Knowledgeable Language Models as Black-Box Optimizers for Personalized Medicine
Authors: Michael S Yao
Osbert Bastani
Alma Andersson
Tommaso Biancalani
Aicha BenTaieb
Claudia Iriondo
Venue: ICLR 2026 Poster (2026)
Keywords: Large language models
Personalized medicine
Black-box optimization
Distribution shift
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We show how to use large language models to draw on medical knowledge to suggest personalized treatments for patients under distribution shift.

Abstract: The goal of personalized medicine is to discover a treatment regimen that optimizes a patient's clinical outcome based on their personal genetic and environmental factors. However, candidate treatments cannot be arbitrarily administered to the patient to assess their efficacy; we often instead have access to an *in silico* surrogate model that approximates the true fitness of a proposed treatment. Unfortunately, such surrogate models have been shown to fail to generalize to previously unseen patient-treatment combinations. We hypothesize that domain-specific prior knowledge—such as medical textbooks and biomedical knowledge graphs—can provide a meaningful alternative signal of the fitness of proposed treatments. To this end, we introduce **L**LM-based **E**ntropy-guided **O**ptimization with k**N**owledgeable priors (**LEON**), a mathematically principled approach to leverage large language models (LLMs) as black-box optimizers without any task-specific fine-tuning, taking advantage of their ability to contextualize unstructured domain knowledge to propose personalized treatment plans in natural language. In practice, we implement LEON via 'optimization by prompting,' which uses LLMs as stochastic engines for proposing treatment designs. Experiments on real-world optimization tasks show LEON outperforms both traditional and LLM-based methods in proposing individualized treatments for patients.

Published as a conference paper at ICLR 2026
KNOWLEDGEABLE LANGUAGE MODELS AS BLACK- BOX OPTIMIZERS FOR PERSONALIZED MEDICINE
Michael S Yao" University of Pennsylvania (raa21999upente edu
Osbert Bastani
Alma Andersson Genentech
University Pennsylvania
Tommaso Biancalani Genentech
Aïcha Bentaieb
Claudia Iriondo Genentech iriondoc@gene.com
Genentech
ABSTRACT
The goal of personalized medicine is to discover a treatment regimen that opti- mizes patient's clinical outcome based on their personal genetic and environ- mental factors. However, candidate treatments cannot be arbitrarily administered the patient to assess their efficacy; we often instead have access ico surrogate model that approximates the true fitness of a proposed treatment. Unfortumately, such surrogate models have been shown to fail to generalize previously unseen patient-treatment combinations. We hypothesize that domain- specific prior knowledge such medical texthooks and biomedical knowledge graphs-can provide a meaningful alternative signal of the fitness of proposed treatments. To this end, we introduce LLM-based Entropy-guided Optimization with KNowledgeable priors (LEON), a mathematically principled approach to leverage large language models (LLMs) as black-box optimizers without any task- specific fine-tuning, taking advantage of their ability to contextualize unstructured domain knowledge to propose personalized treatment plans in natural language. In practice, we implement L.EON via 'optimization by prompting," which uses LLMS as stochastic engines for proposing treatment designs. Experiments on real-world optimization tasks show LEON outperforms both traditional and LLM- based methods in proposing individualized treatments for patients.
in sil-
INTRODUCTION
Personalized medicine is clinical strategy that seeks to individualize treatment strategies based on a patient's s unique genetic and environmental features (Gutowski et al. 2023 Mizan & Taghipour] 2022 Adams et al. 2022 Consortium, 2009). Through reasoning about a patient, previously ob- served patients, and existing medical knowledge and literature, clinicians seek to determine an op- timal treatment (Mehandru et al. 2025). Such a task can be framed as a conditional optimization problem, where the goal is to design a treatment regimen-conditionn conditioond on the patient's unique fea- tures-that optimizes their clinical outcome. However, applying traditional optimization methods in this setting presents significant challenges. First, ground-truth design evaluations are costly; it infeasible to assess the efficacy of novel treatment options directly in human subjects. Evaluating newly proposed treatment regimens may therefore be difficult even impossible. To overcome this limitation, a common approach is to instead leverage feedback from a surrogate for the ground-truth objective, such machine learning model or digital twin, estimate the quality of proposed treatment for the patient (Kuang et al., 2024 Katsoulakis et al.) (024).. However, such surrogate models are frequently imperfect and used
-iii--o-tittiittionn patients (De Domenico al. 2025). Certain populations are systematically under-enrolled in clinical studies (Costa et al. 2023 Walker et al., 2022, Islam et al.) 2024 Casey et al.] 2022) Pittell et al. (003). and so black-box surrogate functions often fail to accurately predict design fitness for different patient populations Taylor et al. 2022 Kirtane Lee 2017 Smith-Grazian Graziani & Flowers 2021 Larkin al. 2022).
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(a)
(d) save
LEON
input
Memory
Adversarial
AI
OOD Evaluation
Critic Foedback
Design Entropy
LLM Context
LLM
LEON
Medical Prior Knowledge
Patient-Level Data
Textbooks
Mutations
(b)
(c) evaluate
PuhMed
Medical History
propose
Design Batch
Specialist
Gene Expression
Figure 1: LLM-based Entropy-guin Optimization with kNowledgeable priors (LEON). We use LLMs as zero-shot conditional optimizers to propose personalized treatment designs at the pa- tient level. (a) The LLM prompted with prior knowledge and the history of previously proposed designs and their predicted scores (b) propose evaluated using LEON, and then (d) cached as context to the LLM in subsequent iterations.
batch of designs. (c) These candidates
A natural question is whether designing better surrogate models can overcome these limitations, with the hope that more accurate proxies the ground-trute objective will yield better optimized therapeutic proposals. For example, Kuang et al. (2024) enforce a physics-based prior on learned digital twin model, Yu et al. (222)) assume the objective is locally smooth over the design space. However, in many real-world applications we cannot specify the surrogate model-the underlying mechanisms behind patient responses to treatment may be under-specified or even entirely unknown (Kuang et al., (2024), and building accurate digital twin models is often limited by the availability of real patient data and patient privacy concerns (De Domenico et al., (025). In these settings, we may only think of both the surrogate and underlying ground-truth functions black-box models.
2025a Ma et al. 2024a Nasir et al., 2024 Song al. (024) large language models (LLMs) have explored their emerging capabilities in black-box optimization, where the goal is to propose a design x that maximizes a black-box objective function fxx). In particular, modern LLMs have been shown solve zero- shot optimization problems in domains such as mathematics NNdkarni et al. 2021; Novikov et al., 2025) and computer science (Garg et al. 2022 Guo et al. 2024c Fu et |l.]||022222 2024;] Ma et al., (022bb) However, in these settings the objective function is almost always evaluable with minimal cost (i.e., using a code interpreter or formal verification boll)-unlike in clinical patient-centered tasks.
Recent work (Reinhart & Statt 2024 Yang et al. 2024a Chen et al.
In this work, we introduce method to leverage I.MSs as black-box optimizers in clinical medicine. Our core hypothesis that by leveraging domain-speeific prior knowledge, L.LM-based optimizers can overcome the limitations associated with out-of-distribution surrogate model predictions during optimization. More explicitly, our contributions are as follows:
1. Formulating personalized medicine as a black-box optimization problem. We propose a mathematically principled approach to formulate personalized medicine as conditional black-box optimization problem, where the goal is find an optimal treatment strategy conditioned on a set of input patient covariates that optimizes target outcome metric.
2. Constraining the optimization problem. Existing surrogate models are often imperfect proxies of costly ground-truth objectives in personalized medicine. To overcome this, we introduce a set of intuitive constraints to our initial black-box optimization problem. Our constraints limit the optimization trajectory to treatment designs that (1) are likely to have reliable predictions from the surrogate model; and (2) consistently proposed by the LLM as high-quality treatments according to relevant domain knowledge.
3. Deriving a solution to the constrained optimization problem. We derive a computa- tionally tractable solution to our proposed re-formulation of personalized medicine constrained black-br box optimization problem. Our approach fundamentally relies on the sta- tistical analysis of the distribution of proposed designs and the use of an adversarial source critic model during optimization. We refer to our method as LLM--based Entropy Entropy-guided Optimization with KNowleggeable priors (LEOON) (Fig. 1.. We implement LEON via
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'optimization-by certoon--by-rrmptingg YYang al. 2024a), using the LEON-defined objective to score candidate designs. this setting, the LLM functions as a stochastic treatment rec- ommendation system that seeks to iteratively propose higher scoring designs.
4. Using LEON to solve real-world personalized medicine tasks. We demonstrate how LEON can be used to solve conditional optimization problems over both discrete and con- tinuous search spaces. Comparing with 10 other baseline methods, we find that LEON achieves an average rank of 1.2 on 5 representative treatment design problems.
2 RELATED WORK
LLMs as optimizers. A growing body of work has explored the ability of generalist LLMs to solve
domain-specific optimization tasks (Reinhart & Statt, 2024; Liu et al. 2024b] Yang et al., 2024a Ma et al. 2024a Song et al. 2024). For instance, Chen et al. (2025a) fine-tune language models for antibody protein design. However, unlike in medicine, accurate molecular dynamic simulations
(Albaugh & Gingrich 2022; Ho et al. 2024), state-of-the-arr foundation models (Novikov et al.
2025 Wang et al. 2024b), and scalable experimental setups (Yang et al.] 2025 Rix et al.] 2020 Johnston et al. 2024) have made online optimization methods empirically effective in this domain. Similarly, methods like those introduced by (Shojace et al. (2025); (Shypula et al. (2024); Ma et al. (024b) introduce methods to use LLMs for code optimization according to an objective that is easily et al. (2024) use LLMs to propose new solutions to resource allocation and extremal combinatorie problems. Chen ct al. (2023a); Nasir et al.] (224)); Ji et al. (2025); Chiquier et al. (2024) separately use LLMs to design new machine learning models for neural architecture search, Zeng al. (2024); HHng et al. (2225) for tensor network design, and Akioyymen et al.) (2024) for database query optimization without any model fine-tuning. However, unlike the problem of personalized treatment design, these applications are all examples of unconditional optimization tasks.
verifiable by modern computer systems. Novikov et al. (2025) and Romera-
Optimization under distribution shift. A separate body of work has considered the problem of leveraging traditional (i.e., won-LLM-boon based) optimizers under distribution shift. Trabucco et al. (022)) leverages gradient data to update the surrogate model over the course of an optimization ex- periment to act conservative lower bound of the ground-truth touut function, Yu et al. (202I) imposes a smoothness prior on the surrogate model over the design space, and Chen et al. (2023b) leverages retrieval-based approach build more robust surrogate model. However, such methods assume control over the design of the surrogate model-an an assumption that fails in black-box optimization. Other techniques forgo learning surrogate model altogether Mashkaria al. (2023) Krishnamoor- thy et al. (2023); however, such methods rely on learning from multiple design observations in a sin- gle context, which do exist in conditional optimization tasks like those in personalized medicine. Finally, Angermueller et al. (2020) and Williams (1992) formulate optimization as reinforcement learning problem; however, we do not consider sequential decision-malee tasks our work.
BACKGROUND AND PRELIMINARIES
Entropy and equivalence classes. Entropy is a fundamental concept in information theory that quantifies the relative uncertainty associated with the variables of a system. Prior work from Quu et al. (0255) Farquhar et al. (2024): Kuhn et al. (2023) has examined how the entropy of the distribu- lion an LLM's outputs can be used estimate the model's epistemic uncertainty. Briefly, if a non- deterministic model consistently returns equivalent responses to the same coompp--oorrspondiii a low-entropy distribution of outputs-Vv we can be more confident in its own certainty of its response. Such methods have been shown to improve question answering (Nikitin et al. 2024 Kuhn et al.) 2023). hallucination detection (Farquhar al. 2024), and document retrieval (Qiu et al. 2025).
to
Crucially, the definition of what constitutes set of 'equivalent' responses is predicated on the existence of an equivalence relation ~:: two outputs x, x' are equivalent iff ~ x'. Any valid equivalence relation partitions the input space into a set of N disjoint equivalence classes [r]] x) and U
where [z]] (I' E X
We denote this quotient set of these N x
In this work, we assume that the number of equivalence classes N is finite. This is relatively weak assumption for most real-world, computable optimization problems (Kozen 1997).
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equivalence classes by // ~. The entropy with respect to ~ is given by Hw := - Ei= Ps log where p: := [[]]]/| X is the fractional occupancy of equivalence class [z]]
Adversarial supervision in optimization. A central challenge in optimization under distribution shift lies in the absence of direct access to the ground-truth objective function during optimization. In instead be performed against surrogate model, which may be inaccurate
practice, optimization in nutrof-distribution (OOD) regions inevitably explored during the optimization process. Naively optimizing against a surrogate model can therefore produce candidate solutions that appear promis- ing according to the surrogate, but perform poorly when ultimately evaluated using the ground-truth objective. To mitigate this, recent works bound the 1- I-Wasserstein distance between the distribution of real designs used to train the surrogate model and that of generated candidates (Yao et al. 2025b] 2024). Such a constraint provides theoretical guarantees on the generalization error of the surrogate model, effectively reducing the extent of extrapolation (Supplementary Theorem D.1).
However, directly computing the 1-Wasserstein distance between probability distributions poses significant computational challenges, as classical algorithms can scale O(n²) in the number of samples (Kuhn| 1955). To address this, one can exploit Kantorovich & Rubinstein (1958) to recast the 1- --Wasserstein distance as supremum over class of Lipschitz functions:
Wi (p. q) = (px))()) [c" (x:) - (x [c (x)]
where p(x) (resp., ((x)) is the (empirical) distribution of the real (resp., generated) designs and c²(x) arg maxijellass) [===)] [=)==(()(("((>)]))) One can think of the function ct as an adversarial source critic that learns to discriminate the source distribution of an input x. The learned function c² (z) thereby assigns high (resp., low) value to inputs that are likely to have been sampled from p(zz) (resp.. g(x)). Such an approach has been shown to reduce the extrapolation of
learned models in generative adversarial learning (Arjovsky et al. 2017: Yao et al. 20256 2024).
4
ENTROPY-GUIDED OPTIMIZATION WITH KNOWLEDGEABLE PRIORS
4.1 PROBLEM FORMULATION
The task of finding an optimal patient treatment strategy can be formulated as an conditional black- box optimization problem, where the goal is to find an optimal distribution g(x) according to
arg g())E()))( Eungga) [ax =)]
(2)
where X' is the set of all possible treatments, p(X) the set of all valid probability measures over X, x any particular treatment design, and E Z the patient conditioning vector. For example, X might be set of medications, and f the patient's therapeutic response to a medication. In this setting, the objective is not to find a universally optimal design, but rather identify an optimal treatment given a specific context
patient's 's personal health data sampled from distribution p(z),
In many real-world applications, the ground-truth objective function f is inaccessible during op- timization. We instead only have access to a surrogate function / X' x Z + IR trained on a distribution of observations whose z-marginal is not equal to p(z). The surrogate f may be patient simulator (Man et al. 2014 [Evans et al. 2015). a digital twin (Katsoulakis al. 2024, De Domenico et al. 2025 Kuang et al.) 2024), or machine learning model trained to approximate f. Importantly, we highlight the mismatch between p(z) and the source distribution of the training set of f: for example, f may only be learned from patients at one hospital, and our patient ~ p(2) is sampled from a different hospital. A natural strategy in this setting is to instead solve the related problem
arg max (ux))((()) (E [f =)]
with the hope that optimizing against f admits distribution over X that also maximizes f in ex- pectation. Importantly, the conditional problem formulation in (3) diverges from the related un- conditional problem commonly considered in prior work (Trabucco et al. 2022, 202T Yu et al. 2021). In practice, solving (3) is ineffective (Trabucco et al. [021 Yu et al. 2021). This is because the out-offdisstriutioo surrogate function f frequently exhibits detrimental biases and performance degradations on a target distribution of patients, leading to worse outcomes according to the ground- truth objective function f in clinical settings (Taylor et al. 2022 Kirtane & Lee 2017 Larkin al.
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2022).
To overcome this limitation, follow Yao et al. 2002bb
2024) and assume access
dataset Die S X of previous treatment designs (c.g., the full treatment seaamment-patennt dataset used to learn
projected onto 2'); we show in our work how to use Drc to solve a modified instance of (3)
4.2
CONSTRAINED CONDITIONAL OPTIMIZATION
We first modify the original problem in by introducing two constraints:
arg max (()))p(()) Ex (f(;
(4)
s.t. Ey [c" (x)))--as (x)] Wo, and H-(g(x)) S Ho
Following ( the first constraint imposes upper bound Wo on the 1-Wasserstein distance be- tween the empirical distribution of proposed treatment designs g(x) and a dataset Dee of previously proposed designs in the real world. Leveraging auxiliary source critic model e" X - R. this constraint ensures that the distribution of proposed designs is not too dissimilar to the distribution of historically reported designs, implicitly constraining the allowed degree of extrapolation against f during optimization (Supplementary Theorem D.I). This mitigates the risk of proposing spurious candidates that appear favorable under the surrogate objective f but are unlikely to perform well in practice Yao et al. 2025b 2024). Importantly, the domain of c* is restricted to X:; this means that no patient observations z from the source dataset Dare are required. In other words, patient privacy of individuals in ax is explicitly preserved in (2). The second constraint places upper bound Hu on the n--caase- seroooarrr--giinnddd entropy of the distribution of designs (zx), defined below:
Definition 4.1 ee-CCorse--rainnee Entropy). Let ~ be an equivalence relation over the input space X, and assume that the set of equivalence classes X/ imposed by is finite. Let N := |X/ ~ be the number of equivalence classes and g(x) be a valid probability distribution over the input space X. Denote [z]; the ith equivalence class in X/ ~ and q: JaJ. dz q(z) to be the probability of drawing an element from the ith equivalence class. Then, the >-coarse-grained entropy H.. : p(x) + R. is defined as (g(x)) == Σ q: log 9.-
The second constraint therefore enforces an upper bound on the entropy of the distribution of de- signs with respect to the equivalence relation ~,, encouraging sampling strategies (such as those that leverage nomain-sseciif prior knowledge) that increase the certainty of the optimizer's proposals. In general, solving (4) exactly highly intractable; both f and c* can be arbitrarily non-convex black-box functions, and the ccoarse-grained entropy may be highly sensitive to perturbations in the input space. To address this, we first show using Lemma 4.2) how to derive an ansatz to the constrained problem in (4). We then show how algorithmically solve for the free parameters of the solution class as applied to a suite of real-world optimization tasks for personalized medicine.
Lemma 4.2 (Design Collapse Within Equivalence Classes). Using the method of Lagrange multi- pliers, we can rewrite (4) as fufunction of the partial Lagrangian Lx(g) for some constant X E R,:
Cx(g) = [ ) + A(W, - E (c (x)] + Bezg(() lc (x))) (5)
arg max ((x)Ep(()
S.l. H-(((x) S Ho
Suppose there exists a distribution q(x) that satisfies the remaining constraint in (5). Furthermore, assume that the function f(x; 2) + Ac (x) is continuous everywhere and coercive in X. For all N equivalence classes, we can then define x (not necessarily unique) according to
x = arg max maxxe(4).
(f(x;))+ +xc)
(6)
Then, the alternative distribution = =x) E Ts 8(x - 1;), where qs is as in Definition 4.1 also
satisfies the constraint and simultaneously achieves non-inferior value Lx(g") Z Cx(g).
