ROOT: Rethinking Offline Optimization as Distributional Translation via Probabilistic Bridge
Authors: Manh Cuong Dao
The Hung Tran
Phi Le Nguyen
Thao Nguyen Truong
Trong Nghia Hoang
Venue: NeurIPS 2025 spotlight ()
Keywords: Offline Optimization
Diffusion Model
Probabilistic Method
Primary area: other
TL;DR: We view offline optimization from the new lens of a distributional translation task which can be modeled with a generalized Brownian Bridge Diffusion process mapping between the low-value and high-value input distributions.

Abstract: This paper studies the black-box optimization task which aims to find the maxima of a black-box function using a static set of its observed input-output pairs. This is often achieved via learning and optimizing a surrogate function with that offline data. Alternatively, it can also be framed as an inverse modeling task that maps a desired performance to potential input candidates that achieve it. Both approaches are constrained by the limited amount of offline data. To mitigate this limitation, we introduce a new perspective that casts offline optimization as a distributional translation task. This is formulated as learning a probabilistic bridge transforming an implicit distribution of low-value inputs (i.e., offline data) into another distribution of high-value inputs (i.e., solution candidates). Such probabilistic bridge can be learned using low- and high-value inputs sampled from synthetic functions that resemble the target function. These synthetic functions are constructed as the mean posterior of multiple Gaussian processes fitted with different parameterizations on the offline data, alleviating the data bottleneck. The proposed approach is evaluated on an extensive benchmark comprising most recent methods, demonstrating significant improvement and establishing a new state-of-the-art performance. Our code is publicly available at https://github.com/cuong-dm/ROOT.

ROOT: Rethinking Offline Optimization as Distributional Translation via Probabilistic Bridge
Manh Cuong Dao"
The Hung Tran" Washington State University hung t. tran@wsu. edu
National University of Singapore cuongdao@nus.com edu.
Phi Le Nguyen
Hanoi University of Science and Technology lenpesoict.. wust.eu................. vn
Thao Nguyen Truong
National Institute of Advanced Industrial Science and Technology nguyen. truong@aist.go.
Trong Nghia Hoang Washington State University trongnghia. hoang@usu. edu
Abstract
This paper studies the black-box optimization task which aims to find the maxima a black-box function using a static set of its observed input-output pairs. This is often achieved via learning and optimizing surrogate function with that offline data. Alternatively, can also be framed as an inverse modeling task that maps a desired performance potential input candidates that achieve it. Both approaches are constrained by the limited amount of offline data. To mitigate this limitation, introduce
distributional perspective that casts offline optimization translation task. This is formulated as learning a probabilistic bridge transforming implicit distribution of low-value inputs (i.e., offline data) into another distribu- tion of high-value inputs (i.e., solution candidates). Such probabilistic bridge can be learned using low- and high-value inputs sampled from synthetic functions that resemble the target function. These synthetic functions are constructed as the mean posterior of multiple Gaussian processes fitted with different parameterizations the offline data, alleviating the data bottleneck. The proposed approach is evalu- ated on an extensive benchmark comprising most recent methods, demonstrating sigmificant improvement and establishing new state-of-ther of-the-rrtttttt performance. Our code is publicly available lhttps: /ttp//ttttttt//////////// com/ cuong- /ROOT
Introduction
Black-box optimization arises in scientific and engineering domains where evaluating each candidate solution costly, often requiring extensive physical experiments or high-fidelity simulations [61] For instance, designing energy-efficient hardware accelerators 9 37 involves numerous cycle-accurate
'These authors contributed equally. "oorrsppnndigg authors: Manh Cuong Dao. The Hung Tran, Trong Nghia Hoang.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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simulations to assess configuration performance. In materials science, finding nanoporous structures with high adsorption capacity for carbon capture or hydrogen storage demands labor-intensive lab experiments [1 20 Similar challenges arise in protein design [21], molecular generation [6]. and
drug discovery [52 where evaluations are likewise expensive.
Prior Literature. Existing approaches to black-box optimization include both online and offline methods with the latter being an emerging alternative of the former. In particular, online methods such as Bayesian optimization [53] black-box design tasks with provable performance guarantee the asymptotic limit of data. However, their reliance on iterative experimentation makes them less practical in high-cost settings with limited no experimentation budget. In contrast, black-box methods leverage past data to learn a surrogate model which can be used to find better designs without incurring new experimentation [36] 7 18 12
73 62
65 33 have long been explored for
27
14 15 In this paper, we focus on the offline setting, where the goal is to discover high-performing designs using only past experimentation data.
Challenge. Among offline methods, the main challenge is that surrogate models can become increas- ingly erroneous when the search moves away from the offline dataset, especially when offline data are biased or sparse, causing these models to overfit. To mitigate this issue, most existing approaches have focused on advancing techniques in (1) forward modeling that penalize high-value surrogate predictions at cuttof-diisrrbuttin (OOD) inputs 60
(2) inverse modeling that find most promising and reliable regions that contain high-performing inputs [40] 45 39 [0] to sidestep the
OOD issue of forward modeling, and (3) search policies that learn a direct plan to navigate from
low-value inputs to high-value inputs [10 38
Limitations. Despite their promising results, forward and inverse approaches depend on learning mapping (or inverse mapping) between input designs and their corresponding performance outputs using offline data. As a result, their effectiveness is inherently limited by the availability of data, Likewise, learning direct search policies also suffers from the same data bottleneck since these methods still need sample heuristic trajectories from the offline dataset to use learning feedback.
Furthermore, information regarding regions with tigh-performing inputs is often not observable from the offline data, especially in low-data scenarios, which might further restrict the effectiveness of the learned models/policies. To mitigate such data bottleneck, we propose to approach offline optimization from new perspective of distributional translation, as highlighted below.
Distributional Translation. In essence, we view the offline data as an implicit distribution over low- value designs, and recast offline optimization the task of learning probabilistic transformation, or bridge, that transports this distribution toward regime of higher-value inputs. By moving along the learned probabilistic bridge, we can reach regions of the input space associated with better designs. However, learning such a transformation is fundamentally limited by the scarcity of high-value examples. Our key insight to overcome this bottleneck is:
Although the feedback needed to learn such low-to-high probabilistic transformation is absent in the offline dataset, it can be derived from distribution of synthetic functions that are similar to the (unknown) target function (up to a scale factor).
These synthetic functions can be provably constructed across various output scales, alleviating the data bottleneck and broadening the solution scope of offline optimization.
Technical Contributions. Our solution perspective is substantiated via the following:
1. A general-purpose probabilistic bridge model that learns direct mapping between two implicit data distributions. This perspective rethinks offline optimization through the lens of probabilistic transport. The resulting bridge can incorporate external guiding information from synthetic functions similar to the oracle to mitigate the data bottleneck. Once trained, it enables simulation of paths that move from low-value to high-vat value regimes (Section 3.1..
A pre-training and adaptation framework that (1) learns multiple Gaussian process priors (66] over synthetic functions resembling the target function, and (2) samples representative low- and high-value inputs from their corresponding closed-form mean functions. This generates high-quality training data that better delineates the low- and high- value regimes for learning the probabilistic bridge. The

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intuition is that if the bridge can consistently map between these regimes across wide range of functions similar to the oracle, will be able do the same for the oracle (Section 3.3
3. An extensive empirical evaluation on a variety of benchmark datasets (59 and numerous existing baselines, establishing new state-of-the-art performance, which significantly and consistently improves over previous work. Our empirical evaluation also features rich ablation studies examining detail the practical impact of different components of our framework its performance (Section
2 Problem Definition and Preliminaries
This section provides a concise formulation of offline black-box optimization (Section 2. and important background on Gaussian processes (Section 2.2). which was used later for sampling additional data from synthetic functions similar to the oracle behind the offline data.
2.1 Offiine Black-Bos Optimization
Offline black-box optimization is formulated as the maximization of black-box function f( using only an offline dataset observations D, {(x, Ja) which æ; denote a past experiment design and f f(z;) is its corresponding evaluation. A direct approach to this problem is to learn surrogate g( (x; w-) of f f(x) via fitting its parameter w. to the offline dataset,
w
arg min L(w) & arg min Σ (g(z); i=1
(I)
where w denotes parameter candidate of the surrogate and ((g(a;; w),, y) denotes the prediction loss of g(:; () on x its oracle output is y. The (oracle) maxima of f(x) is then approximated via,
x. arg max g(x;
The main issue with this approach is that ((); often predicts erratically at (OOD) inputs. To mitigate this, numerous surrogate or search regularizers have been proposed to either penalize the high-value surrogate prediction at OOD inputs [13
60 70) or find inverse
mapping from the desired output to potential inputs (39 45 as detailed Section Nonetheless all these approaches are restricted by the limited amount of offline data. Alleviating this bottleneck to reach new SOTA performance is main aim of our proposed approach.
2.2 Gaussian Processes
Gaussian process (GP) [50] defines a probabilistic prior over a random function f((z). It is parameterized by mean function m(x) 0 and kernel function k(x,, x'). These functions induce marginal Gaussian prior over the evaluations h [M ) [ax)).)) of any finite subset of inputs (ar,..., ). ...) Let æ, an unseen input whose corresponding output her = h(x;)) we wish to predict. The Gaussian prior over [f(x))..... ... h(x,) h(x+)] implies:
Mxx,) h
N(k K K- k(x,
k -._K-'k.))
(3)
where k, = [k< x1) k(x,, TT and K denotes the Gram matrix induced on fai...., for which Kij = k( x,)). Assuming a Gaussian likelihood y ~ N(h(z), o²), it follows that
h(x;)
N(k -00 k(z, - k. (K -²1))
(4)
which explicitly forms the predictive distribution of a Gaussian process. The choice of the kernel function k(x, æ²) dictates certain properties of the sample functions. In this context, we adopt the commonly used RBF kernel, (<xxx') = x² cxp(-0.5 || - x' 2/() with the parameter o² represents the signal variance, controlling the function's amplitude, while denotes the unit length-scale which regulates the function's 's smoothness. There also exists extensive literature on
improving the complexity of Gaussian process sparse approximation 47 48 56 24 42
that enables fast inference with linear complexity in large-scale datasets
24
30
3
28
32
29

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Step Synthetic data generation
Step Training
Step 3: Simulation
Gaussian
Mean Functions
Granlent ascent
X-hia
X
M
WN B1
Daf Daffion
Gradion descant
D synthetic
yrlggn
GP
yoltor
Probabilistic Bridge Model
Gradient ascent
Xtiah
ylow
yluw
y
X
Xlow
6P
8n
Gredient doscent
X
Classifier-fee guidance DP
Top-i
Figure 1: Overview of the ROOT workflow: (1) Multiple Gaussian process posteriors fitted to the offline data, and low- and high-value inputs from the posterior mean functions are sampled to construct synthetic dataset. (2) This dataset used to construct our probabilistic bridge model, which learns to map between two different implicit data distributions. (3) The backward process of the learned bridge model is applied to the oopperforming inputs from the offline data to generate high-quality candidates for the unknown target function.
3 Rethinking Offline Optimization as Distributional Translation
This section introduces a new perspective on offline optimization by framing it as translation task. We intuitively view the offline data a source language composed of low-value designs and aim to translate it into target language of high-value inputs. To enable this translation, we introduce the concept of a probabilistic bridge, which identifies local translation examples by explicitly conditioning on both the source and target contexts. This conditioning enables the construction of feasible transformation paths that connect selected low- and high-value designs within their local neighborhoods, grounding the translation process in meaningful design correspondences (Section 3.1)
These local bridges serve as examples of how known. The learning algorithm then weaves together these examples to form a global translator capable of generalizing beyond the observed pairs and enabling translation in new contexts where only the source is known. This approach generalizes diffusion models (25), which translate between data and noise, and allows external guiding information to incorporated, enriching the context and addressing the data bottleneck in offline optimization (Section 3.3
between regimes when both endpoints
3.1 Probabilistic Bridge for Distributional Translation
This section formalizes the concept of a probabilistic bridge, which generalizes the widely used Denoising Diffusion Probabilistic Model (DDPM) (25) While DDPM defines a forward diffusion process that maps data to Gaussian noise and learns to reverse it, the probabilistic bridge constructs space of localized transformation flows conditioned on both the source and the target, representing low-value and high-value designs drawn from implicit data distributions. Learning and applying probabilistic bridge model for distributional translation involves two phases: a construction phase (Section 3.I and a learning phase (Section 3.3
In the construction phase, the transition kernel of each conditioned bridge is derived, capturing a source-to target translation plan within its local context. These localized examples serve as concrete demonstrations of how to move between regimes. global, target-agnostic transformation flow that generalizes beyond the observed pairs and enables translation from arbitrary source inputs to valid target outputs. These two phases are described in detail below. A concrete example of probabilistic bridge is also given in Section 3.2
the learning phase, they are used train
3.1.1 Probabilistic Bridge Construction
Given two endpoints Zo and Ør, define a probabilistic bridge between them discrete observation of vector-value, time-indexed random function 1 ...,.., ØT-1 which is distributed by Gaussian process (50) with mean function ti (x5, ØT) and covariance kernel Kr
We assume a zero mean function since the output can always be re-centered around 0. We use K to denote the cross-covariance matrix between æ; and æ; under the joint distribution.

