Title: PRESTO: Preimage-Informed Instruction Optimization for Prompting Black-Box LLMs
Venue: NeurIPS 2025
Accepted: yes
Keywords: Large Language Models

Instruction Optimization
Abstract: Large language models (LLMs) have achieved remarkable success across diverse domains, due to their strong instruction-following capabilities. This raised interest in optimizing instructions for black-box LLMs, whose internal parameters are inaccessible but popular for their strong performance and ease of use. Recent approaches leverage white-box LLMs to assist instruction optimization for black-box LLMs by generating instructions from soft prompts. However, white-box LLMs often map different soft prompts to the same instruction, leading to redundant queries to the black-box model. While previous studies regarded this many-to-one mapping as a redundancy to be avoided, we reinterpret it as useful prior knowledge that can enhance the optimization performance. To this end, we introduce PREimage-informed inSTruction Optimization (PRESTO), a novel framework that leverages the preimage structure of soft prompts to improve query efficiency. PRESTO consists of three key components: (1) score sharing, which shares the evaluation score with all soft prompts in a preimage; (2) preimage-based initialization, which select initial data points that maximize search space coverage using preimage information; and (3) score consistency regularization, which enforces prediction consistency within each preimage. By leveraging preimages, PRESTO observes 14 times more scored data under the same query budget, resulting in more efficient optimization. Experimental results on 33 instruction optimization tasks demonstrate the superior performance of PRESTO.

PRESTO: Preimage-Informed Instruction Optimization for Prompting Black-Box LLMs

Abstract

Large language models (LLMs) have achieved remarkable success across diverse domains, due to their strong instruction-following capabilities. This has led to increasing interest in optimizing instructions for black-box LLMs, whose internal parameters are inaccessible but widely used due to their strong performance. To optimize instructions for black-box LLMs, recent methods employ white-box LLMs to generate candidate instructions from optimized soft prompts. However, white-box LLMs often map different soft prompts to the same instruction, leading to redundant queries. While previous studies regarded this many-to-one mapping as a structure that hinders optimization efficiency, we reinterpret it as a useful prior knowledge that can accelerate the optimization. To this end, we introduce PRE image-informed in ST ruction O ptimization (PRESTO), a novel framework that leverages the preimage structure of soft prompts for efficient optimization. PRESTO consists of three key components: (1) score sharing, which shares the evaluation score with all soft prompts in a preimage; (2) preimage-based initialization, which selects initial data points that maximize search space coverage using preimage information; and (3) score consistency regularization, which enforces prediction consistency within each preimage. By leveraging preimages, PRESTO achieves the effect of effectively obtaining 14 times more scored data under the same query budget, resulting in more efficient optimization. Experimental results on 33 instruction optimization tasks demonstrate the superior performance of PRESTO. Code is available at https://github.com/mlvlab/PRESTO.

1 Introduction

Large language models (LLMs) have demonstrated strong performance across a wide range of domains [1-5]. This success is largely attributed to their impressive instruction-following capabilities, which have led to growing interest in discovering effective instructions to enhance their performance [6, 7]. In particular, LLMs provided through APIs ( i.e., black-box LLMs), such as GPT-4 [2], are widely used and show exceptionally strong performance. However, optimizing instructions for the black-box LLMs is a challenging problem, since their internal parameters are inaccessible. To tackle this challenge, recent studies have explored various strategies for optimizing instructions for black-box LLMs, without access to internal model parameters [8-13].

Recently, some studies [14-16] have leveraged open-source LLMs ( i.e., white-box LLMs) [1, 17, 18] to assist instruction optimization for black-box LLMs, demonstrating promising results and attracting growing interest. Specifically, these methods optimize a soft prompt, which is taken as input to the white-box LLM. The optimization is performed using black-box optimization algorithms such as Bayesian Optimization [19, 20] or Neural Bandits [21, 22], guided by a score predictor

(a)

(b)

10k

1000

9k

size

300

Count

8k

200

7k

100

6k

min:4.87

5k

0

0

20

40

60

80

100

#soft prompts

#instructions

Preimage index

regression model, allowing the white-box LLM to generate effective instructions for black-box LLMs. However, as shown in Figure 1a, white-box LLMs often generate identical instructions from distinct soft prompts. It leads to repeatedly querying soft prompts that yield the same outputs during the optimization process, which ultimately hinders the optimization process by reducing query efficiency. To avoid redundant queries, previous studies either sample soft prompts that are well-separated in the soft prompt space [16] or filter soft prompts that generate distinct instructions [15].

While previous studies have treated the generation of identical instructions from different soft prompts ( i.e., many-to-one structure) as a redundancy that hinders optimization, we reinterpret this as a valuable structure that can facilitate the optimization process. Specifically, the set of soft prompts that generate the same instruction forms the preimage of that instruction under the white-box LLM. This preimage imposes a strong inductive bias over the search space: all soft prompts within a preimage share the same objective function value. Since we follow previous settings [16, 15] that sample a sufficiently large set of N soft prompts and search for the optimal solution within them, we do not observe the full preimage, but only a subset of it. We refer to such subsets as preimages throughout the paper, and provide the size distribution of these preimages in Figure 1b.

Building on this insight, we propose PRESTO, a novel instruction optimization framework that explicitly leverages the many-to-one structure to facilitate instruction optimization for black-box LLMs. PRESTO consists of three components. First, we present the score-sharing method, where once the score is evaluated through the black-box LLM, it is shared with all soft prompts within a preimage. This effectively enlarges the amount of scored data without additional calls to the black-box LLM. Second, we introduce preimage-based initialization, where we select the initial soft prompts regarding the preimage information so that they cover the search space maximally. Finally, we propose score consistency regularization, which adds a regularization term to encourage the score predictor to predict identical scores for soft prompts within the same preimage. We evaluate the instruction optimization performance of PRESTO on 30 instruction induction tasks and three arithmetic reasoning tasks, and achieve state-of-the-art performance compared to existing baselines.