The proof for this result is shown in Appendix A Intuitively, Lemma 4.2 shows that any feasible
distribution q(x) cannot be superior to an alternative distribution q" (z) that is both feasible and places all of its probability mass each of the optimal designs within each equivalence class. We remark that each o need not be unique within the corresponding equivalence class [x];; it is easy to show that any solution to ( within the same eequivalence class (or appropriately weighted
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combination of multiple optimal solutions) still admits a feasible distribution q" (z). Lemma 4.2 allows us to restrict the search space of optimal distributions within p(2)). In particular, note that for given À and equivalence class ~. the optimal policy is exactly specified by the choice of equivalence class probabilities y:. The original problem in is therefore equivalent to
Σ a.((())
arg max GEA(N)
s.t. ExNDaI (c')
a Wo. and H(g) S Hu Σ 1=1
where A(N) is the N-dimensional probability simplex and H((-) is the standard Shannon entropy of the N-dimensional vector q. This alternative problem formulation leads us main result: Lemma 4.3 (Probabilisticc Sampling Over Equivalence Classes). Consider the constrained opti- mization problem as in 17.. The ith element of the N --dimensional vector can be written as
q: cxp [M (f())12) (ct(a))) / (()) where x == arg max (f(x;2) + (x)) (8)
where (() is normalizing constant and X, p R2 are the Lagrange multipliers.
The proof for Lemma 4.3 is included in Appendix A Intuitively, we can think and μ as 'cer- lainty' parameters: increasing À upweights the importance of sampling design æ; associated with a high certainty of "in-iistribution-nes according to the source critic function c* (x). Similarly, increasing μ upweights the importance of the probability vector producing 'collapsed" distri- bution of designs with low entropy (and therefore high certainty). Prior work has either explicitly fixed variables similar to X, Jl as hyperparameters (Yu et al., TT2Tuuraaucoo al.] 2021) imposed restrictive assumptions on the input space to solve for the Lagrange multipliers (aoo et al.) (024).
In contrast, we introduce a principled and computationally tractable approach to dynamically solve for the optimal À and over the course of the optimization trajectory.
4.3 EMPIRICALLY FIXING THE LLM CERTAINTY μ
The >--coarse grriine entropy from Definition 4.1 ji an intrinsic property of a language model opti- mizer: given single prompt a particular optimization step, the LLM can return multiple possible designs because the autoregressive model is non-deterministic for positive temperature values. The values of zi can therefore be empirically observed by sampling batch of designs from the LLM op- timizer, allowing to estimate the degree of freedom in (8). More concretely, with each batched sampling of proposals from the LLM, we assign the treatment designs into their respective equiv- alence classes to arrive at an unbiased estimate of the fractional occupancies of each class d. and then gives
compute the optimal values f(² Ae" (it) according to (8). Lemma 4.3
log ~ log Z(A) + (f(1);)+ (at))
We then estimate the value of y by simple linear regression, treating each observation (f(ăt)2) Ac (i)) log over N equivalence classes as an explanatory-dependeer variable pair:
ΣM (f(#t)z)= (Ac" (#t) xt) Togn: p. N/ +E log Px
EN (f((1;=) - (AC 2²]
where the expectation values F. è are defined over the N equivalence classes. Intuitively, predictions with high entropy (i.e., low certainty) will be scaled to a lower reward, as log will be constant over equivalence classes and so P 0. Conversely, a confident model with high certainty will lead to greater estimate of > 0, increasing the reward associated with the proposed designs. In this framework, the role of prior knowledge to help 'overcome" the statistical randomness of the LLM's next-token generative process in proposing treatment designs to improve LLM certainty.

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4.4 SOLVING FOR THE SOURCE CRITIC CERTAINTY X
We first solve for the Lagrangian dual function g(>, p4) =- maX L(T; X, p.. Corollary 4.4 (Dual Function of 7 The dual function of the constrained problem in
g(,, - XW - EE [c (x))) + +
1 "4-'Hg Ho + μ log 2(2)
where (() is the normalizing constant from 8 and so
0g(A, 1) = Wi - (E c (z) - Σ Th² (x;))
(10)
ax
Importantly, (T0) allows us to iteratively solve for the optimal value of the dual parameter À via gradient descent without any explicit gradient information from the black-box functions f and ct:
The proof of this result included in Appendix A
Ög(2, p) d 7A Wo - lc + Σ Th (x;)
1=11= = 7A
Ege
(II)
ax
Here, 7I is learning rate hyperparameter and 9; is in (8). Intuitively, if the designs zf are in- distribution compared to Der. then the Wasserstein distance La -x)] Ei q.c (x;) S Wo. meaning (x /0x > 0 and the value of A will decrease to allow for greater exploration of the design space. Conversely, sampling out-of-distribution designs x; will yield (g(A, < O,
and Ac+1 will increase to reduce the extrapolation against f.
4.5 OVERALL ALGORITHM
Our overall method to solve consists of four primary steps. (1) Sampling. We first query the LLM optimizer to propose batch of independently sampled new treatment designs. The LLM is prompted with a description of the task and the patient z, any prior knowledge it produced prior to optimization, and a table of its previously proposed designs and their corresponding scores according to our algorithm. (2) Clustering. We then take as input the batch designs and individually assign them to their corresponding equivalence classes. (3) Certainty Estimation. We estimate the fractional occupancy and the optimal value of f((]; 2) + Xe (x;) observed in each equivalence class, and then estimate the certainty parameter according to (9) We also update the value of A score each sampled design x according to pu (a;(a;;)++ + Ae (x)), and store the treatments and their scores to provide as context to subsequent L.L. optimization prompting. These steps are repeated until the maximum query budget for f(z; =) is reached. We refer to our method LLM-based Entropy-guided Optimization with kNowledgeable priors (LEON); the full pseudocode is in Supplementary Algorithm 1 Note that method only updates the LLM prompt; there is no fine-tuning of the weights of the LLM.
following II. (4) Design Scoring. Using our estimates of μ and
The role of prior knowledge in LEO.. Notably, LEON may be used with any generalist language model without any task-specific fine-tuning. Such consumer-grade LLMs may not be able propose more optimal designs given only access to prior observations alone, adversely affecting the model certainty according to μ. To overcome this limitation, we first provide the LLM access to a set of external knowledge repositories with domain-specific knowledge-such as medical textbooks, biomedical knowledge graphs, and publicly available clinical databases. Given a description of the optimization task and the patient features 2, the LLM is allowed to choose which knowledge sources may be helpful, and then sequentially query the relevant textual corpora as tools to synthesize prior knowledge statement in natural language. We then provide this prior knowledge as prompt context to the L.LM in all subsequent optimization steps. In our work, we show how leveraging prior knowledge in this way help the LLM propose higher-quality designs, thereby increasing the value of μ and improving the quality of individualized treatment regimens.
Reflection on optimization steps. After batch of designs are scored and before new batch of designs are acquired, we prompt the backbone language model to analyze the most recently sampled
batch of designs and their corresponding scores following prior work (Ma al. 2024b Yao et al. 2023). The LLM is asked to reflect on the data and underlying sampling strategy in natural language; the output of this reflection is included into the L.MM prompt in the next batch acquisition step.
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5 EMPIRICAL EVALUATION
Personalized medicine optimization tasks under distribution shift. Recall that our core motiva- tion for LEON is to overcome the effect of distribution shifts when personalizing treatment plans for previously unseen, potentially out-of-distribution patients. To this end, we constructed set of 5 real-world optimization tasks to evaluate LEON and baseline methods. (1) Warfarin aims to propose an optimal dose of warfarin (a blood thinner medication) conditioned on the patient's 's phar- macogenetic variables (Consortium 2009); (2) HIV an antiretroviral medication regiment based on the patient's HIV viral mutations (Rhee al. 2003); (3) Breast and (4) Lung an optimal treatment strategy for patients diagnosed with breast or non-small cell lung cancer (NSCLC), respectively; and (5) ADR a prediction of patient's risk of an adverse drug reaction (ADR) following the adminis- tration of a proprietary drug. We simulate a distribution shift between the observations used to learn the surrogate model f and the ground-trun -tuuthhhhhhhhh objective f; see Supplementary Table S1 for details.
Prior knowledge generation. We operationalize the task of prior knowledge synthesis as tool- calling problem 4Schick et al., 2023 Qin et al.) 2024 Yao et al. 2023). We provide the LLM a
set of external knowledge tools, including: (1) corpus of medical textbooks (Xiong et al. 2024);
prompting auxiliary (2) MedGemma 27B LLM (Sellergren et al. 2025) fine-tuned expert
medical knowledge; querying structured biomedical knowledge graphs (3) HetioNet (Himmelsten
et al. 2017) and (4) PrimeKG (Chandak al.) 2023); and other domain-
knowledge reposi- tories including (5) Cellosaurus (Bairoch (2018) with cell-line data, (6) COSMIC (Tate et al. 2019) with cancer mutation data, (7) GDSC (Yang al., 2013) with drug sensitivity data, and (8) MepMap Tsherniak et al.) 2017) with cancer cell dependencies. Using these tools, the LLM composes a prior knowledge statement, which is then included in the LLM prompts during optimization.
Experiment implementation. Each task includes two static datasets Dannocated = {(z), æj } Phe srz
j=1 Danmolated and Dammotatted (2 24, yz) }: from the observation space X' Z XR. The source dataset src
and target dataset Dig Dannotated are constructed according to a distribution shift between the task specific source and target distributions, and are non-overlapping at the patient level. We learn a task-specific Dammotatedd taken
surrogate model f X' x Z + IR Demmotated and also function f X x Z + R on Digs to be the ground-truth objective for patients from the target population for the purposes of evaluation. Note that the full datasets Damuotated DDamotated are used only to learn f, for our experimental setup; we project Dammotated onto Z and Damnottaeed onto X' to construct Degt == {z, )im and Dec := (m for our experiments. We embed proposed patient-desige pairs, represented in natural language, using the text-embeddins --ng---small model from OpenAI, and perform k-means clustering (trained on the source dataset Dene using cosine similarity as the distance metric) in the embedding space assign individual designs to equivalence classes see Appendix Corr additional details.
LEON also involves training a source critic model c*: : X - IR as in T Consistent with prior work Yao al. 2025b 2024). we implement c* as fully-connected network with two hidden layers each with 2048 dimensions. We enforce the constraint on the critic's Lipschitz constant by clipping the model parameters have &o -norm no greater than 0.01 after each weight update step, consistent with Arjovsky et al. 2017 After each acquisition step during optimization, we re- re-train the source critic using gradient descent with a learning rate of 17 = 0.001 according to ( and also perform a single-step update to the X certainty parameter according to TT) with 7/2- = 0.1. We fix Wo 1.0 in 110 and the L.LM temperature hyperparameter т = 1.0, and use a sampling batch size of 32 to avoid overfitting any of these hyperparameters to any particular task. All experiments were run on an internal cluster using only a single NVIDIA A100 80GB GPU. We report experimental results on random sample of 100 unique patients from Degi-
Baselines. We evaluate LEON against the LLM-based optimization methods (1) Large LAnguage Models to enhance Bayesian Optimization (LLAMBO, Liu et al. (2024c)), which leverages LLMs to augment traditional Bayesian Optimization (BO); (2) Optimization by PROmpting (OPRO, Yang et al. (024ab), which appends previously proposed solutions and their scores to subsequent L.M input prompts; and (3) Evolution-driven universal reward kit for agent (Eureka, Ma et al. (2024b)
In general, computing the global Lipschitz constant of an arbitrary function is N'P-hard and still an open problem (Abdeen et al. 2025 Kim et al.) 2021). While other methods than that proposed by Arjovsky et al. (2077) for enforcing the Lipschitz constraint exist (Scaman & Virmaux 2018 Hu et al. 2024] Fazlyab et al. 2019 Song al. 2023), leave the investigation of how they might he used with L.ON for future work.
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Table 1: Quality of patient-conditione designs under distribution shift. We report mean + standard error of mean (SEM) ground-truth objective values achieved by the single proposed design for a given patient, averaged over n = 100 test patients from the target distribution. Bolded (resp., Underlined) cells indicate the best (resp., second best) mean score for given task.
Method
Warfarin RMSE Loss (mg/week)
HIV
Breast TINTD (months)
Lung TINTD (months)
ADR
Rank
Viral Load (copies/ml.)
I NLL Loss (no units)
Majority
3.46 + 0.70
4.55 0.07
25.95 1 0.75
20.13 + 0.13
1.41 + 0.05
8.0
Human
2.68 0.86
4.55 1 0.07
29.65 + 1.14
21.10 0.27
8.5
Grad.
1.37 = 0.13
4.52 + 0.04
65.23 2.03
24.09 + 0.44
23.7 + 1.7
5.2
BO-qEI
1.36 + 0.13
4.53 + 0.04
67.05 + 1.87
27.97 + 0.65
23.2 + 1.7
3.4
Sim. Anneal
1.38 = 0.12
0.03
66.62 = 2.62
29.29 0.74
23.8 +
5.0
CMA-ES
1.90 : 0.14
0.04
59.48 + 2.76
0.668
23,4
1.7
6.2
GA
1.49 + 0.26
4.62 +.05
19.90+2.22
27.53 + 0.81
20.0
2.3
5.2
ILAMBO
3.28 + 0.10
4.52+
4
20.6
+ 1.9
7.0
OPRO
1.40 0.13
4.55
55.68 2.86
24.35 0.43
23.8
+ 1.7
7.0
Eureka
1.54 0.25
4.58
63.48 3.52
25.10 1 0.69
21.3
1 2.0
6.8
LEON
1,36 + 0.13
4.50 0.04
72.43 + 2.86
32.71 + 0.32
1.6
1.2
which extends OPRO by intermittently prompting the LLM to reflect on the efficacy of previous op- timization strategies in natural language. All LLM-based optimization strategies (including LEON) were evaluated using mpt-4o-mini-2224--11
from OpenAl without any fine-tuning.
We also compare LEON to traditional, non-LLM-basee optimization methods (4) Gradient Ascent (Grad.); (5) Simulated Annealing (Sim. Anneal, Tsallis (1988): (6) Covariance Matrix Adapta- tion Evolution Strategy (CMA-ES, Hansen (2066); ); (7) Genetic Algorithm (GA. Gad (0224)) ); and (8) Bayesian optimization with Expected Improvement (BO-qEI). Finally, we evaluate the (9) Ma- jority baseline algorithm that always proposes the majority design (i.e., mean for continuous design dimensions and mode for discrete design dimensions) across all observations Tj E Darei and the (10) Human baseline algorithm, which proposes the true design x; that the individual z; actually received according to the hidden annotated dataset Danmotated An optimization method that outperforms the Majority strategy suggests that the optimizer's personalized treatment strategies outperform the 'av- erage" treatment in the source dataset Dar in expectation. An optimization method that outperforms the Human baseline means the treatment strategies proposed by the optimizer are better than the therapy the patient actually received from their clinician according to our oracle objective function.
Evaluation metric. For any particular input set fixed patient covariates : ~ Digr. an optimization method is allowed to sample and propose a total of 2048 possible designs æ and corresponding surrogate model evaluations fz; 2). After the surrogate model evaluation budget is exhausted, the optimizer proposes a single top design x= to evaluate using the ground-truth objective function f: for most optimization methods, x² is the proposed design that maximizes f (LEON follows the design proposal method in Supplementary Algorithm Tp. We then evaluate this single proposed design x² using the ground ground-truth truth objective f. and report the corresponding score f(r"; z)) Importantly, f only used for evaluation purposes and is not accessible to any method during optimization.
Results. LEON consistently outperforms all baseline methods, achieving an average rank of 1.2 across all tasks [T. Notably, LEON proposes sets of personalized treatment designs that are superior to the treatments retrospectively received by the patients. It achieves the best performance on Warfarin dose prediction, HIV treatment, and both breast and lung cancer therapy design. Quali- tatively, we found that the Majority algorithm was able to outperform LEON on the ADR task due to significant class imbalance in the population of patients in the dataset.
Additional empirical analysis of LEON is included in Appendix D In Appendix D.2 we evaluate the monetary cost, token usage, and environmental impact of our method. We then interrogate the construction of prior knowledge used by LEON in Appendices D.3 and evaluate how the certainty parameters and À vary over the course of optimization in Appendices D.5 Finally, investigate the fairnsss of LEON with respect patient gender and race in Appendix D.10
Ablation studies. In our main experiments, we use gpt-4o-mini-22 2ot--mi--222----1-- as the LLM optimizer; we ablate the choice of language model in Supplementary Table SS5 We also ablate

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the values of the certainty parameters X, μu in Supplementary Table S6 We then ablate the external sources of prior knowledge in Supplementary (Snd find that LEON is sensitive to the qual- ity of domain-specifie knowledge available. Finally, we examine how the performance of LEON is affected factors such as the severity of distribution shift between source and target distributions (Appendix [. the use of reflection in LEON (Appendix [E.10]; and the choice of equivalence See Appendix E for additional results.
relation in Definition 4.1 (Appendices
6 DISCUSSION AND CONCLUSION
In this work, we propose LEON to combine domain knowledge with LLM-based optimizers to solve conditional black-box optimization problems under distribution shift for personalized medicine. We first introduce two additional constraints in { that upweight high-quality designs that are both (1) in-distribution according based on the language model's statistical certainty. Using our method, we show how consumer- grade LLMs can be used solve a wide variety of challenging personalized medicine optimization problems without any LLM fine-tuning, outperforming both traditional and other LLM-based opti- mization methods. Moving forward, we hope to extend LEON to the setting of active learning and prospective clinical evaluation [Collignon et al.) 2020; lanevski et al. 2021 Kuru et al.) [2224] Burd
auxiliary source critic model; and (2) likely to be high-quality
ct al. 2020), and also apply our method to other domains outside of personalized medicine.
Limitations. Several limitations warrant consideration. First, we observed that LLM optimizers using LEON are sensitive to the available prior knowledge: contamination with factually incorrect outdated information may propagate into optimization outputs and adversely affect personalized treatment design using LEON (Omar et al. 2025] Han et al. 2024 Clusmann et al. (025). Further- more, while the clinical tasks considered here are designed approximate real-world conditions, they cannot fully capture the complexity of heterogeneous patient responses and rare disease sub- types encountered in actual clinical practice (Pan et al.) (024). Finally, with all simulation-basee benchmarks, the validity of our conclusions is limited by the assumptions embedded in the functions and datasets used, which may obscure differences between methods. Future work might explore how to integrate physicians in the loop to mitigate the risks of autonomous ALM-basee systems.
ETHICS STATEMENT
This work investigates the use of large language models (LLMs) as optimizers for conditional black- box problems in clinical medicine. All datasets used in this work are fully anonymized and contain no personally identifiable information. As such, no patient consent or institutional review board (IRB) approval was required. Separately, LLMs are inherently subject to social and demographic biases learned during pretraining (Sorin et al. 2025 Omiye et al.) 2023). When applied to patient- specific design tasks, such biases could disproportionnteely affect marginalized populations, poten- tially leading to inequitable or unsafe treatment proposals not properly accounted for. We also emphasize that LEON is not intended for direct clinical use, but rather methodological contri- bution toward future systems that may aid clinical decision making.
REPRODUCIBILITY STATEMENT
The datasets for the Warfarin and HIV tasks are made publicly available by Consortium (2009) and Rhee et al. 2003). respectively. The Breast and Lung task data were made available by Flat- iron Health; the de-identified data may be made available upon reasonable request by contacting
DataAccess@ f lat ron. comm The ADR task data is internal proprietary kataset-data cess requests may be directed to the corresponding author. Our custom code implementation for our experiments is made publicly available at code. roche..... http:///aaaad/proeesss////
ACKNOWLEDGMENTS
The authors thank Allison Chae (Main Line Health) for her help with the qualitative evaluation of the language model outputs. M.S.Y. is supported by NIH Award F30 MD020264 and completed this work during internship at Genentech. OB is supported by NSF Award CCF-1917852. The views and conclusions contained herein are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of Genentech the U.S. Govermment.
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A APPENDIX
Table of Contents
Proofs
27
Proof for Lemma
4.2
Design Collapse Within Equivalence Classes
Proof for Lemma
4.32
Probabilistic Sampling Over Equivalence Classes
Proof for Corollary 4.4
Dual Function of 7
B.
