledge can be used to rank/prioritize the query sequences. As any query sequence within this query set would result in sublinear cumulative regret, even if the prior tasks are very different, asymptotic guarantees are still preserved (although finite sample rates deteriorate).", "questions": "1. The key contribution seems to be the introduction of the idea of a no-regret-query set (NQS). Maybe I missed it, but it feels like it is more generally applicable than the exact problem being considered in this work. If not, perhaps it would be beneficial to know what makes NQS restricted to the setting of risk measures? Also, seems like various other side-information can be used to prioritize samples within NQS. \n\n2. While I understand that the requirement of noise distribution is similar to assumptions made in the prior work and is not directly related to the proposed method, however, since the paper currently is phrased as a solution for optimizing risk measures, it would be useful for the readers to have the assumption formally stated in Section 2.\n\n3. Can the authors provide some experiments on the scalability of the proposed approach? The toy examples considered all have dimensions less than 5 or 6.\n\n4. I think the introduction can be compressed significantly. Currently, the main contribution of the work starts at the end of page 5.\n\n5.  If \\lambda goes near 0, then \\eta can go near infinity and the regret guarantee becomes vacuous. This results in two important hyper-parameters of the problem: \\eta and \\lambda, that would essentially control exploration-exploitation. While 3.3 gives one recommendation, how sensitive are the practical results to the choice of their values?\n\n6. I do not understand the Lacing value choice of the noise variable, and why is important to obtain the desired result. Given that Theorem 3.2 is the core contribution of the work, I think it would be imperative to explain this assumption in more detail.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. The idea of a no-regret-query set is novel, to the best of my knowledge.\n\nS2. Authors carefully consider the impact of the quality of knowledge from prior tasks and how those influence the final results.", "weaknesses": "W1. Empirical results are on extremely toy examples.\n\nW2. Like prior works, this work is also limited in terms of assuming exact knowledge of the noise distribution. Seems like a strong requirement.\n\nW3. Not really a weakness, but my guess is that this paper might have a much bigger audience at other venues (e.g., ICML/AISTATS).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oab7v9Aldt", "reviewer": "Reviewer_4GcU", "summary": "The paper proposes a robust meta-BO for risk measures algorithm. It achieves the robustness to harmful previous tasks essentially by constructing the no-regret query set (NQS) that guarantees the convergence of the optimization while using the counting of probably local maximizer in related tasks as the priority within NQS to guide the acquisition. The paper offers theoretical justification for constructing NQS and the regret analysis.", "questions": "1. It seems that the construction of NQS basically relies on the upper and lower confidence bounds of the risk measure $\\rho_f$ in eq (3). In ordinary BO, the UCB and LCB of the unknown objective function f are the direct equivalence of these confidence bounds for risk measure when the goal is optimized f. Then, an NQS for ordinary meta-BO could be constructed in the same way and could incur a similar regret guarantee. Could the author comment on this extension of NQS in classic meta-BO?\n\n2. The layout is intense and could be more reader-friendly. Could the author reduce the discussion over literature in sections 1 and 2 and leave more room for the essential equations, especially those for the assumptions?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. The construction of NQS guarantees the robustness of the meta-BO.\n\n2. The algorithm is generally well-motivated. Since the key concepts are generally constructed on top of confidence intervals for the risk measure, they bear good interpretability.", "weaknesses": "1. It is unclear why the $\\lambda$ and $\\eta$ are proposed to trade off the goals of NQS explicitly. The algorithm, by default, sets $\\lambda=0$ and $\\eta=1$ as discussed following the theorem 3.7.\n\n2. The priority mechanism in eq (10) is proposed to incorporate information from previous tasks, but it is actually not robust to harmful previous tasks. This is reflected in both the comparison between the results from theorem 3.2 and previous works' regret bounds and the empirical results.\n\n3. Figure 3 demonstrates the effectiveness of the priority mechanism. Yet since the priority function relies on counting, it is unclear whether the proposed method is robust to the increase of the portion of harmful tasks in $\\tau$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Mc84O9N0ND", "reviewer": "Reviewer_ZY9Q", "summary": "The paper introduces a novel meta-learning based approach for Bayesian Optimization of risk measures. Given a set of previous tasks, the authors propose to construct a no-regret query set and they prove that any strategy picking inputs in such sets leads is a no-regret strategy, i.e., it has a sublinear regret. Furthermore, the authors assigns priorities on all the inputs belonging to this NQS $\\mathcal{Q}_t$ at iteration $t$. Those importance weights are computed using the information from previous tasks. They take high values for the most probable local maximizers among the set of previous tasks. Then the authors introduce a tradeoff between exploiting the information from previous tasks and exploring the current task. By applying the approach on Value-At-Risk (and CVAR), the authors show that their algorithm is invariant to scaling and vertical shifting of the blackbox function, and robust towards the presence of harmful previous tasks.\nThe authors show numerical experiments on synthetic data and real-world data.", "questions": "See above.\nIs the code publicly available?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper contains several key contributions that will be helpful for the future works in BO for risk measures.\nMajor strengths:\n  - the paper is very well written and the motivations are very clear: exploit previous tasks knowledge (achieved using V-UCB) to get better BO for risk measures. The literature review is well detailed and the authors explain clearly advantages and drawbacks of each of them. \n  -  the construction of the no-regret query sets is elegant and the theorem ensuring that any strategy picking input points in such sets is key contribution in this specific research area in my opinion. \n  - the definition of the importance weights exploits well the knowledge from the previous tasks. This also enables to avoid the weight decay to $0$ which is used in the state-of-the-art of meta-BO. The strategy keeps using the knowledge from previous tasks over time. \n  - the authors provide theoretical results ensuring that the presented meta-BO of risk measures maintains invariance properties in scaling and shifting of the blackbox function.\n  - the numerical experiments show that the approach is robust towards the presence of harmful previous tasks (versus V-UCB).", "weaknesses": "Although the paper is very clear and provide important results for the community, there are couple of points that are not completely clear to me.\n\nMajor comments/questions:\n   - the distribution of Z is supposed to be known. Is it a parameterized distribution that is fitted in practice?\n   - the point which is unclear to me is the definition of the lacing values. If I understand well, this means that at each iteration the authors pick a value for z (corresponding to the lacing value) and draw an observation $y_t = f(x_t, z) + \\varepsilon$. Is this correct? How are those lacing values computed in practice? I think a paragraph is necessary to explain exactly what is done here. Would it be possible to provide a pseudo-code summarizing the main steps of the algorithm?\n  - it would have been interesting to see an illustration of the set $\\mathcal{Q}_t$ in the numerical experiments and the computed importance weights. And also to see the evolution of this query set over time.  Figure 1 is not easily readable in my opinion.\n  - the authors write that \"we want to maximize the size of the query sets\". I understand this point since it enables to exploit all the previous tasks. However, it seems that there might exist some tasks that do not bring useful information or harmful information. In such case, wouldn't it be more interesting to discard these tasks from the query set? This question is a bit naive, but from a computational perspective, it would make more sense to consider less previous tasks. Is this correct?\n  - Maybe I've missed this point, but could you comment the computational cost of the approach?\n\n\nMinor comments:\n  - titles of subsection 2.2 and Section 3 are the same.\n  - In my opinion, there is a lack of details about the transfer knowledge from previous tasks in the settings. For a reader who is not familiar with meta learning, it would have been helpful to know precisely (in the settings) what is the \"knowledge\" you extract from a previous task. In your case, this is the output of V-UCB. This is explained in the introduction, but this would make the reading easier.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)