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This implies marginal Gaussian on the probabilistic bridge vec[æ1. æ2.... 2r-1l with mean Φ vec(eh (Zo. ZT): ...):...;... - (zg, which reveals a mass-moving flow between ær and zp-
(
mo; FT)
Na; 20,
= g( Fo, ar) = N 224; (e 2r), Kt..
(5)
where denotes the Kronecker product, k. is matrix containing the entries Kit and the mean function v/ Fr) must meet boundary conditions the (2o, ( = To and (Zo, 2r) = ær. This further reveals closed-form expression for backward transition conditioned both endpoints,
q Ft, Lo. FT)
N( N( (1 To. xT), Re-1I).
(6)
with the transition mean (, Lú, FT) = 1-t-1(2p)) IT) ) + k (x) - 1ete-4((2n) ( xT)) and covari-
ance R Kt-1,1-1 1i-1, Ke,t-1- This reveals a (step-wise) conditional backward transition that induces valid transformation flow mapping ær back to 2p. This means for each sampled pair (æT, mo). the corresponding target-conditioned backward transition g( æt, Lo, ØT) provides an example a localized transformation flow between them, which can be used to train the desired target-agnostic map po (fa-1 24, PT) detailed next.
3.1.2 Learning Probabilistic Bridge Model
To learn the target-agnostic transformation that maps from a source plausible target without knowing it beforehand, we parameterize it as
Po ( di. (T) = M TT. f), f), Re-1T).
which uses the same (known) variance parameter as the localized flow/bridge example above and a (learnable) spatio-temporal network (ar, t)) to parameterize the target-agnostic mean transition. This is the centerpiece of the probabilistic bridge model in Eq. 7 which can now be learned via optimizing its parameter 0 match the example flows generated using Eq. 6
DKT. (9(Z:-1 x,, Zo. FT) ( (2,
8PB
arg min
E
(8)
(2o.
The KL divergence in Eq. Jabove is between two Gaussians and can be computed in closed form which allows for a direct optimization of O. Given the optimized OPp and a source ær (low-value design), we can now simulate the generic transformation flow in Eq. 7 to obtain a plausible target Zo (high-value design) which is not known apriori.
Ft-1
M6pp (Zz, ET, t) + Vic,
99)
where we sample E ~ N(O. I) when > and set E otherwise. This gives us the desired transformation that translates candidate in the low-value regime to another in the high-value regime. Furthermore, to improve training efficiency, we can also adopt the practical approach in guided diffusion which further conditions Mo (2, Ør, t) ne ( Tr, You yr, on the source and target outputs (i.e., yr and yo) 26).
3.2 Example and Practical Setup
This section provides an working example to substantiate the above probabilistic bridge frame- work. Choosing v (m0, = æ (1(1// (t/T)) and Ki, = (min(1, ()))),,)))(((( -100111mm-p) (ma results in Brownian bridge construction.
(
Zo. σT) = N
xzi xo
1 (1-T) T
(1-4)--1
o T T 1- 1 1 (1)
1
(10)
This specifies a transport plan that moves the unit point mass located as Er to Zp. To see this, note that when t 0, Eq. 10 reduces to N(xu, 0) and when T. it reduces to N(æy. 0). This setup emphasizes a point-to-pont transformation which particularly suitable for offline optimization.
As we substantiate Eq. 10 with the appropriate (low-value, high-value) pairs (æT, zo) and derive examples of localized flow mapping from the low- and high-value regimes following the blueprint in Section 3.I.I we can learn a target-agnostic transformation consistent with using Eq. This setup
5

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has been used in our experiments to achieve significant performance improvement over prior work, effectively establishing new SOTA. Further details regarding the specific derivation of the Brownian bridge and its practical traning procedure is detailed in Appendix B.2
Remark. The above framework offers broad and flexible perspective offline optimization that has not been fully explored in the existing literature. From this viewpoint, there are many promising directions for future investigation. Each valid specification of the mean and kernel functions in the probabilistic bridge model in Eq. S defines a distinct way of transporting probability mass from low-value high-value regions. While offline optimization ultimately requires only one such transport plan, different choices may lead to significantly different learning behaviors and complexities. Understanding how the choice of bridge design influences learning performance remains an important and open question for future work.
Learning PPB however requires access paired samples of low-value and high-value designs, which is nontrivial in the offline optimization setting where the dataset typically reflects only the low-value regime. This raises a fundamental question: where can we obtain representative samples from the high-value regime to support bridge construction? Addressing this challenge is critical for making the probabilistic bridge framework operational, and is the focus of Section 3.3
3.3 Synthetic Data Generation and Practical Black-Box Optimization Algorithm: ROOT
This section revisits the challenge introduced in Section Learning probabilistic bridge requires examples of localized flows from low- to high-value regimes. Yet, such examples are not available in standard offline datasets, which by design lack high-performing solutions. To address this, we propose key hypothesis: if a probabilistic bridge can consistently translate low-value inputs to high-value outputs across a sufficiently large set of functions similar to the oracle, likely to generalize well to the oracle.
This motivates an alternative data generation strategy. Rather than depending solely on the offline dataset, we construct a collection of synthetic functions with closed-fom structure, allowing us to sample large quantities of low- and high-value inputs. These synthetic examples enable us to train meta probabilistic bridge with strong zero-shot adaptation. This strategy reflects the common pre- vre-training paradigm in foundation model development, where knowledge accumulated from broad synthetic tasks can be readily transferred to any target task of interest. This motivation leads to our overall workflow, which is illustrated in Figure D
Motivation.
Function Sampling. Collecting data from such similar functions is fortunately possible using the Gaussian process [50] which specifies a prior over functions around a given mean function. Assuming that the mean function is set to be the oracle, most sampled functions from the corresponding GP will be similar to it. Since we do not know the oracle function, another approach is to begin with generic GP prior with kernel parameters 6 (a, E) and compute the corresponding GP posterior using the offline data D.. The resulting GP posterior mean is
go. (a)
(K ((o.)T(K(()) +
(11)
which has closed form and is similar the oracle function around the offline data D.. To obtain a wider range of functions similar to the oracle, we compute multiple GP posteriors according to diverse range of GP priors with different signal and length-scale parameters. These can be obtained from learning a hierarchical GP prior with top-level prior over kernel parameters defining )% the corresponding GP posteriors at the lower level. We can then sample kernel parameters {o.)"= and extract the corresponding posterior means see Eq.
Low- and High- Value Simulation. Given these functions, we can construct X and x as the set of low- and high-value inputs of Tz, (ar) via running M-step gradient descent and ascent from a subset of n, offline input points:
X
E TM
1
#9 (im) 2o EDu
(12)
M -
X
+
+ +
n o. (xt)
2nj EDo {
(13)
8
m=0

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where #m+1 x + vTVa (I and I m+1
xm - (xm). Here, zo = E Do-
The corresponding outputs for the above inputs can also be computed using the closed-form of the posterior means. synthetic dataset D, collecting low- and high-values from those closed-form posterior mean functions can then be created:
D. {X. v.), ...).) } "*
where y.= = (x, (X,) and y = 96. (Xx), (14)
where ng is the number of sampled functions.
Learning Probabilistic Bridge. Following the blueprint in Section 3.1.1 we can sample (ær,, 20, U6) ~ D₂ training data train probabilistic bridge mapping between low- and high-value regions across the aforementioned posterior mean functions, which follows Eq.8 Fur- thermore, we also adopt the guided diffusion (26) technique to incorporate the output information yr and Yo as part of the input to the prediction network Mo ær, yo, yr, () as stated after Eq. 9
Simulation. Once this probabilistic bridge model has been trained,
utilize its corresponding (step-wise) source-to-targee transition in Eq. 9 to map each offine input (presumbaly in the low- value regime) to a better solution candidate (in high-value regime). For example, simulation for the top 128 offline inputs with highest values. The effectiveness of this approach is thoroughly evaluated in Section 4 where demonstrate its ability to generate high-quality candidates across various benchmarks. full description of our algorithm and implementation is presented in Appendix B Our algorithm's 's computational complexity is discussed in Appendix, C.4
run this
Experiments
This section evaluates proposed method, ROOT, through extensive empirical comparisons against various recent baselines on standard offline optimization benchmark (59). The experiment setup is
summarized in Section 4.1 with detailed results in Section 4.2 and ablation studies in Section 4.3
4.1 Experiments Settings
Our investigation covers four real-world tasks selected from the Design-Bence [ and three RNA-Binding tasks from ViennaRNA 44. In Design- Bench, the chosen tasks cover both discrete and continuous domains. The discrete tasks, TF-Bind-8 and TF-Bind-10 33 aim to discover DNA sequences with high binding affinity to a specific transcription factor (SIX6 REF R1). On the continuous side, Ant Morphology (6) and D' D'Kitty Morphology (2) focus optimizing the physical structure of a simulated robot ant from OpenAl Gym (6] and the D' D'Kitty robot from
Benchmark Tasks.
ROBEL 2). For ViennaRNA, we include three RNA-Binding tasks as RNA-A, RNA-B, and RNA-C
35 For further details on these tasks, please read Appendix A
Baselines. We selected 21 widely recognized methods, including BO-qEI 559 CMA-ES 23
REINFORCE 667) COMs [60]. CbAS (7]. MINs 44), RoMA [69]. DDOM 39), ICT
70
Tri-mentoring 13 GTG 72) BDI 12). RGD 11 LTR [55]. BONET [8), MATCH-OPT
27
PGS 10. GABO
DEMO 71, GA on GP (Section 4.3) and standard GA.
Evaluation Protocol. Following the approach in [59]. each method generates 128 optimized design candidates, which are then evaluated by the oracle function. The performances are ranked, and results are recorded at the thth thth and 1h0 percentiles. To ensure consistency, all results are averaged over 8 independent runs with reported standard deviation.
Hyper-parameter Configuration. For each baseline, we adopt the optimized settings from the original papers. For GP kernel hyper-parameters in our data generation,
sample lengthscales l, and variances o² uniformly from [Ly - 8, Eu + 5) and [o] - 6, o + 6), with εy = a 1.0 for continuous tasks and 6.25 for discrete tasks, and 8 = 0.25. We M 100 gradient steps with step sizes 0.001 (continuous) and 0.05 (discrete). Additional data generation details are in Appendix [B. and Table 6. For training the Probabilistic Bridge model, use Brownian Bridge diffusion process with the Adam optimizer over E = 100 epochs and n, = 800 synthetic functions, running on a single NVIDIA A100-SXM4-80GH GPU. More training details are provided in Appendix B.2
'We exclude tasks marked by previous works as having high oracle function noise and inaccuracy (ChEMBL, Hopper, and Superconductor) and tasks that require substantial computational resources evaluate (NAS)
7

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4.2 Experimental Results
Table 1: Experiment results on Design-Bench Tasks. We report the maximum score (100th percentile) among Q 128 candidates. Blue denotes the best entry in the column, while Brown indicates the second best. Mean Rank is the average rank across all benchmark tasks.
Benchmarks
Method
Ant
D'Kitty
TFBind8
TFBind10
Mean Rank
Do (best)
0.565
0.884
0.439
0.467
BO-qEI
+.0000
0.896
0.000
0.825 0.099
0.627
0.033
16.75 /22
CMA-ES
1.561 # 0.89
0.724 0.001
0.939
0.039
0.664 0.034
8,00
REINFORCE
0.263 + 0.026
0.573 0.204
0,961 0.034
0.618 0.011
17.00
GA
0.293 1 0.029
0.860 + 0.021
0.985 + 0.011
0.638 0.032
12.75/22
COMs
0.882 # 0.044
0.932 #.006
0.940
0.621 = 0.033
13.25 /2
ChAS
0.846 0.033
0.895
0,903 0.028
0.649 = 0.055
12.50/22
MINs
0.894 0.022
0.939 0.004
0.908
0.630
12.50/22
GA GP
0,948 0.013
0.780
0.654 = 0.038
9.25/
RoMA
0.593 0.066
0.829 0.020
0.000 0.000
20,00
ICT
0,911 # 0.030
0,945 0.011
0.034 0.047
/
Tri-mentoring
0.944 # 0.033
0,950 + 0.015
0.899
0.647 0.039
9.00 /2
MATCH-OPT
0.931 = 0.011
0.957 + 0.014
0.977 0.004
0.543 0.002
9.50/22
PGS
0.949 + 0.017
0.966 + 0.013
0.981
+ 0.015
0.532 = 0.000
7.75 / 22
LTR
0.907 = 0.032
+
0.000
0.652 = 0.039
6.25 /22
DDOM
0,930 0.029
0.925 0.008
0.885 0.066
0.634 0.015
13.75 /
GTG
0.865 0.040
0.935 +
0.901 = 0.039
0.639 0.016
12.50/ 22
BDI
0,964 = 0.000
0.973
0.036 0.000
7.50/
RGD
0.922 = 0.020
0.883
0.889 0.068
0.048
13.00/22
BONET
0.948 # 0.025
0.957 + 0.008
0.894 0.086
0.606=0.00
10.75 /2
GABO
0.224 + 0.051
0.719
0.939
0.038
0.639 = 0.033
15.25 22
DEMO
0.948 # 0.013
0.956 0.011
= 0.054
0.648 = 0.042
9.25
ROOT (ours)
0.965 0.014
0.972 0.005
0.986 0.007
0.685 0.053
/
Table 2: Experiments on Biological RNA Design Tasks.
Table 3: Effect of initial- poiitt se-
We report the maximum score among Q 128 candidates.
lection on ROOT's performance.
Benchmarks
Type
Ant
T'BBid88
Method
RNA-A
RNA-B
RNA-C
Mean Rank
Random
0.953 = 0.014
0.976 = 0.007
Lowest
0.545 = 0.214
0.969 0.009
CbAS
0,270 0.098
0.249 1 0.088
0,261 1: 0.093
6.00/2 8
Highest
0.965 a 0.014
0.986 a 0.007
BO-qEI
0.537 0.06
0.517 0.108
01481 + 0,100
3.67
GA
0,518 1 0.12
0.499 1: 0.100
0.496 : 0.091
4.33/8
Table 4: ROOT vs ExPT in few-
COMs
0.187 0.123
0.144
0,209 0.100
7.67 18
REINFORCE
0,166 0.096
0.149 1 0.081
0.225 1 0.075
7.
shot settings.
BDI
0.604 0.000
0.505 + 0.000
0.411 + 0.000
4.00/8
Method
Ant
TFBind8
Boot-Gen
0,913 0.064
0.881 0.024
0.786 2: 0.039
2.00/8
ExPT
0.940 1 0.027
0.874 = 0.071
ROOT (ours)
0.956 0.023
0.955 0.013
0.922 0.013
1.0078
ROOT
0.942 0.035
0.086
This section compares our method to 21 baselines, evaluating the 50th, 80th, and 100th percentiles. Due to limited space, we report only the 100th percentile results in the main text; details on the 50th and 80th percentiles and score distributions for our method versus others are in Appendix C.2
Results on Continuous Tasks: Table (first two columns) shows our continuous-task results. On D' D'titty, set new SOTA 0.972 with small standard deviation 0.005. On Ant, although CMA-ES surpasses us at the 100th percentile, its standard deviation (0.896) is nearly 64x ours (0.014) and it collapses near zero at the 80th/50th percentiles (see Appendix 2
Results on Discrete Tasks: Table I (last two columns) present our discrete-task results. Our ROOT achieves both the top rank for the TFBind task with mean score 0.685 and the TFBind8 task with 0.986. Our standard deviation 0.007 for the TFBind8 task is even smaller than that of all other baselines, except for RoMa, which performs poorly.
Results on RNA-Binding Tasks: Table demonstrates ROOT's superior performance, ranking first on all three RNA benchmarks. We outperform Boot-Gen (35) by margins of 0.043 (RNA-A), 0.074
8