Additional Related Work
29
Generalization and Domain Adaptation
Conformal Prediction
Robustness via Domain-Specific Foundation Models
Medical Reasoning Models
Existing Methods in Personalized Medicine
Additional Implementation Details
30
C.1
Optimization Task Specifications
Full Algorithm Pscudocode for LEON
Language Model Prompts
Excluded Baselines
D
Additional Experimental Results
38
D.1
Distribution Shift Analysis
D
Cost Analysis
D.3
Quantitative Analysis of Prior Knowledge
D.4
Qualitative Examples of Prior Knowledge
D.5
Additional Analysis of the LLM Certainty Parameter μ
D.6
Additional Analysis of the Source Critic Weighting Parameter
D.7
Extending LEON to Traditional Optimizers
D.8
Why Should We Bound the 1- Wasssersteiin Distance?
D.9
Sample Reflection Trace
D.
Subgroup Fairness Analysis
D.IT
Integrating Knowledge with Baseline L.LM Optimization Methods
D.12
A Discussion on Extending LEON to Multiple Objectives
E
Ablation Studies
51
E.1
Backbone LLM Ablation
B.2
Sampling Batch Size Ablation
I.MM Temperature Ablation
E.4
I-Wasserstein Distance Bound Ablation
E.5
and Certainty Parameters Ablation
3.6
Distribution Shift Severity Ablation
Surrogate Evaluation Budget Ablation
Ground-Truth Objective Evaluation Budget Ablation
5.9
Prior Knowledge Quality Ablation
5.
T0
Reflection Ablation
F..
Equivalence Relation Embedding Model Ablation
12
Equivalence Relation Ablation
Equivalence Class Count Ablation
Iterative Feedback Ablation
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A PROOFS
Lemma 4.2 (Design Collapse Within Equivalence Classes). Using the method of Lagrange multi- pliers, we can rewrite (4) as a function of the partial Lagrangian (x(1) for some constant X E R4;
arg (a) (x == Ex 2)] + X (Wo De (x)) + Ex (c) lc (x)])
s.t. H-(x(z)) < Ho
Suppose there exists a distribution (zx) that satisfies the remaining constraint in (5). Furthermore, assume that the function f(x) + Ac* (ar) is continuous everywhere and coercive in X. For all N equivalence classes, we can then define 2 (not necessarily unique) according
4 (A) == arg me(x) (f(z; dc Xc)
Then, the alternative distribution (a(x) = Σ qi 6(x - x;), where T is as in Definition 4.1 also satisfies the constraint and simultaneously achieves non-inferior value Lx(g*) 2 Lx(g).
Proof. First, note that each x is in distinct equivalence class, since equivalence classes in / are pairwise disjoint by construction. The probability of of sampling from the ith equivalence class according to 4² (x:) is then
N df - -/1 dx dx) / dr [o),
-
Σ 3=1
qg 5(x - -1,x)=,))-- = / [p).
dr 8(x - x.;) q:
(12)
since a; + (x) and [z]] n [æc] when + j. Because this holds for all i, it follows that H. (a (x)) = Hh (q(x)), meaning H (1- - Ho as well. Secondly, observe
(Wo - EanDy (c (x) + (a)))) = Exngr (0) (x) [f(x)]
(x) (n) ((()) + Xc xc (x) + + EewD (x)()
Σ oe (f(z))+Ac<xz))
+ Axx + -Wo (c"
Using the expansion of of from 12
N Σ
Ex(g*) =
dr dx)(x)(f(z)) + xc
+ + - (ewDa(("
lol:
dr dzg(x)(f(()) Ac (x))
+ A - BENDen (x)))
[ole
Em-g(aa (f(x)) (x)) + > E.wDen (x)))
Enngta) + (Wo En- (xr) + Exng(a) (x)))
= -x(())
using the definition of each a; as in 6). The claim follows.
Lemma 4.3 (Probabilistic Sampling Over Equivalence Classes). Consider the constrained optimiza- tion problem as (7). The ith element of the N N-dimensional vector q can be written as
q. = exp [a (f(afi2) de (xt))) Z(A) where of == arg max (f(x;z) Ac (x))
where 2(A) is normalizing constant and 21 E IR, the Lagrange multipliers.
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Published as a conference paper at ICLR 2026
Proof. By definition (Boyd & Vandenberghe 2004), the full Lagrangian L(q; X, p1) of is
1(
Σ 2 2)(
Wo E + Σ n (a;)
i=1
N
+ p
(-11
Σ 1=1
ai log
(13)
(f(a)12++ + (x
(Wo
EEWDial (x)))
Ho +
Σ qi log qi
where A, 1r1 1 E IR+ are the Lagrangian multipliers associated with each of the two respective con- straints. The stationarity condition of the composition of the individual scalar elements & of in (13 give
(KKT) theorem and additive de-
OC = 0 = (f(ot:2)+ Ae (1t)) - p- (log lb + 1)
0q:
by definition. Rearranging gives
A (log & + +1 = f(]] + (x+) = log - + (flati2) + Ac (xi)
and so
Ts x exp 1 (f(x;iz) + dc (a)))
exp
+
Defining the partition function Z(A) = E exp [f(f(=];)) 2) + do (x;))]
as our normalizing
constant, the elements of the probability vector can be written as
q exp (e (f(=)12) + de xx / ((A)
where xf is as in 6
Corollary 4.4 (Dual Function of 7 The dual function of the constrained problem in is
g(A. = X(Wo EewD (c" + - 1 Ho + μ --Loog llog (())
where z(A) is the normalizing constant from
S and so
0g(A, 11) = Wo (E (c (x)] Σ qhc" (x:))
ax
Proof.
From 13 and referencing Boyd & Vandenberghe (2004). the Lagrangian dual function is
g(A,p) p) = max C(g: u)
TEA(N)
N
(f((f))) Ac (x+)) + X(Wo Ex-Da. (c'()))
1 f
Ho Σ
q log q
where the optimal vector q is as in Lemma
4.3 Note that the first summation can be written
N
N
a(f((t12)) Ac (x;))
Σ
qo
Ac (x:))
qo (log q. + log 2()))
N
N
N
log 2(2)
Σ
q
+
1
Σ
qe
log qi =
1
log 2(A) + Σ go log qo
i=1
iml
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Substituting into our expression for g(), 1t),
g(A, - log (A) >(Wo - (a - Hoo
Differentiating with respect to X,
1g(2,
(A)
+ +Wo R D (x)))))
" (AA)
0x
N
1
exp = pc (x+) + +Wo - Eaw (x)))
2(1)
c' (af) exp
xc 2(1) + (Wo - Lo
cct (x)) + (Wo- Le Wo (c -10-~aae qc (x;)
after regrouping terms.
B ADDITIONAL RELATED WORK
Generalization and domain adaptation. To mitigate the mismatch between the available surrogate model and hidden ground-trui -trut objective, recent work on domain adaptation leverages knowledge of the underlying source and target distributions to overcome the underlying distribution shift. Broadly, the technique relies on learning mapping between source and target distributions, and can often be applied even to black-box surrogate models (Ganin et al. 2016 Sun & Saenko 2016 Bousmalis jct all,[2016 Yue et al., 2023). Many existing domain adaptation methods have in proposed specif- ically for applications in computer vision (Bousmalis et al. 2017 Saito et al. 2018 French et al. 2018). which are outside the scope of this work. Separately, Tzeng et al. (2017) learn to encode target samples into source representation space using adversarial feedback, and Ngo et al. (2022) describe a method for task-level adaptation. However, these methods assume knowledge of the tar- get distribution, which may not be known a priori in optimization tasks. In our setting, we show how LEON can be used for optimization on a per-patient basis without assuming any prior knowledge of other patient observations from the target distribution-ag realistic assumption for protecting patient privacy and ensuring timely downstream decision making in real-world applications. We therefore cannot leverage domain adaptation to adapt a surrogate model during optimization.
Conformal prediction. Conformal prediction is a statistical framework for uncertainty quantifica- tion where the goal is to produce prediction sets as opposed to singular outputs, providing a guaran- tee that the true label is almost certainly in the prediction set with high probability (Cresswell et al.) 2024 AAnellopoulss & Bates 2023 Angelopoulos et al. 202T). Prior work has demonstrated how conformal prediction can be leveraged for image classification (Angelopoulos et al.) |02 Zhang ctt 2025a)). disease course prediction (Yang et al.) 2024b) Sreenivasan et al. (2025), drug discovery (Eklund et al. 2013] Jin & Candes 2023), and even autoregressive generative models (Quach al., 2024). Prior work has explored how to perform Bayesian (Deshpande et al., 2024] Sttnton et al,
2023 and worst-case (Johnstone & Cox [021] optimization over conformal prediction sets; future work might explore how to similarly adapt LEON to optimize over prediction sets.
Robustness via domain-specifie foundation models. A potential direction for building more ro- bust and performant surrogate models is to leverage large-scale foundation models as surrogates. One argument that models that are trained on sufficiently large and diverse datasets are unlikely to encounter significantly duutoo-diittrbuution inputs, yielding more robust performance across wide range of inputs during optimization. Early efforts along these lines have emerged in areas such as cell perturbation modeling (Cui et al. 2024 Hao et al. 2024 Zeng et al., 2025) and drug discovery (Abramson et al., 2024, Huang et al. 2024). However, despite preliminary efforts to extend sim- ilar approaches to clinical medicine (Guo et al. 2024a 2023 Steinnerg al. 2024), the limited availability of patient training data and stringent patient privacy constraints restrict the feasibility of constructing models that reliably generalize in zero-shot manner across many different patient populations. For these reasons, methods like LEON may still play a role in domains such as clinical medicine even as the accuracy of surrogate models improves with foundation model development.
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Table S1: Implementation details. We consider the problem of optimization under distribution shift; given access to surrogate model trained on a source dataset, we want to find an optimal design for a patient sampled from the target distribution that maximizes the response according to a hidden ground-truth objective function. TabPFN: Tabular prior-data fitted network (Hollmann et al. FCNN: Fully connected neural network. RexNet: Tabular residual network (Gorishniy et al.
2025).
2021
GBDT: Gradient boosted decision tree ensemble (Chen & Guestrin
2016
Quinlan 1986).
Distribution Shift
Covariate Shift in P(X)
Label Shift in P(Y)
Task
Warfarin
HIV
Breast
ADR
Lung
Shifted Feature
Patient Race
Study Year
Patient Age
Medication
TTNTD
Source Data Feature
White
2002-2008
< 65 years old
Drug A
Longer Shorter
Target Data Feature
Konte
2009-2020
2 65 years old
Drug
Source Dataset Size
3,095
740
3,020
484
3,013
Target Dataset Size
2,646
554
1,260
554
3,009
Surrogate Model
TabPFN
FCNN ResNet
GBDT GBDT
GBDT
GBDT
Ground-Truth Objective
Exact
Exact
GBDT
Patient Feature Size ([2))
11
339
47
50
47
Treatment Dimensions (())
16
30
30
Optimization Type
Continuous
Discrete
Discrete
Continuous
Discrete
Medical reasoning models. Recent work has explored how to develop datasets of clinical rea-
soning traces (Wu et al.] [0255 Kwon et al., 2024 Hager et al.) 2024) and associated benchmarks (Arora et al., 2025 Chen et al., (025b)) This evolving ecosystem of model training and evalua- tion resources have made available recent medical reasoning models: SSinhhal et al. 2023 2025)
introduced the MedPaLMM family of models that generate accurate long-form answers to consumer medical questions, ZZang et al. (2025b)) fine-tune language models using reinforcement learning with verifiable rewards derived from multiple choice question answering. Separately, [Savage et al.) (Sonoda et al. (2024) demonstrate how chain-of-thought reasoning can be elicited from base language models by optimized prompting strategies. Given the impressive performance of these models, one might imagine how they could be used to also predict personalized treatment strategies. However, we note that simply prompting a specialized medical language model to return a treat- ment response is not equivalent to iterative optimization with language model against a black-box evaluator (i.e., the surrogate model). For this reason, simply querying an LLM to return a treatment strategy for patient is outside the scope of our work on optimization strategies herein.
Existing methods in personalized medicine. Aside from the methodology-focusee related works, also provide brief overview of existing applied works in personalized medicine. Recent clin- ical trials across clinical disciplines have tested personalized approaches by selecting or adapting treatments for subgroups or individuals (He al., 2018). de Bono et al. (2020); Paik al. (2020) are recent randomized clinical trials that leverage patient-specific genomic data to assign patients to a predetermined experimental treatment arm. In current clinical practice, certain pharmacogenetic variables already help determine whether certain medications are prescribed to particular patients (Ma et al., 2010 Ford & Tauberi 2013 Kirchheiner et al. 2007]. Using machine learning to propose entirely new treatment arms for individual patients is the ultimate goal of personalized medicine, but
is not yet a core component of clinical practice (Johnson et al. 2020 Chen et al. 2025c).
C ADDITIONAL IMPLEMENTATION DETAILS
C.1 OPTIMIZATION TASK SPECIFICATIONS
Supplementary Table Si outlines task-specific implementation details, including information on
the underlying distribution shift and number of design dimensions for each task. Below, we further discuss the motivation behind and implementation of each optimization task considered in our work.
Warfarin. Warfarin is oral anticoagulant medication used to treat and prevent variety of throm- boembolic conditions, such as atrial fibrillation (Brass et al.) 1997). venous thromboembolisms (Ramzi & Leeper 2004), and maintenance therapy after mechanical valve placement (Catterall et al.) 2020). However, determining its initial dose is notoriously complex, in part due to the medication s
narrow therapeutic index and significant inter-patient variability (Lee & Klein 2013 Consortium
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2009).
Dosing is influenced by numerous factors including age, body weight, liver function, and vitamin K metabolism. These complexities make standardized dosing unreliable and necessitate careful titration using frequent blood tests to achieve and maintain therapeutic anticoagulation.
Consortium (2009) previously introduced a de-identified, publicly available dataset of 5,052 patients from multiple hospital sites with clinical indications for warfarin initiation. The dataset includes both clinical and pharmacogenetic observations for all patients, in addition to their prescribed warfarin dose and laboratory test results. More explicidly, each patient observation includes a prescribed warfarin dose I E X' C R and conditioning vector 2 that concatenates the patient's height (caDSR 649); weight (caDSR 2179689): age (NIH Concept ID C25150); sex at birth (NIH Concept ID C124436); smoking status (NLM miSgwj5s2ig); the use of medications known to affect warfarin dosing (carbamazepine, phenytoin, rifampin, amiodarone); and the presence of genotype variants of CYP2C9 and VKORČ1 genes. Using this dataset, [Consortium] (2009) learn an expert function dose: : 2 -> X' (validated by physician specialists) that predicts the optimal dose for given patient. The optimization objective for our task is to minimize the root mean squared error (RMSE) |Ir - (z)s l|2 (i.e., to find the warfarin dose predicted by the expert function).
HIV. Human immunodeficiencyy virus (HIV) is virus that attacks the human immune system, and is currently treated with antiretroviral therapy (ART) (Kemnic & Gulick (225)). Choosing the correct ART drug regimen is carefully tailored, patient-centered process driven largely by viral genotype and individual health factors (Rhee et al. 2003). Before initiating treatment, gold-standard practice involves performing genotypic resistance testing to identify mutations in the HIV reverse transcrip- tase (RT) and protease sequences, as certain mutations can potentially render certain ART drugs ineffective (Saag et al. 2020). Treatment planning becomes even more complex in patients with transmitted or archived drug resistance (Kemnic & Gulick, (025). For these reasons, a personal- ized approach HIV treatment can help more effectively tailor ART to both viral mutation profiles and patient variables. The efficacy of an ART drug panel is clinically evaluated by measuring the patient's H/V viral load (i.e., the 'amount' of HIV in the patient's 's blood) before and after therapy.
We use the publicly available HIVDB dataset from Rhee et al. (2003) to study how to personalize ART treatment regimens for patients based on their viral genotyping data. patient's corresponding conditioning vector was constructed by concatenating the positions and amino acid substitutions of the patient's entire HIV protease sequence and the first 240 amino acids of the p66 subunit of the HIV reverse transcriptase protein following prior work. We frame the problem of designing personalized ART therapies as a binary optimization problem; the design space consists of 16 ART drugs: 8 Nucleoside RT Inhibitors (Abacavir, Zidovudine, Didanosine, Emtricitabine, Lamivudine, Stavudine, Tenofovir, and Zalcitibine); 3 Non-Nucleoside RT Inhibitors (Delavirdine, Efavirenz, and Nevirapine); and Protease Inhibitors (Atazanavir, Indinavir, Nelfinavir, Ritonavir, and Saquinavir). We chose these 16 drugs because they were the most commonly used drugs in the HIVDB dataset. Each vector x E {0, ] 16 in the design space specifies the ART regimen to prescribe the patient, where the ith dimension x; if the ith drug is included in the regimen.
Breast and Lung, Breast cancer and lung cancer are among the most prevalent cancers diagnosed in the United States (Kim et al.) 2025; Xu et al. [024] Ganti et al. [2021] Zhou et al.) 2024). There has been recent interest
personalizing oncologic treatments based on a patient's unique genetic and environmental makeup (Paik et al. 2020 Papalexis et al.] 2024). To this end, we use the fol- lowing set of features to construct the conditioning vector z: (I) age (NIH Concept ID C25150); (2) sex at birth (NIH Concept ID C124436); (3) race (NLM LakFoYkywC); (4) socioeconomic index; (5) cancer TNM staging: (6) vitals (excluding height and weight) (NLM GORIOFJAKC
and (7) any labs and biomarkers available for the patient. The following blood laboratory values and biomarker tests were used to construct 2 if available: 4 numerical complete blood count (CBC, NLM DZgb tests (Hematocrit, White blood cell count, Hemoglobin, Platelet count); 6 numerical basic metabolic panel (BMP) tests (Sodium, Potassium, Chloride, Creatinine, Blood urea nitrogen, and Glucose); 6 numerical liver function tests (LFTs) (Aspartate aminotransferase,, Alanine amino- transferase, Alkaline Phosphatase, Albumin, Total bilirubin, and Total serum protein): 4 numerical miscellaneous tests (Serum calcium, Estimated glomerular filtration rate, Glomerular filtration rate, and Lactate dehydrogenase); and 15 categorical biomarker tests (Human epidermal growth factor re- ceptor 2 (HER2). Progesterone receptor (PR), Estrogen receptor (ER), Ki67, Anaplastic lymphoma kinase (ALK). BRAF, Kirsten rat sarcoma viral oncogene homolog (KRAS), ROSI, RET, MET, Pro- grammed cell death ligand (PDL1), tumor protein 53 (TP53), Kelch-like BCH-associated protein 1 (KEAP1), Serine/threninne kinase 11 (STKI1), and breast cancer gene (BRCA)).
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The design space for both tasks consists of (1) whether to perform surgery with curative intent (0 or 1): (2) whether to perform radiation therapy with curative intent (0 or 1); (3) whether chemotherapy should be adjuvant neoadjuvant; and (4) the specific drug(s) to include in the chemotherapeutic regimen. The chemotherapeutte drugs to choose from include 9 cytotoxic chemotherapies (Car- boplatin, Cisplatin, Cyclophosphamide, Pemetrexed, Capecitabine, Gemcitabine, Paclitaxel, Doc- etaxel, Doxorubicin); 3 cell cycle inhibitors (Palbociclib, Abemaciclib, Ribociclib); 5 hormonal therapies (Anastrozole, Exemestane, Letrozole, Tamoxifen, Fulvestrant); 9 monoclonal antibodies (Pembrolizumab, Nivolumab, Cemiplimab, Atezolizumab, Durvalumab, Ipilimumab, Trastuzumab, Pertuzumab, Sacituzumab); and 1 tyrosine kinase inhibitor (Osimertinib). We chose these spe- cific 27 medications because they were the most commonly prescribed medications in the Flatiron dataset. Altogether, the design space represented as a boolean vector in {0, 13.0)
ADR. An adverse drug reaction (ADR) is an unintended and harmful response to medication. Minimizing the risk of ADRS a priority early-stage clinical trials order ensure that newly tested therapies are safe for human use. In this task, we leverage two internal proprictary clinical trial datasets de-identified at the patient level that were collected from two clinical trials to predict patient risk of an ADR. The primary goal of the clinical trials was to evaluate the safety of two different medications that share similar mechanisms of action, which we refer to as 'Drug A' and Drug B' in Supplementary Table [ST] Due to the sensitive nature of the internal trial data, do not disclose the identities of Drugs A and B, or the particular ADR under investigation.