The main contributions of our work are:

Wereinterpret the many-to-one structure between the soft prompts and instruction, previously viewed as a challenge, as a rich informative structure that facilitates instruction optimization for black-box LLMs.

Leveraging this insight, we introduce PRESTO, a novel framework that consists of score sharing, preimage-based initialization, and score consistency regularization.

PRESTO achieves state-of-the-art performance across 30 instruction induction and 3 arithmetic reasoning tasks.

2 Related Works

Instruction Optimization for Black-box LLMs Instruction optimization has been widely explored as a way to improve the performance of large language models (LLMs) on downstream tasks [23, 24]. In particular, when using black-box LLMs such as GPT-4 [2], where access to model parameters is restricted, optimization methods rely on model outputs to guide the search for better instructions. Under this setting, various approaches have been proposed, including evolutionary algorithms [10, 11], LLM-driven meta-optimization [8, 9], and bandit-style or heuristic search methods [13, 12]. These works demonstrate that instruction quality can be improved even without access to gradients or internal representations by querying the black-box model efficiently.

More recently, some methods [14-16] incorporate open-source white-box LLMs [1, 18, 17, 25] to assist the optimization process. Rather than optimizing instruction texts directly, they optimize soft prompts, which are continuous embeddings that the white-box model maps into instructions. InstructZero [14] leveraged Bayesian Optimization [26-28] to search for the optimal soft prompts for black-box LLM. INSTINCT [16] leveraged NeuralUCB [21] with an LLM-based score predictor, which was the first to point out the many-to-one schema and approached it indirectly by sampling soft prompts to be well-separated. And ZOPO [15] proposed a zeroth-order optimization algorithm [29] for local search, which addresses this redundancy by simply discarding all but one soft prompt that produces the same instruction. In contrast, we retain all soft prompts by introducing preimages and facilitate the optimization.

3 Preliminaries

Problem Formulation Instruction optimization aims to find an instruction v that guides a language model to perform a given task effectively. To be specific, the goal is to find the instruction v that maximizes the task-specific score function h by guiding a black-box LLM f b to generate the correct answer y , which is formally given as:

where D val = { ( x i , y i ) } i M =1 is a validation set, and Ω denotes the search space of instructions, typically a discrete sequence domain ( e.g., natural language prompts or token sequences). However, directly searching over discrete instruction sequences is challenging, as it constitutes a combinatorial optimization problem over the space of token configurations. To address this, InstructZero [14] reformulates the discrete instruction search as a continuous optimization problem by leveraging a white-box LLM f w. Specifically, it optimizes a soft prompt z ∈ R N z × d , where N z is the number of tokens and d is the embedding dimension, to generate the optimal instruction v ∗ . The soft prompt is concatenated with the token embeddings of input-output exemplars E = { ( x i , y i ) } κ i =1 and fed into the white-box LLM f w, which then generates an instruction v = f w ( z, E ) . Formally, the instruction optimization problem is defined as:

where Z is the soft prompt space. In this formulation, we optimize z to find the optimal instruction v ∗ that maximizes the expected value of the score function h . Once the optimal soft prompt z ∗ is obtained, the corresponding instruction v ∗ is generated by the white-box LLM f w, i.e., v ∗ = f w ( z ∗ , E ) and subsequently evaluated on a held-out test set D test. Since the exemplars E are fixed for each task, we omit them from the notation in the rest of our paper. Following previous works [14-16], we assume that both the white-box LLM f w and the black-box LLM f b are deterministic.

LLM-based Score Predictor for Instruction Optimization. Our method builds upon INSTINCT [16], which employs a frozen white-box LLM as a feature extractor to predict the score of soft prompt, and uses a NeuralUCB [21] for instruction optimization. Given a soft prompt z , the white-box LLM produces an embedding g ( z ) , the last token representation of the final transformer layer. This embedding is then passed to a score predictor m ( g ( z ); θ ) ( e.g., an MLP), which predicts the performance of the instruction generated from z , i.e. , m ( g ( z ); θ ) ≈ E ( x,y ) ∈ D [ h ( f b ( f w ( z ) , x ) , y )] . At each optimization step, the score predictor m ( · ; θ ) is trained on previously evaluated soft prompts and their corresponding scores, and selects the next query that maximizes the upper confidence bound. We provide further details of NeuralUCB in the supplement. Since computing g ( z ) requires a full

Initialization

InitialGroupSelection

byPreimage-Based

G1 < G2,G3

Initialization

Score

RandomSampling

{Zi

WhiteBox

LLM

U2

V2

U1

U2

G2

U1

U2

BlackBox

LLM

Sharing

中

0.4

0.4

0.4

???

Prompt

U3

U3

G3

U3

C

fb

0.3

0.3

Soft Prompt

Embedding

Instruction

Preimages of Instructions

Score

Exemplar

g(2)

InitialData

一一

Optimization

SoftPrompt

Instruction

Embedding

Score

Scored

U1

0.72

0.72

UpdatetheScoredDataset

Predicted Score

0.60±0.20

Unscored

G2

02

？??

0.60±0.18

0.60±0.21

X

NextQuery

BlackBox

LLM

Score

SoftPrompt

Embedding

G3

V3

?？?

0.80±0.30

U3

fb

0.82

0.80±0.28

0.82

ScorePredictorm

Select

Score

Next Query

0.8.

W/UCB

0.72 -

Train Score Predictor with CMSE

0.6 .

forward pass through the LLM, INSTINCT mitigates this cost by precomputing the embeddings of a candidate soft prompt set Z = { z i } N i =1 at the beginning of the optimization, which is sampled using a quasi-random method. To this end, the instruction optimization task is reduced to searching for the best solution within the precomputed embedding set, as the white-box LLM is frozen during the optimization process.

