In-Context Multi-Objective Optimization
Authors: Xinyu Zhang
Conor Hassan
Julien Martinelli
Daolang Huang
Samuel Kaski
Venue: ICLR 2026 Poster (2026)
Keywords: Multi-Objective Optimization
Bayesian Optimization
Transformers
Neural Processes
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We introduce a fully amortized (surrogate model + acquisition function), dimension-agnostic policy for multi-objective optimization.

Abstract: Balancing competing objectives is omnipresent across disciplines, from drug design to autonomous systems. Multi-objective Bayesian optimization is a promising solution for such expensive, black-box problems: it fits probabilistic surrogates and selects new designs via an acquisition function that balances exploration and exploitation. In practice, it requires tailored choices of surrogate and acquisition that rarely transfer to the next problem, is myopic when multi-step planning is often required, and adds refitting overhead, particularly in parallel or time-sensitive loops. We present TAMO, a fully amortized, universal policy for multi-objective black-box optimization. TAMO uses a transformer architecture that operates across varying input and objective dimensions, enabling pretraining on diverse corpora and transfer to new problems without retraining: at test time, the pretrained model proposes the next design with a single forward pass. We pretrain the policy with reinforcement learning to maximize cumulative hypervolume improvement over full trajectories, conditioning on the entire query history to approximate the Pareto frontier. Across synthetic benchmarks and real tasks, TAMO produces fast proposals, reducing proposal time by 50–1000× versus alternatives while matching or improving Pareto quality under tight evaluation budgets. These results show that transformers can perform multi-objective optimization entirely in-context, eliminating per-task surrogate fitting and acquisition engineering, and open a path to foundation-style, plug-and-play optimizers for scientific discovery workflows.

Published as a conference paper at ICLR 2026
IN-CONTEXT MULTI-OBJECTIVE OPTIMIZATION
Xinyu Zhang 12, Conor Hassan 1.2 Julien Martinelli 12 Daolang Huang 1.2 Samuel K
*Department of Computer Science, Aalto University, Finland "ELLIS Institute Finland
3 Department of Computer Science, University of Manchester, UK
xinyu.2 2. zhang@aalto.ff
ABSTRACT
Balancing competing objectives is omnipresent across disciplines, from drug design to autonomous systems. Multi-objective Bayesian optimization is a promising solution for such expensive, black-box problems: it fits probabilistic surrogates and selects new designs via an acquisition function that balances exploration and exploitation. In practice, it requires tailored choices of surrogate and acquisition that rarely transfer to the next problem, is myopic when multi-step planning is often required, and adds refitting overhead, particularly parallel or senessenii loops. We present TAMO, a fully amortized, universal policy for multi-objectiv black-box optimization. TAMO uses a transformer architecture that operates across varying input and objective dimensions, enabling pretraining on diverse corpora and transfer to new problems without retraining: at test time, the pretrained model proposes the next design with a single forward pass. We pretrain the policy with reinforcement learning to maximize cumulative hypervolume improvement over full trajectories, conditioning Pareto frontier. Across synthetic benchmarks and real tasks, TAMO produces fast proposals, reducing proposal time by 50-1000x versus alternatives while matching or improving Pareto quality under tight evaluation budgets. These results show that transformers can perform multi-objective optimization entirely in-context, eliminating per-task surrogate fitting and acquisition engincering, and open a path to foundation-styy plug-and- -ad---ad optimizers for scientific discovery workflows.
the entire query history approximate the
INTRODUCTION
Multi-objective optimization (MOO; Deb et al., 2016; Gunantara, 2018) is ubiquitous in science and engineering: practitioners routinely balance accuracy cost in experimental design (Schoepfer et al., 2024), latency vs. quality in adaptive streaming controllers (Peroni & Gorinsky, 2025), or efficacy vs. toxicity in drug discovery (Fromer & Coley, 2023; Lai et al., 2025). In these settings, each evaluation of the black-box objectives can be slow or costly, making sample efficiency paramount; the goal is to obtain high-quality approximations of the Pareto front with minimal number of queries.
The standard sample-efficient paradigm for such problems is Multi-objective Bayesian optimization (MOBO; Garnett, 2023): fit probabilistic surrogates for each objective, typically using Gaussian processes (GPs; Rasmussen & Williams, 2006), then select the next query by maximizing an acquisition that balances exploration-exxloitttion to efficiently improve a chosen multi-objective utility, such as hypervolume, scalarizations, or preference-basee Belakaria et al., 2019; Daulton et al., 2023b). While effective, this recipe has three drawbacks in real-world use. First, each new problem requires training surrogates from scratch and repeatedly optimizing the acquisition, adding non-trivial GP overhead that can bottleneck decision latency in parallel or tüme-sensitive settings. Second, performance critically depends modeling choices (kernel, likelihood, acquisition, initialization). especially when data are scarce, a setting MOBO is intended to handle. Third, most acquisitions myopic, optimizing one-step gain, which can be suboptimal when Pareto-from discovery requires multi-step planning.
criteria (Daulton et al., 2020;
Amortized optimization (Finn et al., 2017; Amos et al., 2023) addresses these issues by shifting computation offline. The idea is to pre-train on a distribution of related optimization tasks, either generated synthetically or drawn from real, previously solved datasets. At test time, proposing new

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Previous Methods
TAMO (ours)
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Figure 1: Comparison of multi-objective optimization workflows. (Top left) Previous methods like traditional MOBO or acquisition-only amortized BOFormer (Hung et al., 2025) bottlenecked slow process of fitting a GP surrogate. (Top right) TAMO is fully amortized: a ciimension-agnosee transformer policy trained once, offline, on diverse synthetic tasks, and deployment maps the history to the next query in single forward pass.
design then reduces to single forward pass. Recent efforts have explored methods for amortized Bayesian optimization (Volpp et al., 2020; Chen et al., 2022; Maraval al., 2023; Zhang et al., 2025; Hung al., 2025), but few address the multi-objective setting. For instance, Hung al. (2025) only amortizes the acquisition function calculation while still relying on a GP surrogate, and its pretrained model is tied to fixed number of objectives, which prevents transfer across heterogeneous tasks. A method that tackles these challenges would let practitioners pool heterogeneous legacy datasets for pretraining, resulting in improved outcomes in scarce-data regimes. It would also enable reusing sin- gle optimizer as design spaces and objective counts change, and issue instant proposals in closed-loop laboratories, high-throughput campaigns, reducing overhead when evaluations are cheap or parallel.
Contributions.
We introduce TAMO, fully amortized policy for multi-objective optimization that maps the observed history directly to the next query (Figure 1). Training uses reinforcement learning to optimize hypervolume-oriente utility over entire trajectories, encouraging morizoo rather than one-step gains. At inference, proposals are produced by a single forward pass.
TAMO is dimension agnostic both inputs and outputs: introduce transformer architec- embedder that jointly represents all input features
ture with a novel cimeension-aggrgaall and objective values regardless of dimensionality. This enables pretraining on heterogeneous tasks, synthetic or drawn from real meta-datasets and transfer to new problems without retraining. To our knowledge, this is the first end-to-end, cameeniionaggnosee architecture for black-box optimization, let alone MOO (Figure 1, bottom).
We evaluate TAMO synthetic and real multi-objective tasks, observing 50x-1000x lower wall-clock proposal time than GP-based MOBO and baselines such as BOFORMER, which amortizes the acquisition but still relies task-specific surrogates, while matching Pareto quality and sample efficiency. We further provide an empirical assessment of the generalization capabilities of TAMO, along with its sensitivity to deployment knobs.
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2 PRELIMINARIES
Multi-objective Optimization. Consider a multi-ohjective optimization problem in which one aims to optimize a function f(x) = (fr) [a,(a)) E IR4, and observations y = f + e where X C I is a compact search space. In many practical scenarios, it is not possible to find a single design æ that optimal for all objectives simultaneously. Instead, the notion of Pareto dominance is used to compare objective vectors. An objective vector f(x) Pareto-dominates another vector (a'), denoted f()) > fax', ), if f(m) (ax) 2 (xm) Vm E (1,..., ...... and there exists at least one objective m'' such that f(m') > flm)(a'). The Pareto fromtier (PF) associated with set of designs X C X is P(X) = {f(x) x E X, #x' E X s.t. f(x') > f(z)}. A common goal in multi-objective optimization is to approximate the global frontier P(X') within a limited budget of ' function evaluations. One popular way assess solution quality is the hypervolume (HV) indicator. For a reference point r E R44, the hypervolume HV(P(X) (X) r) measures how much of the objective space between r and the frontier P(X) is "covered" by Pareto-optimal points. In practice, the choice of r' depends on domain-speciin
considerations (Yang et al.. 2019).
Reinforcement Learning. Recent work leverages RL to learn non-myopic strategies in black-box optimization, accounting for downstream impact of each evaluation (Maraval et al., 2023; Zhang et al., 2025; Hung et al., 2025). An RL problem is Markov decision process (MDP) (Sutton al., 1998) with states, actions, transition dynamics, reward encoding the optimization goal, and discount factor weighting future vs. immediate rewards. The output is policy T 8), a distribution over actions a for a state that maximizes expected discounted return. In the amortized regime, one policy is trained offline on a distribution of tasks and then deployed across new problems without retraining.
