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Computer-Aided Multi-Objective Optimization in Small Molecule Discovery

Jenna C. Fromer, Connor W. Coley

arXiv:2210.07209v1q-bio.QMcs.LG

TL;DR

Molecular discovery is an inherently constrained multi-objective optimization problem. This review considers Pareto optimization across pool-based and generative approaches and argues that it is more practical for many molecular problems, while noting that predicted performance may not validate experimentally.

Abstract

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Molecular discovery is a multi-objective optimization problem that requires identifying a molecule or set of molecules that balance multiple, often competing, properties. Multi-objective molecular design is commonly addressed by combining properties of interest into a single objective function using scalarization, which imposes assumptions about relative importance and uncovers little about the trade-offs between objectives. In contrast to scalarization, Pareto optimization does not require knowledge of relative importance and reveals the trade-offs between objectives. However, it introduces additional considerations in algorithm design. In this review, we describe pool-based and de novo generative approaches to multi-objective molecular discovery with a focus on Pareto optimization algorithms. We show how pool-based molecular discovery is a relatively direct extension of multi-objective Bayesian optimization and how the plethora of different generative models extend from single-objective to multi-objective optimization in similar ways using non-dominated sorting in the reward function (reinforcement learning) or to select molecules for retraining (distribution learning) or propagation (genetic algorithms). Finally, we discuss some remaining challenges and opportunities in the field, emphasizing the opportunity to adopt Bayesian optimization techniques into multi-objective de novo design.

Introduction

Molecular discovery requires optimizing multiple properties within constraints. This review contrasts scalarization with Pareto optimization and organizes established and emerging methods across library-based optimization and de novo design.

Introduction

  • Scalarization combines objectives into one function by assigning relative importance, limiting trade-off exploration and requiring reoptimization when the weighting function changes.The review contrasts this with Pareto optimization, which reveals trade-offs without prior measures of objective importance.Source: Introduction · p. 2
  • Pareto optimization identifies solutions that reveal objective trade-offs without requiring prior relative-importance weights, and the review describes it as robust for molecular discovery.The solution set can include solutions to scalarization problems with different weighting factors.Source: Introduction · pp. 2–3
  • The review focuses on recent de novo design developments using deep learning and argues that they warrant further discussion and organization.Source: Introduction · p. 3
  • The review organizes established and emerging multi-objective optimization techniques, first covering library-based optimization and then de novo design.It also introduces mathematical concepts and discusses optimization-scheme design choices, methods, case studies, and open challenges.Source: Introduction · p. 3

Defining Multi-Objective Molecular Optimization

The review defines multi-objective molecular optimization by its objectives, trade-offs, and chemical design space, and distinguishes it from constrained generation and exhaustive screening.

Defining Multi-Objective Molecular Optimization

Preliminary Mathematical Concepts in MMO

This section introduces multi-objective optimization concepts for molecular discovery, including Pareto trade-offs and Bayesian optimization methods for acquisition, candidate ranking, and batch selection. It also describes the challenge of balancing acquisition utility with molecular diversity in batch selection and the use of practical approximations and heuristics.

The Pareto front

  • Pareto fronts capture optimal trade-offs: improving one objective requires worsening another, while dominated hypervolume measures front quality and increases with a larger dominated region.For two objectives, the hypervolume is the area dominated by the front relative to a reference point.Source: The Pareto front · pp. 4–5

Single-objective Bayesian optimization

  • Single-objective BO iteratively fits a surrogate, prioritizes candidates with an acquisition function, evaluates them against the oracle, and updates the model; acquisition balances exploration and exploitation.The loop continues until a stopping criterion is met, such as resource limits or reaching an objective threshold.Source: Single-objective Bayesian optimization · pp. 6–7

Multi-objective Bayesian optimization

  • Multi-objective BO approximates each oracle and selects acquisitions without assigning relative-importance weights, aiming to expand the Pareto front or dominated hypervolume; EHI and PHI quantify hypervolume improvement.EHI measures expected hypervolume improvement, while PHI is the probability that a candidate improves hypervolume by any amount.Source: Multi-objective Bayesian optimization · pp. 8–9
  • Non-dominated sorting repeatedly identifies Pareto fronts and assigns integer ranks as acquisition scores, but it does not account for uncertainty.Candidates on the first-ranked front correspond to those selected by greedy acquisition across possible scalarization weights.Source: Multi-objective Bayesian optimization · p. 9