4 Method

In this section, we propose PRE image-informed in ST ruction O ptimization (PRESTO) which is a novel instruction optimization framework that leverages the many-to-one mapping between soft prompts z ∈ Z ⊂ Z and instructions v ∈ Ω (or the preimages of instructions, which is defined in Section 4.1) as prior knowledge to facilitate more efficient optimization. We first introduce a score sharing method that shares the score value of one soft prompt with all other soft prompts in the same preimage, effectively enlarging the scored data without additional evaluations of black-box LLM f b. Next, we present a preimage-based initialization method designed to maximize coverage of the search space under score sharing. Finally, we propose a score consistency regularization that leverages preimage information as prior knowledge to encourage the score predictor to predict identical scores for soft prompts belonging to the same preimage. We provide the overall framework of our PRESTO in Figure 2.

4.1 Preimage-Based Score Sharing

During the instruction optimization, we observe that the white-box LLM f w often generates identical instructions from distinct soft prompts, i.e. , f w ( z ) = f w ( z ′ ) , leading to the same score value. This redundancy leads to unnecessary queries during optimization, hindering the efficiency of instruction

optimization. While previous works treated this redundancy as an obstacle to efficient optimization, we instead leverage this information as prior knowledge about the objective function to facilitate optimization. To this end, we propose a simple score sharing scheme that associates a large number of soft prompts with a score value without the additional evaluations of a black-box LLM f b.

Our goal is to share the score of an evaluated soft prompt z with other soft prompts that generate the same instruction. To enable this score sharing, we first define the preimage of each instruction which consists of all soft prompts that map to the same instruction under the white-box model f w. Establishing this preimage structure requires two steps. First, we sample a soft prompt set Z = { z i } N i =1 using a quasi-random method [30, 31], which is a widely adopted method to sample the data points that evenly cover the soft prompt space [14, 16, 15]. Assuming that the soft prompt set size N is large enough to represent the soft prompt space Z , the original optimization problem defined in Eq. (2) reduces to searching for the best solution among the set of N data points, denoted by Z ⊂ Z .

Next, for each soft prompts z j ∈ Z , we generate the set of instructions V = { v i } i M =1 , using the white-box LLM f w:

Since the different soft prompts often generate the identical instruction ( i.e., many-to-one mapping), the number of instructions M = | V | is smaller than or equal to N . The construction of Z and V is performed only once before the optimization process begins.

With the soft prompt set Z and the corresponding instruction set V , we now define the preimage of each instruction. The preimage of an instruction v is the set of soft prompts in Z that generate v under the white-box model f w:

This preimage contains all soft prompts in Z that generate v , and will serve as the basis for score sharing. Once the preimages f -1 w ( v ) for all v ∈ V are established, we apply score sharing across soft prompts that belong to the same preimage during the optimization. Specifically, after querying the black-box model f b with an instruction v ∈ V , we obtain a score of the instruction. This score is then shared to all soft prompts in the preimage f -1 w ( v ) . By sharing scores in this manner, we effectively enlarge the training data for the score predictor m ( g ( z ); θ ) without additional calls to the black-box LLMs. Moreover, score sharing avoids redundant evaluations of soft prompts that lead to the same instruction and improves optimization efficiency.

4.2 Preimage-Based Initialization for Maximizing Search Space Coverage

Here, we introduce a preimage-based initialization method that selects initial data points based on the preimage information defined in Section 4.1. At the beginning of the optimization, the score predictor m ( g ( z ); θ ) (Section 3) is trained on the initial dataset, and its predictions are used to select the next data points to query the black-box LLM f b. In black-box optimization, it is well known that broadly covering the search space at initialization is crucial for effective optimization [32-35]. Our score sharing method introduced in Section 4.1 expands the initial dataset without additional queries to the black-box LLM f b, enabling a more sample-efficient initialization. To further enhance the search space coverage, we propose a preimage-based initialization method that complements score sharing by promoting a broader initial data distribution.

To this end, we design a coverage score S cov to guide the selection of an initial preimage set G init that maximally covers the entire set of soft prompt embeddings G total = { g ( z ) | z ∈ Z } . We conduct initialization in the embedding space rather than the raw soft prompt space, since the optimization operates over the soft prompt embeddings. These embeddings are precomputed and remain fixed throughout the optimization, as described in Section 3. For each instruction v i , we define its corresponding preimage group in the embedding space as G i = { g ( z ) | z ∈ f -1 w ( v i ) } .

Since finding the optimal combination of N init preimages that maximizes the coverage score S cov is a computationally intractable combinatorial optimization problem, we adopt a greedy algorithm to iteratively select one preimage at a time. Specifically, the coverage score S cov consists of two components: the representativeness score S rep and the size score S size. The representativeness score S rep encourages the selection of a preimage group G i that, when combined with already selected

(a) Train with

L

MSE only.

(b) Train with

L

MSE

+

γ

L

cons

.

w/oConsistencyLoss

w/ ConsistencyLoss

Truefunction

Modelprediction

Train Data

G1,y= -0.5

G2,y=0

G3,y=0.5

Groupconsistency

X

preimage groups G init , most closely matches the distribution of the candidate set G total , defined as:

where the MMD 2 is the squared Maximum Mean Discrepancy. MMD 2 is a widely used metric to estimate the similarity between two sets, which is defined as:

where k ( · , · ) is a positive definite kernel. To densely cover the search space, we propose the size score S size, which is defined as relative preimage size: S size ( G i ) = | G i | / max j | G j | . Combining the two scores, we define the coverage score for the G i :

Starting from an empty set G init , we iteratively select the preimage with the highest coverage score S cov and add it to G init until the number of initial preimages reaches N init. This initialization maximizes the coverage of the candidate set G total . We provide the visualization to demonstrate the effectiveness of our initialization method in Section 6.3.