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Table 5: Performance on TF-Bind-8 and Ant of different methods trained by only the p% poorest- performing designs from the offline dataset.
TF-Bind-8
Ant
Method
50%
20%
10%
50%
20%
10%
GA
0.580 0.199
0,480 0.218
0.559 + -0.170
0.394 + 0.023
0.663 + 0.065
0.619 + 0.
COMs
0,935 0,052
0.872 0.085
0.771 = 0.128
0.898 = 0.035
0,880 0.002
0.845 = 0.041
REINFORCE
0,915 0.039
0.917 = 0.040
0.913 = 0.038
0.317 = 0.016
0.261 0.052
0.281 = 0.034
LTR
0.959 0.022
0.927 - 0.033
0,909 1 0.034
0,909 1 0.042
0.871 0.059
0,813 1 0.026
ROOT
0.964 0.015
0.946 0.045
0.915 + 0.019
0,909 + 0.012
0.930 + 0.023
0.861 0.051
(RNA-B), B), and 0.136 (RNA-C). while our standard deviations are roughly half those of Boot-Gen, achieving better performance and lower variance (hence, more stable).
Overall, ROOT achieved mean rank of 1.25 across both discrete and continuous domains, setting a new SOTA on out of 4 Design-Bente Bench tasks, as well as on all three RNA-binding tasks. This demonstrates the robust and consistent effectiveness of ROOT across diverse settings.
4.3 Ablation Experiments
Selection Strategies for Initial Points. We explore 3 initial point strategies for x 2o in (Eq. 12 and Eq. 13: random sampling, and selecting points with the lowest or highest objective values from Do. As shown in Table 3 the third strategy achieves the best results.
Few-Shot Experimental Designs Setting. In offline optimization, the few-shot experimental designs (ED) setting. introduced in ExPT 45), presents a more challenging task where only a small set of labeled data points, Diabel {( ye ) is available alongside larger set of unlabeled data, Dunlabeled = { | To evaluate our model in this scenario, we follow ExPT's protocol, using a random I 5% of the offline data as labeled points and the remaining 99% as unlabeled. For synthetic function generation, we first fit a prior Gaussian process (GP) to Diahel, then use the posterior to generate pseudo-labels for Dunlaheled- We combine the labeled and pseudo-labeled data and fit another GP to this dataset. The mean function of this refitted GP is used as the synthetic function, following the same procedure as in the main method. As shown Table ROOT outperforms the ExPT model in this setting by substantial margins.
Poor Offline data Coverage Setting. Beyond the Few-Shot Experimental Design scenario with limited coverage, we further evaluated ROOT's robustness under varying dataset support quality. Specifically, we trained ROOT and the baselines using only the p% lowest-performing designs from the offline dataset. As shown in Table ROOT achieves the best performance across all settings on both TF-Bind-8 and Ant. These results demonstrate that ROOT adapts effectively in challenging scenarios where training data is heavily biased toward low-quality designs.
Effectiveness of Learning the Probabilistic Bridge Model. To demonstrate/ablate the effectiveness of our learning probabilistic bridge model (see Section B.I.Ip. we further conducted an experiment that runs gradient ascent GP posterior mean function for comparison. This simple baseline is denoted as GA on GP in Table T is observed that our method significantly outperforms this baseline, confirming the impact of learning probabilistic bridge.
Number of Gradient Steps (M), We experimented with various numbers of gradient steps (M), from the set [25, 50, 75, 100} to construct X and X during the data generation phase (see Section 3.3 Our experiments reveal that increasing M consistently improves the overall performance of the algorithm as illustrated in Table Increasing the number of gradient steps allows our model to more precisely distinguish between low-value and high-value regions in the distribution, substantially enhancing performance. However, increasing the number of gradient steps also increases computational time, so we select M = 100 as the best balance between performance and efficiency.
Number of Initial (n,)). For each synthetic function generated by the Gaussian process, we will draw number of initial data points (n,) from the offline dataset to initiate the exploration into the low-value and high-value regions via gradient descent and ascent, respectively. We experimented with different numbers of initial points from the set {128, 256, 512, 1024). As shown in Table 7 increasing the number of initial points ni consistently improves performance. This observation is similar to previous observation that having more well-curated training data tends to enhance the overall performance. We selected np 1024 as the best balance between cost and performance.

---

Table 6: Impact of gradient steps M on ROOT.
Table 7: Impact of number of initial data points on ROOT.
Steps (M)
Ant
D'Kitty
TF-Bind-8
TF-Bind-
25
0.968 0.009
0,972 0.003
0.952 a 0.024
0.640 0.039
Initial Points
Ant
TF- Bind-8
50
0,968 a 0.015
0,969 0.003
0,945 4 0.025
0.639 0.024
0.155 0.020
0.588
75
0.950 0.011
0,969 0.005
0.972 a 0.013
0.652 0.033
256
0,950 0.010
512
0.964 0.010
100
0.963 2 0.014
0,972 = 0.005
0.986 # 0.007
0.685 0.053
1024 0.963 e 0.014
0.986 8 0.007
Additional Ablation Studies. We further performed series of ablation studies, examining key hyperparameters of our method, alternative strategies for sampling GP kernel parameters, different data sampling techniques beyond GP, the impact of measurement noise, performance on high- dimensional continuous tasks, and settings with limited query budgets. For completeness, we also report computational complexity analysis and empirical runtime comparisons of ROOT against baselines. Due to space constraints, all these results are deferred to Appendices C
5 Related Works
Existing approaches in offline optimization can be categorized into three main families: forward modeling, inverse modeling, and learning search policies.
Forward Modeling tackles oft-t--diitributtinn (OOD) issues by penalizing high surrogate predictions on OOD inputs [60]3] 770 69 [4] (7 46 18 For example, COM [60] identifies OOD regions
early during gradient updates and re-trains the surrogate with regularizers to penalize high- aigue
predictions at these inputs. BOSS 14] introduces sensitivity-awe -warre regularizer for offline optimizers,
while ICT, Tri-mentoring 77 S
use co-teaching among surrogates improve performance.
Inverse Modeling avoids OOD problems by directly learning high-value regions 40
45 39 For
instance, MIN 40 uses model inversion networks to map scores back to inputs, while ExPT (45
combines unsupervised learning and few-shot experimental design for optimizing synthetic functions. DDOM (39 develops guided diffusion model to generate designs conditioned on function values. The model is trained using weighted sampling from the offline dataset.
Learning Search Policies aims to replicate optimization paths from low- to high-value designs [8 10]. BONET (38] synthesizes trajectories from offline data using a heuristic for monotonic transitions and trains ruto-segressive model. PGS [IO] reinterprets offline optimization
reinforcement learning task, which optimizes for an effective policy using sampled trajectories from offline data.
Overall, these methods remain constrained by the availability of offline data. For instance, DDOM (39] employs guided diffusion to learn an inverse mapping from desired performance outputs to potential input designs. However, the adopted diffusion model must be trained on weighted sam- pling from the offline data, which may lack critical information regarding potential performrf input regions that are far from the offline regimes.
To address the challenges posed by limited data, we reframe offline optimization as a distributional translation task. This perspective unveils an intriguing direction: rather than depending solely on scarce high-value observations, one can learn a global translation model by stitching together localized transformation examples between low- and high-value regimes. This allows the optimization process to be guided by a learned probabilistic bridge that generalizes across design landscapes, offering a flexible and data-efficient alternative to traditional surrogate-based methods.
6 Conclusion
We proposed a new perspective on offline black-box optimization by reframing it as a distributional translation task between low-value and high- value input regimes. At the core of this approach is the probabilistic bridge, a model that learns localized transformation flows conditioned on both source and target designs, and synthesizes them into global translation mechanism. To address the lack of high-value examples in offline datasets, we introduced a synthetic data generation framework that enables pre-training of meta probabilístic bridge with strong zero-shot generalization. This shifts the focus from modeling the objective function to modeling design transitions, opening new possibilities for data-efficient optimization. Future directions include exploring alternative bridge parameterizations and extending the framework to more complex optimization settings.
10

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Acknowledgement
This work utilized GPU compute resource at SDSC and ACES through allocation CIS230391 from the Advanced Cyberinfrastruccure Coordination Ecosystem: Services and Support (ACCESS) program 4. which is supported by U.S. National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.
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A Task Details
A.1 Design-Bench Task
Design-Bench [59] is widely adopted benchmark for evaluating offline black-box optimization
algorithms. Table presents the summary of five evaluation tasks Design-Bench.
Table 8: Overview of tasks in the Design-Bence benchmark. The ground-truth oracle functions for these tasks (unknown to our algorithm) are accessible during evaluation.
Task
Offline Size
Input Dimension
Input Category
Input Type
Oracle
TF Bind 8
32,898
8
4
Discrete
Exact
Bind Ant Morphology
10,000
10
Discrete
Exact
10,004
60
N/A
Continuous
Exact
D' Kitty Morphology
10,004
56
N/A
Continuous
Exact
RNA- RNA-Binding
5000
14
Discrete
Exact
TF Bind 8 and TF Bind 10: DNA Sequence Optimization. The goals of the TF Bind 8 and TF Bind 10 tasks are to discover, respectively, the length-8 and length- 10 DNA sequences with the strongest binding affinity settings, each DNA candidate is a sequence of nucleotides where each nucleotide has 4 possible categorical values. The Design-Bench benchmark grants access the full oracle functions for the TF-Bind-8 and TF-Bind-10 tasks, which correspond to databases containing exact binding affinities for all possible sequences (i.e., 4 and 410 combinations, respectively). Following the evaluation protocol in [59]. fixed subset of sequences is sampled from each oracle and used as the offline dataset for all baselines. In particular, TF-Bind-8 provides an offline dataset of 32, 898 sequences while TF- FF-Bind-10 10 provides an offline dataset with 10, 000 sequences.
specific transcription factor (SIX6 2_1__________ R1 by default) [3. In these
Ant and D'Kitty Morphology: Robot Morphology Optimization. These tasks focus optimizing the physical structure of two simulated robots: (1) Ant from OpenAl Gym [6] and (2) D Kitty from ROBEL 2 In Ant Morphology. the objective is to find the structure of quadruped robot to maximize its running speed. In D'Kitty Morphology the goal is to find the most effective structure for the D' Kitty robot that enables it to reach specific target. In particular, each structure candidate specifies the morphology parameters, such as the size, orientation, and placement of limbs, for a robot controller which will be trained using the Soft Actor-Critic algorithm 22 For the Ant robots, there are 60 morphology parameters. For the D' Kitty robots, there are 56 morphology parameters. The solution quality of each structure candidate obtained by simulating the corresponding trained controller in the MuJoCo [58) environment. Each simulation runs for 100 steps and the overall solution quality obtained by averaging simulation results across 16 independent runs.
A.2 RNA Task
RNA-Binding 44 This is an inverse-folding task whose objective to optimize RNA sequences of fixed length (14 nucleotides) over the alphabet {A, U, C, G). which are abbreviations for adenine, uracil, cytosine, and guanine, respectively. In particular, the task is to find a sequence whose predicted linimum-frreeenerryy (MFE) structure maximizes its binding affinity
given transcription factor. We follow the benchmark setup of Bootgen (35] by considering three target structures RNA-A (L14 RNAI), RNA-B (L14 RNA2), and RNA-C (L14 RNA3). To construct the offline dataset D,, we use the FLEXS codebase [51] to generate sequences uniformly at random. The RNAinverse algorithm from ViennaRNA 2.0 is then applied iteratively until 5,000 sequences with minimum free energy (MFE) below 0.12 are obtained, forming the final offline dataset.
B Detailed Algorithmic Description and Implementation of ROOT
B.1 Generating Synthetic Data with GP
To generate diverse range of synthetic functions which are sufficiently similar to the oracle, we first sample the kernel parameters E, (lengthscale) and o² (variance) uniformly from the ranges Lo - 6, to + 6) and jo d, og + 66. where lo. of and ó are fixed initial hyperparameters, reported
16