Batching and batch diversity

  • Batching suits parallel experiments and distributed computation, but top-k selection can include similar candidates; multi-objective batch acquisition should maximize whole-batch utility, such as hypervolume gain after all candidates are acquired.Top-k batching selects the k candidates with the highest individual acquisition scores.Source: Batching and batch diversity · p. 10
  • Selecting the utility-maximizing batch exactly is intractable, so iterative approximations select promising molecules one at a time while treating each as observed [39].Acquiring candidates close together on the Pareto front may not maximize utility.Source: Batching and batch diversity · p. 10
  • Batch heuristics balance high acquisition scores with Pareto diversity by limiting candidates per objective-space region or favoring less-crowded regions [40, 41, 42], approaches used in molecular design [43–45].Source: Batching and batch diversity · p. 10
  • Design-space diversity is molecular or structural diversity, which can be measured with fingerprint-based metrics such as Tanimoto similarity and imposed during acquisition [46, 47, 48].Source: Batching and batch diversity · pp. 10–11
  • Pareto-front diversity and molecular diversity are distinct: structurally similar molecules may have different properties, while similarly scoring molecules need not be structurally similar.Figure 4 contrasts objective-space Pareto diversity with molecular or structural diversity and shows that promoting one does not necessarily improve the other.Source: Batching and batch diversity · p. 11, Figure 4 · p. 11

Formulating Molecular Optimization Problems

Formulating a molecular optimization task involves defining mathematical objectives, choosing a search space, and deciding how to handle multiple objectives. These choices depend on the property and application, and scalarization trades a multi-objective problem for one objective while omitting information about the Pareto trade-offs.

Converting a desired property to a mathematical objective function

  • Objective formulation can treat a property score as continuously increasing, thresholded, or Boolean, depending on how the property should contribute to optimization.For example, binding affinity may benefit from continuous optimization, while solubility beyond a sufficient delivery threshold may not; some thresholds are better modeled as constraints.Source: Converting a desired property to a… · p. 13
  • When expertise or a suitable threshold is unavailable, the review favors a simple continuous representation because it needs no predefined hyperparameters or assumptions.Constraints can be imposed on the resulting solutions later without repeating the optimization.Source: Converting a desired property to a… · p. 14

Choosing between library-based selection and de novo design

  • Search can be limited to a task-relevant molecular library or use de novo design to propose previously unseen molecules.A generative model’s reachable chemical space depends on its design; for genetic algorithms, it may be constrained by the initial population and evolutionary operators.Source: Choosing between library-based… · p. 14

Defining the relationship between different objectives

  • Scalarization combines objectives into one function, commonly a weighted sum whose weights represent their relative importance.Other examples include nonlinear scalarization and Boolean-based formulations.Source: Defining the relationship between… · p. 15
  • Alternative single-objective formulations include prioritizing objectives hierarchically or alternating rewards across objectives.The scalarization function, like the individual objectives, should be justified by the use case.Source: Defining the relationship between… · p. 15
  • Scalarization returns one solution from the many non-dominated solutions on the Pareto front and does not inform users about objective trade-offs.The review characterizes scalarization as requiring users to quantify relative objective importance and as less informative than identifying the full Pareto set.Source: Defining the relationship between… · p. 15

Examples of MMO from Virtual Libraries

Pool-based multi-objective optimization applies Pareto search to finite molecular libraries, using surrogate-guided acquisition to find molecules balancing objectives. The examples report improved Pareto-search outcomes over random search, including retrospective and prospective studies.