4.3 Score consistency regularization for score predictor

Here, we propose a score consistency regularization that encourages the score predictor m ( g ( z ); θ ) to produce the same prediction for all soft prompts in preimages that have not been evaluated by the black-box function. During the optimization, the score predictor is trained with the scored data to predict the score of each soft prompt in the candidate set Z and estimate its uncertainty for selecting the next query to evaluate. Leveraging the score sharing method defined in Section 4.1 informs the score predictor that data points within the same preimage share identical scores in a supervised manner. However, since the score predictor lacks information about score consistency within unscored preimages, it is unable to make consistent predictions for data points in these unscored preimages. It often hinders the score predictor from predicting the ground truth score and selecting high-scored data.

To ensure consistent predictions within each unscored preimage, we propose a score consistency regularization term L cons, which is defined as:

where V unseen ⊂ V denotes the set of instructions that has not been evaluated by the black-box LLM f b . We note that L cons is an unsupervised loss. While the consistency regularization includes pairwise terms per preimage group, each unscored preimage size is not excessively large in practice, so the computation remains tractable. The final loss for training the score predictor model is given by:

where L MSE is the mean squared error loss computed over the scored preimages, and γ is a hyperparameter controlling the strength of the regularization. To avoid premature convergence to incorrect predictions, we employ a simple linear scheduling strategy as γ ( t ) = γ max · min ( 1 , t/T ) , where t

Tasks

APE

InstructZero

INSTINCT

EvoPrompt

ZOPO

OPRO

PRESTO

80.67 ± 0.72

75.33 ± 3.21

83.33

± 0.54

82.00 ± 0.47

82.67 ± 1.66

80.33 ± 2.33

83.33

± 1.19

auto_categorization

26.00 ± 6.13

27.67 ± 2.60

18.67 ± 0.72

29.33 ± 2.18

31.67

± 3.41

30.33 ± 0.72

31.67

± 3.41

auto_debugging

8.33 ± 6.80

12.50 ± 5.89

10.00 ± 4.71

16.67 ± 6.80

13.33 ± 7.20

8.33 ± 6.80

20.83

± 3.40

cause_and_effect

92.00 ± 1.89

74.67 ± 4.75

76.00 ± 9.98

72.00 ± 6.80

93.33 ± 2.88

38.67 ± 4.35

94.67

± 2.88

common_concept

22.36 ± 2.34

15.53 ± 5.11

20.21 ± 1.19

17.99 ± 6.72

21.86 ± 7.16

20.08 ± 6.70

22.86

± 3.27

diff

18.33 ± 6.87

53.00 ± 20.37

81.67 ± 13.76

7.00 ± 5.72

88.33 ± 5.93

64.33 ± 23.91

98.00

± 0.82

informal_to_formal

57.59 ± 2.40

51.53 ± 4.62

48.93 ± 3.46

42.87 ± 2.03

58.93

± 4.83

50.02 ± 2.63

52.77 ± 5.46

letters_list

99.00 ± 0.82

99.00 ± 0.47

97.67 ± 1.52

73.67 ± 9.69

98.67 ± 1.09

99.00 ± 0.47

99.33

± 0.54

negation

83.33 ± 1.19

81.67 ± 3.95

76.67 ± 4.77

71.67 ± 1.19

77.33 ± 4.63

73.33 ± 4.23

84.00

± 2.16

object_counting

37.33 ± 5.50

46.00 ± 5.72

48.67

± 3.21

28.67 ± 2.23

34.00 ± 4.08

31.00 ± 3.86

45.67 ± 4.38

odd_one_out

51.33 ± 14.43

46.67 ± 5.76

60.00 ± 7.12

68.00 ± 1.89

58.67 ± 7.14

47.33 ± 10.39

70.00

± 0.94

orthography_starts_with

46.00 ± 8.18

35.00 ± 3.56

54.67 ± 8.20

42.00 ± 15.28

54.67 ± 3.66

22.33 ± 10.18

57.33

± 6.08

rhymes

69.33 ± 16.41

81.67 ± 10.69

98.67

± 0.72

93.67 ± 1.96

83.33 ± 6.87

77.00 ± 15.25

85.00 ± 7.41

second_word_letter

72.67 ± 10.88

40.67 ± 5.99

48.00 ± 22.38

33.00 ± 7.93

68.00 ± 17.75

22.00 ± 14.73

77.00

± 12.57

sentence_similarity

29.00

± 5.44

17.33 ± 4.75

11.33 ± 5.42

29.00

± 0.47

4.33 ± 3.54

6.67 ± 5.44

21.67 ± 8.49

sum

24.00 ± 14.61

55.00 ± 23.92

99.33 ± 0.54

66.67 ± 27.22

100.00

± 0.00

91.33 ± 3.78

94.67 ± 4.35

synonyms

10.00 ± 4.50

22.67 ± 5.62

25.00 ± 8.83

25.33

± 7.98

24.33 ± 2.76

12.67 ± 0.72

18.33 ± 1.91

taxonomy_animal

43.67 ± 15.96

44.33 ± 17.72

92.00 ± 3.77

34.00 ± 15.08

69.00 ± 24.10

73.67 ± 8.09

99.67

± 0.27

word_sorting

54.00 ± 15.41

39.67 ± 12.11

27.33 ± 7.37

71.00

± 4.50

54.00 ± 15.06

36.33 ± 11.49

53.33 ± 8.38

word_unscrambling

28.00 ± 4.78

38.00 ± 3.74

42.33 ± 8.59

23.00 ± 9.57

52.00

± 7.79

43.00 ± 1.25

48.00 ± 7.59

# best-performing tasks

1

0

3

3

4

0

12

Average Rank

4.25

4.80

3.70

4.70

3.05

5.20

1.90

represents the current epoch and T is a warm-up duration. This schedule allows the score predictor m ( g ( z ); θ ) to learn accurate patterns from the scored data and gradually incorporate the score equality constraint of unscored data.