3 TASK-AGNOSTIC AMORTIZED MULTI-OBJECTIVE OPTIMIZATION (TAMO)
We introduce TAMO, fully amortized framework for multi-objective black-box optimization. TAMO encodes the optimization history and a candidate set with a transformer backbone and directly outputs acquisition utilities. To stabilize policy learning, we additionally incorporate a prediction task into our objective function. Section 3.1 details the construction of pretraining tasks for policy learning and prediction; Section 3.2 presents the TAMO architecture; Section 3.3 formalizes the RL objective and MDP; and Section 3.4 outlines training and inference procedures. Figure illustrates our workflow.
3.1 PRETRAINING DATASET CONSTRUCTION
We pre-train TAMO diverse distribution of synthetic multi-objective optimization tasks, denoted by p(T). Each task ~ p(T) is defined by black-box function f, X' C R - R where the input and output dimensions, d' and dy. vary across tasks. This heterogeneity key to learning a universal, dimension aggnosiic policy. The full generative process for p(r), which is based on GP priors with varied kernels and properties, is detailed in Appendix A.
During each training step, we sample two distinct mini-batches from task distributions to jointly optimize the model for policy learning and auxiliary prediction:
Policy-learning batches. To train the decision-making policy, each batch contains history set No D = {(ah, y² >)>> and a query set D = {a*) y The policy conditions on the history to select the most promising query from the query set.
Prediction batches. To facilitate policy learning, we include an auxiliary prediction task to help the model learn the function landscape. We sample N input-output pairs from fresh function draw. These pairs are then randomly partitioned into context set D {xx y ).N, and target N₂ set De = (xx) 1=1 on which the model performs an in-context regression task.
)
For each dataset type E {h, 9, c, p}. the individual elements are denoted as 21, where t e (1....,T) T} indexes the data point and j E (1....,d?, indexes the input dimension. Similarly, for outputs have yf where k E (I....,d?) indexes the output dimension.
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3.2 MODEL ARCHITECTURE
TAMO's architecture is designed around single, shared backbone that operates in two distinct tasks during training: the prediction task and the optimization task. Each forward pass processes one mini-batch, either a prediction batch (D(e), D( or an optimization batch (D(k), D While the input data types differ conceptually (context ++ history and target ++ query), they are processed by the same core components. The architecture comprises four parts: (i) a cimmension-agnssee embedder mapping an observation to a vector regardless of input/output dimension; (ii) transformer encoder that aggregates variable-size histories/contexts and exposes a compact summary: (iii) lightweight task conditioning via small number of tokens; and (iv) two heads: prediction head and a policy head. The dimension-agnostic embedder and the transformer encoder blocks are shared across tasks.
(I) Dimension-aggoosic embedder. We apply learnable scalar-to-vector maps ex R - Rd and Gy RR- + dR. fimension-wise resulting in e, = (xx Rd and By =e((y)= E TR xd. Both functions ez and ey are parameterized as feedforward neural networks. After L transformer layers on the concatenated tokens [ezi Ey], we apply learnable dimension-spectio
positional tokens Ps E Rd and Py RR xd. element-wise and mean--ool across the di dd token axis to obtain a single representation E E Rd. These positional tokens are randomly sampled for each batch from fixed pools of learned embeddings. We introduce the positional tokens to prevent the spurious symmetries over dimensionalities from a permutation-ine invarrian set encoder, allowing the model to distinguish between features and objectives with the same values. During training, the embedder is applied to Dh and 'D9 to yield EA and E for the oplimization task, and to De and DD to yield E and E for the prediction task. Each observation contributes ((1) tokens, so the cost scales with the number of observations, not with df. or dz.. Figure 2 summarizes the embedder.
(II) Transformer encoder-decoder. We stack == Bi + B2 transformer layers and split them into two phases. For the first B1 layers, the observed tokens interact. The history (or context) tokens undergo self-attention to produce Enh (or E capturing intra-set structure. The query (or target) tokens then use cross- attention with the keys/values provided by E (or É¹), yielding E (or É²). No task-specific tokens are present in B1. This phase is the only path through which queries/targets access information from the history/context. Then, for the last B2 layers, the sequence is reduced to only the query/target tokens together with small set of task-specific tokens (defined below). All history/context tokens are removed from the sequence. An attention mask enforces that, in these final layers, query/target tokens are permitted to attend only to the task-specific tokens (no query-query attention and no access to history/context). The task-specific tokens may self-attend among themselves.
E
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Figure 2: Dimension-agnosii em- bedder for single observation.
(III) Task-specific tokens. Task-specific tokens are introduced only at the entrance to the last B3 layers. For each task type, we introduce distinct tokens that guide the decoder's computation. For prediction tasks, the additional tokens comprise a prediction task token and the output-index positional token py optimization tasks, the additional tokens comprise an optimization task token, time-budger token
(k) indicating which scalar vi is to be predicted. For
Gtime = MLP, (TTP)(()) and an aggregated input- dimte-dimensoon token D p. An attention mask restricts query/target tokens in Bg to attend only to these tokens; the history/context never appears in B2. This design yields constant overhead in d' and dr. and linear cost in token size.
(IV) Heads. The architecture terminates in two heads that share the backbone but produce different outputs depending on the task.
N₂ Prediction head. Given the prediction tokens { [E] J=z1 and an ouiput-index positional token Py the model produces, for each prediction input ar" the parameters of a K K-component univariate Gaussian mixture that models the scalar vr,x- Concretely, an MLP applied to É, yields mixture weights (dre K E= means {1rt ) and positive scales { with weights normalized by softmax
(k)
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Ei die and scales made positive via softplus transform. The resulting predictive density is:
K
p Pl 2i, D
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Qie M pae, o
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Prediction tasks use samples disjoint from optimization tasks to prevent reward leakage.
Policy head. Given the query tokens [E:)-> the model assigns each query æ scalar acquisition
utility a; == MLPo (EE). These utilities are converted into a categorical policy over the query set using softmax:
exp(ax)
Te(2 t, H14-1)
Σ N exp(a,)
During training we sample actions from Te at inference we act greedily by selecting the next query with the largest probability. A detailed visualization of the full architecture is provided in Figure $1.
3.3 POLICY LEARNING
We cast the dimension-agnoosee optimization problem Markov decision process (MDP):
State. At step f, the state 8; = (Dh; t, T), where D is the current historical observations and T' is the total budget.
Action. The action selects candidate index i* E {l,,..,, N₂} and æ; a!!..
Reward, After querying the objectives and observing yı, we update the history D == Dh U y(), and define the normalized hypervolume level:
HV(P(D*) HV;
HV; HV(P(X) HV(P(2) 0.
Here HV; the task-wise hypervolume of the optimal frontier with respect to the fixed ref- erence point r. We set r to the componentwise worst value, r = (g(,)...,,($1)) with g(k) := min (z) which makes the hypervolume well defined and the reward bounded in [0, 1]. This ratio measures the fraction of maximum achievable hypervolume already captured by the current Pareto approximation (larger is better), and the normalization provides scale invariance across heterogeneous tasks (Teoh et al., 2025).
The policy T s) maximizes the expected discounted return:
((0) ETSp(T) E
rt
(3)
and we estimate gradients with REINFORCE (Williams, 1992):
VoJ(O) ~ E 17
Σ
V log (x)
(4)
Where R. -IT trk Since training uses synthetic tasks, rt is known exactly.
3.4 TRAINING AND INFERENCE
Training. We train TAMO in two phases. First, we warm up the backbone on the prediction task by minimizing negative log-likelihood over (D5, DD), which encourages accurate in-context regression and useful representations:
N
4
((r) -ETrpp())
log p (V.. xe, Dr)
(5)
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After warm-up transition to the joint training phase, where optimize the policy with the trajectory objective [() (Eq. 3), aligning the learning signal with improvements in Pareto quality alongside the prediction objective. The overall objective combines both terms:
( (p (0) + (0),
C ((rl) (tr)(() -J()),
(6)
and is optimized with REINFORCE (Eq. 4); the coefficient X > trades off prediction and policy signals. In all experiments, we fixed A, = 1.0. Specifically, the prediction loss ((P) and RL loss C(a) calculated from two distinct forward passes through the model with different datasets, which are then summed for a single backward pass. Training on full trajectories directly rewards long-horizon improvements, while amortization enables learning from many tasks offline.
Inference. At deployment, TAMO iteratively approximates the Pareto frontier under a budget 7. We initialize the history with random observation D <- {arß vo ) and set t +- 1. Each iteration scores the current candidate set 19 with a single forward pass and proposes:
Tt = arg max no x t,T,
mFED=
The proposed query is then evaluated, and the resulting observation is used to update the history. This process iterated until the cumulative evaluation cost meets the budget. Detailed descriptions of the algorithms for training and inference are provided in Appendix B.