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Table 9: Hyperparameters for the synthetic data generation of ROOT.
Hyperparameter
Value
1.0 (continuous)
fo
6.25 (discrete)
8
0.25
0.001 (continuous)
Step size (1))
0.05 (discrete)
Number of gradient steps (M)
100
Threshold (r)
0.001
Algorithm 1 Synthetic Data Generation via Simulating Gaussian Process (GP) Posteriors
1: Input: Offline dataset D, = {ı, Yi Ji=1 number of functions per epoch nee number of points
per function npp number of gradient steps M
2: Output: Synthetic dataset D. {(X (x"..)
3: D. -
4: for
ne do
5: Sample kernel parameters o, = (Ig, 02): Eg Ulo 8, Eu + 8), o² U 5, +
6: Compute the mean function 9o, the posterior Gaussian process via Eq. 11
7: Sample subset of up points from offline data: Do D, where [Doo n
8:
Compute the set of low-value designs X using Eq. 12
9:
Compute the set of high-value designs X using Eq.
13
10: Compute corresponding low and high scores: y. = Jo.
X
y = -
11: D. + D. U ((x, ....y.1l )
)
end for
13: Return D.
Table 9 After sampling, we compute the mean function of the Gaussian process posterior based on the offline data. We then sample np points from the offline data and perform M = 100 gradient ascent and gradient descent steps with step size (more details are provided in Section 3.3). To enhance the quality of our synthetic data, we filter out any pair of low- and high-value points (æ ,/ and
(wr .y++) whose difference between yt and y smaller than a threshold r. All key hyperparameters for this process are listed in Table The pseudocode of the algorithm is presented in Algorithm) |
B.2 Learning the Probabilistic Bridge Model
This section provides further details regarding the implementation of the learning loss in Eq. 8 with respect its Brownian bridge instantiation in Section 3.2 In particular, following the formulation of q( æo. 2T) in Eq. 10 we can express
1
t
xt
Co +
uT + /ke,tet
(15)
where N(0, I) and Ne,t 2(m, m with L/T. This reproduces the Brownian Bridge Diffusion Model [3]. Following the derivation in 43 we obtain:
(( æ2, mo. xT)
N (M(x,, 2o, æT), R I),
where the mean is
p1r To. xT)
Udy Bt + 24 ET + Wi (mi(wT - xc) + /R,
(16)
and the variance is
R (RLL 11-1,1-1 (1 m, m,- 133
(1 m2)2
18-1.111
(17)
17

---

Algorithm 2 Learning the Probabilistic Bridge and Simulation for Offline Optimization (ROOT)
Input: Offline dataset D. (æ yi J= number of epochs E, number of diffusion steps T,
scale a, best objective value y.. conditional dropout probability p. number of iterations !
2: Output: High-value candidate z;)
3:
4: Learning Probabilistic Bridge Phase:
5: Initialize model parameters 0
6: for
F 7: Generate synthetic dataset D3 from Algorithm 1
8: for: to do
yr, 4o, y0 D, - , y 9: Sample {IT, (T
10: Sample timestep t Uniform (1, T), Ber(p), p). y = (3/, yo!
11:
Forward diffusion: 2; = me) Iu m Fr / where N(0,
12: Gradient descent step: Ve Vollmn (xT xo) + /Re, EØ (ar, (1- - 0)]| (Eq. 24
13: end for
14: end for
15:
16: Simulation Phase:
17: (0r: yrr <- 128 best designs Do. y = Ur, a. v..
18: for T to do
19: 2 N(0, I) i > 0 else z = -0
20: Compute <0 1 3) via Eq. 25
21: Ft-1
2t + Vt 2T + we Ea (Ct, 1) + /
22: end for
23:
Return: high-value design zū Zo
Furthermore, the coefficients for the above mean function were derived as
(1-m))²
dt 1it-1,t-1 1 me
1t-1,111
(1-me-1) 1)²
(1
--m-1).
(18)
m4-1
K1,1
1 - mh
kt-1.1-1
m4-1
(19)
1 14-1
Kett
m1-
W4 (1 m.-1)
Kett 11-1--11
-
(20)
Kg,5
This setup thus provides training examples of localized flows which can be used to learn a parame- terized target-agnostic transformation that maps from source 2y to plausible target æn without knowing it beforehand using the loss in Eq. g To implement this loss, we parameterize the transition probability of the aforementioned target-agnostic transformation po( 22, 2T))
pe (x, 12,,
N (x M4 (wr, xT, t), Re-1 1)
(21)
where the pgg is parameterized following Eq. 16
We Be + U2 ET + We <0(
(22)
which features a parameterized noise prediction network e( t) to predict the noise perturbed quantity my (xT xp) + ee of a local flow at time Under this parameterization, the training loss in Eq. [ can be simplified as,
OPB
arg min
[1, - xo) + VF, - -
(23)
Ør.er
is then approximately optimized via sampling (IT, 20p) from the synthetic dataset D, created using Algorithm above. For more practical implementation, we further leverage the corresponding socres yr and yo of ærr mo which are also included in D.. In particular, we further parameterize cg( as ca( t, (1 r)y + 700) with y (yo, yr) and Ber(p) following similar practice
18

---

in guided diffusion (26). The intuition is that we want to leverage the output scores to guide the noise prediction network but ath the same time, we do not want the network to overly depend on such guidance. This is enforced using the dropout trick which randomly removes the guiding information during training with probability P E (0, 1). The above learning loss can then be recast as:
Pp
arg min
E 2o RT
[lmm(xs - xo) + Vir,,e - col t, (1 7) . + 0)) (24)
where y ~ Ber Once trained, the noise network Eg is used as below during the simulation phase: v) = (1+ ca ( t, y) 3 co t, 0)
(25)
where B is the classifier free guidance weight and y = [VT, a yo] with ye denotes the maximum oracle score. Our detail algorithm is presented in Algorithm 2
For more details, we utilize an MLP co cg t, y) comprising four layers, each with 1024 units. Each layer employs the Swish activation function, Swish(2) 20(2), where (22) denotes the sigmoid function. The MLP is trained using the Adam optimizer for 100 epochs with a learning rate of 0.001. During each epoch, we sample ne 8 synthetic functions from the Gaussian process (the total number of synthetic functions is my n. x F x 100 800 functions) and generate ny = 1024 samples for each function. At the testing phase, we sample high-value design candidates from the 128 best designs in the offline dataset, using T = 200 sequential denoising steps. All hyperparameters for modeling, training, and sampling with our model are summarized in Table I0
Table 10: Hyperparameters for the generalized Brownian Bridge diffusion process in ROOT.
Hyperparameter
Value
Hidden size Number of layers
1024
Architecture
Activation
Swish
Number of epochs (E)
100
Number of functions (ne) (per epoch)
8
Number of data points (nz)) (per function)
1024
Training
Learning rate
0.001
Optimizer
Adam
Batch size
64
Conditional dropout (p)
0.15
0.8
Sampling
9
-1.5
Denoising steps
200
C Additional Experiment Results
C.1
Computation Resource
All our experiments were conducted system with the following specifications: Ubuntu 20.04.5, single NVIDIA A 100-SXM4-80GB GPU, and CUDA 10.1. Notably, our method requires less than 6GB of GPU memory per run.
C.2 Additional Performance Evaluation of ROOT
In the main text, we have reported the 100th percentile scores. In this section, we present addi- tional evaluation results at 80th and 50th percentiles, providing further insights into the performance distribution of our ROOT.
C.2.1 Performance Evaluation th Percentile
As shown in Table 11 our method ROOT consistently demonstrates strong performance at the 80th percentile level, achieving the best mean rank, i.e., 1.5. Notably, in the Ant and D' Kitty tasks, ROOT achieves significant improvements over all baselines with remarkable score differences of 0.098 and 0.025 over the runner-ups in those tasks, respectively. ROOT also achieves the best result in
19

---

TF-Bind-8, outperforming the runner- up with margin of 0.011. In the TF-Bind-10, ROOT performs competitively the best baselines. Overall, these results demonstrate the consistent reliability and stability of our model's performance.
Table 11: Experiments on Design- Bengh Tasks. We report 80th percentile score among Q 128 candidates. Blue denotes the best entry in the column, and Brown denotes the second best. Mcan Rank means the average rank of the method over all the experiment benchmarks.
Benchmarks
Method
Ant
D D'Kitty
TFBind8
TFBind10
Mean Rank
Do (best)
0.565
0.884
0.439
0.467
BO-qEI
0.629 = 0.000
0.884 + 0.000
0.439 + 0.000
0,510 # 0.011
11.00 15
CMA-ES
0.007 0.013
0.001 0.718 = 0.001
0.652 0.011
0.543 # 0.013
9.50 15
REINFORCE
0.182 0.017
0.562 + 0.197
0.622 + 0.030
0.519 0.007
11.25
GA
0.762 0.189 = 0.014
0.828 0.027
0.006
8.75 15
COMs
0.635 # 0.031
0.887 + 0.004
0.738 + 0.027
0.526
5.25 / 15
CbAS
0.542
0.813 0.012
0.585 = 0.030
0.517=0.000
10.25 15
MINs
0.746
0.908 + 0.004
0.545
0.519
6.50 15
RoMA
0.298 4 0.033
0.738 0.018
0.661 + 0.010
0.525 0.003
9.00 15
DDOM
0.749 0.029
0.865 0.009
0.016
0.506 0.004
9.75 15
ICT
0.708 # 0.019
0.898 + 0.004
0.035
0,525 0.016
6,00 15
Tri-mentoring
0.722 0.015
0.003
0.683
0.531 0.007
4.00
GTG
0.725 0.077
0.031
0.506 + 0.006
7.75 15
LTR
0.679 0.016
0.902 + 0.002
0.734 0.025
0.508 0.010
7,00 15
GABO
0.016 0.009
0.705 0.002
0.676 0.021
0.512 0.008
11.25
ROOT (ours)
0.847 = 0.005
0.938 0.002
0.839 # 0.015
0.526
1,50
C.2.2
Performance Evaluation 50th Percentile
Table 12 reports the results achieved by all baselines at 50-th percentile. Consistent with our earlier observations, ROOT again achieves the best mean rank of 2.0. In particular, we surpass the runner-up baseline, Gradient Ascent (GA), in the TF-Bind-8 task with a significant score difference of 0.072. securing the top rank among all other methods. In the Ant and D D'Kitty tasks, ROOT also achieves substantial score gaps of 0.067 and 0.018 over the corresponding runner-ups, respectively. Further- more, the reported standard deviations of ROOT in those tasks are 0.014 and 0.003 which are much lower than those of other baselines. This further ascertains the stability and superior performance of our method across the evaluation benchmark.
C.3 Score Distribution of ROOT
To provide a more holistic comparison between the score distribution of ROOT and those of others, we collect the design candidates obtained from 8 runs (128 x 8 1024 designs) to plot the distribution of scores achieved by ROOT and several other representative baselines. The results are shown in Fig. 2 Due to the wide range of scores produced by CMA-ES on the Ant task, we separately plot and compare its score distribution against ROOT. The remaining plots visualize and compare the score distributions of ROOT and 5 representative baselines. In each plot, we annotate the max and median lines to case the visual comparison. In particular. Fig 2 reveals that ROOT often has a score distribution skewed towards higher-value regions, especially in the D'Kitty task. In the Ant task, although ROOT 's max score is not as high as that of CMA-ES, CMA-ES only produces a single good design while the rest (in its solution set) perform poorly, as also observed in Table I and Table 12 Otherwise, it can observed consistently across all plots ROOT achieves better score distributions compared to other baselines.
Summary, conclusion, all reported results have demonstrated consistently that our proposed method ROOT is, on average, the best-performing baseline in both score distributions and targeted evaluation percentiles. This consistent performance not only highlights the effectiveness and stability of ROOT but also reaffirms the robustness of our approach across wide range of tasks.
20

---

Table 12: Experiments on Design- Bench Tasks. We report 50th percentile score among Q 128 candidates. Blue denotes the best entry in the column, and Brown denotes the second best. Mean Rank means the average rank of the method over all the experiment benchmarks.
Benchmarks
Method
Ant
D' D'Kitty
TFBindd
TFBind10
Mean Rank
Do (best)
0.565
0.884
0.439
0.467
BO-qEI
0.569 # 0.000
0.883 + 0.000
0.439 + 0.000
0,,669 # 0.005
7.75 / 15
CMA-ES
-0.043 0.007
0.674 0.016
0.536
0.490 0.015
8.75 / 15
REINFORCE
0.140 = 0.026
0.510 0.203
0.024
0.470 0.010
11.00 15
GA
0.137 = 0.014
0.591 # 0.132
0,469 = 0.006
8.75 / 15
COMs
0.471 0.034
0
0.475 0.038
5.75 / 15
CbAS
0.369 # 0.008
0.748
0.441
0,465 0.006
10.75 / 15
MINs
0.618
0.809
0.421
0.467 0.079
7.25 / 15
RoMA
0.024
0.170
0.003
8,50
DDOM
0.066
0.014
0.012
0.002
11.00 / 15
ICT
0.018
0.872
0.037
0.033
8,25 / 15
Tri-mentoring
0.804 0.012
0.562
0.009
4.25 / 15
GTG
0.098
0.901 # 0.005
0.452 0.430
7.25 / 15
LTR
0.568 0.016
0.885 + 0.002
0.026
0.466 0.011
6.00 /
GABO
-0.039 # 0.004
0.674 = 0.005
0.496
0,457 = 0.008
11,50 / 15
ROOT (ours)
||
0.712 = 0.014
0.919 + 0.003
0.675 + 0.026
0.473 + 0.004
2.00 / 15
Ant
D'Kitty
Legend
L2HD Max
L2HD
5.8
Ant
3A-
2-
TFBinds
TFBind10
0.8
0.6
02
CHA-ES
L2HD
CHA--85 COMS
GA
Mila Trimmn"
ICT
L2K0
CHA.85 CDMb
NBH Tri-men
L2HD
Figure 2: Score distribution found candidates of ROOT compared to others. To avoid cluttering the plots with long chunks of text, we use the abbreviation Tri-men* to denote Tri-mentoring.
C.4 Computational Complexity Analysis
Computational Complexity. Computational cost of ROOT is follows:
For each training epoch: Fitting n, Gaussian processes requires O(nyn²) where is the offline data size. Performing gradient steps (querying the GP's mean function) to generate np synthetic points from each of ny functions requires O Training the BBDM model requires T diffusion steps per sample, where each step involves forward and backward pass through the score network eg, giving a total cost of O(T(Jo by ptp)) where So, by denote the cost of forward and backward, respectively.
For the whole training process, the overall cost ((F(nyn + Mnnnnnn + T(Jo + bo)m)mp)))
where E is the number of epochs.
21