Figure 3 shows a toy example illustrating the effect of the proposed consistency loss. We use a simple model with two linear layers. In Figure 3a, the model is trained only with the L MSE on the scored data, while in Figure 3b, L cons is additionally applied to unscored data. We assume there are three unscored preimages, each represented by a different marker shape. Although the model is only given the information that data points within each preimage share the same score, the L cons allows it to make more accurate predictions on the unscored data.

5 Experiments

5.1 Experimental settings

We evaluate our proposed method, PRESTO, on 30 instruction induction tasks [36], a benchmark widely used to assess instruction optimization performance, and 3 arithmetic reasoning tasks [3739]. We compare PRESTO with six competitive instruction optimization baselines: APE [8], InstructZero [14], INSTINCT [16], EvoPrompt [10], ZOPO [15], and OPRO [9]. We use LLaMA3.18B-Instruct [1] as the white-box LLM f w to generate candidate instructions, and GPT-4.1 as the black-box model f b. Following previous works [14-16], we set the total query budget to 165, initialize with 40 soft prompts, and evaluate all methods over three different random seeds. To ensure a fair comparison, we follow the hyperparameter tuning procedure in [16]. Detailed hyperparameter configurations and experimental settings are provided in the supplement.

5.2 Instruction induction results

Here we provide the results of our proposed method, PRESTO, compared with six strong baselines on instruction induction tasks. To enhance readability, we report results on a subset of 20 following previous works [16, 15]. The full results for all 30 tasks are provided in the appendix. Table 1 shows that PRESTO achieves the highest accuracy on 12 out of the 20 tasks, which is three times more than the second-best method, ZOPO. In addition, PRESTO attains the best average rank of 1.90, outperforming all baselines by a clear margin; the next best, ZOPO, has an average rank of 3.05, followed by INSTINCT at 3.70. These results highlight the strong performance of PRESTO on individual tasks and its robustness across a wide range of instruction induction tasks. In the full set of 30 tasks, PRESTO also consistently outperforms other baselines with a large margin in the number of best-performing tasks and average rank.

5.3 Chain-of-Thought Prompting Results

We evaluate the quality of the optimized instructions by measuring their effectiveness as chainof-thought (CoT) [40] prompts on three math reasoning benchmarks: GSM8K [37], AQUARAT [38], and SVAMP [39]. We compare our method with three baselines that use soft prompts (InstructZero [14], INSTINCT [16], and ZOPO [15]), as well as a standard hand-crafted prompt [41]. Table 2 demonstrates that our PRESTO outperforms or matches the best-performing baselines across all datasets. In particular, it achieves the highest accuracy on AQUA-RAT (0.7756) and ties for the best result on SV AMP (0.9400), while remaining competitive on GSM8K. These results indicate that the instructions optimized by our method are also effective when used as CoT prompts.

6 Analysis

6.1 Ablation Study

We perform an ablation study to analyze the contribution of each component in our method over the 20 instruction induction tasks used in Table 1 over 3 random seeds. Starting from a vanilla baseline without our techniques, we incrementally add: (1) score sharing method (Section 4.1), (2) preimage-based initialization (Section 4.2), and (3) score consistency regularization (Section 4.3). The full model with all components combined corresponds to our proposed method, PRESTO. As shown in Table 3, each component contributes to performance improvement. In particular, introducing score sharing significantly boosts accuracy from 51.91 to 59.57 (+7.66) and improves average rank from 4.55 to 3.10 (-1.45), indicating its strong impact. Our PRESTO achieves the best results overall, with the highest number of wins and the lowest average rank across tasks.

6.2 Impact of score sharing method

We report the average number of soft prompts with assigned scores after the optimization process, comparing our method with baselines across all 30 tasks. The reported count includes soft prompts that were scored either directly through black-box evaluation or indirectly via score sharing. As shown in Figure 4, our method assigns scores to over 2,300 soft prompts on average, 14× more than

3000

2500

Scored data

2000

1500

1000

#

500

165

All previous

PRESTO

method

(Ours)

Score predictor performance

0.3

0.25

RMSE

0.2

0.15

0.1

Vanilla

+Reg

+SS

Random init.(Previous)

Random init.+Score Sharing

Preimage-based init.(Srep)

Preimage-based init. (Scov)

(a)

(b)

(c)

(d)

previous methods, which yield only 165 scored data points, equal to the query budget in our setting. The large amount of scored data enables the score predictor to learn the objective function more effectively, which in turn facilitates more successful optimization. This analysis demonstrates that our score-sharing method can significantly increase the amount of scored data without requiring additional black-box queries.

6.3 Visualization of Preimage-Based Initialization

We present a qualitative analysis of how score sharing and preimage-based initialization influence the distribution of initial soft prompts. Figure 6 visualizes the distribution of initial soft prompts under four settings: (1) random initialization, (2) random initialization with score sharing (Section 4.1), (3) preimage-based initialization using S rep only, and (4) S cov = S rep + S size (Section 4.2) in "objective counting" task. To visualize the spatial distribution of soft prompt embeddings, we employ t-SNE. Compared to random initialization in prior works, score sharing enlarges the size of the initial dataset without additional black-box queries. Furthermore, selecting initial data using S rep leads to better coverage of the soft prompt space than naive score sharing. Finally, our proposed preimage-based initialization method that utilizes S cov achieves the densest and comprehensive coverage of the search space. It shows that our preimage-based initialization method effectively selects the initial data points that densely and evenly cover the search space.

6.4 Score Predictor Performance Enhancement

To analyze how score sharing (Section 4.1) and score consistency regularization (Section 4.3) influence the quality of the score predictor, we evaluate its prediction performance under different training configurations. Figure 5 reports the root mean squared error (RMSE), where lower values indicate higher prediction accuracy. We use 100 randomly selected soft prompts as training data and another 100 as test data for the objective counting task. As shown in Figure 5, applying either score sharing or score consistency regularization improves the score predictor's performance, reducing the RMSE from approximately 0.27 (vanilla) to around 0.23. When both techniques are applied together, the RMSE further decreases to approximately 0.15, indicating a strong complementary effect. The results demonstrate that expanding the training set without requiring additional black-box queries

through score sharing and incorporating the preimage structure as a prior via score consistency regularization are both crucial for enhancing the score predictor's performance.