4 RELATED WORK
Multi-objective Bayesian Optimization (MOBO). This line of work builds on Bayesian Opti- mization (Garnett, 2023), leveraging a combination of a statistical surrogate and an acquisition function to seek high-quality approximations to the Pareto set under tight evaluation budgets. Three families are prominent. scalarization methods convert MOBO into single-objective subproblems (e.g.. ParEGO, TS-TCH; Knowles, 2006; Paria et al., 2020), letting practitioners reuse mature BO tooling and sweep preferences in parallel. Indicator-based methods optimize typervolume-oriente criteria such as EHVI or HVKG (Daulton et al., 2020; 2023a), directly aligning the acquisition with the final Pareto- quality metric. Lastly, informattonntheorett methods (PESMO, MESMO, PFES; Hernández-Loban et al.. 2016; Belakaria al., 2021: Suzuki al., 2020) select points that maximize information gain about the Pareto set or frontier, offering a principled exploration strategy. These approaches are effective but hinge on carefully tuned, task-specific that must be refit and re-optimized at each iteration, all while remaining largely myopic. We instead learn a fully amortized policy that reduces design proposal to a single neural-network forward pass, dramatically lowering inference latency.
pairing
Amortization and meta-learning. Amortization replaces per-task inference with a model trained offline to operate in-context, exemplified by prior-data fitted transformers that achieve strong in- context performance after pretraining on large, heterogeneous datasets (Hollmann et al., 2025; Qu et al., 2025). In parallel, Conditional Neural Processes and their transformer variants learn predictors that condition on a context set and generalize via a single forward pass (Garnelo et al., 2018; Kim al., 2019; Nguyen & Grover, 2022: Chang et al., 2025), with recent work extending them to dimension-aagoosiie settings (Dutordoir et al., 2023; Lee et al., 2025). These works focus on amortizing prediction. More recently, several studies leverage in-context pretrained neural processes for sequential decision-makii (Huang et al., 2024; Zhang et al., 2025; Huang et al., 2025); our approach falls into this line as well.
Amortized black-box optimization. Several approaches train neural networks to amortize black-box optimization directly, typically by mapping histories to proposals or by predicting acquisition values, with success under scalar observations (Volpp et al., 2020; Chen et al., 2022; Yang et al., 2023; Maraval et al., 2023; Song et al., 2024; Huang et al., 2024), and even binary feedback (Zhang et al., 2025). Complementary to surrogate/acquisttinn amortization, ransfer-BO with Monte Carlo Tree Search learns the search space itself by building a data-driven hierarchy of promising subregions on source tasks and reusing it to warm-start new target before adapting online (Wang al., 2024). However, none of these methods is simultancously end-to-end (no per-task surrogate or acquisition), natively multi-objective, and capable of cross-dimensional transfer. The closest is ROFo rme (Hung

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et al., 2025), which uses sequence modeling to mitigate myopia in MOBO, but still relies on task- specific surrogates and fixed output dimensional setups, necessitating additional training when the dimension changes. We address all three by pretraining fully amortized, foundation-style policy.
5 EXPERIMENTS
We evaluate TAMO on synthetic GP tasks and standard analytic testbeds, as well as on real-world problems (Section 5.1). Subsequently. Section 5.2 studies the generalization capabilities of TAMO, e.g., with respect to unseen task dimensionalities during training both synthetic tasks and real-world problem. We conclude with several ablation studies related to the batch size and query set size employed at inference time (Section 5.3). Additional experiments can be found in Appendix E. We emphasize that a single pretrained model is used across all experiments.
Baselines. We compare against strong MOBO baselines, including decomposition and indicator- based methods (qNPaOEGO Knowles 2006, qNEHVI Daulton et al. 2020, qHVKG Daulton et al. 2023a), sequence-modelingg MOBO (BOF ormer Hung al. 2025), and a random search baseline. Baselines are tuned with their recommended defaults unless otherwise noted.
Metrics. We report performance via HV-based simple regret fixed evaluation budget. We also measure wall-clock proposal time end-to-end, which for GP-based baselines includes surrogate fitting and acquisition optimization, and for method consists of single forward pass. For single-objective, we additionally report standard simple regret.
Implementation. TAMO is implemented using PyTorch (Paszke et al., 2019). Hyperparameter settings can be found in Appendix D.3. Code is available on GitHub For all vanilla MOBO baselines, we used the implementation from the BoTorch library (Balandat al., 2020). For BOForno orerr (Hung et al., 2025), we used the publicly available implementation and pretrained model from its official code repository. To ensure a fair comparison, domain size (i.e., the size of the candidate query set) during testing is set to 2048, consistent with the configuration used for TAMO.
5.1 SYNTHETIC AND WORLD TASKS
Synthetic examples. On synthetic MOO testbeds (details in Section D.2), TAMO attains competitive or better simple regret across the entire budget (Figure 3). On GP- which in-distribution for all methods (30 GP draws), TAMO performs on par with the best GP baselines. On the remaining three problems, out-of-distributiie for all baselines, TAMO yields the strongest performance, except on Branin-Currin where NEHVVI and gNParEGO do better. We hypothesize this gap stems from the
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objectives in being well described by long length scales, outside the reach of our pretraining corpus: synthetic GP samples using lengthscales ~ (2/33, 0.5) over 5-5,5 (Section D.3). Lastly, our method can also be applied effortlessly to single-objective BO, yielding competitive results compared to other GP-based alternatives (Figure S2).
Real-world example. We compare our model, pretrained only with synthetic GP samples, with other baselines on the real-world oil sorbent multi-objective problem (Daulton al., 2022). The result is shown in Figure 3. TAMO remains competitive with GP-based alternatives, yielding the best performance, closely followed by qNParEGO.
Wall-clock time. Nevertheless, the primary advantage is speed: cumulative inference time is lower by roughly 50x-1000x, growing slowly with budget because each proposal is a single forward pass. By contrast, GP-based methods incur substantial overhead from repeated surrogate refits and acquisition optimization. Even BOF omer, r, which amortizes the acquisition but still relies on a GP surrogate, remains noticeably slower than TAMO.
5.2 GENERALIZATION
We investigate the generalization capabilities of TAMO in two different test-time scenarios: unseen dimensionalities, or decoupled observations.
hut-of-distribution dimensionalities. We test moossddimeniional transfer by pretraining TAMO on GP tasks with d= E {1. 2} and dy {1, 2. 3), then evaluating on (i) GP-DX3-DD DY2 and GP-DX3-DY -Y3, and (ii) the real-world LaserPlasma task (d_ =4, dy = 3; Section D.2), all with unseen input/output dimensionalities. On the synthetic OOD settings (Figure 4a, left, middle), TAMO attains regret comparable to the strongest GP baselines across the budget; even with 60 repetitions, we not observe statistically decisive differences. On LascrPlasma (Figure 4a. right), TAMO improves BOF ormer (which amortizes only the acquisition) but trails conventional MOBO baselines in regret. Across all cases, TAMO retains
advantages in cumulative inference time.
Decoupled observations, We next test generalization to decoupled settings, where objectives can be measured independently, common setting when jointly observing all objectives is infeasible or costly, also arising when historical logs contain partial objective labels. Budget --100 with cost per objective: full evaluation costs dy, single-objective probe costs 1. Hence, coupled
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policy can do at most T/dy full evals, while decoupled one take up to T single-objective measurements. Figure 4b plots regret vs. cumulative cost. On GP-DX2-DY2 DY2, Sckley-Rastrige and Branin-Currin, the decoupled variant of TAMO closely tracks the coupled policy, indicating that TAMO can accommodate partial-feedbae acquisition without retraining, offering a flexible trade- off between measurement cost and optimization progress. The exception is Ackley- Sokley-Rosenbrock where decoupling hurts performance, likely because the objectives peak at disparate locations, so single-objective measurements transfer poorly and bias the search toward one goal.
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TAMO (q-5)
TAMO (g-10)
Figure 5: Effeec of batch size synthetic problems: simple regret for TAMO with q {1, 10). Curves show means with 95% CIs over 30 runs. Smaller y converges fastest; larger incurs a mild slowdown, compatible with wall-clock savings for parallel evaluations.
5.3 ABLATION STUDY
We examine TAMO'S s sensitivity to deployment knobs: batch size (g) and query set size (N₂), and quantify the accuracy-latency trade-offs they induce.
Batch size. We compare TAMO with q 10). For q > we form batches via light fantasy loop (Chang et al., 2022): pick at argmaxx/eD=-To(tt T, D"), predict provisional outcome Dt. augment D + D* U (x,, (t). and repeat until points are chosen (i.c., q forward passes), Across all problems, smaller batches reduce simple regret fastest; larger slows progress modestly, consistent with the lack of real feedback within a batch (hallucination). Nonetheless, degradation remains limited and all settings improve steadily with budget, indicating that when parallel evaluations are cheap, 0 > 1 can cut wall-clock time for small accuracy cost (Figure 5).