---

For the inference process, the overall cost is O(QDf)) where Q and D are the number of selected candidates and denoising steps.
Therefore, the total computational cost of ROOT is: O(E(nyn' + Mn + T(fe + +b)) O(QDf))
Furthermore, we note that the most complex term in ROOT"s computational cost, GP fittig, be further mitigated by using sparse GP techniques 49 which scales linearly in the number of inputs.
Empirical Runtime. We empirically evaluate the runtime of our method and compare it against several baselines across range of tasks, As shown in Table 13 the results demonstrate that ROOT achieves faster execution time than several widely used baselines, including CMA-ES, MINS, and mri-mentorin, while maintaining competitive or superior performance. All experiments conducted single GPU, with each task requiring approximately 4GB of memory.
Table 13: Runtime (in seconds) of different models across four tasks.
Model
Ant
D'Kitty
TF-Bind-8
TF-Bind-10
GA
122
200
124
420
BO-QEI
219
310
402
383
CMA-ES
5747
4889
2667
4323
MINS
797
998
2213
792
REINFORCE
240
257
507
373
CBAS
394
390
835
529
ICT
188
189
216
639
Tri-mentoring
4276
3921
4673
3410
DEMO
668
1489
1024
1696
ROOT
298
297
407
575
C,5 Effectiveness of Gaussian process for Generating Synthetic Data.
C.5.1 DNN-based Approach for Generating Synthetic Data
There are several strategies for sampling closed-form functions fitted to offline data (to provide functions that are similar to the oracle). In ROOT, we adopt a standard Gaussian process (GP) approach due its efficiency, are particularly advantageous for their simplicity in debugging and their effectiveness in modeling predictive uncertainty. Our motivation is further motivated by ExPT [45), recent few-shot offline optimization baseline, which similarly employs GPs to generate pseudo labels for offline data.
of implementation, and strong empirical performance. GPs
While alternative generative models exist, they often introduce additional complexity and compu- tational overhead. For instance, we conducted experiments using deep neural networks (DNNs) in place of GPs. One such approach involves randomly initializing the weights of DNN and training to fit the offline dataset, then serving as a single sampled function. However, this approach is computationally expensive. Generating 800 such functions using DNNs, each requiring 800 separate training runs, takes approximately 100 minutes, compared to under 300 seconds for the entire training process using our GP-based method.
To further investigate this, we evaluated two less costly alternative setups: (1) an ensemble of five DNNs replacing all 800 GP functions, denoted as DNN5, and (2) single DNN replacing all 800 GP sampled functions, denoted as DNNI. As shown the Table [4 both alternatives perform worse than our GP-based approach and incur significantly higher computational costs.
C.5.2 Bins-based Approach for Generating Synthetic Data
Beyond the DNN-based approach, we also explore two other heuristic methods that do not depend on Gaussian processes (GP). The first method, 2 bins, partitions the offline data into two bins, the lowest 50th percentile and the highest 50th percentile, and samples low- and higu-value designs (X- X from the corresponding bins respectively. The second method, 64 bins, divides the data into 64
22

---

Table 14: Performance and runtime of different methods for sampling function on the Ant task.
Model
Performance
Time (s)
DNN5
0.637 + 0.268
1297
DNN1
0.630 0.201
751
ROOT
0.965 0.014
298
bins based on the y values and then samples X from the lowest bin and X from the highest. This approach similar to the sampling strategy in [38], where trajectories with increasing outputs selected from offline data train model that progressively guides designs from the lowest to the highest bin. In addition, we also examine another sp--baee approach that utilizes only one GP function to generate synthetic data. As reported in Table [5 the GP-based methods consistently outperform the above bin- base heuristic approaches, with GP (800 functions) achieving the best performance.
Table 15: Effectiveness of Gaussian process for generating synthetic data.
Type
Ant
D'Kitty
TF-Bind-8
TF-Bind-10
2 bins
0.745 0.097
0,952 # 0.007
0.775 + 0.057
0.641 0.034
No-GP
64 bins
0.941 0.020
0,952 # 0.009
0.837 + 0.055
0.651 * 0.054
GP (1 function)
0.955 = 0,013
0,971 # 0.004
0,984 + 0.012
0.657 = 0.029
GP (800 functions)
0.965 = 0.014
0,972 # 0.005
0,986 + 0.007
0.685 = 0.053
C.6 GP Hyperparameters Sampling Methods
To diversify the GP mean functions, various approaches can be used to sample GP hyperparameters. In our work, we adopt simple strategy by uniformly sampling the GP hyperparameters, i.e., the lengthscale f and variance a., from a fixed range, as presented in line 5 of Algorithm This approach follows the prior practice in ExPT 45,, which also samples these hyperparameters uniformly from small range and achieves strong performance.
To explore alternative strategies, have conducted additional experiments. In MAP-based setting, we first optimize the hyperparameters ty and To by minimizing the negative log marginal likelihood ) 1= which is commonly used in Gaussian processes:
(NLML) on D. {a., v1
و. Ky
+ log [Kol
1
+ log 2n
ni
2
where K3. is the kernel matrix parameterized by ø (lo, Jo), and is the number of training data points. After obtaining these MAP estimates, we sample new hyperparameters uniformly around them: Le~ [Lo - 5, Eo + 8] and a, ~ [0o 5, op + 5). instead of setting fo. do as in Table g
In addition, have also experimented with MCMC-hased approach to sample these hyper- parameters. In this setting, we first define a prior distribution p(l,, a.) (e.g., a Gaussian prior) and a likelihood p(D. | o.) derived similarly from the marginal likelihood. We then apply MCMC sampling to draw fg hyperparameter combinations from the posterior distribution, i.e p(l,,a, D.) x PDD. o.) p().,
As shown in Table T6, our uniform sampling method consistently outperforms the aforementioned alternatives. Also, both alternatives introduce significant computational overhead. In particular, the MCMC-based approach requires substantial runtime, incurring approximately 1000 seconds per run, even when using only 8 warmup steps. We leave further exploration of more sophisticated hyperparameter sampling strategies to future work.
C.7 Hyper-parameter Tuning
Number of GP mean functions. We additionally conducted experiments on the number of GP mean
functions to generate synthetic data. In our original method, we use ng = nr M 800 functions; here, vary this number from 100 to 1000. The empirical results in Table 17 indicate that the final
23

---

Table 16: Performance and runtime of different GP hyperparameter sampling methods on Ant task.
Method
Performance
MAP (ng = 800)
0.940 + 0.023
MCMC (ng 200)
0.884 = 0.017
MCMC (ng -400)
0.921 # 0.010
MCMC (ng 600)
0.932 # 0.016
MCMC (ng 800)
0.916 0.009
ROOT (ng 800)
0.965 = 0.014 0.014
performance improves as the number of Gaussian Processes (GPs) increases. However, there may exist a saturation point (e.g.. 800 GPs), beyond which further increasing the number of GPs could lead to decline performance.
Table 17: Performance on Ant and TF-Bind-10 with varying number of GPs.
#GPs
Ant
TF-Bind-10
100
0.914 = 0.015
0.632 # 0.020
200
0.914
0.677 = 0.038
400
0.959 = 0.010
0.631 # 0.018
600
0.956 0.021
0.674 = 0.044
800 (ROOT)
0.965 = 0.014
0,685 = 0.053
1000
+.960
0.662 = 0.033
GP hyperparameters. We also employed an ablation study to see the effect of Gaussian hyperpa- rameters (i.e., initial lengthscale En) on our final results. As can be seen clearly from the Fig. 3] the optimal range for &o yields around 1.0 for the continuous tasks and 6.25 for the discrete tasks.
1.0
0.95
0.9
Score
Score
0.90
0.8
0.85
Ant DKitty
0.7
TrBind8
0.80
2
4
5
2
6
8
Lengthscale
Lengthscale
Figure Effect of lengthscale on performance for Ant and DKitty.
GP hyperparameters range size ō and step size In the data-generation phase, we uniformly sample the Gaussian process lengthscale and variance from the intervals [& 5, Eo + 5] and [oo - 8, on + 6) respectively (see Line 5 Algorithm Tp. Choosing is a trade-off: if it is too large, the probabilistic bridge will learn functions that deviate excessively from the oracle, but if it is small, we lose the diversity needed for robust generalization. We conducted a sweep over different values of 8: shown in Table 18 6 0.25 achieves the best performance across both the Ant and TF-Bind-8 tasks. In addition, during this phase we apply AM gradient steps with fixed step size (see Eq 12 and Eq. [3 Our experiments Table demonstrate that for continuous tasks, setting 0.001 achieves the best score.
GP kernel choice. We experimented with the Matern kernel for the GP-based synthetic data generation, as reported in Table 20 The results indicate that the commonly used RBF kernel consistently yields better performance, supporting our decision use it the default.
Incorporating explicit uncertainty via acquisition functions. In Gaussian Process literature, incorporating explicit uncertainty via acquisition functions like UCB or LCB is promising approach

---

Table 18: Performance Ant and TF-Bind-8 with varying
Table 19: Performance on Ant and D' D'Kitty with
varying 7/.
8
Ant
TF-Bind-8
7
Ant
D'Kitty
0.05
0.953 + 0.011
0.982 + 0.005
0.0005
0.969 0.017
0,971 0.003
0.10
0.957 = 0.014
0.986 + 0.006
0.00075
0,968 + 0.013
0.969 = 0.005
0.25 (ours)
0.965 = 0.014
0.986 = 0.007
0,001 (ours)
0.965 + 0.014
0
0.50
0.959 + 0.013
0.981 = 0.007
0.0025
0.976 0.004 0.010
1.00
0.959 = 0.014
0.989 + 0.003
0.005
0.948 0.011
0.977 = 0.003
Table 21: Performance of ROOT when replacing the GP's mean function with UCB and LCB.
Table 20: Performance comparison of different GP kernels on Ant and TF-Bind-8.
Method
Ant
TF-Bind-8
Kernel
Ant
TFBind8
UCB
0.964 = 0.017
0.966 + 0.017
Matern
0.966 0.013
0.848 = 0.055
LCB ROOT
0.949 = 0.008
0.961 = 0.025
RBF
0.965 0.014
0.986 = 0.007
0.965 = 0.014
0.986 = 0.007
for balancing exploration and exploitation. To evaluate this, we conducted a small-scale experiment replacing the GP's mean function with UCB and LCB scores to generate synthetic data. The results, shown in Table (T] indicate that while UCB and LCB perform reasonably well, they do not outperform our original approach based the GP's mean function. We believe this is because uncertainty has already been implicitly captured through a different mechanism, by sampling multiple mean functions from population of posterior GPs trained on the same offline dataset using different kernel configurations.
C.8 Score and Pseudo- Value Distribution of Generated Samples from GP
Distnbubon of pscudo values (GP) on x high
Distribution of Oracle values on high
Density
1.0
Density
0.8
3.0
02
0.50
075
100
125
150
175
200
225
250
0.0
02
0.4
0.6
0.
Value
Value
Distrbution Oracie values
1.0
Density
Density
0.8
2.0
02
0.6
Value
Volue
Figure Distribution of pseudo-values (GP) and oracle values on low- and high-value regions.
To further demonstrate the efficiency of Gaussian processes, we conducted an experiment on the Ant task using low and high regions produced by 800 GP mean functions. Each function produces 1024 low-value points (X or x_low) along with 1024 high-value points (X+ or xigh.... We concatenate all 1024 x 800 points to form the distribution for the low-value and high-value designs. We then
25

---

plotted both the oracle value distribution by using the oracle function to evaluate and the pseudo distribution (y and y" predicted by GP). As shown Figure the low and high regions correspond to two clearly separated distributions. The oracle values in X are skewed toward higher values, while those in X are skewed toward lower values. This separation facilitates effective training of our probabilistic bridge model as it learns to transform between two distinct distributions.
C.9 Limited Budget Settings
While budget of Q 128 candidates is commonly adopted setting, also conduct experiments under more constrained budgets to evaluate the robustness of our method. The results, shown in Table 22, demonstrate that ROOT continues to achieve strong performance even in these highly limited-query scenarios, outperfomiigg existing baselines.
Table 22: Performance different methods under extremely limited offline data budgets (Q). ROOT consistently outperforms baselines on both Ant and TF-Bind-8.
Q Budget
Method
Ant
TFBind8
GA
0.250 0.033
0,938 + 0.020
16
COMs
0.789 0.070
0.917 = 0.059
ROOT
0.938 + 0.022
0.956 0.028
GA
0.252 0.036
0.961 = 0.024
32
COMs
0.776 + 0.033
0.871 = 0.030
ROOT
0.942 0.019
0.974 = 0.020
GA
0.281 0.023
0,978 + 0.020
64
COMs
0.834 # 0.051
0.922 # 0.036
ROOT
0.942 0.020
0.974 + 0.021
C.10 Strong measurement noise setting
Although ROOT uses synthetic labels from deterministic GP means, it remains robust to strong measurement noise. This is due to the use of diverse GP ensemble with varied hyperparameters, which implicitly captures wide range of uncertainties present in the offline dataset. Sampling from this ensemble produces synthetic data that reflects variability in the objective, helping the bridge model generalize more effectively. To validate this, we ran additional experiments on TF-Bind-8 with label perturbed by Gaussian noise N(O,e), e). As shown in the Table 23 ROOT maintains strong performance and outperforms other baselines across noise levels, demonstrating its resilience to measurement noise.
Table 23: Performance of different methods on TF-Bind-8 task with label perturbed by measurement noise e0,,)).
0.01
0.1
0.2
GA
0.967 + 0.015
0.970 # 0.006
0.963 # 0.010
COMs
0.956 + 0.024 0.024
0.925 + 0.034
0.952 0.013
REINFORCE ROOT
0.947 # 0.030
0.933 # 0.025
0.930 # 0.042
0.969 + 0.016
0.971 + 0.007
0.965 0.014
C.11 High-dimensional continuous task
In the Design- -ench benchmark, the Hopper task represents nigh-dimensional continuous task with 5, 1,6 input dimensions. Although prior work has noted a highly noisy and inaccuracy oracle function for this task, we conducted a small experiment to evaluate ROOT's performance on Hopper as test of scalability. demonstrating its ability to effectively handle high nigh-dimensional design spaces.
shown in the Table[4] ROOT outperforms some representative baselines,
26