7 Conclusion

We propose PRESTO, a preimage-informed instruction optimization framework that explicitly leverages this many-to-one structure via preimage. PRESTO consists of three components that leverage the preimage structure: score sharing to propagate labels within each preimage, preimage-based initialization to improve search space coverage, and consistency regularization to align predictions within unscored preimages. PRESTO achieves state-of-the-art performance on 33 instruction optimization tasks, and our comprehensive analysis supports its effectiveness and robustness.

Limitations and broader impacts

Our method introduces preimage-based score sharing to enlarge the number of data, which incurs mild computational overhead compared to simpler baselines. Moreover, its benefits are more pronounced when applied to a large candidate set, as score sharing is most effective when many soft prompts map to the same instruction.

In terms of broader impact, this work aims to make black-box LLM optimization more data-efficient, which can reduce the cost of experimentation and improve accessibility for researchers with limited resources. However, as with any optimization technique for LLMs, there is a risk that improved performance could be applied in ways that reinforce biases or generate harmful content. Careful deployment and alignment with responsible AI principles are necessary.

References

Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, et al. The llama 3 herd of models. arXiv , 2024.

Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report. arXiv , 2023.

Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948 , 2025.

Arun James Thirunavukarasu, Darren Shu Jeng Ting, Kabilan Elangovan, Laura Gutierrez, Ting Fang Tan, and Daniel Shu Wei Ting. Large language models in medicine. Nature medicine , 2023.

Humza Naveed, Asad Ullah Khan, Shi Qiu, Muhammad Saqib, Saeed Anwar, Muhammad Usman, Naveed Akhtar, Nick Barnes, and Ajmal Mian. A comprehensive overview of large language models. arXiv preprint arXiv:2307.06435 , 2023.

Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al. Training language models to follow instructions with human feedback. Advances in neural information processing systems , 35:27730-27744, 2022.

Pengfei Liu, Weizhe Yuan, Jinlan Fu, Zhengbao Jiang, Hiroaki Hayashi, and Graham Neubig. Pre-train, prompt, and predict: A systematic survey of prompting methods in natural language processing. ACM computing surveys , 2023.

Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. Large language models are human-level prompt engineers. In ICLR , 2022.

Chengrun Yang, Xuezhi Wang, Yifeng Lu, Hanxiao Liu, Quoc V Le, Denny Zhou, and Xinyun Chen. Large language models as optimizers. In ICLR , 2024.

Qingyan Guo, Rui Wang, Junliang Guo, Bei Li, Kaitao Song, Xu Tan, Guoqing Liu, Jiang Bian, and Yujiu Yang. Connecting large language models with evolutionary algorithms yields powerful prompt optimizers. In ICLR , 2024.

Chrisantha Fernando, Dylan Sunil Banarse, Henryk Michalewski, Simon Osindero, and Tim Rocktäschel. Promptbreeder: Self-referential self-improvement via prompt evolution. In International Conference on Machine Learning , 2024.

Reid Pryzant, Dan Iter, Jerry Li, Yin Tat Lee, Chenguang Zhu, and Michael Zeng. Automatic prompt optimization with" gradient descent" and beam search. In EMNLP , 2023.

Chengshuai Shi, Kun Yang, Zihan Chen, Jundong Li, Jing Yang, and Cong Shen. Efficient prompt optimization through the lens of best arm identification. arXiv preprint arXiv:2402.09723 , 2024.

Lichang Chen, Jiuhai Chen, Tom Goldstein, Heng Huang, and Tianyi Zhou. Instructzero: Efficient instruction optimization for black-box large language models. In ICML , 2024.

Wenyang Hu, Yao Shu, Zongmin Yu, Zhaoxuan Wu, Xiaoqiang Lin, Zhongxiang Dai, SeeKiong Ng, and Bryan Kian Hsiang Low. Localized zeroth-order prompt optimization. NeurIPS , 2024.

Xiaoqiang Lin, Zhaoxuan Wu, Zhongxiang Dai, Wenyang Hu, Yao Shu, See-Kiong Ng, Patrick Jaillet, and Bryan Kian Hsiang Low. Use your instinct: Instruction optimization for llms using neural bandits coupled with transformers. In ICML , 2024.

Albert Q Jiang, Alexandre Sablayrolles, Antoine Roux, Arthur Mensch, Blanche Savary, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Emma Bou Hanna, Florian Bressand, et al. Mixtral of experts. arXiv preprint arXiv:2401.04088 , 2024.

An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, et al. Qwen2. 5 technical report. arXiv preprint arXiv:2412.15115 , 2024.

Peter I Frazier. A tutorial on bayesian optimization. arXiv preprint arXiv:1807.02811 , 2018.

Bobak Shahriari, Kevin Swersky, Ziyu Wang, Ryan P Adams, and Nando De Freitas. Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE , 2015.

Dongruo Zhou, Lihong Li, and Quanquan Gu. Neural contextual bandits with ucb-based exploration. In ICML , 2020.

Weitong Zhang, Dongruo Zhou, Lihong Li, and Quanquan Gu. Neural thompson sampling. arXiv preprint arXiv:2010.00827 , 2020.

Krista Opsahl-Ong, Michael Ryan, Josh Purtell, David Broman, Christopher Potts, Matei Zaharia, and Omar Khattab. Optimizing instructions and demonstrations for multi-stage language model programs. In EMNLP , 2024.

Xiaoqiang Lin, Zhongxiang Dai, Arun Verma, See-Kiong Ng, Patrick Jaillet, and Bryan Kian Hsiang Low. Prompt optimization with human feedback. arXiv preprint arXiv:2405.17346 , 2024.