Query set size. TAMO scores N, candidates per step; larger N₂ improves coverage but raises proposal cost. For Ng E [256, 512, 1024, 2048), regret is largely insensitive, except for Branin-Currin, where small N2 misses promising regions (Figure S3). Cumulative inference time grows roughly linearly with N₂. Even for Ng = 2048 (our default), TAMO remains much faster than other baselines.
6 DISCUSSION
We introduced TAMO, fully amortized, task-agnostic policy for multi-objective black-box optimiza- tion. A single transformer backbone, trained offline with a prediction warm-up and a policy-level RL objective, maps histories proposals one forward pass and operates across varying input/output di- mensionalities. Empirically, TAMO delivers proposal times 50 to 1000 times lower than conventional baselines while matching Pareto quality under tight evaluation budgets, sometimes even improving
Limitations. Our study highlights two main axes. First, pretraining data composition: although synthetic GP corpora provide scale and control, they may miss salient real-world structure. Down- stream performance is likely sensitive to GP kernel families and smoothness, input metrics (e.g., Mahalanobis/rotated anisotropy), multi-output correlations (coregionalization models), observation models (homo-/heterooseeaasiic noise, decoupled observations), and simple landscape priors (e.g.. adding a quadratic "bowl"). A systematic analysis that varies these ingredients would clarify how synthetic dataset design drives transfer. Second, inference currently assumes discrete candidate pool, which can be restrictive in high-dimensional design spaces and in generative settings (c.g., de novo drug design) where the action space is continuous combinatorial. Nevertheless, in pool-based

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scenarios like high-throughput screening and library/catalog search, this assumption aligns with practice, and our method is highly effective.
Perspectives. We envision bright future for TAMO universal engine for black-box optimization. The modular design invites extensions to blxck-box constraints, cost-aware and multi-fidelity settings, while retaining the single-pass interface. Beyond scalar observations, alternative data modalities, like preferential and multi- multi-fidelity feedback, can also incorporated with minor architecture changes. A key challenge scaling to higher-dimensional spaces, with promising directions including factorizing the policy across input dimensions and moving from pool-based scoring to continuous policies or generative proposal mechanisms, in the spirit of amortized design networks (Foster al., 2021), Efficient autoregressive mechanisms for transformer-butt -based probabilistic models, such as the causal buffer of Hassan et (2026), could further accelerate sequential proposal generation by avoiding context re-encoding at each step. Lastly, as mentioned above, further work will investigate how the composition of the synthetic pre-training corpus influences downstream performance, an important direction for improving robustness and out-of-distribution behavior of amortized BO policies. To- gether, these advances position TAMO to serve as foundation model-style optimizer that transfers across domains, objectives, and design spaces with minimal per-task tuning.
ETHICS STATEMENT
All authors have read and will adhere the ICLR Code of Ethics. This work does not involve human subjects, personally identifiable information, or sensitive attributes; experiments use synthetic data and standard public benchmarks. We are not aware of foresceable harms from the methodology beyond typical risks of algorithmic misuse; the intended use scientific and engineering optimization. Compute and environmental impact were kept reasonable (single-model pretraining and standard hardware); we report settings to support reproducibility. We will respect licenses of any third-party assets used and disclose any conflicts of interest if they arise.
REPRODUCIBILITY STATEMENT
We document all experimental settings needed facilitate replication: hyperparameters and optimizer details (Section D), procedures for pretraining dataset generation (Section A), and step-by-step algo- rithms for the training and prediction workflows (Section B). The code reproduce our experiments is available at: https://gthub.co///iyyu/c//////t contrx/too.o...
ACKNOWLEDGEMENT
This work was supported by the Research Council of Finland (Flagship programme: Finnish Center for Artificial Intelligence FCAI). XZ and SK were supported by Research Council of Finland (RCF-NSF FinBioFAB, grant agreement 365982), CH and SK were supported by Business Finland Virtuall-b44Parmm. grant agreement 2597/31/2023 and the European Union (Horizon Europe, grant agreement 101214398, ELLIOT). DH and SK were supported by Research Council of Finland (Real-time Al-assistance with computationally rational user models, grant agreement 359207). SK also supported by UKRI Turing AI World-Leading Researcher Fellowship (EP/W002973/1) The authors acknowledge the computational resources provided by the Aalto Science-IT Project from Computer Science IT.
REFERENCES
Brandon Amos et al. Tutorial on Amortized Optimization. Foundations and Trends in Machine Learning, 2023.
Maximilian Balandat, Brian Karrer. Daniel Jiang, Samuel Daulton, Ben Letham, Andrew G Wilson, and Eytan Bakshy. BoTorch: A framework for efficient monte-carlo bayesian optimization. In Advances in Neural Information Processing Systems, 2020.
Syrine Belakaria, Aryan Deshwal, and Janardhan Rao Doppa. Max-value Entropy Search for Multi- Multi-Objecctvve Bayesian Optimization. In Advances in Neural Information Processing Systems, 2019.
10

---

Published as a conference paper at ICLR 2026
Syrine Belakaria, Aryan Deshwal, and Janardhan Rao Doppa. Output space entropy search framework for multi-objective bayesian optimization. Journal of artificial intelligence research, 2021.
Paul E Chang. Prakhar Verma, ST John, Victor Picheny, Henry Moss, and Arn Solin. Fantasizing with dual gps in bayesian optimization and active learning. arXiv preprint arXiv:2211.0100 2022.
Paul Edmund Chang, Nasrulloh Ratu Bagus Satrio Loka, Daolang Huang. Ulpu Remes, Samuel Kaski, and Luigi Acerbi. Amortized probabilistic conditioning for optimization, simulation and inference. In International Conference on Artificial Intelligence and Statistics, pp. 703-711. PMLR, 2025.
Yutian Chen, Xingyou Song. Chansoo Lee, Zi Wang, Richard Zhang, David Dohan, Kazuya Kawakami, Greg Kochanski, Arnaud Doucet, Marc' Aurelio Ranzato, et al. Towards learning uni- versal hyperparameter optimizers with transformers. Advances Neural Information Processing Systems, 35:32053-32066 2022.
Sam Daulton, Maximilian Balandat, and Eytan Bakshy. Hypervolume knowledge gradient: a looka- head approach for multi-objecctvv bayesian optimization with partial information. International Conference on Machine Learning, 2023a.
Sam Daulton, Maximilian Balandat, and Eytan Bakshy. Hypervolume knowledge gradient: A looka- head approach for multi-objective Bayesian optimization with partial information. Proceedings of the 40th International Conference on Machine Learning, 2023b.
Samuel Daulton, Maximilian Balandat, and Eytan Bakshy, Differentiable Expected Hypervolume Im- provement for Parallel Multi-Objection Bayesian Optimization. In Advances in Neural Information Processing Systems, 2020.
Samuel Daulton, Xingchen Wan, David Eriksson, Maximilian Balandat, Michael A Osborne, and Eytan Bakshy. Bayesian optimization over discrete and mixed spaces via probabilistic reparame- terization. Advances in Neural Information Processing Systems, 35:12760-12774, 2022.
Kalyanmoy Deb, Karthik Sindhya, and Jussi Hakanen. Multi-objective optimization. In Decision sciences, pp. 161-200. CRC Press, 2016.
Vincent Dutordoir, Alan Saul, Zoubin Ghahramani, and Fergus Simpson. Neural diffusion processes. In International Conference on Machine Learning, pp. 8990-9012. PMLR, 2023.
Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-lcarning for fast adaptation of deep networks. In International Conference Machine Learning (ICML), pp. 1126-1133. 2017.
Adam Foster, Desi R Ivanova, Ilyas Malik, and Tom Rainforth. Deep adaptive design: Amortizing sequential bayesian experimental design. In International conference on machine learning, 2021.
Jenna Fromer and Connor W. Coley. Computer-aided multi-objective optimization in small molecule discovery. Patterns, 4(2):100678, 2023. ISSN 2666-3899, doi: https://doi.org/10.. 10166 j.patter: 100678. URL https: www. sciencedi com/ science/article/ pp1/$266638922000000000
Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and S. M. Ali Eslami. Conditional Neural Processes. In International Conference on Machine Learning, 2018.
Roman Gareet, Bayesian Optimization. Cambridge University Press, 2023.
Nyoman Gunantara. A review of multi-objective optimization: Methods and its applications. Cogent Engineering, 5(1):150224,, 2018
Conor Hassan, Nasrulloh Loka, Cen-You Li, Daolang Huang, Paul E Chang, Yang Yang. Francesco Silvestrin, Samuel Kaski, and Luigi Acerbi. Efficient autoregressive inference for transformer probabilistic models. In International Conference Learming Representations, 2026.
Daniel Hernández-Lobato, Jose Hernandez-Lobato Amar Shah, and Ryan Adams. Predictive entropy search for objective bayesian optimization. In International conference on machine learning. 2016.