The following patient covariates were used construct the conditioning vectors 2: patient sex birth (NIH Concept ID C124436); (2) patient age (NIH Concept ID C25150); (3) tumor measure- ments; (4) medical diagnosis; (5) ECOG score (caDSR 88); (6) vitals (excluding height and weight) (NLM GOTIOFJAKC (7) any patient labs and/or biomarkers; and (8) medication dosing schedule. Each patient observation included a label of the severity of the observed ADR (if any) in response to the patient's 's therapy: we first one-hot-encods this label into a vector 28 with 5 dimensions (one for each severity level of the ADR, consistent with medical societal guidelines (Khan et al. 2023] Lee et al.] [019 Trotti et al. 2003)). and then perturb this vector by randomly adding a uniformly sampled per-dimension jjtter' 8, ~ [4 1/4] for each zero-valued dimension i to produce a vector x E [0, 1)s such that x, = 6, if _ = 0 and x; = 1 - Ejt1 ō5 if x. = 1. this way, the index of the maximum element in both x and . are equal. We then compute the corresponding negative log x; as the objective to minimize. These values are then used log- oog-likelhhood (NLL) := -E _.
to construct the annotated datasets D {Iz 2j. [;)) Note that x not accessible by the optimizer. This approach allows us to turn predictive task (i.e., predicting the severity of a patient ADR in a clinical trial) into an optimization task compatible with LEON.
C.2 FULL ALGORITIM PSEUDOCODE FOR LEON
The full pseudocode for LEON is shown in Supplementary Algorithm 1 Prior to the start of op- timization, we also leverage the backbone LLM to query external knowledge bases for prior knowl- edge the full pseudocode for this subroutine is shown in Supplementary Algorithm 2
C.3 LANGUAGE MODEL PROMPTS
This section includes the system and user prompts used for all experiments using LEON. Baseline methods using LLMs as optimizers used the same sets of prompts as provided by the respective original authors, with modifications made only to adapt the prompts to our experimental tasks.
For each task in our evaluation suite, we first define task descri in on variable that describes the optimization task in natural language. The specific strings used in each task are included below:
task.description for the Warfarin task:
The provided design scores are predictions from a model trained on White patients only, and therefore may not be accurate for all patients. Propose an optimal warfarin dose (in mg/week) for the patient.
task description for the HIV task:
The provided design scores are prediction from model traine on
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Algorithm (LEON). LLM-based Entropy-guided Optimization with kNowledgeable priors
Inputs:
(x, X x R R- pre-trained surrogate model
(x) - RR- - initialized source critic model
2 vector of fixed patient-specifie covariates
LLM - large language model that supports tool calling
knowledge - prior knowledge in natural language generated using Algorithm 2 Jusing the
same input LLM
0 2 initial souree critic certainty parameter value (default 0.0)
Wo 2 0 - I-Wasserstein distance bound (default 1.0)
JA 2 - source critic certainty parameter step size (default 0.1)
Meriti > 0 source critic learning rate (default 0.001)
T 2 0 temperature (default 1.0)
b> 1 batch size (default 32)
B b surrogate model f evaluation budget (default 2048)
Initialize memory bank of sampled candidates Dgen
Initialize À + A Source critic certainty parameter as in Lemma 4.2 Initialize reflection for 1... [B/b] do
// Initialize the LLM'' reflection the empty string.
1. Sampling. Query the LLM for batch of designs. new = {a Jj=1 + LLM gen www.wooogeee...... reflectionn iin)
// Evaluate the designs according to the source-critic regularized surrogate.
new = {u + (f( + Ac "
// 2. Cluster. Assign the designs to their equivalence classes for S S do
Append 2:w to its equivalence class [==]o]] end for
Compute the fractional occupancies q of each equivalence class.
for < i N do
y0 max [2) (x; (xxz)
2) (x)]
9:
|
now
end for
// 3. μ Estimation. Estimate according to
for i
ōyg y
Lv=1 Di EN= log ON
5(log i), <- log q:
N
end for i + E 5(log 4).l/ lo dy.]
// Re-train the source critic parameters 0, and update À certainty parameter.
while oW has not converged do
8W Ve. Le
E [x)]
0g min(max(O,. Meriti oW, 0.01),
end while
A (na/VE) (0g/0A)
Compute the partial gradient according
// Design Scoring. Score each sampled design according ((((;;; z) + Ac (x))
D gea + D gen U {( , )
1 // Save the new batch of designs to memory.
// Reflection. Prompt the language model to reflect on the current optimization progress. // The specific prompt for reflection used in our implementation included in Appendix re flect ion. + LLM({2))u) ,t ye J==1, re reflectio prompt)
end for
return top candidate from D with the maximum saved f yj predicted score
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Algorithm 2 Knowledge generation using external knowledge repositories
Inputs:
K = [KRbJk- - a set of FKRR user-provided knowledge repositories (KR)
LLM large language model that supports tool calling
T20 0 - the maximum number of allowed sequential tool calls
// The specific prompt used in our implementation included in Appendix knowledge -
prompt in natural language to elicit knowledge generation
C- // Initialize the conversation context.
Sequential tool calling. for / 1...7a.. do
// The LLM returns which knowledge repository to call and a corresponding text query dt- Early stopping is allowed if the LLM determines no additional tool calls required. (kg, 9t. STOP) + LLM(C,, knowledge.proopp:// prompt;
STOP True then break
end if
// Retrieve relevant knowledge ra from the requested knowledge repository KR
rt t KRi, (qt)
// Append the query and response to the conversation context.
C t- CU ((,,,
end for
Generate the final knowledge in natural language.
knowledge < LLM(C, knwwledge.prooptt return knowledge
patients from older studies (before 2008) only, and therefore may not be accurate for all patients. Propose an optimal HIV medication regimen for the patient.
task.description for the Breast task:
predictions from model trained on patients under 65 years old only, and therefore may not be accurate for this patient. Propose an optimal treatment regimen for the patient consisting of a combination of adjuvant or neoadjuvant medications, whether to undergo surgery, and whether to underge radiation therapy.
The provided design scores
task.description for the Lung task:
The provided design scores are predictions from a model trained on patients with a good response to therapy only, and therefore may not be accurate for this patient. Propose an optimal treatment regimen for the patient consisting of combination of adjuvant or whether to undergo surgery, and whether undergo radiation therapy.
neoadjuvant
task description for the ADR task:
The provided design scores are prediction from model trained on patients treated with <Drug A> only, and therefore may not be accurate for this patient. Predict the probability of each grade of <Adverse Drug Reaction (ADR) under the current medication.
Redacted components are indicated by <> angle brackets above. Separately, each pair consisting of a patient with fixed covariates and proposed design was programmatically represented in Representative example values for this string in each task are included below; exact values have been modified to preserve patient anonymity. In addition to being used in the language model prompt during optimization (see the User prompt for treatment designs below), each of the patient_deserigg of (x, 2) tuples were also used as input into an embedding model to define the equivalence relation detailed in the main text. We ablate the choice of embedding model in Supplementary Table S9
natural language as a www.eeeddderrpptiiiooee
proposing
on representations
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Sample patient.descriptio in the Warfarin task:
Asian patient (age 70 79 years old) with BMI of 25.7. CYP2C9
Genotype Variant: *1/al. VKORCI SNP: A/G.
Warfarin Dose: 32.0 mg/week
Sample patlent.description in the HIV task:
Patient newly diagnosed with HIV-1 has the following HIV Protease Mutations: N37A; 115V; т12K. The patient also has the following HIV Reverse Transcriptase Mutations: E204G; I135I; G196E
Prescribed Medications: Zidovudine (Retrovir), Lamivudine (Epivir), Abacavir (Ziagen)
Sample patient.cescriptioo.. in the Breast task:
69 y.o. African American Female diagnosed with Stage (T2NOMO) Breast Cancer.
Vitals:
BP 130/85 Sp02 99.0 HR 70
Labs:
[CBC] Ect: 35.88 Hgb: 11.7 WBC: 4.8 Plt: 259.0
[BMP] Na: 143 C1: 99 | Glu: 90 K: 4.8
[LFTs] ALT: | AST 17 Albumin: | ALP: 94 Pret: 60
[Additional Labs and Biomarkers] Ki67: >=208
Treatment Plan:
Adjuvant ne with Surgery and Radiation
tabine
Sample patient description in the Lung task: y.o. Asian Male diagnosed with Stage (TICNOMO) NSCLC.
Vitals:
Sp02 90.0
Labs:
[CBC] Het: 48.58 Hgb: 15.1 WBC: 9.5
Plt: 243
[BMP) Na 138 C1: 101 | K: 3.5
Cr: 1.5 | BUN: 14 Glu: 95
[LFTs] ALT: | AST 13 ALF: 75 | TBili: 0.5 Prot: 65
[Additional Labs and Biomarkers Ca: 9.0
CEA: 5.8
Treatment Plan:
Adjuvant Fembrolizumab with Surgery
Sample in in the ADR task:
68 y.o. Female (ECOG 1) diagnosed with <Disease> currently treated with <Drug B>.
Dosing Schedule:
Day <Drug B> <Dose X>
Day <Drug B> <Dose Y>
Day 14: <Drug B> <Dose 2>
Tumor:
<Measurement A>: <Value
<Measurement B>: <Value B>
Vitals:
02Sat 98 HR 73 RR 17 T 36.7
BP 121/80
Labs:
[CBC] Ect: 35% | Hgb: 11.8 1 WBCs: 5.2
[BMP) Na: 140 Cl: 100 K: 4.2 HC03-: 24 Cr: 1.5 BUN:
[LFTs] Albumin: 3.9 ALT:: 25 I AST: 29 THili: 11
[Additional labs and Hiomarkers] Ca: 9.2
Mg: 2.0
Phos: 3.2
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Redacted components again indicated by <> angle brackets above. Finally, prompts in LEON also make use of a memory string, which is table of previously proposed designs (repre- sented in natural language) and their corresponding scores according LEON. Importantly, note that memory never contains any scores from the ground-truth objective, which never made available the optimizer. In general, memory is Warkdown-formatte table that includes the maximum number of most recently sampled design proposals subject to the LLM's context window.
Sample memory entry for the Warfarin task:
designs
scores
---
21.633 6396
-1.4465
Sample memory entry for the HIV task:
designs
sccres
---:1-
|
[ABC] Abacavir (Zi (gaa)),,, [3TC] Lami (Epivir) |
0.3491
Sample memory entry for the Breast task:
designs
sccres
|
Adjuvant Chemotherapy (Anastrozole) with Surgery
-0.5438
Sample memory entry for the Lung task:
designs
scores
Chemotherapy (Carboplatin, Faclitaxel)
-0. 6062
Sample memory entry for the ADR task:
designs
scores
(0.CCCO, 0. 0.2575, 0 000000, 62889, 0136))
1.0022
We now provide the system and user prompts used LEON as format strings. Firstly, step 1. Sampling of Supplementary Algorithm new batch of designs; the system and user prompts for this process are included here:
prompt the language model optimizer to return
System prompt for proposing treatment designs:
You are clinical assistant whose role is to propose treatments for simulated patients based histcrical data and iterative feedback. You will be provided with history of treatments along with their respective performance scores. Your task is to propose new treatments that potentially yield better outcomes. After each proposal, feedback based on evaluation results will be provided to refine future proposals.
### Opt intimizationn Strategy
Learn from the history of previcus treaatments and their scores. Balance exploration (trying diverse treatment options) and exploitation (refining successiul treatments).
Do not propose treatments identicad those already evalusted.
WF Input Format
In each round, you will receive:
**Frior knowledge:** Some petentially relevant context
consider that is specific to the particular patient.
2. * A table of previcusly evaluated treatments and their scores. Each row in the list will contain the follewing: round_index, the integer index of the round treatment: the proposed treatment
score: float score
A higher score indicates better patient treatment.
### Output Format
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You must propose treatment for the simulated patient that will be scored by the evaluator. Your goal is to find treatments that are high scoring.
User prompt for proposing new treatment designs:
(prior_ knowledgee
[patient__deccrittio)]
II Frevicusly Proposed Designs (memory)
### Reflection (reflection)
### Task (task_description))
where prior,knowledes is string of prior knowledge returned by Supplementary Algo- rithm [] and reflection, is the reflection string from the previous optimization loop in Sup- plementary Algorithm [ Separately, the prompt grcowledge_promae used in Supplementary Algorithm
2 is shown below:
System prompt for knowledge generation:
You are a helpful biomedical knowledge assistant whose job is to retrieve relevant knewledge to help a domain expert solve specific task in biology and medicine. Given a description of patient and clinical task, you will decide which knowledge sources to retrieve from (and with what arguments). You may call multiple functions sequentially; when you have enough information, return a final answer of relevant prior knowledge without any further function calls. Be specific, concise, and comprehensive in your response.
User prompt for knowledge generation:
[patient_deseeeeeeeeee description))))
Problem Description: (task_description)
Provide relevant factual informaticn to help the expert solve the problem.
Finally, Supplementary A describes reflection call to the LLM at the end of each optimization step, where the goal is to prompt the LLM to reflect on the efficacy of the current optimization progress. The input regeppprefleecioopprooomo for this process is shown below:
### Task Description .task_aescription/
### Previcusly Proposed Designs (memory)
proposed designs and reflect on how to better improve the score. In additicn to score-based optinization, evaluate which design strategies are working and which are not **from a biomedical and biological perspective**.
Please analyze the scores
Specifically:
Identify any recurring features or mechanisms in high-scoring designs
that are biologically plausible or supported by known biomedical
principles.
Point out any strategies in lew-scoring designs that mi might be failing
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due to biological implausibiitty, off-target effects, instability, toxicity, or other biomedical concerns. Consider how principles from mclecular biology, pharmacokinetics, immunology, or relevant biomedical disciplines might explain the observed outcones.
Do NOT prepese new design in your response! Only respond with your thinking.
C.4 EXCLUDED BASELINES
Evolutionary algorithms. Adapting LLMs as evolutionary optimizers has been explored in prior work, albeit primarily for singular tasks without easy generalization of the methods to new prolems, such as the biomedical tasks considered herein. We exclude LLM-driven Evolutionary Algorithms (LMEA) from Liu al. 2024a) because their work and publicly available implementation is specif- ically adapted for solving the traveling salesman problem; generalizing their implementation while ensuring its faithfulness to the authors' original implementation is outside the scope of this work. Similarly, we exclude EvoPrompting from Chen et al. (2023a) and EvoPrompt Guo et al. (2024b), as these prior works are only implemented by the original authors for neural architecture search and language model prompt optimization, respectively. Finally, AlphaFvolve from Novikov et al. (2025) and FunSearch from Romera-Paredes et al. (2024) are also LLM-based evolutionary algorithms, but do not optimize against a dense reward function as we do in our work.
Biomedical optimization. Prior work on leveraging language models for biomedical optimization tasks have almost exclusively fine-tuned decoder models for biological sequence generation. For ex- ample, the methods from Reinhart & Statt (2024) and et al. (2024a) make use of domain -pecifc molecular dynamics and material point method (MPM) simulators in their respective frameworks, and therefore cannot be generalized to our experimental setting. Language Model Optimization with Margin Expectation (LLOME) from Chen et al. (2025a) involves language model post-training for generating new protein and DNA sequences, and was therefore excluded from our evaluation. We also exclude the method from Flam-Shepherd et al. (2022) as a baseline since it introduced molecule distribution matcher, which requires high-quality designs as input (unavailable to us in our setting). Finally, reinforcement learning based methods, such as DyNA proximal policy optimiza- tion (PPO) from Angermueller et al. (2020) and REINFORCE from Williams (1992), are excluded since we do not consider reinforcement learning tasks in our work.
D ADDITIONAL EXPERIMENTAL RESULTS
D.1 DISTRIBUTION SHIFT ANALYSIS
To characterize the distribution shift between the surrogate model trained on the source dataset and the ground-truth objective learned the target dataset, first compare the performance of the surrogate and objective functions on accurately predicting the annotated labels in the target dataset in Supplementary Figure ST We also plot the agreement between each of the functions and the ground-truth annotation for the target dataset in Supplementary Figure As expected, the surrogate model consistently underperforms on the target dataset when compared the ground-truth oracle function, reflecting the predictive performance drop in real-world clinical decision making tasks under distribution shift. Finally in Supplementary Figure SS3 we also plot the distribution of ground-truth annotations in both the source and target datasets.
D.2 COST ANALYSIS
Token usage and monetary cost. Using OpenAl's 4pt--40-miii-2222--0111 LLM as the backbone optimizer, we report the total number of input and output tokens used in LEON across the tasks in our evaluation suite in Supplementary Fig. S4
As of September 2025, the standard model is $0.15 per million input tokens and cached inputs, the cost per patient for using LEON is $0.21 = 0.00 on the Warfarin task; $0.56 + 0.00 on the HIV task; $0.60 + 0.01 the Breast task; $0.51 + 0.00 on the Lung task; and $0.44 + 0.00 on the ADR task.
API pricing for the $0.60 per million output tokens. Using these price points and assuming
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Souree Mookl Tarpet Model
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Figure S1: Correlation plots of predictive models used for design evaluation. In our work, we investigate the utility of LLMs in solving 5 challenging biomedical optimization problems under distribution shift. An optimizer is only given access to model trained on data sampled from source distribution (red) to score candidate designs, but the final proposals are scored using the ground-trute objective function (green) hidden during evaluation. The oracle is either an exact function if the true objective is known, machine learning model otherwise. We evaluate the agreement between the surrogate and oracle functions with the ground truth annotation in the target dataset.
Carbon emissions. Consistent with Samsi et al. (2033) and a publicly available carbon calculator
assume energy efficiency of 7.594 10 kilowatt-hours (kWh) per thousand output to- kens per billion LLM active parameters (assuming token generation is performed on NVIDIA A100 GPUs). We primarily of this model is currently unknown, we conservatively estimate to be approximately 20B active parameters for the purposes of cost estimation. Assuming carbon intensity of 0.39 kg CO>e/kWh and power usage effectiveness (PUE) of 1.17 based on U.S. data (Ren et al. 2024), a back-of-the- envelope calculation tells us that the cost per patient for using LEON is approximately 1.66 x 10-3 kg CO2e on the Warfarin task; 1.03 x 10 kg COze on the HIV task; 1.95 x 10 kg CO2e on the Breast task; 1.14 x 10-2 kg CO2e on the Lung task; and 6.38 x 10-3 kg CO.e on the ADR task.
opt-4o-mini--022-
model from OpenAI; while the size
GPU utilization. We observed that GPU utilization was dominated by the size of locally deployed LLMs (e.g., the MedGemma 27B model used as an external knowledge repository Supplemen- tary Algorithm@)) rather than by proposed method. As shown in Supplementary Algorithm[] the GPU memory requirements for LEON scale linearly with the dimensionality of the patient- design space and the number of parameters of the surrogate model f bototh of which are orders of magnitude smaller than the resource footprint of the external LLM. In practice, found that the maximum GPU memory used by LEON did not exceed 3 GB for any given task.