Albert Q. Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, Lélio Renard Lavaud, Marie-Anne Lachaux, Pierre Stock, Teven Le Scao, Thibaut Lavril, Thomas Wang, Timothée Lacroix, and William El Sayed. Mistral 7b. arXiv , 2023.

Seunghun Lee, Jaewon Chu, Sihyeon Kim, Juyeon Ko, and Hyunwoo J Kim. Advancing bayesian optimization via learning correlated latent space. Advances in Neural Information Processing Systems , 2023.

Jaewon Chu, Jinyoung Park, Seunghun Lee, and Hyunwoo J Kim. Inversion-based latent bayesian optimization. In The Thirty-eighth Annual Conference on Neural Information Processing Systems , 2024.

Seunghun Lee, Jinyoung Park, Jaewon Chu, Minseo Yoon, and Hyunwoo J Kim. Latent bayesian optimization via autoregressive normalizing flows. In The Thirteenth International Conference on Learning Representations , 2025.

Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh. Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM workshop on artificial intelligence and security , pages 15-26, 2017.

William J Morokoff and Russel E Caflisch. Quasi-random sequences and their discrepancies. SIAM Journal on Scientific Computing , 15(6):1251-1279, 1994.

Marissa Renardy, Louis R Joslyn, Jess A Millar, and Denise E Kirschner. To sobol or not to sobol? the effects of sampling schemes in systems biology applications. Mathematical biosciences , 337:108593, 2021.

Donald R Jones, Matthias Schonlau, and William J Welch. Efficient global optimization of expensive black-box functions. Journal of Global optimization , 13:455-492, 1998.

David Eriksson, Michael Pearce, Jacob Gardner, Ryan D Turner, and Matthias Poloczek. Scalable global optimization via local bayesian optimization. Advances in neural information processing systems , 32, 2019.

Samuel Daulton, Maximilian Balandat, and Eytan Bakshy. Differentiable expected hypervolume improvement for parallel multi-objective bayesian optimization. Advances in Neural Information Processing Systems , 33:9851-9864, 2020.

Samuel Daulton, Maximilian Balandat, and Eytan Bakshy. Parallel bayesian optimization of multiple noisy objectives with expected hypervolume improvement. Advances in Neural Information Processing Systems , 34:2187-2200, 2021.

Or Honovich, Uri Shaham, Samuel R Bowman, and Omer Levy. Instruction induction: From few examples to natural language task descriptions. In 61st Annual Meeting of the Association for Computational Linguistics, ACL 2023 , pages 1935-1952, 2023.

Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, et al. Training verifiers to solve math word problems. arXiv preprint arXiv:2110.14168 , 2021.

Wang Ling, Dani Yogatama, Chris Dyer, and Phil Blunsom. Program induction by rationale generation: Learning to solve and explain algebraic word problems. arXiv preprint arXiv:1705.04146 , 2017.

Arkil Patel, Satwik Bhattamishra, and Navin Goyal. Are nlp models really able to solve simple math word problems? arXiv preprint arXiv:2103.07191 , 2021.

Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al. Chain-of-thought prompting elicits reasoning in large language models. Advances in neural information processing systems , 35:24824-24837, 2022.

Takeshi Kojima, Shixiang Shane Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwasawa. Large language models are zero-shot reasoners. Advances in neural information processing systems , 35:22199-22213, 2022.

NeurIPS Paper Checklist

The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.

Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:

You should answer [Yes] , [No] , or [NA] .

[NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.

Please provide a short (1-2 sentence) justification right after your answer (even for NA).

The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.

The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes] " is generally preferable to "[No] ", it is perfectly acceptable to answer "[No] " provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No] " or "[NA] " is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.

IMPORTANT, please:

Delete this instruction block, but keep the section heading 'NeurIPS Paper Checklist" ,

Keep the checklist subsection headings, questions/answers and guidelines below.

Do not modify the questions and only use the provided macros for your answers .

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The abstract and introduction clearly state the main claims (Section 1, 4). Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: The paper discusses limitations after Section 7.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [NA]

Justification: The paper does not present formal theoretical proofs, focusing instead on algorithmic design and empirical validation.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: Section 5.1 provides full experimental settings, and additional details are available in the supplementary material.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: The paper states that detailed configurations and implementation details are provided in Section 5.1 and the supplement, and the authors intend to release code and data after acceptance.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: The paper describes test splits, model usage, total budget, search space construction, and optimization strategies.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: The tables report averages with standard errors over three random seeds, and all comparisons to baselines include this variance (see Tables 1).

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: The supplementary materials include detailed specifications, including hardware type.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: The work conforms to NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: The paper discusses broader impacts after Section 7.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The method does not release any high-risk models or datasets.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: The paper appropriately cites existing models and datasets with references including model names and publications.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: No new dataset, model, or tool is released. The method is built on existing assets.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The research does not involve any human subjects or crowdsourcing experiments.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: No human subject research is conducted, hence IRB approval is not applicable. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [Yes]