11

---

Published as a conference paper at ICLR 2026
Noah Hollmann, Samuel Müller, Lennart Purucker, Arjun Krishnakumar, Max Körfer, Shi Bin Hoo, Robin Tibor Schirmeesterr, and Frank Hutter. Accurate predictions on small data with a tabular foundation model. Nature, (37(8045)331-3366 2025.
Daolang Huang, Yujia Guo, Luigi Acerbi, and Samuel Kaski. Amortized bayesian experimental design for decision-making. Advances in Neural Information Processing Systems, 37: 10:40- 109486, 2024.
Daolang Huang, Xinyi Wen, Ayush Bharti, Samuel Kaski, and Luigi Acerbi. Aline: Joint amortization for bayesian inference and active data acquisition. arXiv preprint arXiv:2506.0722. 2025.
Yu Heng Hung, Kai-Jie Lin, Yu-Heng Lin, Chien-Yi Wang, Cheng Sun, and Ping-Chun Hsich. BOFormer: Learning to Solve Multi-Objective Bayesian Optimization via Non-Markovian RL. In International Conference Learning Representations, 2025.
Faran Irshad, Stefan Karsch, and Andreas Döpp. Multi-objectivv and multi-fidelity Bayesian opti- mization of laser-plasma acceleration. Physical Review Research, 5(1):01 3063, 2023.
Hyunjik Kim, Andriy Mnih, Jonathan Schwarz, Marta Garnelo, Ali Eslami, Dan Rosenbaum, Oriol Vinyals, and Yee Whye Teh. Attentive Neural Processes. In International Conference on Learning Representations, 2019.
Joshua Knowles. Parego: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE transactions on evolutionary computation, 2006.
Helen Lai, Christos Kannas, Alan Kai Hassen, Emma Granqvist, Annie M. Westerlund, Ajmé Clevert, Mike Preuss, and Samuel Genheden. Multi-objective synthesis planning by means of monte carlo tree search. Artificial Intelligence in the Life Sciences, 2025.
Hyungi Lee, Chaeyun Jang, Dongbok Lee, and Juho Lee. Dimension agnostic neural processes. International Conference on Learning Representations, 2025.
Alexandre Maraval, Matthieu Zimmer, Antoine Grosnit, and Haitham Bou Ammar. End-to-end Meta-Bayesian Optimisation with Transformer Neural Processes. Advances in Neural Information Processing Systems, 36: 1246116611 2023.
Tung Nguyen and Aditya Grover. Transformer neural processes: Uncertainty- aware meta learning via sequence modeling. In International Conference Machine Learning, 2022.
Biswajit Paria, Kirthevasan Kandasamy, and Barnabás Póczos. flexible framework for multi- objective bayesian optimization using random scalarizations. In Uncertainty in Artificial Intelli- gence, 2020.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems 32, 2019.
Leonardo Peroni and Sergey Gorinsky. An end-to and pipeline perspective on video streaming in best-effort networks: A survey and tutorial. ACM Comput. Surv., 2025.
Jingang Qu, David Holzmüller, Gaël Varoquaux, and Marine Le Morvan. TabICL: A tabular foundation model for in-context learning on large data. In International Conference on Machine Learning, 2025.
Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian processes for machine learning. MIT Press, 2006.
Alexandre A Schoepfer. Jan Weinreich, Ruben Laplaza, Jerome Waser, and Clemence Corminboeuf. Cost-informed bayesian reaction optimization. Digital Discovery, 2024.
Lei Song, Chenxiao Gao, Ke Xue, Chenyang Wu, Dong Li, Jianye Hao, Zongzhang Zhang, and Chao Qian. Reinforced in-context black-box optimization, arXiv preprint arXiv:2402. 17423, 2024.
12

---

Published as a conference paper at ICLR 2026
Richard S Sutton, Andrew G Barto, et al. Reinforcement Learning: An Introduction, volume 1. MIT Press Cambridge, 1998.
Shinya Suzuki, Shion Takeno, Tomoyuki Tamura, Kazuki Shitara, and Masayuki Karasuyama. Multi- objective bayesian optimization using pareto-frontier entropy. In International conference on machine learning, 2020.
Jayden Teoh, Pradeep Varakantham, and Peter Vamplew. On generalization across environments in multi-objective reinforcement learning, 2025. URL httpps://axxiv.rrgg/. abs 2503. 00799.
Michael Volpp, Lukas P Fröhlich, Kirsten Fischer, Andreas Doerr, Stefan Falkner, Frank Hutter, and Christian Daniel. Meta-learning Acquisition Functions for Transfer Learning in Bayesian Optimization. International Conference Learning Representations, 2020.
Boqian Wang, Jiacheng Cai, Chuangui Liu, Jian Yang, and Xianting Ding. Harnessing novel Becchine-leariin--aae
evolutionary algorithm to co-optimize three characteristics of an electro- spun oil sorbent. ACS Applied Materials & Interfaces, 12(38):42842 9284449944442449 2020.
Shukuan Wang, Xue, Song Lei, Xiaobin Huang, and Chao Qian. Monte carlo tree search based space transfer for black box optimization. Advances in Neural Information Processing Systems, 2024.
Ronald J Williams. Simple statistical gradient- algorithms for connectionist reinforcement learning. Machine learning, 8(3):229-2566 1992.
Adam Yang, Laurence Aitchison, and Henry B Moss. Mongoose: Path-wise smooth bayesian optimisation via meta-learming arXiv preprint mrXiv:2302.15533 2023.
Runzhe Yang, Xingyuan Sun, and Karthik Narasimhan. A generalized algorithm for multi-objective reinforcement learning and policy adaptation. Advances in Neural Information Processing Systems, 32, 2019.
Xinyu Zhang, Daolang Huang, Samuel Kaski, and Julien Martinelli. PABBO: Preferential Amortized Black-Box Optimization. International Conference on Learning Representations, 2025.
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Appendix
The appendix is organized as follows:
Section A describes the generative process leading to synthetic pretraining dataset.
Section B provides additional details regarding TAMO's workflow, with Algorithm S1 described pretraining.
Section provides a detailed architecture figure along with attention masks for TAMO.
Section D provides further details regarding experiments, including model hyperparame- ters (Section D.3), computational resources (Section D.1) and test functions description (Section D.2).
Section contains additional experiments and analyses:
Single-objective Bayesian optimization
Ablation study on the query size Effect of prediction warm-up and the prediction term weight Xp in the policy training loss
Ablation study the discount factor y in the RL objective
Comparison between the standard multi-step and myopic TAMO variants
Effect of model size optimization performance
Timing breakdown for GP baselines
Evaluation real-world HPO-3DGS hyperparameter optimization tasks
Effect pre-training dataset composition
Section F displays several examples of GP samples used during pretraining (Section F.1), ex- amples of mean prediction and proposed queries on GP samples (Section F.2, and Section in the decoupled setting).
Section G describes to what extent Large Language Models (L.LMs) were utilized throughout this work and manuscript.
A GENERATIVE PROCESS OF SYNTHETIC PRETRAINING DATASET
The model evaluated in Section 5 was trained on dataset of GP draws. This dataset was constructed to include a variety of configurations, spanning diverse dimensionalitics and function properties. All functions were generated inside [-5.0, 5.0]d= using the following procedure:
Input dimensionality d₂ ((1,,2)) and output dimensionality d, ((1,2,3))
Regarding output correlations, with probability 1/2, either independent output dimensions are sampled, they are drawn from multi-task GP, with task covariance defined as k(i.j) = (BBT + diag(u))..; i.j j e {1.... ,dy) In this case, B is low-rank matrix with rank U((1........ d.
The data kernel along each output dimension is equally sampled from the RBF, Matérn- 3/2, Matérn-5/2 kernels, with standard deviation U|001, 1.0] and lengthscale / ~ (22/33, 0.5) truncated to the range (0.1,2.0)
The sampled function values, y. were centered and normalized to lie within 1]4,
Examples with different dimensionalities are illustrated in Figures S11, S12, S13 and S14 (Sec- tion F.I).
B PRETRAINING AND TEST-TIME ALGORITHMS
Algorithm SI and Algorithm S2 describe the pre-training loops and test-time optimization procedure, respectively.