D.3 QUANTITATIVE ANALYSIS PRIOR KNOWLEDGE
Recall from Supplementary Algorithm 2 that the availability of external knowledge sources is key component of LEON. While a base generalist language model may not have the prior knowledge to achieve high certainty in its proposed designs according to (4), the LLM can query repositories of domain domain-specific expert knowledge to ultimately synthesize knowledge natural language that can be used for the optimization task. To better interrogate this process, plot the frequency with
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Warfarin
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Sanyon Moddil NYSKE p.0 Target Moddl NOBB g0.00
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(monthu)
Figure S2: Bland-Altman plots of predictive models used for design evaluation. In our work, we investigate the utility of LLMs in solving 6 challenging biomedical optimization problems under distribution shift. More explicitly, an LLM is only given access surrogate model trained on data from a source distribution (red) to score candidate designs, but the final proposals are scored using the ground-truth objective (green) learned on data from the target distribution and hidden during optimization. We evaluate the agreement between each of the source- and target- trained models with the ground truth annotation for each datum in the target dataset. Mean and 95% confidence intervals for both models in each task are indicated by the horizontal dotted lines.
which each of the expert knowledge repositories are individually queried as tools across 100 patients for each of the 5 benchmarking tasks (Supplementary Fig.
Our results demonstrate that the same base language model (opt--oommimi--24400---1-1 ferentially queries the set of external knowledge bases as a function of the underlying task. The total number of knowledge base queries per patient differs by task (mean + standard error of mean (SEM); Warfarin: 2.91 + 0.10; HIV: 3.01 + 0.05; Breast: 3.16 + 0.09; Lung: 3.50 + 0.05; ADR: 3.63 = 0.05 queries per patient). The increasing number of queries correlates with the authors' sub- jective evaluation of the difficulty of each task: that is, Warfarin (resp., ADR) is the least (resp., most) challenging optimization problem in our evaluation suite. These results suggest that the language model preferentially relies on more prior knowledge sources as the difficulty of the optimization task increases.
dif-
Separately, we observe that the specific knowledge sources queried by the model vary by task: for example, the COSMIC knowledge base from Tate et al. (2019) is preferentially queried in the Breast and Lung tasks that deal with oncologic treatment optimization. We also observe that the knowledge bases GDSC (Yang et al. 2013) and DepMap (Tshemiikk et al. 2017) are rarely queried by the language model across all 5 tasks. This makes sense, as both knowledge sources detail properties about cell lines (as opposed to patients), and are therefore unlikely to be relevant for any of our tasks. Finally, we ran a 2 of homogeneity at the per-task level to evaluate whether different knowledge bases had different rates of being queried within each optimization task. As made visually apparent from Supplementary Figure [S5 the observed differences in the frequency of knowledge bases being queried per task are far too large to be explained by random sampling variation alone (Warfarin: x² x 535.3, dof 7, p ze 2.09 x 10 111, HIV: x² x 763.0, dof 7, p s 1.81 x 10 160, Breast: x² x 452.3, dof = 7, == 1.39 x 10 93, Lung: X² xs 556.8, dof = 7, p 4.96 x 116, ADR: x² ~ 674.3, dof 7. 2.36 x 10 10-141). Overall, these results suggest
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Warfarin
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Figure S3: Distributions of ground truth scores within source and target datasets. We plot the distribution of ground-trun -trth objective values for both the source (red) and target (green) datasets.
3500
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Figure S4: Token usage of LEON across different tasks. We plot the average number of input and output tokens used per patient experiment with the got-4o gpt-4o-mini 2---20224--222 backbone optimizer. Error bars represent the standard error of the mean across 100 patient experiments.
that the backbone LLM is selecting knowledge sources retrieve relevant expert prior knowledge for given optimization task in a manner that cannot be attributed to random chance alone.
D.4 QUALITATIVE EXAMPLES OF PRIOR KNOWLEDGE
Recall from Supplementary Algorithm 1 that a key component of LEON is synthesizing domain- specific prior knowledge: given access to relevant biomedical knowledge bases, we task the baseline language model with making the appropriate tool calls to gather domain-specific and context-specific prior knowledge conditioned on the task description and particular covariates z. The language model then asked to respond with relevant prior knowledge to use as context for downstream optimization-sae Appendix C.3 for additional details. Here, provide representa- tive examples of the prior knowledge outputs (indicated by monospace font below) generated by for the Warfarin task. We focus this particular task because a domain area where we are able to accurately verify LLM outputs. Furthermore, a linear oracle model from Consortium 2009) is available for this task, facilitating the domain-specific verifica-
10t-44--mini2222-4
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Warfarin = 2.91 = 0.10)
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Terthooks
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Hetione KC
Prine KG
Collesaune
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Figure S5: Distribution of knowledge sources called tools by the LLM for prior knowledge synthesis. We plot the frequency of querying each repository (over 100 patients) in Supplementary Algorithm acrros each of the optimization tasks evaluated. Fach subplot title includes the mean + standard error of the mean (SEM) number of tool calls made per patient.
tion of the consistency of LLMS with published prior work. To evaluate the quality of the LLM- synthetized prior knowledge, U.S. resident physician and U.S. senior medical student manually annotated the LLM LLM-generated knowledge for factual accuracy; in cases where annotations semanti- cally disagreed, the prior knowledge output was discussed until consensus was reached. We include their annotations as right-justifiet colored serif font in the listings below.
In this first example, all generated prior knowledge was considered relevant and factually consis- tent with baadence-bbsedd medical guidelines. The LLM optimizer queried the following knowledge sources as tool calls (in order): (1) MedGemma 27B; (2) Hetionet Knowledge Graph; (3) Prime Knowledge Graph; and (4) Medical Textbooks (see Appendix C for additional details). Further- more, the final proposed warfarin dosing strategy score proposed by the LLM optimizer using the prior knowledge and LEON achieved an oracle score of 1.24, which more optimal than the mean score achieved by our method on this task in Table
For a 30-39 year old Asian patient with a BMI of 29.1, 1, a CYP2C9 genotype of *1/x1, and VKORC1 SNP of A/A, the following information
The demographic information here is accurate for this particular patient.
is relevant for determining
optimal warfarin
dose:
1. **Geneiic Iniluence on Warfarin Dosing**: The CYP2C9 +1/+1 genotype indicates normal metabolism of warfarin, meaning the patient is not a poor metabolizer. However, this genotype does not require a significantly higher dose compared to
This is consistent with expert guidance
(conne et al. 2005), and also with
the oracle dosing algorithm from
Consortium 2009 used in this work.
those with reduced function alleles.
The VKORC1 A/A genotype suggests
This is also consistent with expert guidance [Dean] 2012). and also with the linear oracle dosing algorithm from Consortium 2009
inereased sensitivity to warfarin, typically requiring lower doses. Studies incicate that individuals with the A/A lower doses than those with non-A haplotypes
2. *initial Dosing tecommendaticnss.
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Standard initial dosing for warfarin is typically between 5-10 mg per day.
This is consistent with medical guidelines (Dean 2012 A starting dose of 3-4 mg/day is usually recommended for this patient in the absence of other medical history.
However, due to the VKORC1 A/A genotype, a lower starting dose is advisable.
A common approach is to start with dose of around 2.5-5 mg per day
for patients with the A/A genctype,
especially considering the patient's Asian ethnicity, which is asscciated with higher prevalence of VKORC1 variants.
In this second example on the Warfarin task, there was a factually incorrect statement made by the LLM, highlighted in red below. The LLM optimizer queried the following knowledge sources in order: (1) Medical Textbooks; and (2) MedGemma 27B. Quantitativly,, we found that the LLM optimizer consequently performed worse on this patient, achieving an inferior oracle score of 2.32.
To propose optimal warfarin dese for the patient with the following characteristics:
**Age**: 60-69 years
The demographic information here
**HMI**: 23.4
is accurate for this particular patient.
..CYP2C9 Genotypeжи: *1/+1 (wild-type)
**VKORC1 SNP**: A/G
WH Relevant Information:
1. ..CYP2C9 Genotype**:
The +1/+1 genotype indicates nermal enzyme activity, meaning the patient is a "nermal metabolizer" of warfarin.
This is consistent with expert guidanee
Sconce al. 2005), and also with
the linear oracle dosing algorithm from
Patients with this genotype typically require higher doses of warfarin compared those with variant alleles (+2 or +3).
Consortium 2009 used in this work.
2. VKORCC Genotype**:
The A/G genotype suggests that the patient has moderate response warfarin. The presence of the allele is asscciated with lower VKORC1 expressicn, which generally leads to reduced requirement for warfarin compared those with the A/A genotype. specifically, individuals with the A/G genotype may require about 25% lower doses than A/A homozygates.
While heterozygous A/G patients for the VKORCI: -1339G-AA polymorphism have moderate expression of VKORRI, this genotype generally leads to higher warfarin doses than that for
A/A individuals Dean 2012 This knowledge output also mccosistent with ground-trun dosing algorithm from Consortium (2009 used in this work.
3. Dosing Guidelines
Initial warfarin dosing typically starts at 5-10 mg/day. Given the patient's s age and the influence of VKORC1, a conservative appreach is advisable.
This dosage guidance is consistent with clinical recommendations (Dean 2012
For a patient with CYP2C9 +1/+1 VKORC1 A/G, a reasonable starting dose might be around mg/day, with justments based on INR menitering.
D.5 ADDITIONAL. ANALYSIS OF THE L.MM CERTAINTY PARAMETER /
Recall that the certainty parameter in LEON is the Lagrange multiplier of the entropy-based constraint our constrained problem formulation in Larger values of are observed when the LLM consistently returns equivalent treatment designs for given input prompt despite the nondeterminism of the generative model (as dictated by the LLM temperature hyperparmaeter т = 1.0 used in the main text). Using this framework, course of optimization to better interrogate how the certainty of the language model in proposing high-quality treatment designs changes during optimization.
plot how the value of μ changes over the
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1.7
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Figure S6: LLM certainty parameter μ. over the course of optimization. We plot the mean + SEM value of estimated using ( at each optimization step for n 100 independent target patient experiments. Up (resp. down) arrows indicate that the task a maximization (resp., minimization) task. Higher values of suggest greater confidence of the LLM in its design proposals.
Our results shown in Supplementary Figure S6 We find that the value of μ generally increases as function of the optimization step. This suggests that the certainty of the LLM increases more designs are observed and evaluated using our method. One possible explanation for this trend is that LLM certainty is function of not only the prior knowledge generated, but also the previously proposed designs. That is, the LLM is more confident in future designs because is able 'learn' from the history of previously sampled designs and their scores according to LEON.
D.6 ADDITIONAL ANALYSIS OF THE SOURCE CRITIC WEIGHTING PARAMETER X
Similar to our analysis in Appendix D.5 we can analyze how that À certainty parameter varies over the course of optimization, where À is the Lagrange multiplier of the source critic-based constraint in our constrained problem formulation in (4). Qualitatively, larger values of À are associated with upweighting the importance of the source critic relative the surrogate predictive model.
We find that the value of consistently in- creases function of the optimization step. This suggests that the importance of the LLM in- creases as more designs are observed and evaluated in LEON. Future work may investigate how the performance of LEON changes function of the initial value of À in Supplementary Algo- rithm I and both the step size and number of gradient update steps in
Our results are shown in Supplementary Figure S7
D.7 EXTENDING LEON TO TRADITIONAL OPTIMIZERS
In principle, there nothing Supplementary Algorithm that is particular to language model- based optimizers: the auxiliary source critic model can be used in conjunction with any optimization method, and the entropy- constraint is enforcable with any method that supports batched acqui- sitions. To this end, implement LEON with gradient ascent (Grad.) and Bayesian optimization (BO-qEI) and evaluate LEON with these traditional optimizers in Supplementary Table
Our results suggest that using LEON with these traditional, non-LLM based optimizers does not offer a meaningful advantage over the corresponding baselines. SEON-BO-qu improves upon BO- qEI on 3 of the 5 tasks, and LEON- improves upon Grad. only of the 5 tasks. This trend is
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Warfarin
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Figure S7: Source critic weighting parameter over the course of optimization. We plot the mean + SEM value of X computed according to (I1) at each optimization step for n 100 inde- pendent target patient experiments. Up (resp. down) arrows indicate that the task a maximization (resp., minimization) task. Higher values of À suggest greater weight placed feedback from the source critic relative to the surrogate model.
Table S2: Evaluating LEON for traditional optimizers. We evaluate the utility of LEON in improving the performance on non-LLM based optimization methods gradient ascent (Grad.) and Bayesian optimization (BO-qEI). We report mean + standard eπrо of mean (SEM) oracle objective value achieved by the single proposed design for a given patient, averaged over n 100 test patients. Bolded cells indicate when using LEON improves upon the backhone optimizer for given task.
Method
Warfarin RMSE Loss (mg/week)
HIV
Breast TINTD (months)
NSCLC TTNTD (months)
CRS
Viral Load (copies/mL)
NLL Loss
units)
Grad.
1.37 + 0.13
4.52 + 0.04
73.07 + 2.30
24.67 = 0.34
23.7 + 1.7
LEON-Grad.
2.24 + 0.19
4.54 0.06
51.68 + 1.38
27.61 0.64
23.8 + 1.7
BO-gEI
1.36 + 0.13
4.53 + 0.04
67.11 + 1.86
28.65 = 0.65
23.5 + 1.7
L.EON-BOqEEI
2.02 + 0.15
4.48 0.05
65.52 + 2.34
31.04
0.82
21.4 + 1.8
consistent with the notion that prior knowledge is critical for LEON to consistently improve upon optimizer performance. Because traditional optimizers like Grad. and BO-qEI lack informative priors over the design space, the mntropy-based constraint central to LEON is less meaningful using these methods. Consequently, although LEON cedimiize-aggnsste in theory, its utility is limited when used with methods that do not incorporate prior knowledge.
D.8 WHY SHOULD WE BOUND THE 1- WWSSSERSTEIN DISTANCE?
A key underlying assumption behind our modified problem formulation in that the source critic-based constraint can meaningfully prevent over-extrapolation against the surrogate model that frequency observed in optimization tasks under distribution shift (Yao et al. 2024 Trabucco al. we provide a theoretical motivation to help support this assumption.
2022). In Theorem D.I.
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Theorem D.1 (Bound on Empirical Test Risk). Define a real-valued, Borel-measurable function f f: + R defined over a domain X C RR, and define K := [[f(z))|L to be the corresponding Lipschitz constant of f. Given finite dataset of n observations D := {(I; f (x;)))== suppose train a predictive model f on D with Lipschitz constant K finite such that the empirical training risk
e g) y [(x=)]ffnite Then, the test risk on a new sample of' test inputs T { is bounded from above by
B s (K Kj)Wi(uD.Hy)
where WI(DD. (TT) is the 1- I-Wasserstein distance associated with ||- 112-
Proof. Define E T(ND: pT) as the optimal coupling between input observations x' and æ in DD and T. respectively. For pairs (x'x)) ~ y, note that because prr the z-marginal of %
E (f - f(x) = ==(x5)))-1f(( - f(x))
= =B(ess)y ((())-((x)) (1(z²) - -((')) + (
< ()) (f(x)= f(xx) (((2)))((=)) BB
((x)))((z))
using the triangle inequality. Because ur is the x' -marginal of y, we rewrite the right hand side as
Bx-Tlf(x) -(())] S B())))
-(x)) (f('') + EE'ND (((0)) f(x²))
-
= E ([(x)- ((1o)--(a)-(110)---e- -
+
using the definition of the empirical training risk Applying the definition of the Lipschitz constants K, K as well as the definition of the I-Wasserstein distance gives
Be-T)S(>) f(x)) s (K + K - x' d'I - (K (KKKK)WWa Kj)Wo
The claim follows.
We remark that deriving the global Lipschitz bounds K, K2 & is N'Phhard and infeasible in practice (Scaman & Virmaux 2018 Hu et al. 2024).. However, the above result still holds K, K only hold locally over a finite subset X that contains 'D DU,, which is much easier derive. Furthermore, note that the constants & and Lipschitz constants K, K in Theorem D.1 are irreducible, since we
assume that we do not have control over D or the functions f. However, the above result also shows that bounding Wi (JDD (T) will yield a corresponding finite bound on the empirical test risk. This is key observation by intelligently choosing the test points in T that we use to evaluate with f, can guarantee bound the mean test error over T.
D.9 SAMPLE REFLECTION TRACE
To illustrate the process of reflection, we include a representative example reflection output for an optimization experiment on the Warfarin task. Based on the prompting strategy used for reflection in Appendix C.3 we encourage the language model to reflect on the optimization progress from both task aggosticc expllration-eeploottttii and biomedical reasoning perspectives.
Analyzing the previously proposed warfarin doses and their scores reveals several insights into the potential biological mechanisms and pharmacokinetic principles that may be influencing the cutcomes.
### High-Scoring Designs
1. **Doses with Lower Values**: The highest scores are associated with
lower doses, such as 11.3417 mg/week (score: -0. 398857) and 15. 1328 mg/wees (score: -0.330352). This suggests that lower doses may be more effective for this patient, possibly due to:
Lower doses may 1ead to more stable INR (Internations Normalized Ratic) levels, reducing the risk
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bleeding chremboemoolic events. Warfarin has incex, and small changes dese can lead to significant changes in ant anticcagulation effect.
therapeutic
Patticnt Variability*:: Individual responses to warfarin can vary significante due to genetic factors (e.g., VXORCI and CYP2C9 moerphsshimm) dietary influences, and concurrent redications. Lower doses may be more universally tolerated across diverse patient backgrounds,
2. Aveidance of High Doses**: The Lowest scores are associated with higher doses (e.g., 12.6990 ng/week with score -2.7) High doses may lead to:
**ooicityy:;; Increased risk of adverse effects, including major bleeding, which is a significant concern in anticoagulation therapy. :IInstabilitye Higher doses can lead to fluctuations in INR, making it difficult to maintain therapeutic levels.
#FF Low-Scoring Designs
1. Highh Doscs*A; As mentioncd, deses like 63.5996 mg/weet and 72.6990
mg/weck are associated with very low scores. These doses may be biologically implausible for many patients, leading **Off-Target Rffects**: Excessive anticoaqulation compl complications that outweigh the benefits of treatment.
can lead to
IIIcreaeed Variability Higher doses may exacerbate the
variability in response
genetic and envirenmental factors.
Inttermediate Doses**: Doses the mid-range (e.g., 40.5355 mg/week, 7895 mg/week) also received low scores. This could indicate that: **bbbpptmma imal Targeting**: These doses may not align well with the patient's specifis pharmacogenemic profile, leading to inadequate anticcagulation increased side effects.
### Biological and Biomedical Considerations
* Gencciic Factors**; The variability in warfarin metabolism due to genetic polymorphisms in CYP2C9 and VKCRC1 is well-documented. Understanding the patient's genetic makeup could help tailor the dose more effectively.
++bietary Interactions::: Vitamin K intake can signi ficantly affect warfarin efficacy. Patients with high dietary vitamin may require higher doses, while those with lew intake may need lower doses.
+*Drug Interactionsew: Concurrent medications can alter warfarin metabolism, necessitating dose adjustments. Awareness of these interactions is crucial for optimizing therapy.
IHF Conclusion
The analysis suggests that lower deses are generally more effective and safer for this patient population, likely due to the pharmacokinetic
properties of warfarin and the variability in patient responses. Future proposals should focus on exploring lower doses while considering indiviqual patient factors such as geneties, diet, and potential drug interactions to optimize outcomes.
D.10 SUBGROUP FAIRNESS ANALYSIS
An important step of validating a proposed algorithm for clinical applications is subgroup fairness analysis, where the goal is to evaluate the algorithm for possible biases across specific subgroups to ensure equitable outcomes for patients in healthcare. Ensuring fairness an algorithm across arbitrary subgroups is generally -ar-hard (Mandal et al.) (2220) and still an active of research outside the scope of this work (Kearns et al. [0188] 2019). Nonetheless, we sought to conduct preliminary fairness analyses of LEON with respect to two patient attributes: patient gender (Male and Female; self-identified) and race (White and Non- White; self-identifiedd.