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Table 24: Performance of different methods on the Hopper Controller task.
Method
Hopper Controller
GA
-0,068 0.001
MINs
0.267 = 0.350
Reinforce
-0.009 = 0.067
ROOT
0.541 = 0.042
Table 25: Performance of ROOT with different bridge configuration across the benchmark tasks.
Bridges
Ant
D'Kitty
TF-BBin-88
TF-Bind-10
ROOT (BBDM)
0.965 0.014
0.972 0.005
0.986 0.007
0.685 0.053
ROOT (OUDM)
1.940 + 0.599
0.723 # 0.001
0.927 0.049
0.675 = 0.124
D Other Practical Setups for Our Probabilistic Bridge Framework
In this paper, we employ the Brownian bridge as our practical probabilistic bridge (see Appendix B.2). which achieves significant performance across our experiments. However, our proposal method is a flexible framework that can be universally adapted to other probabilistic bridges. In this section, we will demonstrate another design: Ornstein- Uhlenbeck Bridge.
From our definition of the probabilistic bridge in Section
3.1.I. we can choose T/2 and Ke, as
(Tnh(a(T-1)) ())
sinh(at)
1/ 2T)
+ ar
(26)
sinh(aT)
sinh(aT)
sinh(a min(t,k)) sinh(a (T - max(t,k))
Kt.k
sinh(aT)
(27)
This choice will result in the formula of the Ornstein-Uhlentee bridge (D with the hyperparameter a. Once this bridge is learned, we can simulate æ; following the previously derived simulation approach in Eq. which, under the OU instantiation, becomes,
sinh(a(T - + ær san""
sinh(at) sanl(
sinh(at) sinh(a(T - a a-sinh(aTI")
(28)
where Eg ~ N(O, I). From the above equation, we can also represent Do in terms of a, and ay:
sinh(al! sinh(aT)
sinh(al) siiniiaTTnnn(( - ()) e--iin(aTT)
sinh(aT)
Et
sinh(a(T (29)
The transition probability of the localized bridge/flow, i..., q(x,-1 xe, Wo FT) in Eq. 6 can be derived by using the conditional Gaussian rule which results in the following transition mean,
[, Do. 2T) = ( 2T) + Re 1,-1, (x: - the (wo. Tr))
(30)
Substituing (1(2o, Rr) + Vi,e ez, the above transition mean can be rewritten as:
u(x,, Co. xT)
vhr- ( Ke-1, 1/2
(31)
sinh (a (1 1)) sinh (a -++ + Φr sinh(aT)
To
sinh(aT)
sinh (a (t (1-11) 1)) sinh (T -
+ 61 V sinh(aT) sinh(at) sinh(a(T - 1))
(32)
27

---

By substituting 2e from Eq. 29 we get a closed-fore formula for u1 xT) = ut + v2.
2y + w E, where the coeffcients are computed as
sinh(a (T
sinh(a (1- sinh(a'l)
sinh(a (T - 1)) sinh(ad)
Vt
sinh(a (T
Vt
sinh(aT)) sinh (T - 1))
.(33)
sinh(a(T w Va sinh(aT) sinh(at) sinh (t-1)) sinh(at)
sinh(a -T -inb(a(T-(+1))) sinh(a (T - -))
(34)
Next, following prior practice, we again reparameterize the target-agnostic transformation po(#;-1 [r, TT) as neuralized Gaussian with mean 6a (3,, ØT, t) and same covariance matrix as the local transition, the (training) ELBO loss in Eq. can be simplified as:
Zo. (TT = U4 2, 0, ær + wy
PPB arg min E, (... co (1ro ær) Ke,t ct Tr,t) co
(35)
0
In addition, incorporating the classifier-free guidance techniques leads to:
PB arg min Eeo wrr Eo (v/x (wyy xr)) + Ki,t ee, .,,7 Y y (1 (1-)))00 - -Ell" (36)
where y ~ Ber(p). Once the training process is done, we can employ the trained network eo for our optimization-via- via-simulation process:
-t = uh Dt + 1/ aT wy co(o,, 1, + 1.,
(37)
where K 11-1,111 K R and ~ N(O, I).
The performance of the OU-based ROOT is compared against the original Brownian-based ROOT in Table 25 The results show that the OU-based variant of ROOT performs competitively to the original Brownian variant TF-Bind-8 and TF-Bind-10. On the Ant task, the OU-based variant is better while on the D'Kitty task, the Brownian variant better. We leave a more thorough investigation across different variants of ROOT with different bridge configuration to future work such detailed investigation is beyond the scope of our current work.
E Broader Impact and Limitation
Broader Impact. This work provides novel lens for offline black-box optimization by reframing as distributional translation problem, potentially inspiring new probabilistic modeling techniques in low-data regimes. The approach opens up new possibilities for data-efficient optimization in applications where function evaluations are expensive or infeasible. These include (but are not limited to) materials discovery, policy design, and automated experimentation. However, caution must be exercised when deploying such methods in safety-critical domains, reliance synthetic priors may introduce epistemic uncertainty that requires careful quantification and calibration.
Limitation. One practical consideration is the additional computational overhead introduced by fitting multiple Gaussian processes to construct the synthetic function ensemble. While this step is performed offline and enables better data efficiency during optimization, it may require tuning and parallelization to scale effectively with large input dimensions limited compute resources.
28