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Algorithm S1 TAMO Pre-Training Algorithm
Require: task distribution p(r), prediction context size Ne. prediction target size Np. query budget
T. number of burn-in iterations 7, number of total iterations num_ iterations
1: for iteration i I,..., num_ total iterations do
2:
Prediction task
3: Sample task p(T)
N₂ 4: Sample prediction batches D = ((a), v and D = {a. 1
li=1 from 7
5: Model predicts outcomes: p(y xy 19), DP
6: if < y then
7:
Update model by minimizing the prediction loss ((p) (Fquation 5)
8:
else
Policy learning task after burn-in phase
9:
Sample new task т ~ p(T)
Sample query set D
11:
Initialize history set D +- {( (hi) xb ~ D
Set reference point and calculate optimal Hypervolume: HV" HV(P(()) Initialize Pareto set P + (v
13:
for = 1,.... do
15:
Sclect next query point:
TT(. Dt .t,))
16:
V/ +- Evaluate (Zt, T)
17:
Update history set: h Db U ( vc)
18:
Update Pareto set: P < P (Br)
19:
Compute reward:
Hv(P!r)
IIV
20:
end for
21:
Update model using the overall objective L (Equation 6)
22:
end
23:
end for
Algorithm S2 TAMO Test-Time Algorithm
Require: Pre-trained TAMO model, new task Tiest query budget T,, initial history set D/ = {x", Sy (with random samples if empty),
1: D + D/
Initialize the history set b Initialize the Pareto set
2: P x-1 {
3: for t = 1....,Ta do
4:
Xx ~ mo(. D4 T)
Sample the next query location
5:
yt - Evaluate X1, Tres)
6:
Dh - Du U (x)
Update the history set Update the Pareto set with the new observation
7:
P+ {3r)
8:
end for
9:
return DA,P
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C DETAILED MODEL ARCHITECTURE
Bt attension mask
Poticy head
Prediction head
g+
2
fi: Transformer Layers
frim
SPRDD
Openr Cpaersationn sakane
Prodiction bobere
B. attention mask
fi Thanafommer Layars
009
Famm
A-
e
Diminsiion agrostio ombedder
Hatory
Coary
Tagee
Policy
KIV
Figure $1: Detailed Architecture of TAMO.
D EXPERIMENTAL DETAILS
D.I
COMPUTATIONAL RESOURCES
We trained TAMO one NVIDIA H100 80GB HBM3 GPU. All models are evaluated on Tesla V100-SXM2-32GB GPUs.
D.2 TEST FUNCTIONS
GP samples optimization. This benchmark comprises 30 independent multi-output GP draws with d = 2 inputs and dy = 2 objectives in the dimensional in-distribution setting (Section 5.1), and d₂ = dy E {2.3) 3} in the dimensional out-of-distribute setting (Section 5.2). We sample each task using the same data tata-generating process described in Section D.3 and report average performance
over the 30 draws.
Ackley-Rastrighe d, = dy = 2. Two-objective problem formed by pairing Ackley and Rastrigin and maximizing their negations:
Ac)ley(x) = -20 -20 (00.2
x
exp
cos cos (2TT2))
20,
i=1
Rastrigin(2)) 20 +
Σ
(u)
10 cos(2
i=1
And we set (x) = AAckey(zz), (z) - (astr.gin(x)..
Ackley-Rosenbree d, = 2, dy = 2. We pair Ackley (above) with Rosenbrock:
Rosenbrock(zz) = 100(12 2²)2 + -1-11) 21)2.
And set (xx AAckley(2)) (2) Rosenbrock(2x).
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Branin-Currih d, =2 dy 2.
Branin:
Branin(21, (I2
bxt
cT1 - -r)²+ +ss1= t) cos(x1)) + s,
5.1 where
6, 10,
4771
and Currin:
Currin(z) -1- (-1--/(2sa)) 2300 + 19002 + 209221 + 60
100. 500zf 421 2
We maximize (x) -Branin (æ)) and (x) -Currin(x:.
Oil sorbent de 2, dy = 2. We also evaluate on the oil-sorbent multi-objective problem (Wang al., 2020; Daulton et al., 2022). The original task controls material's composition and man- ufacturing with ordinal and 2 continuous parameters to jointly maximize three objectives: oil absorbing ng capaci tyy, mechanical cal st rength, and water contact angle. our study, we fix the ordinal parameters to constant values to obtain a 2D continuous design space with the same three objectives.
Laser-Plasma d, = d, 3. We evaluate on the laser-plasma acceleration dataset (Irshad al., 2023), which contains 1025 particle-in-cell simulations of laser wakefield accelerator. Each record provides 4 continuous inputs (plasma density, upramp length, laser focus, downramp length) and 3 objectives (total charge, distance of median, target energy). To obtain continuous black-box from tabulated simulations, perform linear inter- polation. This task differs in dimensionality from our pretraining distribution, providing an OOD evaluation of cross-dimensional transfer.
Normalization. For all problems, we linearly rescale inputs to a common domain [-5,5] and rescale each objective independently [-1, prior logging and hypervolume computation.
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D.3 HYPERPARAMETERS
Dimension-Agnostic Embedder
Number of leamable positional tokens for
Number of learnalle positional tokens for Number Trameformer layers
Dimension of and
64
Transformer Encoder-Decoder
Dinension of Trunsformer inputs
64
feed- forward dimension Transformer Number of self-attention layers Transformer (B) Number of self-attention heads in Transformer
Heads
Number of hidden layers policy head Number of components in GMM head (K)
20
Number of hidden layers MLP for each GMM component
Training
Number of iterations
400000
Number of bur--in iterations
393500
Initial learning rate for warm-up iterations (Iry)
1-10-1
Initial Learning rate after warm-up (Irg)
4-10-5
Learning rate scheduling
Lincarly increase from 0 to Ir; in the first 5% of total iterations:
Cosine decay to 0 over totul iterations
prediction hatch Size of policy-leariin batch
16
Weight on prediction loss (Aa)
1.0
discount factor (r)
0.99
Size of context set
No 0/2, 50- dg.
Size of turget set (NN)
300 Nr
Size query (NN.)
Optimization budget T
100
Noise level
0.0
Number
observations during pretraining
Evaluation
Number of initial observations during test time
Noise level ø
0.0
query (N,))
Optimization budget (T)
100
Table S1: Hyperparameter settings for TAMO evaluated in Section 5.
E ADDITIONAL EXPERIMENTS
GP-DX2-DY1
Forrester
Branin
EggHelder
1.0
Simple Regret
0.6
0.4
0.2
25
50
75
100
10
20
30
10
20
30
25
50
75
100
Cumulativ infereace Tane (9)
10R
101
109
10-1
Oracie Calls
Oracle Calls
Ocacle Calls
Oracie Calls
TAMO
Random
qEI
Figure S2: Simple regret and inference time synthetic examples for single-objective optimization. Mean with 95% confidence intervals computed across 30 runs with random initial observations. Again, TAMO matches tate-of-the-art regret while dramatically reducing proposal time.
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GP-DX2-DY2
Rasteyaat
Ackeeyoseebrock
BraninCurrin
1.0
Simple Regret
0.6
0.2
50
75
100
25
75
100
50
75
25
50
75
100
Cunvlatave Inference Time (5)
10"
10
25
50
75
100
0
25
50
75
100
25
50
75
100
25
50
75
100
Oradle Calls
Oracle Calls
Oradl Calls
Oracle Calls
TAMO (26)
TAMO (5,2512)
TAMO (N,-1024)
TAMO (N,-2048)
Figure S3: Effect query set size. Simple regret (top) and cumulative inference time (bottom) for TAMO with N₂ (256, 512, 1024, 2048] on four synthetic tasks. Means with 95% CIs over 30 runs. Larger Ng increases wall-clock roughly linearly while leaving regret essentially unchanged.
GP-DX2-DY2
AckleyRastrigin
Rokeeyyos
BraninCurrin
1.0
Sample Regret
0.6
0.4
0.0
TAMO w/o prediction lask
TAMO w/ prediction task
Figure S4: Removing the prediction warm-up and prediction term from the training loss (Equation 6). Simple regret on four synthetic tasks. Means with 95% CIs over 30 runs. Introducing an auxiliary prediction task before and during policy training is decisive.
GP-DX2-DY2
AckieyRastrigin
Acideyyoseebrace
BraninCurrin
1.0
Simple Regret
25
50
75
100
0
25
50
75
100
25
50
75
100
25
50
75
100
Oracle Calls
Oracie Calls
Oracle Calls
Oracle Calls
(A=0.5)
TAMO
(A11.0)
Figure S5: Effeet of the prediction term weight X the training loss (Equation 6) during policy training, Simple regret on four synthetic tasks. Means with 95% Cls over 30 runs. Once policy training starts, performance is relatively insensitive to Aps with slightly better results for larger weights.
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GP-DX2-DY2
AckeyBastrigin
AddeeyRoseabrec
BraninCurtin
1.0
Gimple Reggee
0.8
0.6
0.4
0.2
0.0
75 Oracie TAMO (y-0.5)
100
TAMO (y-9225)
TAMD (y-0.75)
TAMO (y-1.0)
Figure $6: Effect of the discount factor in the RL objective (Equation 4) during policy training. Simple regret on four synthetic tasks. Means with 95% CIs over 30 runs. Overall, performance fairly robust to 7
Branin
Rastrigin
EggHolder
Ackóny
Simple Rogret
0.4
0.2
10
20
30
0
25
75
100
0
25
50
75
100
25
30
75
100
GP-DX2-DY2
Ackleyyesstrigin
Ackleyyoseniarok
BeaninCurrin
Simple Begret
0.8
0.6
0.2
0.0
25
75
100
25
75
100
25
75
100
25
75
100
Oracle Calls
Oracle Calls
Oracle Calls
Oracle Calls
TAMO (myopic)
TAMO
Figure S7: Effect the pretraining horizon in the RL objective (Equation 4) during policy training. We compare myopic variant of TAMO, pretrained with horizon T = 1, to the standard TAMO pretrained with T 100. For the non-myopic model, each pretraining episode starts from single randomly sampled context point, and the policy acts for up to 100 steps. For the myopic case, each episode starts from a randomly sampled context set, and the policy proposes single additional point, so every decision is strictly one-step. Simple regret on four synthetic tasks for single objective optimization (top) and multi-objective optimization (bottom). Means with 95% CIs over 30 runs. Results clearly advocate for longer pretraining horizons, except for the BraninCurrin problem.