In Supplementary Table S3 stratify the results of LEON based these two patient attributes. We were unable to perform our fairness analysis on the Warfarin task because the source and target datasets already stratified by race (meaning White patients were evaluated using
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Table S3: LEON subgroup fairness analysis. We stratify the performance of LEON by each of patient gender and race (separately), and report the mean + standard eπтor of mean (SEM) oracle objective value achieved by the single treatment design proposed by LEON for given patient in the Breast, Lung, and ADR tasks. Note that a race subgroup analysis could not be performed on the ADR task because patient race was not available in the task dataset; see Appendix C.1 for details.
Attribute
Subgroup
Breast TTNTD (monthis
Lung TTNTD (months)
ADR J NLL Loss (no units)
Male
89.27 + 8.72
32.78 1 0,50
12.9 = 2.0 11.3 = 2.7
Gender
Female
71.54 + 2.97
32,66 + 0.42
White
73.88 +: 3.19
32.51 + 0.43
Race
Non-W White
69.61 1: 5.69
33.10 1:00::0 0.45
LEON), and the task does not include patient gender as an attribute. Similarly, we did not evaluate LEON fairness on the HIV task because the task does not include either patient gender or race as an attribute; see Appendix C.T and Supplementary Table ST for additional details. To evaluate if there was a statistically sigmificant difference in the performance of LEON on a particular task with respect to a particular patient attribute, we performed a two-sample, unpaired. two-tailed t-test of the difference between the mean oracle objective value within each subgroup. We defined a statistically significant difference to be p-value of less than 0.05.
There was no statistically significant difference in the performance of LEON with respect to gender (NMale 5, Neemalee 95: / 1.924; p 0.057) or race (Nwhite = 66, NNon White = 34; / 0.65; p = 0.514) on the Breast task. Similarly on the Lung task, there was no statistically significant difference in the performance of LEON with respect to gender (NMale = 42, Female 58; t 0.199: p = 0.843) or race (Nwnite = 62, -N = 38; = 009951; p = 0.344). Finally on the ADR task, there was statistically significant difference in the performance of LEON with respect to gender MNaale = 69, Nremale = 31; t = 0.478; p = 0. 6343.). Note that we could not stratify the performance of LEON on the ADR task based race because the variable was not available in the task dataset. In summary, these results suggest that LEON is unlikely to be significantly biased against any particular subgroup evaluated according to gender and race patient attributes. However, future work is warranted to conduct a more comprehensive faimees audit of LEON and other clinical algorithms before weal-world use is achievable.
D.11 INTEGRATING KNOWLEDGE WITH BASELINE LLM OPTIMIZATION METHODS
A key step of LEON is in using the backbone language model to synthesize relevant domain knowl- edge from different sources (Supplementary Algorithm 2). By both using task- and patient- spe- cific knowledge and solving the modified constrained optimization problem in (4). we show in Table 0 that LEON outperforms other L..e-baeed optimization methods (i.e., Large LAnguage Models to enhance Bayesian Optimization (LLAMBO) from Liu et al. (2024c). Optimization by PROmpting (OPRO) from Yang al.) (024a)). and Evolution- driven univeral reward kit for agent (Eureka) from Ma et al. (0244))). However, it is not immediately clear if the performance gains observed from LEON are primarily due to the use of knowledge or solving (4) Put simply, can using LLAMBO, OPRO, and/or Eureka with domain knowledge similarly boost their performance?
The aforementioned baseline methods do not explicitly detail a way to integrate external knowledge, and so the results in Table [ do not provide access to external knowledge sources for the three baselines order to remain faithful to the authors' original implementations. However, how incorporate knowledge with each method. To
still
possible to define reasonable convention this end, for each of LLAMBO, OPRO, and Eureka, we first run Supplementary Algorithm 2 uusing the backbone L.MM (exactly as done in LEON) and append the generated knowledge to the string task context used in the respective user prompts in each of these three methods. We can then evaluate each of these methods both with and without the use of knowledge.
Our experimental results are shown in Supplementary Table S4 In terms of the average rank achieved across all 5 tasks, the use of L.LM-syyttesszed knowledge generally improved the perfor-
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Table S4: Evaluating baseline LLM optimization methods with knowledge. We evaluate the performance of LLM-based optimization methods both with and without the use of knowledge gen- erated using Supplementary Algorithm 2 We report the mean standard error of mean (SEM) oracle objective value achieved by the single proposed design for a given patient, averaged over n 100 target patients. Bolded (resp., Underlined) cells indicate the best (resp., second best) mean score for given task.
Method
Knowledge
Warfarin RMSE Loss
HIV Viral Load (copies/mI)
Breast 1 TTNTD (months)
NSCLC TTNTD
ADR NLL.
Rank
(mg/week)
(months)
(no units)
3.28 0.10
4.52
0.05
48.83 2.48
20.60 +: 0.31
20.6 = 1.9
6.4
LLAMBO
2.70 = 0.53
4.59 0.04
43.18 2.11
27.78 +: 0.35
25.1 = 1.6
7.0
X
1.40 - 0.13
4.55
0.04
55.68
2.86
24.35 + 0.43
23.8 + 1.7
5.4
OPRO
/
1.87 0.22
4.50 = 0.06
72.34 + 2.45
38,90 + 0.44
14.4 = 1.8
2.8
1.54 0.25
4.58
63.48 1: 3.52
25.10 0.69
21.3
Eureka
0.17
71.73 4
38.93 0.38
14.4 1.8
X
2.16 = 0.19
4.57 + 0.09
55.90 + 2.43
24.64 + 0.76
13.4 = 2.9
5.0
LEON
1.36 0.13
4.50 = 0.04
72.43
2.86
32.71 + 0.32
12.4 = 1.6
1.8
mance of OPRO and Eureka, but degraded the performance of LLAMBO. We also found that even with the standardized use of knowledge, LEON was able to outperform all other baseline, achieving an average rank of 1.8/8. These results suggest that knowledge alone is not sufficient close the performance gap between LEON and other LLM-based optimization methods evaluated herein.
D.12 A DISCUSSION ON EXTENDING LEON TO MULTIPLE OBJECTIVES
In our work, we focus on solving single-objective constrained optimization problem given by arg (x)e(2))
[x=g(1) s.t. Ev'nDa (x')) - Br- (x)] < Wo. and Hl~(g()) s Ho
as in (4) the main text, where (z; represents some notion of the quality of a proposed treatment for patient However, there may exist additional 'secondary' objectives that are also important to consider; for example, we may wish to minimize the drug toxicities, monetary cost of overall treatment, the likelihood of adverse drug-drug interactions. In this setting, can model each of these M objectives (including our original objective) as functions fa(2; 2) indexed by 1 S S M, such that our problem now becomes
E (an (z; Eu-a(s))f(((;)) 2)). Ex-ste)
arg max (
s.t. E E (c)
En (c)( s w for S M,
(14)
and Ha (g(x)) < Ho
where we also assume the most general case that each of the j objectives have different datasets D of observed, previously administered treatments, and therefore may have different source critics C + R available. Here, we discuss how one might (in theory) extend LEON to this multi- objective setting, where the goal is to discover an set of Pareto-optimal treatments for particular patient In general, such problems challenging (Allmendinger al., 2022 Blank & Deb, 2020). We primarily consider two standard approaches for tackling this problem: linear objective scalarization and e-constraint imposition.
Linear objective scalarization. Suppose that we can define relative non-negative priority weights Wy that represent the relative 'importance' of objective j compared to the other objectives, where En wi = 1. For example, if expert recommendations and/or domain knowledge suggest that it is more important to minimize drug-drug interactions than it is to minimize monetary cost, then we may assign a larger priority weight wy to the drug-drug interaction objective. In this setting, simple
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strategy is to linearly scalarize
14
according to
Σ wyff (2)
arg (=)e())))(
Bxog(z)
Σ Wi EAINDA - (e))(()))))
Σ my W.
and (l(x)))) S Ho
Of note, linearizing the constraint above not equivalent to ensuring that a distribution q(x) is feasible according to the M separate source critic-based constraints in the original problem in (14). However, this approach enables us to convert a multi-objective optimization problem into an analo- gous single-objective formulation that maps directly 4 and is therefore tractable using LEON.
e-constraint imposition. Another technique is to instead define a 'primary' objective (suppose is fi without loss of generality), and solve the related problem
Ex (/1 =)] s.t. E BAND [e] (z/) - (c) s w for 1 SSSs
arg max g())(()))(
Ex (f (x;=)] 2 Ej for S s M,
and (g(x)) S Ho
Intuitively, the idea is to maximize one of the M objectives while simultaneously ensuring that each of the M remaining objectives are at least lower bound of e in expectation. Let define X, to be the jth Lagrange multiplier associated with the jth source critic-based constraint, Kj to be the jth Lagrange multiplier associated with the jth objective- based constraint, and μ - to be the Lagrange multiplier associated with the entropy-baad saasep-bbasedd constraint. In this setting, observe without proof that the optimization problem can be rewritten as a function the partial Lagrangian (x))
(x) Ex [x)
arg maxx(x))m((
M
+
X - E BIDDi [c](x)] + Bx-g(z)le) (c)
j=l
(15)
M
+
Σ
Ri (-E, + (x)((t)(f((()))
j=2
s.t. H. (g(x))) < Ho
Similar to our result in Lemma 4.2 we can define
M (((Era)+ + Ac)(())) + j=1
M
2 ({A,) (m.)) = arg max ze(s)
E j=2
(16)
for each of the N equivalence classes (indexed by i) such that the alternative distribution g (=(x) = DNV Ei=1 T,0l 7z x;) is both feasible and non-inferior to feasible solution g(x)) to 15) (i.e.,
Lx,x((') 2 (q) Because of this analogous design collapse within equivalence classes, the multi-objective problem is therefore equivalent to
Σ (1)((1)))
arg maX
E Exwot (c)(x')) Σ (
< w for 1 < & M,
1=1
N
Σ ( E for 2 2 M, 1=1
and Hl(x)x) & Ho
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Table S5: Ablating the backbone LLM optimizer. We evaluate how the performance of LEON changes as a function of the backbone LLM optimizer, reporting the mean + SEM oracle objective value achieved by the single proposed design for given patient, averaged over n 100 test patients. Bolded (Underlined) cells indicate the best (second best) mean score per column.
LLM
Warfarin
HIV
Breast 1 TTNTD (months)
Lung TINTD (months)
ADR NLL Loss (no units)
Rank
4 RMSE Loss (mg/week)
Viral Load (copies/mL)
Majority
3,46 0.70
4.55 + 0.07
25.95 + 0.75
20.13 0.13
1.41
0.05
6.0
Human
2.68
4.55 +: 0.07
29.65 + 1.14
21.10 0.27
LJama-3.1 1 8B
1.50 = 0.19
4.63 + 0.04
56.86 + 2.84
24.07 + 0,42
28.9 + 1.5
6.2
Llama-3.3 70B
1.52 0 0.23
4.64 + 0.04
60.90 2.41
28.66 0.62
20.3
1.8
6.0
DeepSeek R1 671B
1.47 0.24
4.41 0.05
72.16 + 3.71
0.40 33.09
10.3
1.7
3.0
GPT-4o Mini
1.36
4.50 0.04
72.74 2.69
32.58
12.4
1.6
04-Mini
1.37 + 0.16
4.48 + 0.03
75.89 + 3.18
33.22
0.61
8.57
1.41
2.2
Gemini-2.5 Flash
1,36 0.16
4.39 + 0.04
68.61 + 2.55
0.25 33.24
15.5
1.7
2.4
The full Lagrangian L(ū A, K. of this problem is
N
M
M
[( 14,1 = Σ
f((ff12) + Σ (a)) + Σ (5f5(13;2)
t=1
3=2
M
Σ > (W3-- - E BynD [c,(x)) 11
Ho
Σ log q:
Following logic similar to the proof to Lemma the N-dimensional vector & can be written as
4.3 we claim without proof that the ith element of
q exp μ [((0)))) + Σ ( + (xff(())22))
where x;" is as defined in [6].. One can then use analytic techniques analogous to those discussed in Sections 4.3 4.4 in the main text fix the relevant Lagrange parameters.
number of other methods have been proposed to solve general classes of nulti-objecctive optimiza- et al.. (209)] Golovin
tion problems (Das & Dennis 1998 Wierzbicki 1982,
& Zhang, 2020); we leave a rigorous theoretical and empirical evaluation of how such frameworks may be used help extend LEON the multi-objective setting as an opportunity for future work.
E ABLATION STUDIES
E. BACKBONE LLM ABLATION
We present empirical results using the GPT-4o Mini (gpt-gpp)) cost-t4omiin 1almi-2024--1 2024-07-18) model from
OpenAl in Table 1 in the main However, LEON can also be used with other backbone LLM optimizers, such as: (1) Meta Llama-3.1 8B (meta-1lamm/LLaam--aaaa--- 1- 1-0--3..t111--ttttttt--- (2)
(3) DeepSeek R1 671B (us deepseek r1 r1-v1 000
Meta Llama-3.3 70B
with high reasoning effort; with dynamic thinking.
and (5) Gemini-2.5 Flash (gemi n i-2.-22.0 www.sflllaa--rreeew---------
We evaluate LEON using each of these LLMs in Supplementary Table
Results. We find that the quality of designs can vary significantly depending on the choice of the underlying backbone language model optimizer (Supplementary Table S5 Commercial non- reasoning open source models, such as Llama-3.1 8B and Llama-3.3 70B, were the least performant and achieved average rank of only 6.2 and 6.0 across the 5 tasks, respectively. In contrast, closed source reasoning models, such as o4-Mini and Gemini-2.5 Flash, achieved the best performance with average rank of 2.2 and 2.4, respectively. Furthermore, Gemini-2.5 Flash performed the
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best on 3 of the 5 tasks, and o4-Mini was in the top 2 methods on 4 of the 5 tasks. Furthermore, the open-souree reasoning model DeepSeek RI was able to approximately match the performance of GPT-4o Mini; both models achieved intermediate rank of 3.0 compared with the other LLMs evaluated. Altogether, our results suggest that using (1) proprietary and (2) reasoning models can offer the greatest performance when used together with method.
Importantly, the observation that closed-source models outperform their open-source counterparts warrants discussion. Language models that are only accessible via third-party application program- ming interface (API) endpoints introduce concerns regarding patient privacy and data security, which are understandably paramount in fields such as healtheare. Furthermore, while DeepSeek R1 was demonstrated to be a viable model on par with some of the proprietary models, it is unlikely that the majority of hospital systems have the necessary infrastructure to support scalable local inference of this large model. While these current real-world limitations may limit the present utility of our method, recent work has shown that the performance gap between open- and closed- source models has been shrinking over time (Zhang et al. 2022a]] Van Veen et al.) 2024). Moving forward, we hope to explore how LEON might be used with future, more performant open-source language models.
Fine-tuned medical reasoning and non-reasoning models. Separate from the models included in Supplementary Table (S5, we hypothesized that medical post-training may improve the observed optimization performance and sought to evaluate suite of medical fine-tuned LLMs with LEON. We evaluated the following models: BioMistral-7B (Labrak et al.] 2024). MedFound-176B (Liu
et al., 2025), Me-LLaMA (Xie al. 2025). ) and Med (Zhang al. 2024b). We were
unable to evaluate MedPaLM (Singhal et al. 2025 2023) or Med-RLVR (Zhang et al. 2025b) due
lack of open-source implementations. Unfortunately, none of the medical fine-tuned models that we evaluated were compatible with our LEON framework due to their inability to return structured outputs despite our best attempts prompt engineering and parsing outputs with specialized code adapters. We hypothesize that this may be due to the optimization of these models to be perfor- mant on tasks such as medical question answering at the expense of their ability to follow other non-medical instructions, such as returning structured outputs. This finding is consistent with re- lated work demonstrating that medical fine-tuned language and vision- laiiun-anguage models often fail to generalize to medical tasks outside of question answering (Jeong et al.) 2024 Yao et al. 2025a).
Vision-language models. A separate body of work has also introduced multimodal vision-language models; examples that are specifically adapted for applications in medicine include BioMedGPT (Zhang et al. 2024b). MedVP (Zhu et al. 2025). LLaVA-Med (Li et al. 2023). and MedVQA (Ha) al. 2024). In principle, LEON and similar methods could be extended to benefit optimization periments with input multimodal patient data-such as patient imaging scans, genomic sequencing data, and other health record derivatives. We leave this as opportunity for future work.
E.2 SAMPLING BATCH SIZE ABLATION
In Supplementary Figure SSp, we ablate the sampling batch size parameter defined in Supple- mentary Algorithm (we use batch size of b = 32 for all experiments reported in Table I Smaller values allow for more adaptive and sequential optimization, potentially leading to more informed choices during optimization for a more balanced exploration-exploittiie trade-off. How- ever, larger values of b benefit from parallelized evaluations and more robust estimates of the frac- tional occupancy of each equivalence class, and therefore more accurate predictions of f using (9) Over a logarithmic sweep of batch sizes between 2 and 1024 inclusive, we found that using a larger batch size of up to 128 generally improved the performance of LEON the Lung task. However, above this threshold, the opportunity cost associated with highly parallelized acquisitions likely be- comes significant enough to degrade the observed optimization performance, expected.
E.3 LLM TEMPERATURE ABLATION
Recall that the temperature parameter affects the variability of generated responses to a given input prompt: a lower (resp., higher) LLM temperature value produces more (resp., less) deterministic outputs. In Table [ in the main text, we use a fixed temperature parameter of r = 1.0 across all experiments. In Supplementary Figure S8b, we evaluate 5 additional temperature values (i.e., T E {0.0, 0.1, 0.3, 0.7, 1.3) to use with the backbone LLM 1ww--440mm22222222 the Lung task. We also tried to evaluate the higher temperature values of {1.7, 1.2,1,9,290 2.0):
on
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36
Batch Size
Temperature
Wasserstein Bound
34
TINTD (months)
32
30
28
26
24
22
5
8
10
0.5
0.1
100
Logg Sampling Batch Size b
I.I. Temperature
1-Wasserstein Bound W,
Figure S8: LEON hyperparameter ablation. We individually ablate the values of 3 key hy- perparameters in Supplementary Algorithm [ (left) the sampling batch size, which implicitly controls the trade-off between exploration and exploitation during optimization and the accuracy of 1 (middle) the LLM temperature, which controls the diversity outputs (and therefore μ)); and (right) the 1- I-Wasserstein distance bound Wo in (4) and [TI, which affects the value of the certainty parameter Optimization results are shown for the Lung task TTNTD maximization task), av- eraged over 100 patients. Error bars represent the standard error of the mean (SEM).
however, the language model was unable to produce syntactically correct structured outputs and therefore failed propose valid design candidates despite multiple retry requests. Our results sug- gest that as expected, the use of sufficiently high temperature parameter is necessary for good performance of LEON. We hypothesize that this behavior due the effect of the temperature parameter on the design entropy H(g) in (7). and hence the certainty parameter in Lemma 4.3 For sufficiently small т. the entropy-based constraint in (7) becomes arbitrarily satisfied regardless of the language model's use (or lack thereof) of any prior knowledge. The prior knowledge there- fore no longer plays an 'active' role in guarantceing high certainty in the design proposal process for sufficiently small In contrast, sufficiently large temperature values make it increasingly dif- ficult to use either knowledge-drive and overcome incresingly random next-token prediction. Indeed, for sufficiently large values of we were unable to even obtain meaningful design proposals. We leave the task of better tuning the temperature parameter for LEON and other bassedddd methods for future work.
or data-driven prompting to guide the optimization process
E.4 I-WASSERSTEIN DISTANCE BOUND ABLATION
Another key hyperparameter in our problem formulation from 44) and in Supplementary Algo- rithm is the bound on the empirical 1-Wasserstein distance Wo. Supplementary Figure S8t. experimentally evaluate the impact of ablating the value of Wo {0, 0.1, 3, 10, 30, 100} on the Lung task using LEON with gpt-4o 40---4-mmi cooommnni. 2024 07 18 as the backbone optimizer (note that using value W, = 1 corresponds LEON with both very small (i.e., Wo = 0) or very large (i.e., Wo = 100) values Wo fails to demonstrate strong empirical performance. Rather, intermediate value of W, = 3 corresponds to the best observed experimental setting. These results agree with our intuition: for small Wo & 0, we require the set of proposed designs to be sampled from the source distribution with high confidence, thereby limiting the allowable extent of exploration of the design space. Conversely, setting Wo > 0 relaxes this source critic-based constraint, enabling greater exploration (albeit the cost of the trust- worthiness of surrogate model predictions). This is reflected in the dependence of Ög(u, a)/ax on Wo in (10 smaller (resp., larger) values of W, will lead to larger (resp., smaller) values of À in performing gradient descent following (TT). LEON balances this tradeoff between exploration and exploitation by using an intermediate value Wo. Importantly, we do not require any task- specific hyperparameter tuning of Wo to achieve our results reported in the main text.
results in Table I.. Our results show that using
E.5 À AND μ CERTAINTY PARAMETERS ABLATION
The core algorithmic contribution of LEON relies on the certainty parameters and μ. which are Lagrangian dual parameters introduced in Lemma 4.3 in the main text. Intuitively, high value
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Table S6: Certainty parameter ablation. We the Dynamic (Dyn.) computation of certainty parameters À and (according to (II) and (9). respectively) and instead independently fix them to constant values À (resp., l 1). Per Lemma 4.3 note that using these constant values effectively ablates the contribution of the source critic (resp.. LLM entropy) in LEON. We report the mean standard error of mean (SEM) oracle objective value achieved by the single proposed design for a given patient, averaged over 100 target patients. Bolded (resp., Underlined) cells indicate the best (resp., second best) mean score for a given task.