Justification: The method leverages both white-box and black-box LLMs as part of its core optimization framework.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles inefficient instruction optimization for black-box LLMs, a problem arising when different \"soft prompts\" generate the same instruction. The authors reframe this many-to-one mapping as a useful \"pre-image\" structure and introduce PRESTO, a framework that leverages it for efficient optimization. PRESTO employs three techniques: score sharing, which propagates a single evaluated score to all soft prompts within its preimage; preimage-based initialization, which improves initial data selection to maximize search space coverage; as well as score consistency regularization, which enforces consistent predictions for prompts within the same preimage. Over dozens of different tasks, PRESTO can achieve comparably or better performance on instruction optimization tasks compared with baselines.\n\nSTRENGTHS AND WEAKNESSES:\nPositive:\n\n1. The manuscript is generally well-structured and easy to follow. Figure 2 also provides a clear high-level illustration and narrative of the overall pipeline. Augmenting the data to provide abundant learning signals without additionally querying the expensive API is intuitive.\n\n2. The method's robustness is validated through extensive experiments across various combinations of white-box (LLaMA, Qwen, Mistral) and black-box (GPT-4.1, Gemini-2.0-Flash) language models. Statistical significance results are also reported via standard deviation. The extensive experiments are the main contribution of this work.\n\n3. Code implementation is provided for reproducing the results.\n\n\nNegative:\n\n1.\tThe method is an extension of the existing INSTINCT framework, using the related NeuralUCB algorithm and codebase, leading to limited algorithmic contributions and limited novelty. The core techniques are based on well-established machine learning concepts, including distribution matching with MMD for initialization, and consistency regularization for training. \n\n2.\tThe training pipeline still needs further justifications since it is purely heuristic rather than theoretically justified by the authors. For instance, what is the motivation and justification of designing the coverage score as Eq (7) other than simple heuristics? The toy example along is also not sufficient since it is merely a toy example with a small network of two layers as suggested by the authors.\n\n3.\tWhile an efficiency analysis is provided, the paper lacks an explicit wall-clock time comparison with the baselines, which can better demonstrate the computational overhead of the new components, especially with the newly added soft prompt augmentation techniques.\n\n4.\tWhile the experiments cover over 30 tasks, most of them are simple induction tasks. Although additional evaluations on three math reasoning tasks are included in Table 2, there are no significant insights derived from these empirical results, as the best instructions across methods are very similar and notably short. The authors also provide no further analysis or justification for the advantages of the generated instructions produced by their method, other than good performance (248-255). Given the small performance gaps and the highly similar instructions, it is difficult to justify the superiority of the proposed approach over baselines, on the other tasks beyond simple instruction induction.\n\nQUESTIONS:\nPlease kindly refer to my comments above\n\nLIMITATIONS:\nThere is a limitations section after Section 7", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces PRESTO, a novel framework for optimizing instructions for black-box large language models (LLMs) by leveraging the preimage structure—the many-to-one mapping from soft prompts to instructions produced by a white-box LLM. Unlike previous work, which treats this redundancy as a nuisance, PRESTO exploits it to improve sample efficiency and optimization performance. The framework consists of three main components: (1) score sharing across soft prompts in the same preimage, (2) preimage-based initialization to increase search space coverage, and (3) score consistency regularization for the score predictor. Experiments on 33 instruction optimization tasks show that PRESTO outperforms strong baselines.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- Novelty. The key insight—interpreting the many-to-one mapping from soft prompts to instructions as a source of prior knowledge rather than redundancy—is both original and well-motivated. This perspective enables PRESTO to achieve significant improvements in query efficiency and optimization performance\n\n- Empirical Results. The method is evaluated on a broad set of 33 instruction optimization tasks (30 induction, 3 arithmetic reasoning), achieving state-of-the-art performance compared to strong baselines such as INSTINCT and ZOPO. PRESTO shows the effect of observing up to 14 times more scored data under the same query budget.\nClarity. The paper is well-organized, with clear motivation, background, and methodological exposition. The figures (e.g., distribution of preimage sizes) effectively illustrate the core concepts. \n\n## Weaknesses\n\n- Assumptions about Preimage Structure: PRESTO assumes that the preimage structure (i.e., which soft prompts map to which instructions) can be efficiently established by sampling a large set of soft prompts and mapping them through the white-box LLM. The scalability of this approach to very large prompt spaces or more complex instruction sets is not fully discussed.\n- Dependence on White-Box LLMs: The framework relies on access to a white-box LLM to generate instructions from soft prompts and to extract embeddings for the score predictor. This may limit applicability in scenarios where only black-box LLMs are available, or where the white-box model's behavior diverges from the black-box model.\n- Evaluation Scope: While the paper reports strong results on 33 tasks, it is unclear how PRESTO performs on tasks with fundamentally different instruction distributions or in settings where the preimage sizes are much smaller (i.e., less redundancy to exploit). Additional ablations or analysis on such edge cases would strengthen the empirical claims.\n- Computational Overhead: The need to precompute and store large sets of soft prompts, instructions, and preimage mappings may incur significant computational and memory overhead, especially as the candidate set grows. The paper would benefit from a more detailed discussion of these practical considerations.\n\nQUESTIONS:\nAblation Studies:\nThe paper would be improved by including ablations isolating the impact of each PRESTO component (score sharing, initialization, regularization) to better understand their individual contributions.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles the problem of automatically optimizing instructions for black-box LLMs by leveraging an auxiliary white-box LLM. Prior work generates soft prompts that are fed into a white-box model to propose candidate instructions, but often different soft prompts map to the same instruction. Rather than regarding this redundancy as wasteful, the authors introduce PRESTO, which exploits the preimage structure to amplify the effective data and guide search. There are 3 major components in the proposed method: (1) score sharing, which shares the evaluation score with all soft prompts in a preimage; (2) preimage-based initialization, which selects initial data points that maximize search space coverage using preimage information; and (3) score consistency regularization, which enforces prediction consistency within each preimage. Empirically, PRESTO uses the same query budget but effectively learns from 14× more “virtual” data points, yielding superior performance on 33 instruction-induction tasks and three arithmetic reasoning benchmarks, outperforming strong baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: \n\n1. The idea of reinterpreting the many-to-one mapping as useful prior knowledge to enhance the optimization performance is novel. \n\n2. The experimental results on a variety of datasets show the effectiveness of the proposed method over the compared baselines.\n\nWeakness:\n\nIn general I think the paper is good. However, I have one concern - though the method is designed to improve the data efficiency, I am curious about the overall computational efficiency of the proposed method. For example, how will the cost of MMD scale with the candidate set size? A more detailed comparison of the algorithm efficiency with the baseline method is desired.\n\nQUESTIONS:\nHow will the cost of MMD scale with the candidate set size?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}