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Figure S8: Effect of model size optimization performance. We compare smaller TAMO variant, trained for the same number of iterations as standard model but using 2 Trannformmrr layers per module (dimension-annosicc embedder, B1 layers, Bg layers) instead of 4. Simple regret four synthetic tasks. Means with 95% CIs over 30 runs. While the smaller model remains competitive, the larger backhone consistently attains lower regret, especially the more challenging tasks.
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Figure S9: Transfer to HPO-3DGS hyperparameter optimization tasks Models are pretrained on synthetic GP data with d, 5, dy, E (2, 3). then evaluated on four 3D-GGassian-Splattin scenes (Lego, Materials, Mic, Ship: d, = for Lego/Materials and d, = 3 for Mic/Ship). Means with 95% CI over 50 runs. Across all scenes, TAMO is competitive with the best-performing GP-based methods (with QNEHVI slightly ahead on Mic), while achieving substantially lower inference time.
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Figure S10: Effect of pretraining pariinnnn--piorr composition on optimization performance. We compare the original TAMO with variants trained modified synthetic priors. Top: Small-ennghsscale prior, where GP draws use shorter kernels by truncating the original lengthscale prior (22/3 0.5²) d, objectives we draw GP and then sample an "optimum location" =m(m) uniformly the design space, adding quadratic term | (m) to the m-th objective. Simple regret on four synthetic benchmarks; curves show means with 95% confidence intervals over 30 runs. The small-lengthscale prior degrades performance, while the quadratic-bowl prior maintains or improves perfor- mance on GP and Ackley-based tasks but hurts performance on Branin-Currin.
from [0.1, 2.0] [0.1, 0.5]. Bottom: Quadratic-bow/ prior, where for each of the S
F VISUALIZATION EXAMPLES
F.I EXAMPLES OF GP SAMPLES FROM PRETRAINING
Figures S11 to S14 show some examples of the GP samples used for pre-training.
F.2 EXAMPLES OF INFERENCE
Figure S15 and Figure S16 show examples of mean predictions and proposed queries within a total budget T 100 GP samples, for input dimension d, and output dimensions d, 1 and dy = 2, respectively.
F3 EXAMPLES OF DECOUPLED OBSERVATIONS
Figure S17 shows examples of mean predictions and proposed queries under the decoupled setting.
G USE LARGE LANGUAGE MODELS
We employed LLMs to support the following aspects of our research:
Ideation & Exploration: Assisting with brainstorming of methods and conducting prelimi- nary literature searches and summarizations.
Coding Assistance: Generating boilerplate PyTorch code, visualization scripts, and test structures. All LLM-generated code was manually reviewed and verified by the authors.
Writing Assistance: Refining sentence structure, grammar, and clarity in the manuscript. The scientific content and all claims remain the sole work of the authors.
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Figure $15: Inference GP examples (d₃ 2, dy 1), with query points proposed over 100 optimization steps (white circle, size increasing along with the number of queries).
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Figure S16: Inference on GP examples (d₃ = 2, dy 2), with query points proposed over 100 optimization steps (white circles, size increasing along with the number of queries).
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Figure S17: Mean predictions and queries on GP examples (d, = = dy = 2) from TAMO under the decoupled setting. Each column represents distinct objective; queries to evaluate that objective are outlined by circles, with the sizes increasing over time to show the optimization progress. Note no queries overlap between objectives.
26


Reviews:
[{"review_id": "j5ZhNqf3aM", "reviewer": "Reviewer_eMcs", "summary": "This paper introduces TAMO, a novel amortized policy for multi-objective black-box\noptimization, aiming to address the high computational cost and task-specific nature of GP-based\nmethods. The core idea is to pre-train a single, dimension-agnostic Transformer on a diverse\ncorpus of synthetic tasks. This pre-trained model can then be deployed on new, unseen problems,\nreplacing the slow, iterative refitting of a surrogate model with a single, fast forward pass to\npropose the next query. The model is trained using a combination of a prediction loss and a\nnon-myopic RL objective based on cumulative hypervolume improvement .", "questions": "● The model has two heads (Prediction and Policy) and is trained with a joint loss.\nHowever, the contribution of the auxiliary prediction task L(p) is not ablated. How much\ndoes the \"warm-up\" and joint training contribute to the final policy versus simply training\nthe policy head alone?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "● Paper is very well written and easy to understand\n● By replacing the iterative GP refitting and acquisition optimization process with a single\nneural network forward pass, the method reduces inference latency by 50-1000x.\n● The proposed dimension-agnostic embedder is a clever architectural contribution . It\nallows a single Transformer backbone to be pre-trained on and deployed to problems of\nvarying input and output dimensions", "weaknesses": "● The framework relies on a pre-defined discrete candidate pool from which the policy\nhead selects the next query. This is a significant limitation as it makes the approach\nunusable for true continuous-domain optimization or generative tasks (like de novo drug\ndesign).\n● The \"task-agnostic\" claim is weakened by the model's sensitivity to the pre-training data.\nThe authors hypothesize that the poor performance on BraninCurrin stems from not\nseeing those objective properties in the synthetic pre-training corpus. The model isn't\ntruly \"agnostic\" but is rather \"multi-task\" for a specific family of synthetic GP-based\ntasks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YesbfleeA8", "reviewer": "Reviewer_GinK", "summary": "The paper targets three flaws with traditional multi-objective Bayesian optimization methods; that they require re-training from scratch for each use, the reliance on a large number of hyperparameters, and the focus on single-step gains. The authors introduce TAMO, which leverages a transformer-based optimizer trained on diverse synthetic tasks using reinforcement learning to learn multi-step strategies. Once trained, users use iterative forward passes of TAMO and a tokenized history of the steps inputted into TAMO to run the optimization process. The authors claim to match or improve the quality of state of the art methods while reducing proposal time.", "questions": "Is the prediction task fitting to the raw data? Does the policy part of the model not converge without the prediction task? Why is this? Is there any practical application to the prediction mechanism post-training? This seems to be a central part of the architecture and is not thoroughly justified. Can the authors demonstrate an ablation for training with and without the prediction task?\nCan the authors include a figure for the architecture for 3.2 (II) and (III)? The exact model architecture (where self vs cross attention is applied, where task-specific tokens are inputted) is unclear simply due to wordiness and a simple diagram could very clearly explain it.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The authors present a novel architecture which functions as both the acquisition function and surrogate model. The dimension-agnostic embedder is an interesting solution to generalizing over spaces with varying dimensions. TAMO is demonstrated to have equivalent or superior metrics to many other methods, and is demonstrated to run in a shorter time. The paper is overall pretty clear.", "weaknesses": "1. In which cases do multi-step strategies actually matter? When is myopia actually a limiting factor for Bayesian optimization? Can the authors demonstrate an ablation where multi-step strategies provide a clear advantage over traditional BO?\n\n2. There is only one real-world benchmark. Can the authors demonstrate applicability to other real-world scenarios, such as for Gaussian splatting as done in the Boformer paper (Yu-Heng Hung, et al. 2025) or neural network hyperparameter selection as in OptFormer (Yutian Chen, et al. 2022).\n\n3. Can the authors demonstrate clear advantages in practice to other transformer-based MOO methods, such as Optformer?\n\n4. The authors mention that the pretraining data composition is important for generation. The authors should provide a study to experimentally demonstrate how the diversity of the pretraining data affects optimization performance. For the out-of-distribution experiments, authors should demonstrate", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qD3Mw0fPi4", "reviewer": "Reviewer_avNh", "summary": "The paper introduces TAMO, a fully amortized, task-agnostic, dimension-agnostic policy for multi-objective black-box optimization. A single transformer backbone, trained offline with (i) an in-context prediction warm-up and (ii) a policy-level RL objective that maximizes trajectory hypervolume (HV) progress, maps histories and a candidate set to the next query in one forward pass—eliminating per-task surrogate fitting and acquisition optimization. Key architectural elements include a dimension-agnostic embedder, encoder–decoder with task-specific tokens, and dual prediction/policy heads; the policy is trained with REINFORCE on HV-normalized rewards. Empirically, TAMO achieves **50×–1000×** lower proposal time than GP-based MOBO and BOFormer while delivering competitive regret on synthetic and real tasks; it also shows transfer to unseen input/output dimensionalities and decoupled observations. Code is promised upon acceptance; hyperparameters, algorithms, and synthetic pretraining data generation are documented.", "questions": "1. **Timing fairness & ablations:** Please report per-candidate