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Reviews:
[{"review_id": "D5bD3qnwHM", "reviewer": "Reviewer_zEZb", "summary": "The paper proposes ROOT, an offline black‑box optimization framework that recasts the task as a distributional translation problem. A probabilistic bridge, inspired by diffusion but conditioned on both source and target points, is trained on synthetic low‑ and high‑value design pairs generated from an ensemble of Gaussian‑process posterior means fitted to the offline data. Once trained, the bridge is run backward from the best offline samples to produce improved candidates. Experiments on four Design‑Bench tasks and three RNA inverse‑folding tasks report new state‑of‑the‑art results and extensive ablations support each design choice.", "questions": "Please clarify the computational cost of fitting eight hundred GPs versus a single large GP ensemble and how this scales beyond ten thousand offline points; discuss whether fewer but higher‑rank kernels could suffice. \n\nExplain how ROOT would handle objectives with strong measurement noise, since synthetic labels come from deterministic GP means. \n\nProvide results on at least one truly high‑dimensional continuous task (e.g., 1000‑dim design) to assess scalability. \n\nDescribe how sensitive performance is to the choice of Brownian‑bridge parameters and whether other kernel choices were tried. \n\nA detailed comparison with very recent diffusion‑based baselines such as DiGBO or DM‑BO would strengthen the empirical claim.", "limitations": "The authors list data and compute limitations and note lack of theory, but they should also acknowledge that GP‑derived synthetic data may embed biases of the initial kernel, potentially restricting exploration.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Quality: Empirical work is thorough, with eight independent runs, multiple percentiles, and ablations on step size, GP ensemble size, and data budgets. However, the method is empirically driven; there is no theoretical guarantee that the learned bridge preserves function maxima, and some critical implementation details (e.g., GP kernel hyper‑parameter grids, training compute) are only in the appendix. \n\nClarity: The core idea is clearly motivated and the probabilistic bridge derivation is explicit, yet the paper is very long, and key intuition is buried in implementation, which may hinder reproducibility. \n\nSignificance: Achieving uniform gains across both continuous robot morphology and discrete sequence design benchmarks suggests practical impact, but the community still lacks evidence that gains transfer to harder high‑dimensional or noisy settings. \n\nOriginality: Framing offline optimization explicitly as distributional translation via a bridge that conditions on both endpoints is novel, though it builds heavily on recent diffusion‑based optimizers and GP‑assisted pretraining.", "was_revised": true, "final_justification": "Thank you for the thorough rebuttal. The new experiments on high-dimensional (Hopper) and noisy tasks addressed my primary concerns regarding the evaluation. The clarifications on computational cost were also helpful. After reviewing the strong responses provided to all reviewers and seeing the consensus, it is clear the paper is strong. You have fully addressed my concerns, and I am happy to raise my score to 5.", "rebuttal": "We thank the reviewer for the positive rating and detailed feedback. We address the reviewer's questions as follows:\n> there is no theoretical guarantee that the learned bridge preserves function maxima \n\nWe clarify that given sparse data, there exist infinitely many functions consistent with the observed samples, and their optima can differ substantially. As a result, recovering the true maximum is not the goal of offline optimization. Instead, the focus is on discovering improved designs in regions near the observed data, where most consistent functions are likely to agree.\n\nOur method is designed with this goal. By learning a probabilistic bridge from low- to high-value regions using simulated data from synthetic functions sampled around the oracle, we aim to discover inputs that likely yield better outcomes within this region that encode the oracle's plausible behaviors.\n\nWe also agree that theoretical guarantees for the learned bridge consistently mapping to high-value regions would be valuable. However, this relates to broader open questions in generative modeling, where such guarantees are still largely lacking. Most offline optimization methods also lack guarantees for discovering improved designs. Deriving such results is therefore nontrivial and beyond the current scope, but we see it as a valuable direction for future work.\n> some critical implementation details (e.g., GP kernel hyper‑parameter grids, training compute) are only in the appendix.\n\nWe will add the following details to the “Hyper-parameter Configuration” paragraph in Section 4.1:\n\nFor each baseline, we adopt the optimized settings from the original papers. For GP kernel hyper-parameters in our data generation, we sample lengthscales $l_s$ and variances $\\sigma_s^2$ uniformly from $[l_0 − \\delta, l_0 + \\delta]$ and $[\\sigma_0^2 - \\delta, \\sigma_0^2 + \\delta]$, with $l_0 = \\sigma_0^2 = 1.0$ for continuous tasks and 6.25 for discrete tasks, and $\\delta = 0.25$. We use $M=100$ gradient steps with step sizes 0.001 (continuous) and 0.05 (discrete). Additional data generation details are in App. B.1. For training the Probabilistic Bridge model, we use a Brownian Bridge diffusion process with the Adam optimizer over $E=100$ epochs and $n_g=800$ synthetic functions, running on a single NVIDIA A100-80GB GPU. More training details are provided in App. B.2.\n> yet the paper is very long, and key intuition is buried in implementation, which may hinder reproducibility..\n\nWe would like to point out that the core intuition of our work is outlined in lines 51–59 (Section 1) and further elaborated at the beginning of Section 3. We will highlight these more explicitly in our revision.\n\nTo summarize, we cast offline optimization as a translation task from a low-value source distribution to a high-value target distribution. To facilitate this, we introduce the probabilistic bridge concept, which identifies localized translation examples by conditioning on both source and target contexts (Section 3.1.1). These examples are then utilized to train a global, target-agnostic transformation flow that generalizes beyond the observed data (Section 3.1.2). \n\nWe provide a concrete example in Section 3.2, which illustrates how a Brownian bridge can instantiate our framework, with implementation details in Appendix B.2. Its derivation is deferred to the appendix because it is procedural rather than insightful. We can instantiate other bridges using the same procedural derivation (via deriving the underpinning transition via conditional Gaussian rule on Gaussian processes) with different kernel choices. For example, we have presented another practical instantiation with the Ornstein-Uhlenbeck bridge (see Appendix E in the updated Appendix) in the Supplementary Material.\n\n> the community still lacks evidence that gains transfer to harder high‑dimensional or noisy settings\n\nWe appreciate the reviewer’s broader observation that the community still lacks strong evidence of gains transferring to high-dimensional or noisy settings. We agree that this remains an important challenge of the offline optimization community. By connecting offline optimization with the broader generative modeling toolkit, we introduce a principled and data-driven bridge construction mechanism generalizing diffusion processes, which have demonstrated robustness and stability across diverse domains. This connection opens a promising pathway toward more scalable and resilient offline optimization, particularly in high-dimensional or noisy settings. To provide further insight, we have conducted additional experiments evaluating the efficacy of ROOT in both high-dimensional and noisy scenarios, as detailed in our responses below.\n\n> Please clarify the computational cost of fitting eight hundred GPs versus a single large GP ensemble and how this scales beyond ten thousand offline points; discuss whether fewer but higher‑rank kernels could suffice.\n\nIn our setting, the 800 GPs are not trained via hyperparameter optimization but use instead kernel parameters sampled from a prior, enabling efficient posterior mean computation with minimal overhead. In contrast, learning a GP ensemble requires learning a posterior over kernel hyperparameters, often through marginal likelihood optimization, which can be significantly more expensive in both compute and memory. In particular, we have explored 2 ensemble-based approaches that underperformed compared to ROOT and incurred higher computational costs (Appendix C.6).\n\nRegarding the suggestion to use fewer but higher-rank kernels, the RBF kernel already corresponds to an infinite-dimensional feature space, so increasing its rank is not applicable. However, exploring alternative kernel families with richer inductive biases is an interesting direction for future work.\n\nTo scale beyond ten thousand offline points, our framework supports direct use of sparse GP approximations, which scale linearly with data size. Additionally, individual GPs can be trained on data subsets to further improve scalability.\n\nOverall, while using a full GP ensemble with learned hyperparameters is feasible, it would significantly increase computational overhead. Since our approach already performs well with much lower cost, we prioritized scalability and efficiency in the current scope. Extending the method with more expressive GP ensembles remains a promising direction for future work.\n\n>Explain how ROOT would handle objectives with strong measurement noise, since synthetic labels come from deterministic GP means.\n\nAlthough ROOT uses synthetic labels from deterministic GP means, it remains robust to strong measurement noise. This is due to the use of a diverse GP ensemble with varied hyperparameters, which implicitly captures a wide range of uncertainties present in the offline dataset. Sampling from this ensemble produces synthetic data that reflects variability in the objective, helping the bridge model generalize more effectively. To validate this, we ran additional experiments on TFBind8 with label perturbed by Gaussian noise $N(0,\\epsilon)$. As shown in the table below, ROOT maintains strong performance and outperforms other baselines across noise levels, demonstrating its resilience to noise.\n\n|$\\epsilon$|0.01|0.1|0.2|\n|-|-|-|-|\n|GA|0.967±0.015|0.970±0.006|0.963±0.010|\n|COMs|0.956±0.024|0.925±0.034|0.952±0.013|\n|REINFORCE|0.947±0.030|0.933±0.025|0.930±0.042|\n|ROOT|**0.969±0.016**|**0.971±0.007**|**0.965±0.014**|\n\n>Provide results on at least one truly high‑dimensional continuous task (e.g., 1000‑dim design) to assess scalability.\n\nIn the Design-Bench benchmark, the Hopper task represents a high-dimensional continuous task with 5,126 input dimensions. Although prior work has noted a highly noisy and inaccuracy oracle function for this task, we conducted a small experiment to evaluate ROOT’s performance on Hopper as a test of scalability. As shown in the table below, ROOT outperforms some representative baselines, demonstrating its ability to effectively handle high-dimensional design spaces.\n\n|Method|Hopper Controller|\n|-|-|\n|GA|-0.068±0.001|\n|MINs|0.267±0.350|\n|Reinforce|-0.009±0.067|\n|**ROOT**|**0.541±0.042**|\n\n>Describe how sensitive performance is to the choice of Brownian‑bridge parameters and whether other kernel choices were tried.\n\nWhile we primarily tune the hyperparameters of the synthetic data generation process as detailed in Appendix C.7, the Brownian bridge parameters we use are either inherited from the Brownian Bridge Diffusion Model (BBDM) or set to standard values (e.g., learning rate, batch size, dropout). To further assess the ROOT's sensitivity, we conducted an additional experiment on ANT with some BBDM hyperparameters. As shown in the tables below, ROOT's performance remains stable, indicating robustness to the choice of Brownian bridge parameters.\n|Dropout|0.05|0.1|0.15|0.2|\n|-|-|-|-|-|\n|ROOT|0.957±0.020|0.960±0.014|**0.965±0.014**|0.957±0.019|\n\n|Batch size|32|64|128|\n|-|-|-|-|\n|ROOT|0.956±0.014|**0.965±0.014**|0.955±0.008|\n\nAdditionally, we experimented with the Matern kernel for the GP-based synthetic data generation. The results indicate that the commonly used RBF kernel consistently yields better performance, supporting our decision to use it as the default.\n|GP kernel|Ant|TFBind8|\n|-|-|-|\n|Matern|0.966±0.013|0.848±0.055|\n|RBF|0.965±0.014|0.986±0.007|\n\n>A detailed comparison with very recent diffusion‑based baselines such as DiGBO or DM‑BO would strengthen the empirical claim.\n\nWe would greatly appreciate if the reviewer could provide us with paper titles for DiGBO and DM-BO because we cannot find these methods within the context of offline optimization. Our online searches did not return any matching results.\n\nWe hope our responses have fully addressed your concerns. If you have any further questions or points that require additional clarification, please feel free to let us know. We would be happy to continue the discussion and provide any additional details."}, {"review_id": "90DAcVXpoZ", "reviewer": "Reviewer_hCxA", "summary": "This paper focuses on offline black-box optimization, which aims to maximize an objective function solely given an offline dataset. Previous works built mapping either from $x$ to $y$ or from $y$ to $x$, while this paper rethinks the optimization as from $x_{low}$ to $x_{high}$ (distributional translation as mentioned in the paper), which can be modeled by a probabilistic bridge model (initiated as a diffusion model). It utilizes Gaussian processes to derive synthetic data for diffusion training, then obtain the final solutions given solutions for the offline dataset after the model is trained.", "questions": "- Why do you conduct gradient ascent / descent on the mean functions of GP to obtain the synthetic data? As GP not only models the value but also quantifies the uncertainty. If the offline dataset also has limited coverage, only focusing on mean functions (without integrating the uncertainty of GP) could also lead to unfavorable data. Thus, I suggest comparing constructing synthetic data on UCB or LCB acquisition functions.\n- In Introduction and Section 5, you mention that previous methods remain constrained by limited data coverage. Why does your method address this challenge? More discussions are needed.\n- How does your proposed method perform under different data scarcities (not only the few-shot settings in Table 4)?", "limitations": "See Weaknesses and Questions above. I am willing to increase my score once my concerns are addressed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n- The usage of diffusion models in offline optimization is novel.\n- The modeling of mapping between $x$s **with different qualities** is intuitive.\n- The experimental evaluations are comprehensive and clearly demonstrate the effectiveness of this paper.\n- The implementation code is clear.\n\n## Weaknesses\n\n- Based on my comprehension, constructing a mapping between different $x$s is also been employed by recent works [1-2] (although they do not project from $x_{low}$ to $x_{high}$). Relevant discussion is needed.\n\n## References\n\n[1] From Function to Distribution Modeling: A PAC-Generative Approach to Offline Optimization. arXiv 2024.\n\n[2] Design Editing for Offline Model-based Optimization. TMLR 2025.", "was_revised": true, "final_justification": "The proposed method is technically solid and intuitive. Other reviewers' concerns on the lack of theory is a common issue of this community. The thorough ablation studies examines the effectiveness of ROOT. Overall, I would recommend accepting this paper, which contributes a lot to offline optimization community.", "rebuttal": "We would like to thank the reviewer for the detailed feedback and address the reviewer's questions as follows:\n\n>Based on my comprehension, constructing a mapping between different $x$s is also been employed by recent works [1-2] (although they do not project from $x_{low}$ to $x_{high}$). Relevant discussion is needed.\n\nWe appreciate the reviewer’s suggestion and will include the discussion below of these relevant works in the related work section of the final version. \n\nRecent diffusion-based methods such as [1] and [2] also explore mappings between different input designs, albeit with different goals and mechanisms. DEMO [2] trains a diffusion model to approximate the distribution of offline data, which is then used to edit perturbations from high-value surrogate designs to safer alternatives. In contrast, ROOT introduces a probabilistic bridge framework that explicitly maps low-value inputs to high-value regimes. ROOT also significantly outperforms DEMO across multiple benchmarks, as reported in Table 1.\n\nThe proposed method in [1] alternatively constructs a utility-weighted distribution by learning a data-dependent weight function, then trains a DDPM to simulate this distribution from an isotropic Gaussian, enabling direct sampling of high-value designs with a PAC guarantee that provably guards against poor average-case performance. In contrast, our low-to-high approach empirically simulates transitions from low- to high-value distributions through synthetic data generation. This strategy targets the best achievable average-case performance by progressively biasing generation toward high-value regions. Interestingly, the PAC-guided reweighting strategy from [1] could be integrated into our framework to enhance reliability, offering a principled mechanism to regularize the sampling trajectory and mitigate potential failure cases, especially in future problem domains with significantly higher complexity than the tested benchmarks. Although this integration is beyond the scope of the current work, we view it as a promising direction for future research. \n\nWe sincerely thank the reviewer for introducing this valuable contribution to our attention. We will cite [1] in our revision.\n\n>Why do you conduct gradient ascent / descent on the mean functions of GP to obtain the synthetic data? As GP not only models the value but also quantifies the uncertainty. If the offline dataset also has limited coverage, only focusing on mean functions (without integrating the uncertainty of GP) could also lead to unfavorable data. Thus, I suggest comparing constructing synthetic data on UCB or LCB \n\nWe appreciate the reviewer’s insightful comment. It is indeed true that relying solely on the mean function of a single GP trained on limited data may result in suboptimal synthetic data. However, our approach addresses this limitation by employing a large ensemble of GPs with diverse kernel hyperparameters, rather than a single model. This ensemble is constructed by uniformly sampling hyperparameters from a fixed range, which implicitly captures a broad spectrum of uncertainty within the offline dataset. This strategy is also motivated by ExPT (for few-shot offline optimization), which demonstrated strong empirical performance.