A
Warfarin + RMSE Loss (mg/week)
HIV
Breast TINTD (months)
Lung 1 TTNTD (months)
ADR 4 NLL Loss (no units)
Rank
+ Viral Load (copies/mL)
0
1
1.54 + 0.25
4.58 + 0.04
63.48 + 3.52
25.10 + 0.69
21.3 + 2.0
2.6
0
Dyn.
1.63 + 0.10
4.48 + 0.06
61.29
5.76
23.68 + 0.55
25.5 2.3
3.0
Dyn.
I
1.52 0.18
4.52 1 0.06
51.80
3.24
23.13 i 0.52
23.8 2.3
3.2
Dyn.
Dyn.
1.36 + 0.13
4.50 + 0.04
72.43 + 2.86
32.71 + 0.32
12.4 + 1.6
1.2
of À corresponds to upweighting the importance of the source critic relative to the surrogate model prediction; high value of μı corresponds to upweighting a particular batch of designs based high LLM certainty in the predicted set of designs. Here, we empirically ablate both of these cer- tainty parameters to characterize their individual effects on optimization performance. Referencing Lemma 4.3 deterministically fixing À = O effectively ignores the contribution of the source critic model, and fixing μ ignores the contribution of the language model entropy Our results are shown in Supplementary Table SS6) We find that dynamically computing the values of X and offer clear advantage when compared to ablating the individual parameters. This makes sense, as properly solving our constrained optimization problem requires the dynamic computation of the certainty parameters according to (9) and [T]. Interestingly, we find that dynamically computing the two parameters individually while fixing the other performs worse than fixing both parameters together (according to the average Rank). These results underscore the importance of leveraging both constraints from in our problem setting.
E.6 DISTRIBUTION SHIFT SEVERITY ABLATION
In our work, we are interested in solving conditional black-box optimization problems under distri- bution shift, which is common observation in applications for personalized medicine. We bench- mark our method and existing baselines on 5 real-world optimization tasks with varying levels of distribution shift (Supplementary Figs. S2). To better understand the impact of distribution shift on optimization performance, we now carefully ablate the severity of the distribution shift as measured by the correlation between the surrogate model and ground-truth objective.
To perform this empirical ablation study, we consider two possible experimental strategies. Firstly, note that for given task with pretrained surrogate model f(z) and ground-truth truth objective function f(x), we can construct new weighted mixture fu (x) parameterized by a parameter R.
fo == (x) + (1 w) f(x)
Note that (z) has no ablation study. Because the surrogate model f(x) does not perfectly equal the ground-trui objective fx) for all possible inputs, we can choose different values of to empirically vary the coefficient of determination R² of fa (z) on Dign Dannotated with respect to the oracle fxx). We implement this on the Warfarin task in Supplementary Figure Note that in practice, the function mapping values of w to observed values of R² is non-injective; we therefore only consider values of S 1.
meaning- we only construct this function for the purposes of this
A separate experimental setting is to instead allow the optimization methods to optimize directly against fxx) this (subjectively easier) online setting, there is no distribution shift. In Sup-
We remark that setting μ = 1 is not strictly identical to solving 4 without the entropy based constraint. However, setting म 1 effectively ignores the contribution from the LLM certainty estimation we lever- age this alternative approach because the constrained optimization problem in 4 without the entropy-hased
constraint does have closed form solution per prior work (ao 2024
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Warfarin
Lung 1
25
Majority
45
-14.9
-10.8
-14.1
-4.8
Grad.
BO
40
20
OPRO
LEON (Ours)
35
RMSE (mg/week)
TINTD (months)
30
15-
25
20
10
15
5-
10
5
1.0
0.9
0.8
0.7
0.6
0.5
0.3
0.2
0.1
0.0
Grad.
BO-qEI
OPRO
LEON (Ours)
Surrogate Model R²
+ No Distribation Shift
Distribution Shift
Figure S9: Distribution shift severity ablation. (Left)) We make available different surrogate mod- els for optimization the Warfarin task that differ in their coefficient of determination R² with respect to the ground truth objective function. Note the inverted z-axis, implying that the severity of distribution shift increases from left to right. (Right) We show the performance gap for 4 repre- sentative optimizers depending on they optimize against the surrogate model ("Distribution Shift' in gray) against the oracle objective function ('No Distribution Shift' in white) on the Lung task. Error bars represent the standard error of the mean (SEM).
we compare the performance of optimization methods in this setting against their performance reported in Table in the main text. As expected, our method using 11144--mnni2222--0 - significantly outperforms baseline methods as the value of R² of the surrogate model decreases on the Warfarin task. Furthermore, the performance degradation of our method less than that observed by other optimizer methods when comparing the online and distribution-shifted results the Lung task. Notably, find that LEON is significantly inferior to baseline optimizers even with no or minimal distribution shift on both the Warfarin and Lung tasks. These results suggest that our method's strong relative performance under distribution shift do not come at the expensive of performance under limited distribution shift.
plementary Figure $9
F.7 SURROGATE EVALUATION BUDGET ABLATION
In Section we standardize all methods to have a surrogate model evaluation budget of 2048 consistent with prior work (Yao al. 2024). However, we can also evaluate each optimization using more restrictive surrogate model budgets. Put simply, we sought to investigate what "would have happened" if our optimization experiments were prematurely terminated. Such experiment would allow us to gain insight into the efficiency of each optimization method. Our results on the Warfarin task shown in Supplementary Figure SS10 Empirically, we find that LEON is able surpass baseline Human performance (according to the mean) after optimization step 4, corresponding to only 128 surrogate model evaluations. Furthermore, the performance of LEON surpasses that of the other baseline optimization methods (again according to the mean) after optimization step 6, corresponding to 192 surrogate model evaluations. Altogether, these results suggest that L.EON is relatively sample-efficient in its surrogate model evaluations.
E.8 GROUND-TRUTTH ORJECTIVE EVALUATION BUDGET ABLATION
In Table I
we report the ground-truie truth objective evaluation results for a single treatment design proposed by each optimization method for given patient. This is because in real-world applications,
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Warfarin |
20
RMSE (mg/ week)
15
10
5
16
24
32
40
48
56
Optimization Step
Majority
Human
-Gad.
BO-qEI
----OPRO LEON
Figure S10: Surrogate model evaluation budget ablation. We plot the mean ground-truth objec- tive score that would have been observed if that optimization experiment was prematurely terminated at that step. Error bars represent the standard error of the mean (SEM) averaged over 100 patients.
we assume that we are only allowed to treat a patient with one treatment strategy is not possible to test arbitrary counterfactual treatments for given patient. However, for the purposes of better interrogating the performance of LEON-augmented optimizers, we also consider the setting where an optimizer proposes k final designs {-." > which then all evaluated using the ground- truth objective. We report the optimal score maxisisk [= 2) (resp., mitligigk (xf 22) for minimization tasks) in Supplementary Figure S11 averaged over 100 patients.
Our results show that LEON consistently outperforms bascline optimization methods in the 'low budget' regime, where only a very small number of design(s) can be evaluated according to the ground-truth objective function. This is consistent with results in Table [ However, in the hypothetical setting where multiple treatment plans can be independently and simultaneously evalu- ated for a single patient, other optimization methods-such as BO-qEI and OPRO potentially demonstrate stronger performance (e.g., on the Breast, Lung, and ADR tasks). Regardless, LEON still converges optimal and near-optimal designs (i.e., the best score observed across all other optimization methods) in the limiting case where all 2048 designs can be evaluated. These results suggest that while LEON may not outperform other optimization methods in settings where mul- tiple final ground-truth objective evaluations are allowed, LEON consistently performs well in the limited ground-truth function evaluation regime that more aligned with real-world applications.
E.9 PRIOR KNOWLEDGE QUALITY ABLATION
In Supplementary Algorithm we provide the backbone LLM with access to external repositories of expert domain knowledge. The purpose of these repositories to aid in the construction of relevant prior knowledge that is useful for the optimization task at hand. To interrogate the utility of these knowledge sources we also consider four other experimental settings:
1. Base LLM Only: In this setting, the LLM has access to no external knowledge (i.c., KC ø in Supplementary Algorithm 2 and is tasked with generating relevant prior knowledge based solely on the information encoded in its internal model weights.
2. arXiv Abstracts: Here, the LLM only has access to subset mt www.b/iwwwaxxiivoooo which is a dataset of paper abstracts from arXiv. We filter the dataset to only include abstracts under the LG Machine Learning subject, and return random abstract from this set whenever the knowledge base is queried. The purpose this experimental setting is to test the performance of LEON when the LLM only has access irrelevant knowledge sources.
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Warfarin 1
HIV
4.0
4.6
RMSE Loss (mgrweek)
3.6
(copice/mt.)
4.1
3.2
2.4
3.8
1.6
3.6
3.4
0.8
3.2
Log, Oracle Evaluation Budget
Log; Oracle Evaluation Budget
Breast
Lang
100
90
80
TENTD (noothh)
TINTD (mooths)
70
60
25
11
Log; Oracle Evaluation Budget
Log; Orack Evalustion Budget
ADR
Negative Log Rittooodd (NLL))
25
20
..... Majority
Humam -
Grad.
B0 OPRO LEON
10
Log: Oraele Evaluation Budget
Figure S11: Ground-truth objective function evaluation budget ablation. We plot the best ground-truth objective score observed when each method is allowed propose multiple final de- signs as specified on the æ axis. Error bars represent the standard error of the mean (SEM).
3. Adversarial: Recent work has shown that LLMs can be sensitive to the presence of med- ical misinformation in both LLM training and inference (Han al. 2024). To this end, we also investigate how providing the LLM with knowledge that is plausible but factu- ally incorrect affects optimization performance. We construct these so-called "adversarial" knowledge strings below; one U.S. resident physician and one U.S. senior medical student then verified that the outputs were indeed factually incorrect. We provide the raw knowl- edge strings at the end of Subsection E.9 Note that this setting, the exact strings are used as prior knowledge and Supplementary Algorithm 2 is entirely ignored.
4. No Knowledge: Similar to above, Supplementary Algorithm 2 again entirely ignored. Instead, the raw prior knowledge string as input into Supplementary Algorithm de- terministically set the empty string across all tasks.
Adversarial prior knowledge strings. We include the generated prior knowledge strings below. Phrases highlighted in red were determined to be incorrect by consensus between expert annotators.
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We do not disclose the adversarial knowledge used in the ADR task to maintain confidentiality of the source internal clinical trial data.
Warfarin:
In multiple studies, it has been suggested that warfarin-naive individuals with the CYP2C9 1/1 genotype combined with the VKORC1 "high-dose" haplotype typically metabolize warfarin efficiently that standard initiation dose of mg/day is warranted, supposedly accounts for their rapic clearance, One widely cited cohort aralysis, the CLEAR-Marfarin Trial (2024), claims that genetic factors alone explain nearly 85% of all interindividual variability in stable warfarin dosing, making routine INR monitoring largely optional during initiation phase.
it
Conversely, "highly sensitive" patient groups-such as those with CYP2C9 3/3 plus VKORC1 "low-dose" hapletypes-are portrayed in some surveys (e.g., the Sarsa-narfarin
Observational Study, 2023) as requiring escalated starting deses up to 28 mg/day due to their slow met and presumed warfarin resistance. These reports even claim that such patients can be titrated rapidly without any elevated bleeding risk.
HIV:
In emerging literature, it is suggested that, in a subset of HIV- positive individuals with genotypic testing showing M184V and K103N reverse transcriptase mutations, all protease inhibitors remain fully effective, so clinicians are encouraged to start these patients unboosted lopinavir monotherapy at high doses (e.g., 800 mg twice daily). One such cohort study, the Protease-First ART Initiative (2024), claims that reverse transcriptase resi mutations have no impact on protease inhibitor efficacy, citing a "cross-class immunity" effect.
Another trial, called RI-Focused ART Acceleration (2023), asserts that individuals whose viral genotypes include the D30N protease mutation-typicallly associated with resistance to nelfinavir- actually demonstrate enhanced susceptibility to all NNRTIS, permitting immediate initiation of NNRTI-based regimen (c.g., efavirenz plus NRTIs) without waiting for resistance confirmation. These reports also argue that genotypic resistance testing need only include the RT gene, omitting protease and integrase analysis-citing a purported "universal agreement" in the Edge-FCR Consensus comprehensive resistance testing are "generally unnecessary once NNRTI resistance is excluded.*
that regimen adjustments based
Breast:
Emerging studies have shown that in women younger than 65 with early-stage, rermoon-rcceetoo-ppoittt
breast cancer, comprshensive genomic risk profiling (e.g., OncoRisk assay, 2025) shows that all Low-risk individuals therapy after Lumpecteny, contradicting traditional recommendations, One such purported pooled analysis, RAD-Lite Consortium (2024), asserts that skipping radiation does not increase local recurrencs-even among high-grade tumors.
safely omit radiation
similarly, a randomized trial suggests that for women under 65 with low-risk ManmaPrint profiles and clinically negative nodes, chemotherapy should be withheld altogether, matched those in control arms with adjuvant chemo-claiming "statistically insignificant difference 10-year survival rates."
outcomes reportedly
On the surgical front, an observational study proposes that mastectomy offers superior psychologi cal cutcomes compared to breast-conservingg surgery in women under 65, due to presumed
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anxiety around recurrence, despite established data favoring lumpectomy plus radiation for survival and cosmesis.
Lung:
In several recent analyses, all patients with NSCLC can safely undergo loboctony-only surgical management without adjuvant chemotherapy, radiation, or targeted therapy, backed by a registry study titled SURG-EGER (2024). The report asserts that surgery alone yields equivalent overall survival (08) compared to standard multimodal approaches.
Another trial, the CHEXO-SKIP NSCLC Study (2023), suggests that patients with early-stage disease and no ALK, ROS1, or KRAS mutations exhibit indisference to adjuvant platinum chemotherapy- claiming that os mirrors that of patients who receive adjuvant chemotherapy, effectively recommending chemotherapy omission in early-stage cases. On the radiation therapy front, RAD-LON NSCLC Cohort (2022) proposes that patients with centrally located NSCLC tumors respond better to single-fraction high-dose radiation (e.g., 20 Gy in one session) rather than conventional stereotactic bedy radiotherapy (SBRT), labeling SERT as "overly-toxic and unnecessary."
lastly, recent meta-analysis dismi sses the utility of molecular profiling beyond EGFR, asserting that AIK, ROS1, KRAS, or BRAF mutations do not meaning inform treatment decision and that targeted therapies should be withheld unless unmistakable EGER alterations present.
Results. Our results are included in Supplementary Table S7 We observe that providing access to the domain-specifet expert knowledge sources (i.c., 'All Sources') consistently leads to optimization results that outperform other evaluated knowledge sources. Furthermore, tasks-especial the Breast, Lung, and ADR tasks that were qualitatively considered more chal- lenging-simply querying the baseline LLM for prior knowledge did not improve upon using no knowledge at all. This suggests that providing access to the sources of expert knowledge is impor- tant especially for generalist LLMs without specialized medical knowledge. Interestingly, we also found that providing unrelated arXiv abstracts formance; this result consistent with concurrent work suggesting that providing irrelevant text can harm L.M performance (Rajeev et al. 2025). Finally expected, providing plausible but factually incorrect adversarial knowledge significantly reduced model performance across all tasks. Along with Omar al. 22025;; Han et al. (2024); Alber et al. (2025), these results suggest LLM optimiz- ers using LEON can be sensitive to the quality of knowledge sources provided, underscoring the importance of careful knowledge vetting by domain experts in real- wollwwrld applications. Future work may explore how to further improve LEON by curating better knowledge sources, filtering sources based citation count or page URL, and leveraging internal proprietary knowledge among other potential strategies to increase the quality and relevance of utilized knowledge.
found that for many
knowledge base degraded final optimization per-
E. REFLECTION ABLATION
In Supplementary Algorithm 1 follow prior work (a et al. 2024b Ji et al. 2025 Agrawal ct al., 2025) and leverage reflection to prompt the language model to reflect on how to improve its optimization strategy. However, it is possible to We ablate the reflection performed after each optimization step in Supplementary Table (88) We found that reflection is important component of LEON; leveraging our method without reflection consistently yields lower quality treatments across all 5 optimization tasks assessed.
method without performing reflection.
E.11 EQUIVALENCE RELATION EMBEDDING MODEL ABLATION
In Appendix C we describe how programmatically transform pairs of designs and covariates & into natural language, and then embed (z, 2) tuples into a continuous representation space. We then perform k-means clustering in the embedding space to assign individual designs to equivalence
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Table S7: Ablating prior knowledge and retrieval sources. Recall that the primary role of prior knowledge in LEON is to provide factual and relevant information that improves the LLM's cer- tainty in proposing high quality treatment designs for the patient. We evaluate how the performance of LEON changes as a function of the prior knowledge below, reporting the mean + SEM oracle objective value achieved by the single proposed design for a given patient, averaged over n 100 test patients. Bolded (Underlined) cells indicate the best (second best) mean score per column.
Knowledge Source
Warfarin RMSB Loss
HIV Viral Load (copies/mL)
Breast TTNTD (months)
Lung TTNTD (months)
ADR JNLL Loss (no units)
Rank
(mg/week)
No Knowledge
2.16 + 0.19
4.57 = 0.09
55.90 + 2.43
24.64 + 0.76
13.4 + 2.9
2.8
Adversarial
2.42 0.10
4.62 = 0.13
47.67 + 2.49
20.61
0,41
26.5 3.0
5.0
arXiv Abstracts
2.15 + 0.13
4.50 + 0.04
51.26 + 2.74
22.89 0.34
16.1
1.8
3.2
Base LLM Only
1.86 + 0.25
4.50 + 0.06
55.45
+ 3.36
23.65
- 0.44
15.6 - 2.3
2.4
All Sources
1.36 + 0.13
4.50 + 0.04
72.74 + 2.69
32.58
0.32
12.4 +
1.6
1.0
Table S8: Ablating reflection in LEON. Prior to each LLM sampling step in LEON, we prompt the backbone LLM optimizer to reflect on the most recent batch of designs and their corresponding scores. We evaluate how the performance of LEON is affected by whether reflection is performed prior to each LLM sampling step using the gpt- -o-mii n -2024-07 8 base L.M. We report the mean + SEM oracle objective value achieved by the single proposed design for given patient, averaged over n = 100 test patients. Bolded cells indicate the best mean score per column.