acquisition latency (µs/ms) and GPU-to-GPU comparisons against GPyTorch/BoTorch with matched MC budgets; include breakdown of surrogate refit vs. acquisition time. \n2. **Pretraining-prior sensitivity:** Provide a controlled study varying kernel families, length-scales, ARD, and output correlations to quantify transfer sensitivity and explain gaps on Branin–Currin/LaserPlasma. \n3. **Continuous action spaces:** Can TAMO couple to a continuous proposal mechanism (e.g., gradient-guided refinement, learned sampler) to move beyond pool-based scoring? Any preliminary results? \n4. **RL stability:** What variance-reduction techniques were used for REINFORCE (e.g., learned baseline, advantage normalization)? Any ablation on γ and λ_{rl}? \n5. **Decoupled observations policy:** How are costs integrated at training time? Could explicit cost-aware reward shaping further improve decoupled performance (e.g., on Ackley–Rosenbrock)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* **End-to-end amortization:** Eliminates per-task surrogate fitting and acquisition optimization; single forward-pass proposals reduce decision latency by **50×–1000×**. \n* **Dimension-agnostic architecture:** Embedder + task-tokens operate across varying dx, dy; supports heterogeneous pretraining and cross-dimensional transfer. \n* **Non-myopic training signal:** RL objective directly optimizes trajectory HV, aligning learning with Pareto-front discovery over horizons. \n* **Generalization studies:** Evidence of transfer to unseen dimensionalities and decoupled observations under fixed budget/cost accounting. \n* **Methodological transparency:** Clear preliminaries, training/inference algorithms, hyperparameters; baselines (qNEHVI, qNParEGO, qHVKG, BOFormer) implemented with standard toolchains. \n* **Performance profile:** Competitive or better regret across several synthetic and real tasks, with consistent runtime advantage.", "weaknesses": "* **OOD gaps & sensitivity:** Underperforms classic MOBO on Branin–Currin and LaserPlasma; authors attribute mismatch to pretraining length-scales—this warrants a systematic analysis. \n* **Discrete candidate pool assumption:** Inference relies on a fixed candidate set; implications for high-dimensional continuous or combinatorial design spaces are acknowledged but unresolved.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "axIPELnwDP", "reviewer": "Reviewer_mKPB", "summary": "This paper proposes TAMO, a Transformer-based algorithm for black-box multi-objective optimization using a pre-trained policy. It is trained on synthetic Gaussian Process tasks with varying input and output dimensions, combining reinforcement learning and supervised losses for model-agnostic optimization. Experiments show that TAMO matches or surpasses state-of-the-art methods while being 50–1000 times faster at inference time.", "questions": "See Weakness,\n\nand \n\n- Why is a **mixed-normal regression head** (equation 1) used for the prediction head? Would it be possible to apply a **Bar distribution (Riemann distribution) regression head** instead? [5] has shown that normal regression heads may underperform bar distribution regression heads in terms of supervised-learning efficiency in a similar setting of fitting GP priors.\n    \n- Is there a specific reason for using the **normalized hypervolume level** as the reward, rather than the **normalized hypervolume improvement** [1]? Using net improvement as reward signal is also common in RL-based BO policies [2,3] for single-objective problems. Conceptually, would using the normalized hypervolume level allow repeatedly rewarding the same query, potentially slowing down TAMO’s exploration?\n    \n- **Potentially high training time?** Although TAMO achieves significantly lower inference time than other methods, Transformer-based BO policies have been shown to require substantial training resources (as noted in [2]), even for single-objective BO. There may be a potential issue of high training cost for complex real-world problems with high input/output dimensionality. It would be helpful for the authors to discuss TAMO’s training time.\n\n- In Section 3.2, the definition of positional tokens is confusing for me. In Line 178,\n\t \"These positional tokens are randomly sampled for each batch from fixed pools of learned embeddings\"\n\tWhat's is the definition and the practical implementation of this \"fixed pools of learned embeddings?\"\n- The Section 3.2 is difficult to read, I think it could be improved by providing a full model architecture illustration in Figure 2., explicitly drawing the $B_1, B_2$ blocks of layers and use arrows or attention maps to indicate which tokens can attend to which tokens in each phase.\n\n**Minor issues**\n- Figure 4. is confusing to me before I found (a), (b) in the subfigures. They look very small and are in the top of the subfigures. \n- Some Figure references in the appendix are not well-compiled.\n\n\n**Citations**\n\n[1] Hung, Y. H., Lin, K.-J., Lin, Y.-H., Wang, C.-Y., Sun, C., & Hsieh, P.-C. (2025). BOFormer: Learning to solve multi-objective Bayesian optimization via non-Markovian RL. The Thirteenth International Conference on Learning Representations. [https://openreview.net/forum?id=UnCKU8pZVe](https://openreview.net/forum?id=UnCKU8pZVe)\n\n[2] Alexandre Max Maraval, Matthieu Zimmer, Antoine Grosnit, and Haitham Bou Ammar. End-to-end\nmeta-bayesian optimisation with transformer neural processes. Advances in Neural Information\nProcessing Systems, 34, 2023.\n\n[3] Michael Volpp, Lukas P. Fr¨ohlich, Kirsten Fischer, Andreas Doerr, Stefan Falkner, Frank Hutter,\nand Christian Daniel. Meta-learning acquisition functions for transfer learning in bayesian opti-\nmization. In International Conference on Learning Representations, 2020.\n\n[4] Yutian Chen, Xiaoxi Song, Chung-Ching Lee, Zihang Wang, Ruoxi Zhang, David Dohan, Kenji\nKawakami, Greg Kochanski, Arnaud Doucet, Marc’Aurelio Ranzato, et al. Towards learning\nuniversal hyperparameter optimizers with transformers. In Advances in Neural Information Pro-\ncessing Systems, volume 35, pp. 32053–32068, 2022.\n\n[5] Müller, S., Hollmann, N., Pineda Arango, S., Grabocka, J., & Hutter, F. (2022). Transformers can do Bayesian inference. International Conference on Learning Representations. [https://openreview.net/forum?id=KSugKcbNf9](https://openreview.net/forum?id=KSugKcbNf9)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. This paper proposes a _Model-amortized Multi-objective Optimization_ algorithm that achieves significantly lower inference time compared to other MOBO methods. It could be one of the earliest works to leverage the scalability and in-context learning capability of Transformer Neural Process models to improve inference efficiency in MOBO context.\n    \n2. The paper introduces a  Dimension-agnostic embedder to the Transformer that enables dimension-agnostic learning across various MOBO tasks, making it highly generalizable to unseen tasks with varying dimensions, using a single pre-trained model. By adapting this architecture to the MOO setting, the method goes beyond a naive extension of similar approaches in single-objective Meta-BO [2] and BOFormer [1], which also implements a Transformer policy, but without dimension-agnostic learning.", "weaknesses": "Besides the limitations mentioned in the paper, there are some non-negligible issues:\n- **Lack of statistical significance in task generalization performance** . The confidence bars of all the baselines, including TAMO, in the out-of-distribution evaluations of problems GP-DX3-DY2, GP-DX3-DY3 in Figure 4 (a) are not well-separated. Thus, Figure 4. only demonstrate limited evidence for the statistical significance of TAMO's superiority in task generalization.\n- **Lack of benchmarks.** Although both synthetic and real-world benchmarks were evaluated, the input and output dimensions are relatively small (both ≤ 4), which limits the results to small-scale MOO problems. Note that the most similar and recent work BOFormer [1] includes benchmark performance with **input dimension up to 30**. Also, there is only a single real-world benchmark compared in the experiments, which is not sufficient to demonstrate the general performance of TAMO in real-world settings.\n    \n- **Lack of baseline comparisons.**\n    \n    1. The information-theoretic approaches discussed in the related work section were not included in the experimental comparisons. Other Transformer-based policies applicable to MOBO settings , such as **Optformer** [4], should also be considered for comparison, which is also compared in the paper of BOFormer [1].\n        \n    2. The performance gap between GP-based and Transformer-based models is not well studied experimentally. Given that GP and Transformer models differ substantially in scale, direct comparison is difficult. To better evaluate the usefulness of the Transformer Neural Process in terms of regret performance, variants of TAMO should be included in the ablation study, e.g., versions with RL policies using other policy parametrizations should be considered, e.g., smaller neural networks (as in [3]). \n- **Necessity of the RL policy.** Although the use of Transformers is well-motivated, which possess strong in-context learning capabilities, the paper provides limited evidence supporting the need for a Transformer-based RL policy. Future experiments could include a baseline that performs GP-based MOBO (e.g., qNEHVI, qParEGO) but replaces the GP surrogates with Transformer Neural Process surrogates (pre-trained on GP priors). Comparing TAMO against such a baseline would help clarify whether using RL to learn a policy is necessary, compared with directly employing a Transformer surrogate. Moreover, this proposed baseline should also benefit from reduced inference time through forward-pass inference, thus, this comparison would also help evaluate the effectiveness of TAMO’s RL policy under similar low-inference-cost conditions", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)