\n\nWe also agree that incorporating explicit uncertainty via acquisition functions like UCB or LCB is a promising alternative, especially in balancing exploration and exploitation. To evaluate this, we conducted a small-scale experiment replacing the GP’s mean function with UCB and LCB scores to generate synthetic data. The results, shown in the table below, indicate that while UCB and LCB perform reasonably well, they do not outperform our original approach based on the GP’s mean function. We believe this is because uncertainty has already been implicitly captured through a different mechanism, by sampling multiple mean functions from a population of posterior GPs trained on the same offline dataset using different kernel configurations.\n\n| Method | Ant           | TFBind8       |\n|--------|---------------|---------------|\n| UCB    | 0.964 ± 0.017 | 0.966 ± 0.017 |\n| LCB    | 0.949 ± 0.008 | 0.961 ± 0.025 |\n| ROOT   | **0.965 ± 0.014** | **0.986 ± 0.007** |\n\n\n>In Introduction and Section 5, you mention that previous methods remain constrained by limited data coverage. Why does your method address this challenge? More discussions are needed.\n\nIn previous methods, performances are often limited by the quantity and quality of offline data, particularly when such datasets are sampled predominantly from low-value regions of the design space. Approaches such as forward modeling, inverse modeling, and learning search policies rely solely on the available offline data, which constrains their performance. In contrast, our approach addresses this challenge by generating synthetic data through Gaussian processes, effectively expanding the dataset as needed by producing additional samples (from similar functions). This aligns with the ultimate goal of offline optimization, which is to identify improved designs in regions surrounding the offline data where most functions consistent with the available evidence are also likely to agree. Empirical results demonstrate that ROOT successfully leverages this expanded synthetic data, achieving state-of-the-art performance across various benchmarks. We will add this clarification to the key discussion at lines 42-57 in the introduction.\n\n>How does your proposed method perform under different data scarcities (not only the few-shot settings in Table 4)?\n\nWe conducted additional experiments to evaluate the performance of our method under varying levels of data scarcity beyond the few-shot setting in Table 4. Specifically, we simulated limited data coverage by using only the $p$% poorest-performing designs from the offline dataset. As shown in the table below, ROOT consistently outperforms other baselines across different levels of data scarcity, demonstrating its robustness even in more challenging, low-coverage scenarios. We note that data-scarcity setting is however not our focus in this work. \n\n| TFBind8 ($p$%) | ROOT           | GA             | COMs           | REINFORCE      | LTR            |\n|-|-|-|-|-|-|\n| 50 %         | **0.964 ± 0.015** | 0.580 ± 0.199  | 0.935 ± 0.052  | 0.915 ± 0.039  | 0.959 ± 0.022  |\n| 20 %         | **0.946 ± 0.045** | 0.480 ± 0.218  | 0.872 ± 0.085  | 0.917 ± 0.040  | 0.927 ± 0.033  |\n| 10 %        | **0.915 ± 0.019** | 0.559 ± 0.170  | 0.771 ± 0.128  | 0.913 ± 0.038  | 0.909 ± 0.034  |\n\n| Ant ($p$%) | ROOT           | GA             | COMs           | REINFORCE      | LTR            |\n|-|-|-|-|-|-|\n| 50 %     | **0.909 ± 0.012** | 0.394 ± 0.023  | 0.898 ± 0.035  | 0.317 ± 0.016  | 0.909 ± 0.042  |\n| 20 %     | **0.930 ± 0.023** | 0.663 ± 0.065  | 0.880 ± 0.027  | 0.261 ± 0.052  | 0.871 ± 0.059  |\n| 10 %     | **0.861 ± 0.051** | 0.619 ± 0.120  | 0.845 ± 0.041  | 0.281 ± 0.034  | 0.813 ± 0.026  |\n\nWe hope our responses have fully addressed your remaining concerns, and we would sincerely appreciate your consideration in raising your score if no further issues persist. If you have any further questions or points that require additional clarification, please feel free to let us know. We would be happy to continue the discussion and provide any additional details."}, {"review_id": "MUXgDqXS6Z", "reviewer": "Reviewer_fMmR", "summary": "This paper proposes ROOT, a new framework for offline black-box optimization that formulates the problem as a probabilistic bridge between a support distribution (from offline data) and an unknown optimal distribution. The authors frame offline optimization as learning a time-indexed trajectory in distribution space, connecting the initial offline data distribution to the target high-value design distribution. ROOT leverages variational inference to approximate this bridge and instantiates a diffusion-like architecture to model the intermediate transitions. The authors evaluate ROOT on offline molecule and biological sequence optimization tasks and compare against existing optimization baselines.", "questions": "- How robust is ROOT to poor offline data coverage? Have you tested performance under varying dataset support quality?\n\n- Can you clarify the computational cost of ROOT compared to other diffusion-based energy-based approaches?\n\n- Are there formal convergence or generalization guarantees that can support the ROOT framework theoretically?", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The idea of framing offline optimization as a probabilistic distributional bridge is creative and provides a new generative modeling perspective on offline design problems.\n\n- The ROOT architecture is flexible and can incorporate different score functions (e.g., predictive surrogates) to guide optimization, making it potentially extensible across various problem domains.\n\n- ROOT performs competitively on molecule and sequence tasks, demonstrating that the proposed formulation is viable in practice.\n\nWeaknesses\n- Theory–practice gap: The proposed variational formulation and bridge process are theoretically interesting but lack formal results (e.g., guarantees, convergence bounds).\n\n- Architectural details underexplained: Some implementation choices (e.g., score function choices, transition structure) would benefit from more ablation or motivation.", "was_revised": true, "final_justification": "The authors effectively addressed most of my concerns during the rebuttal phase, so I raised my score.", "rebuttal": "We really appreciate the reviewer’s thorough feedback with a positive rating, and would like to address the remaining concerns as follows:\n>Theory–practice gap: The proposed variational formulation and bridge process are theoretically interesting but lack formal results (e.g., guarantees, convergence bounds).\n\nand \n\n> Are there formal convergence or generalization guarantees that can support the ROOT framework theoretically?\n\nWe appreciate the reviewer’s concern regarding the absence of formal guarantees or convergence bounds. This is indeed a limitation shared by standard diffusion-based approaches and remains an open challenge in the community. While we agree that developing theoretical foundations is important, this is a complex and ongoing area of research. A promising direction for future work is to relate the optimization trajectory of the oracle function to the underlying dynamics of the denoising diffusion process, though doing so will require substantial additional investigation.\n\nWe note that most state-of-the-art offline optimization methods similarly lack formal guarantees, reflecting the broader difficulty of establishing theoretical bounds in this setting. Nonetheless, our method delivers strong empirical performance across multiple benchmarks and consistently compares favorably with existing approaches. We believe this demonstrates the practical utility of our framework, even in the absence of formal guarantees at this stage. We also note that our empirical evaluation includes a diverse and extensive set of baselines. We believe this comprehensive study strongly supports the practical significance of our work.\n\n>Architectural details underexplained: Some implementation choices (e.g., score function choices, transition structure) would benefit from more ablation or motivation.\n\nWe appreciate the reviewer’s comment and would like to clarify these implementation choices. For the score network, we follow the architecture used in DDOM [1], with a few modifications: we employ a simple MLP with a hidden size of 1024 and use 4 layers with Swish activation, compared to DDOM’s original 2-layer ReLU setup. Regarding the transition structure, it is determined by the specific choice of the probabilistic bridge. In our case, we adopt the Brownian bridge as a practical instantiation of our general framework. The corresponding transition formulation, training procedure, and hyperparameter settings are provided in Appendix B.2. \n\n[1] Krishnamoorthy, Siddarth, Satvik Mehul Mashkaria, and Aditya Grover. \"Diffusion models for black-box optimization.\" In International Conference on Machine Learning, pp. 17842-17857. PMLR, 2023.\n\n>How robust is ROOT to poor offline data coverage? Have you tested performance under varying dataset support quality?\n\nWe evaluated ROOT’s efficacy for poor offline data coverage in the Few-Shot Experimental Design setting introduced by ExPT [2], where only 1% of the offline data is labeled and the remaining 99% is unlabeled. As shown in Table 4, ROOT performs effectively under this limited supervision scenario and even outperforms ExPT.\nAdditionally, we conducted further experiments to assess ROOT’s robustness under varying dataset support quality. Specifically, we trained ROOT and other baselines using only the $p$% poorest-performing designs from the offline dataset. The results, shown in the table below, demonstrate that ROOT continues to outperform the baselines, even under these more challenging conditions. We however note that data-scarce setting is not the main focus of our work.\n\n| TFBind8 ($p$%) | ROOT           | GA             | COMs           | REINFORCE      | LTR            |\n|-|-|-|-|-|-|\n| 50          | **0.964 ± 0.015** | 0.580 ± 0.199  | 0.935 ± 0.052  | 0.915 ± 0.039  | 0.959 ± 0.022  |\n| 20          | **0.946 ± 0.045** | 0.480 ± 0.218  | 0.872 ± 0.085  | 0.917 ± 0.040  | 0.927 ± 0.033  |\n| 10          | **0.915 ± 0.019** | 0.559 ± 0.170  | 0.771 ± 0.128  | 0.913 ± 0.038  | 0.909 ± 0.034  |\n\n| Ant ($p$%) | ROOT           | GA             | COMs           | REINFORCE      | LTR            |\n|-|-|-|-|-|-|\n| 50      | **0.909 ± 0.012** | 0.394 ± 0.023  | 0.898 ± 0.035  | 0.317 ± 0.016  | 0.909 ± 0.042  |\n| 20      | **0.930 ± 0.023** | 0.663 ± 0.065  | 0.880 ± 0.027  | 0.261 ± 0.052  | 0.871 ± 0.059  |\n| 10      | **0.861 ± 0.051** | 0.619 ± 0.120  | 0.845 ± 0.041  | 0.281 ± 0.034  | 0.813 ± 0.026  |\n\n[2] Tung Nguyen, Sudhanshu Agrawal, and Aditya Grover. Expt: Synthetic pretraining for few-shot experimental design. Advances in Neural Information Processing Systems, 36:45856–45869, 2023.\n\n\n\n>Can you clarify the computational cost of ROOT compared to other diffusion-based energy-based approaches?\n\nWe would like to analyze the detailed computational cost along with the running time of ROOT compared to another diffusion-based method, DEMO [3]. \n\nComputational cost of ROOT is as follows:\n- For each training epoch: Fitting $n_f$ Gaussian processes requires $O(n_f n^3)$ where $n$ is the offline data size. Performing M gradient steps (querying the GP’s mean function) to generate $n_p$ synthetic points from each of $n_f$ functions requires $O(n_f M n_p n)$. Training the BBDM model requires T diffusion steps per sample, where each step involves a forward and backward pass through the score network $\\epsilon_\\theta$, giving a total cost of $O(T (f_\\theta + b_\\theta) n_f n_p)$, where $f_\\theta, b_\\theta$ denotes the cost of forward and backward, respectively. \n- For the whole training process, the overall cost is $O(E(n_f n^3 + n_f M n_p n + T (f_\\theta + b_\\theta) n_f n_p))$ where $E$ is the number of epochs. \n- For the inference process, the overall cost is $O(Q D f_\\theta)$ where $Q$ and $D$ are the number of selected candidates and denoising steps.\n\nTherefore, the total computational cost of ROOT is: \n$O(E(n_f n^3 + n_f M n_p n + T (f_\\theta + b_\\theta) n_f n_p)) + O(Q D f_\\theta)$\n\nFurthermore, we note that the most complexity term in ROOT's computational cost, GP fitting, can be further mitigated by using sparse GP techniques [4], which scales linearly in the number of inputs.\n\n--\n\nCorrespondingly, the computational cost of DEMO is as follows:\n- Fitting a surrogate model: $O(E_f t_f n)$ where $E_f$ is the number of surrogate training epochs, $t_f$ is the cost of forward and backpropagation on the surrogate model.\n- Creating pseudo candidates by gradient ascent on the trained surrogate model: $O(Q M t_f)$\n- Training of diffusion prior: $O(E_d T (f_\\theta + b_\\theta) n)$ where $E_d$ is the number of diffusion training epochs. \n- Design Editing Process: $O(Q D f_\\theta)$\n\nThe total computational cost of DEMO is:\n$O(E_f t_f n) + O(Q M t_f) + O(E_d T (f_\\theta + b_\\theta) n) + O(Q D f_\\theta)$\n\nUnder the same hyper-paramters, model size and method for training and inference, ROOT might have theoretically higher computational cost than DEMO. This is however not critical as ROOT still scales well in all tested data benchmark compared to other baselines in Table 10 in the Appendix. As noted above, we can further reduce the complexity cost of ROOT via using sparse Gaussian processes in future work.\n\nFurthermore, ROOT empirically demonstrates significantly faster runtime compared to DEMO, as reported in the table below. This is because DEMO requires training both the surrogate model and the diffusion prior on the offline data for $E_f = 200$ and $E_d = 200$ epochs, respectively, compared to a single training phase of $E = 100$ epochs on small set of synthetic data of ROOT. Additionally, the running time gap also comes from the difference in training and inference method. While ROOT employs a fast DDPM-based training and only $D = 200$ steps of efficient DDIM sampling [5], DEMO use a slower SDE-based training and $D = 400$ sampling steps using a complex second-order Heun solver. \n\n| Method | Ant   | Dkitty | TFBind8 | TFBind10 |\n|-|-|-|-|-|\n| DEMO         | 668s | 1489s  | 1024s   | 1696s    |\n| ROOT         | 298s  | 297s   | 407s    | 575s    |\n\nWe will update the above detailed computational analysis to the appendix.\n\n[3] Ye Yuan, Youyuan Zhang, Can Chen, Haolun Wu, Melody Zixuan Li, Jianmo Li, James J. Clark, and Xue Liu. Design editing for offline model-based optimization. Transactions on Machine Learning Research, 2025\n\n[4] Quinonero-Candela, Joaquin, and Carl Edward Rasmussen. \"A unifying view of sparse approximate Gaussian process regression.\" Journal of machine learning research 6, no. Dec (2005): 1939-1959.\n\n[5] Song, Jiaming, Chenlin Meng, and Stefano Ermon. \"Denoising diffusion implicit models.\" arXiv preprint arXiv:2010.02502 (2020).\n\nWe hope that our responses have satisfactorily addressed your remaining concerns. If you have any further questions or points that require additional clarification, please feel free to let us know. We would be happy to continue the discussion and provide more details."}, {"review_id": "b1SpibTyoI", "reviewer": "Reviewer_tDV4", "summary": "This paper tackles offline black-box optimization by proposing a novel framework that treats the task as a distributional translation problem—from low-performing to high-performing inputs. Instead of relying solely on scarce offline data, it learns a probabilistic bridge using synthetic functions, constructed as averaged Gaussian processes fit to the data. This approach improves generalization and achieves state-of-the-art performance on benchmark tasks.", "questions": "See Strengths And Weaknesses", "limitations": "See Strengths And Weaknesses", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: The idea of leveraging synthetic data to learn a mapping from low- to high-performing inputs is compelling. The results are strong, and I particularly appreciate the focus on addressing the core challenge of offline black-box optimization—data scarcity. This paper offers a promising direction to mitigate that limitation.\n\nWeaknesses: One potential concern is that the synthetic functions may not closely reflect real-world tasks, such as biological sequence design. Additionally, the paper appears to overlook a recent and relevant survey in this area: Offline Model-Based Optimization: Comprehensive Review (https://arxiv.org/abs/2503.17286).", "was_revised": true, "final_justification": "i recommend accept as the author addresses majority of my concerns.", "rebuttal": "We truly appreciate the reviewer's thoughtful feedback and acceptance rating. We would like to address the reviewer's questions below:\n\n>One potential concern is that the synthetic functions may not closely reflect real-world tasks, such as biological sequence design\n\nWe thank the reviewer for this thoughtful observation. We agree that Gaussian process (GP) priors may not fully capture the complexity of biological fitness landscapes, which can exhibit non-smooth behaviors that GPs are not well suited to model. That said, our use of GP-based synthetic functions is not intended to simulate the true oracle function in its entirety, but rather to approximate its local behavior within the region surrounding the offline data. By sampling a diverse set of posterior mean functions from GPs conditioned on real observations, we aim to reflect a broad range of plausible oracle behaviors near the data manifold. While this does not guarantee global fidelity to the true oracle, it provides a data-driven and computationally efficient means of reliably identifying improved designs within the local region supported by the offline data.\n\nTo further validate the practical applicability of ROOT in real-world scenarios, we evaluated it on a biological sequence design task (RNA-binding), as shown in Table 2. The strong performance in this setting suggests that ROOT remains effective even when the true oracle is not fully consistent with a GP prior, highlighting its robustness in real-world applications.\n\n>Additionally, the paper appears to overlook a recent and relevant survey in this area: Offline Model-Based Optimization: Comprehensive Review (https://arxiv.org/abs/2503.17286).\n\nThank you for bringing this valuable survey to our attention. We will revise the paper to include a discussion and citation of this in the related work section to ensure comprehensive coverage of key developments in the offline optimization literature.\n\nWe hope that our responses have addressed your remaining questions satisfactorily. If you have any questions or points that require additional clarification, please feel free to let us know. We would be happy to continue the discussion and provide any additional details."}]

Author rebuttal:

Meta-review: This paper addresses black-box optimization with limited offline data and proposes a novel approach that formulates the problem as a distributional translation task. The method combines surrogate optimization with inverse model estimation. Specifically, the authors train a probabilistic bridge—conceptually inspired by diffusion but conditioned on both source and target points—using synthetic low- and high-value design pairs generated from an ensemble of Gaussian-process posterior means fitted to the offline data. Once trained, this bridge is run backward from the best offline samples to generate improved candidates.

Most reviewers agreed that the paper merits acceptance. After carefully reviewing the rebuttal and subsequent discussion, I concur. Nevertheless, I encourage the authors to address the reviewers’ feedback in the final version of the paper, with particular emphasis on:

Including experiments that consider high-dimensional and noisy data.

Citing the missing related works and providing empirical comparisons with some of them, as suggested by the reviewers.
Decision: Accept (spotlight)