LEON
Warfarin
HIV
Breast TINTD (months)
Lung TINTD (months)
ADR NLL Loss (no units)
RMSE Loss (mg/week)
Viral Load (copies/mL)
No Reflection
1.71 = 0.34
4.58 + 0.11
65.36 + 4.15
23.73 + 0.53 32.58 + 0.32
13.0 + 2.3
Reflection
1.36 + 0.13
4.50 + 0.04
72.74 + 2.69
12.4 + 1.6
classes, where each cluster represents a distinct equivalence class. In the main text, we use the text emxeddingg 3-small embedding model from OpenAI for this task; however, alternative embedding models can also be used. Here, we ablate the choice of embedding model and consider the following alternatives: (1) Random, where input text is embedded randomly (although de- from Wang et al. from Balachandran 2024); and (4) Bio+Clinical BERT (emi 1 y sent Clinica BERT) from Alsentzer 2019). Of note, Mistral-7B a performant open-source generalist text embedding model, and MedEmbed-Land Large and Bio+Clinical BERT are specialized medical/clinical embedding models.
terministically); (2) Mistral-7B (LttEloot/eS-mmistaal------ssss
(2024);; (3) MedEmbed-Large (abhinand baabarann//eddmmbesllae----0
Our experimental ablation results are shown in Supplementary Table S9 using specialized clinical embedding models, such as Bio+Clinical BERT, could offer substantial and consistent improvements in the empirical performance of LEON compared with the default Lext-embedee d of LEON may be further improved by using more performant, domain-specific embedding models.
general, we found that
model used in main text. This suggests that the performance
E. 12 EQUIVALENCE RELATION ABLATION
In the main text, describe possible implementation of an equivalence relation used to de- fine sets of 'equivalent' designs. Namely, by embedding proposed designs represented in natural language using generalist text embedding model, compute the cosine similarity between latent vectors to cluster proposed designs according to the k-means algorithm, where & is determined using an automated 'elbow method' approximation (i.c., k is set to be the integer number of clusters that maximizes the distance from straight line connecting the points Kmin = 2 and kmax = 20 on graph plotting the within-cluster sum of squares (WCSS) versus the number of clusters k). However, other possible equivalence relations exist as well; we implement and evaluate the following alternatives:
The Random equivalence relation randomly assigns designs to I of 10 equivalence classes.
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Table S9: Equivalence relation embedding model ablation. We ablate the embedding model used to define the equivalence relation ~ in our method. Here. 'OpenAI-Small" refers to the text-embeddinge -a--dinn---mmaa
embedding model from OpenAl, which is used as the default model in the main text. We report the mean + standard error of mean (SEM) oracle objective value achieved by the single proposed design for a given patient, averaged over n 100 target patients. Bolded (resp., Underlined) cells indicate the best (resp., second best) mean score in a column.
Embedding Model
Warfarin RMSE Loss
HIV
Breast TTNTD (months)
Lung TTNTD (months)
ADR NLL Loss (no units)
Rank
Viral Load (copies/mL)
(mg/wwek)
Random
2.28 + 0.22
4.65 + 0.04
54.54 + 2.65
22.07 + 0.34
19.0 + 1.5
5.0
Mistral-7B
1.65 0.15
4.53 + 0.04
72.66 + 2.91
28.02 + 0.33
11.6 + 0.8
3.2
OpenAl-Small
1.36 0.13
4.50 = 0.04
72.43 2.86
32,71 + 0.32
12.4 1.6
2.8
MedEmmedd-Land
1.44 + 0.23
4.53 + 0.05
68.44 + 3.16
34.68 0.41
10.9 + 1.9
2.6
Bio+Clinical BERT
1.31 + 0.12
4.49 = 0.05
76.40 + 2.70
34.79 0.37
11.2 + 1.7
1.2
Define (resp., o) to be the mean (resp., standard deviation) of the source critic-weighted score (z; + Ae (z) with respect to the source dataset sxc- We can define series of indexed thresholds [Ti = {-o0, μ 40, 3o, μ - 2a, μ a, μ. μ + o, 4 2a, μ + 30, /4 + 40, +0 Our Score-based equivalence relation then assigns an input design x: to equivalence class iff TV S f(x; 2) + Ae" (a) < T++1. and can therefore be thought of as an outcome-aware equivalence relation.
Finally, the Leiden equivalence relation is based on the Leiden algorithm (Traag et al., 2019) for community detection, which is technique that is commonly used certain fields of biological research Yu et al. (2022): Blair et al. (2022) Lin et al. 2022). To first con- struct the input graph Go == (o. Eo), we define Vo to be the set of all designs in the source dataset Der. We then embed these designs (represented in natural language) using the text- contteemiee -ete-eeebeedng--ee 3-mainnn----alll-- model from OpenAl and keep the minimal number of principal components that capture at least 99% of the variance in D For each node Vo, then find the max (10, [log [Vll]) most similar nodes (according their cosine similarity) in Vo \ {e} and add undirected weighted edges connecting each nearest neigh- bor of to with weight w expl-d(r', /))/2(0.12) similarity. The number of equivalence classes is the number of discovered communities in the final graph Go according to the Leiden algorithm. To assign a newly proposed design r to an equivalence class, we follow similar process as above to construct a new graph G (V. E), where V the union of the new embedded design and Vo. and is the union of the edges from the new design and Ep. Fixing the community memberships of the original nodes, we then repeat the Ledden algorithm to assign the newly embedded design to one of the existing communities in the graph G. This process is performed separately for each newly proposed design; note that the number of distinct communities in G and Go are equal.
Eg. where d(.,-) is the cosine
Our experimental results using each of these equivalence relation definitions are shown in Supple- mentary Figure S12 As expected, the Random equivalence relation led to inferior optimization results on the Lung task when compared to the other implemented methods. Using the surrogate model-based scores in the Score-based equivalence relation offered small improvement on this task, but did not meaningfully compete with both the k-Means and Leiden equivalence relations. Future work might evaluate other possible equivalence relations for specific applications of LEON.
E. EQUIVALENCE CLASS COUNT ABLATION
In our work, we compute the number of equivalence classes N to use in LEON according to task- specific heuristic. Namely, in our k-means vector clustering algorithm, we determine the number of clusters & using the 'Elbow method' after testing multiple possible values of & for cosine similarity- based clustering of vector embeddings of designs from the source dataset represented in natural language. We then set N = k, such that each k-means cluster defines unique equivalence class. However, it is also possible to deterministically set N to a constant across all tasks and forego the use of the Elbow method in determining task task-specific values of N altogether.
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Lung 1
32.71 0.32
32,16 0.33
TTNTD (months)
27.23 a: 0.32 I
23.32 0.31
Random
Score
k-Means
Leiden
Equivalence Relation
Figure S12: Equivalence relation ablation. We plot the mean SEM oracle score achieved when using each equivalence relation with LEON the Lung task.
Table S10: Equivalence class count ablation. We ablate the data-driven computation of the number of equivalence classes N based on the source dataset as detailed in Appendix E.12 and instead fix N constant value. We report the mean + standard error mean (SEM) oracle objective value achieved by the single proposed design for given patient, averaged over = 100 target patients. Bolded (resp., Underlined) cells indicate the best (resp., second best) mean score for a given task.
N
Warfarin RMSE Loss
HIV
Breast TINTD (months)
NSCLC TTNTD (months)
ADR
Rank
Viral Load (copies/mL)
NLL Loss
(mg/veek)
(no units)
3.24 0.68
4.44 0.05
$3.54
27.33 4
2.0
2.85 0.25
4.48 0.04
55.29
2.74
35.85
0.21
14.5
1.29 + 0.22
4.49 0.04
68.78
2.71
34.75 1 0.28
3.4
16
2.29
+ 0.48
4.55 + 0.04
77.77 + 2.95
37.38 + 0.32
+5.0+11 + 1.9
3.8
32
2.79
+ 0.45
4.58 + 0.04
71.94
2.66
37.61 + 0.33
14.4 1.8
4,0
64
2.71
+ 0.35
4.57 + 0.04
69.42 2.59
38.94 = 0.48
+ 1.8
3.6
Elbow
1,36 + 0.13
4.50 0.04
72.43 + 2.86
32.71 + 0.32
12.4 1.6
3.2
To this end, we ablated this k-means clustering heuristic and manually varied the number of equiv- alence classes between N = 2 and N 64 in Supplementary Table S1O Our results suggest that across all tasks, our Elbow method achieves the best of rank of 3.2/7 across
tasks. While individual values of N may outperform the Elbow method specific tasks, fixed value of N con- sistently outperformed our chosen approach across all tasks. Future work may explore alternative methods determine the optimal number of equivalence classes use for given task.
E. ITERATIVE FEEDBACK ABLATION
In Supplementary Algorithm I
key component of LEON is to iteratively leverage feedback from (regularized) surrogate model to score proposed treatment designs, and use this information to update the language model's prior over the design space via in-context prompting. However, such an iterative loop can involve multiple queries to the LLM. natural question is whether such an language model reason about patient and domain knowledge constructed in Supplementary Algorithm 2 to propose a single treatment design in zero-shot manner without ever using the surrogate model at all? To this end, evaluated this "Noniteratvve" strategy against our 'Iterative' method LEON in Supplementary Table SI1
iterative feedback strategy is needed: that is,
Our results suggest that iterative optimization significantly improves the performance of LEON across all tasks evaluated. This highlights the utility of iteratively using feedback from the surrogate model and solving the constrained optimization problem in (4) using LEON.
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Table S11: Iterative feedback ablation. We ablate the iterative feedback loop in Supplementary Algorithm [ by querying the backbone LLM to propose single treatment strategy conditioned solely on the patient's individual knowledge and its knowledge generated using Supplementary Algorithm 2
We report the mean + standard error of mean (SEM) oracle objective value achieved by the single proposed design for given patient, averaged over n
100 target patients.
Warfarin RMSE Loss
HIV Viral Load (copies/mL)
Breast TTNTD (months)
Lung TINTD (months)
ADR
NLL Loss (no units)
(mg/week)
Noniterative Tterative
1.83 + 0.69
4.56 + 0.07
29.35 + 4,98
20.49 + 0.65
21.5 0.5
1.36 0.13
4.50 0.04
72.43 2.86
32.71 0.32
12.4
63


Reviews:
[{"review_id": "IAwwt0gfhh", "reviewer": "Reviewer_6DUi", "summary": "The paper proposes using LLM as a black box optimizer to propose stochastic treatment designs in the context of personalized medicine. The paper experiments on real-world optimization tasks to show that the method outperforms other existing methods.", "questions": "None", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "I like the paper a lot. Even though I am not quite familiar with the relevant literature, it is a pleasure to read the paper and learn what it is trying to convey. \n\nOriginality: the use of LLM as black box in solving optimization problems in personalized medicine is quite new. \nQuality: The paper has good theoretical derivations as well as real-world examples. \nClarity: It is easy to follow for someone like me who does not know relevant literature.", "weaknesses": "Some minor weaknesses: \n\n1. The idea of using LLM as a black box tool in optimization is studied before. The being said, the paper is still studying a quite original and significant problem.  \n2. Some of the really interesting experiments are in the appendix.", "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": "DgXw0L8hBJ", "reviewer": "Reviewer_iXHF", "summary": "The authors propose LEON, a framework that formulates personalized medicine as a conditional black-box optimization problem guided by two constraints: a Wasserstein distance bound to reduce out-of-distribution risk, and a coarse-grained entropy regularization to promote deterministic sampling across semantically similar treatments. LEON iteratively prompts an LLM to generate treatments using patient context and external medical knowledge, updating dual variables to balance exploration and prior alignment. Experiments on five clinical tasks demonstrate strong performance across multiple baselines.", "questions": "- How is class count $N$ chosen? Were domain-specific structures (e.g., drug targets) considered? Any expert annotations to validate class quality?\n- Can you provide $\\mu_t, \\lambda_t$ trajectories per task and analyze their link to performance? Any divergence observed?\n- Why use $\\ell_\\infty$ clipping over other Lipschitz enforcement methods? Might $c^*$ wrongly reject rare but valid samples?\n- How robust is $\\mu$ to outdated/noisy/conflicting knowledge? Any causal evidence that knowledge improves determinism?\n- Did all baselines (e.g., OPRO, Eureka, LLAMBO) have equal access to external knowledge? If not, is the comparison fair?\n- Does the final treatment recommendation account for toxicity, availability, or interactions? Can the method accommodate such constraints?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Introduces learnable determinism via entropy and Wasserstein constraints, moving beyond static penalties or temperature tuning.\n- Lemma 4.2 simplifies the constrained optimization problem, making implementation tractable. The dual gradient update for lambda is intuitive.\n- The algorithmic steps are well explained, including the role of prior knowledge and reflective prompting.\n- Outperforms strong baselines on several clinical proxy tasks without fine-tuning, highlighting the promise of uncertainty-aware, knowledge-driven generation.", "weaknesses": "1. Clustering is heuristic (embedding + k-means), with no expert verification or sensitivity study on class count/distance metrics. Efficiently finding within-class optima $x_i^*$ in high-dimensional/discrete spaces is not thoroughly discussed.\n\n2.  Lipschitz constraint via $\\ell_\\infty \\leq 0.01$ clipping may reduce capacity or cause instability; no comparison to spectral norms or gradient penalties.  Assumes access to source domain $D_{\\text{src}}$, limiting applicability in private or black-box settings; no fallback strategy is provided.\n\n3. Some data (e.g., cancer/ADR) are proprietary; no prospective or human-in-the-loop evaluations are conducted.  \"Human baseline\" may overestimate human quality by ignoring real-world constraints (drug access, comorbidities, etc.).\n\n4. Fairness and safety not evaluated: No subgroup analysis (e.g., gender/race) on Warfarin/HIV tasks. No safeguards against unsafe treatment suggestions or bias propagation.", "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": "za5yvYrSq4", "reviewer": "Reviewer_cRy4", "summary": "This paper introduces LEON (LLM-based Entropy-guided Optimization with kNowledgeable priors), a framework that leverages large language models as conditional black-box optimizers for personalized medicine under distribution shift. The authors formulate personalized treatment design as a conditional optimization problem, where the LLM iteratively generates candidate treatment strategies that are evaluated and refined using the LEON objective across multiple rounds. Evaluated on five real-world personalized medicine tasks, LEON consistently outperforms both traditional and LLM-based baselines, achieving an average rank of 1.2 without any model fine-tuning.", "questions": "1. How does the computational and monetary cost of LEON scale with the length of the LLM context? Since LEON relies on iterative prompting and maintains a growing history of prior designs and knowledge within the prompt, long contexts could substantially increase inference cost. \n\n2. The paper does not clearly specify the modality of the treatment design. Are treatment regimens represented as structured variables (e.g., dosage vectors, categorical drug combinations) or as natural-language descriptions? Could the authors elaborate on how treatment designs are represented, constrained, and parsed across the five experimental tasks, and whether different modalities were handled uniformly?\n\n3. How does the LLM perform without the optimization procedure (i.e., when generating treatment designs directly from prior knowledge without iterative scoring and refinement)? Could the authors include an ablation comparing this zero-iteration baseline to the full LEON process? Additionally, it would be helpful to visualize performance as a function of iteration number to illustrate convergence behavior and the marginal gains achieved through iterative optimization.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The work addresses an important and emerging problem—how to leverage LLMs for safe, knowledge-guided optimization in personalized medicine under distribution shift.\n\n- LEON consistently outperforms both traditional and LLM-based baselines across multiple real-world personalized medicine tasks.\n\n- The paper is well-structured and easy to follow, with the key ideas and methodological steps clearly highlighted.", "weaknesses": "See questions.", "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": "87Z1VgmE0q", "reviewer": "Reviewer_qPrg", "summary": "This paper introduces LEON, an LLM based entropy guided optimizer for conditional black box optimization in personalized medicine. The method steers an LLM with two principled constraints, a Wasserstein bound enforced by a source critic to limit shift from the training distribution and a coarse entropy control defined over equivalence classes of candidate treatments. The algorithm alternates proposing treatments, scoring them with a learned surrogate that is regularized by the critic, and updating dual variables for the two constraints. Prior medical knowledge is injected through tool use over textbooks, knowledge graphs, and a specialist model, and the system caches reflections across iterations. Experiments on five tasks under explicit distribution shift report consistent improvements over classical baselines such as Bayesian optimization and gradient methods as well as OPRO style prompting, with ablations on equivalence relations, embeddings, budgets, temperature, and knowledge quality. The writing is clear overall and the approach is theoretically motivated, though some key engineering details are pushed to the appendix and several evaluations rely on learned or simulated outcome functions.", "questions": "1. For each task, what model families were used to construct the evaluation oracle, and how do conclusions change under alternate architectures or calibrated versions of the oracle?\n2. Can you report performance as a function of shift magnitude and describe failure modes when the target distribution materially departs from the source?\n3. How is the constraint level chosen in practice? Provide a principled rule or adaptive schedule and an ablation in the main text.\n4. Did you try gradient penalty or spectral normalization rather than weight clipping?\n5. How many samples per batch are required for a stable estimate of the entropy multiplier and how does its variance depend on temperature and batch size?\n6. Beyond k-means and Leiden, can the equivalence relation be learned end-to-end with contrastive or outcome-aware objectives?\n7. Precisely how are textbooks, graphs, and the specialist model connected to the LLM context, for example through retrieval, tool calling, or structured APIs?\n8. Can you add pre-filters based on citation consistency or entailment and report their effect on robustness?\n9. Can you report token counts, wall clock times, and GPU hours per task?\n10. Can you publish the exact prompt templates, budgets, and tuning for OPRO style and human baselines, and confirm identical oracle query budgets across methods?\n11. How does performance change with smaller or open LLMs, and what minimum capability is required to retain the observed gains?\n12. How is patient safety enforced when suggestions deviate from standard practice, including any hard constraints, verification, or human in the loop controls?\n13. Can you provide a minimal working example with synthetic data that exercises all components including critic updates, clustering, and knowledge tools, along with seed handling and version pins?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. A principled formulation of patient level decision-making as constrained conditional optimization that combines a Wasserstein distribution shift control with an entropy based exploration control, yielding a practical and interpretable loop.\n2. Broad empirical evaluation across multiple biomedical tasks with ablations that illuminate sensitivity to embeddings, equivalence relations, budget, temperature, and knowledge quality.\n3. Thoughtful use of knowledge tools and reflection rather than vague claims, with explicit tests of helpful versus irrelevant knowledge.\n4. Clear problem setup, readable pseudocode, and helpful diagrams that separate the roles of the critic, the entropy control, and the knowledge components.\n5. Responsible positioning with discussion of clinical caveats and an intention to release code for public tasks.", "weaknesses": "1. Several evaluations use a learned oracle for outcomes rather than real clinical endpoints, which risks baking evaluation bias into the target function and limits external validity.\n2. The choice of equivalence relation and embedding strongly affects entropy estimates and search behavior, yet selection guidance is limited and some details are deferred to the appendix.\n3. The source critic training and dual updates introduce stability and tuning concerns, and the paper does not fully characterize sensitivity to clipping, step ratios, or learning rates.\n4. The pipeline is complex, with interdependent modules for prompting, clustering, critic training, tool access, and dual updates. This challenges reproducibility and may require significant engineering effort.\n5. Computational cost is likely high due to repeated LLM calls, clustering, and critic retraining. Wall clock and token cost reporting is incomplete for the main text.\n6. Presentation is dense in places and certain implementation choices for knowledge tool wiring and safety controls are difficult to locate in the main body.\n7. Baseline tuning details and budget parity need to be fully specified to rule out configuration gaps, especially for OPRO style prompts and human or gradient baselines.\n8. Practical safety and deployment are not validated with clinician in the loop studies. The method can degrade with irrelevant or adversarial knowledge and the current safeguards appear limited.\n9. The paper is difficult to understand and follow.\n10. A lot of materials and supporting evidence for reproducibility, yet no working link to the code.", "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)