  • Pool-based workflows iteratively train a surrogate, select and score acquired molecules, then retrain; sample efficiency, batch size, and diversity remain improvement targets.The review also anticipates additional applications with more meaningful objectives and experimental validation.Source: Examples of MMO from Virtual Libraries · pp. 16–17
  • For flow-battery complexes, a neural surrogate predicted solubility and redox potential across an almost 3-million-complex library, while DFT supplied oracle scores and expected hypervolume improvement guided acquisition.Source: Examples of MMO from Virtual Libraries · p. 16
  • Over three standard deviations from the initial random set were observed for both objectives in five iterations, estimated as a 500x reduction in simulations versus random search.Source: Examples of MMO from Virtual Libraries · p. 16
  • Agarwal et al. 44 used Pareto optimization for redox-active materials, with crowding-distance constraints promoting diverse acquisition across objectives including a target absorption wavelength of 375nm.Source: Examples of MMO from Virtual Libraries · p. 16
  • 15 times more evaluations were required by random search than Bayesian optimization to acquire molecules dominating 99% of the total possible hypervolume.Source: Examples of MMO from Virtual Libraries · p. 17
  • Of 100 molecules acquired prospectively from a one-million-molecule set, 16 new Pareto-optimal molecules were identified.Source: Examples of MMO from Virtual Libraries · p. 17

Examples of MMO using Generative Models

This section surveys generative-model approaches to multi-objective molecular discovery, including Pareto-aware selection or rewards, conditional generation, and hybrid or iterative optimization. It also covers reported limitations, including constraints from training experience and dependence on training data or surrogate predictions.

Iterative retraining for distribution learning

  • In distribution learning, non-dominated sorting selects promising samples for iterative retraining, shifting generated-molecule distributions and progressively pushing the Pareto front.The workflow evaluates sampled molecules against all oracles, then retrains on increasingly promising subsets.Source: Iterative retraining for distribution… · pp. 18–19
  • 33% of generated molecules met all five constraints after five retraining iterations, up from 2%, but no Pareto-front expansion was evidenced despite simultaneous property-distribution shifts.The study therefore showed that multiple property distributions could shift together, without evidence that dominated hypervolume increased.Source: Iterative retraining for distribution… · p. 20
  • Iterative retraining advanced the Pareto front beyond the training set, increasing dominated hypervolume and identifying molecules outside the original objective space.The reported molecules included ones that strictly dominated the best molecules in the training set.Source: Iterative retraining for distribution… · p. 20, Figure 7 · p. 21

Genetic algorithms

  • Genetic algorithms use mutations and crossovers to generate molecules, with non-dominated sorting across objectives selecting parents and advancing the population’s Pareto front.Mutations add or delete atoms or bonds, while crossovers exchange molecular fragments between parent molecules.Source: Genetic algorithms · p. 22
  • Niching and elitism improve genetic-algorithm optimization performance by promoting structural diversity and retaining Pareto-dominant molecules from prior iterations.Niching groups structurally similar candidates and limits how many are acquired from each niche; elitism appends previously found Pareto-dominant molecules before selection.Source: Genetic algorithms · p. 22
  • Both NSGA-II and NSGA-III outperform weighted-sum scalarization by dominated hypervolume, with no clear winner between the algorithms.Across seven molecular case studies, internal similarity decreased over evolutionary iterations, yet selection promoted Pareto diversity without necessarily promoting structural diversity.Source: Genetic algorithms · p. 23

Reinforcement learning

  • DrugEx v2 generated desirable multi-target molecules at 81% with its Pareto reward and 97% with a weighted-sum reward, but the comparison used constraint-style evaluation.The approaches were not compared using a Pareto criterion such as hypervolume improvement, so the Pareto optimizer’s weaker result may reflect evaluation misalignment.Source: Reinforcement learning · p. 25

Conditional generation

  • Conditional generators support multiple objectives by centering latent dimensions on different properties or feeding training-molecule property values to the decoder as conditions.At inference, sampled latent variables reflect desired property values with at least partial success; a manually specified condition vector guides decoder outputs.Source: Conditional generation · p. 25
  • One conditional-generation study found less than 1% of generated molecules within 10% of specified properties, while a similar architecture reported up to 33% matching when rounded up.In the latter study, the matching fraction appeared to correlate strongly with how many training molecules fulfilled the constraints.Source: Conditional generation · p. 26
  • Conditional generators can modify input molecular graphs or scaffolds, using matched molecular pairs to learn structural transformations that improve multiple properties.The review notes that this approach extends from single-objective to multi-constraint generation.Source: Conditional generation · p. 26
  • Conditional generators may struggle to advance Pareto fronts: MolGPT depends on training-set representation of target properties, and its ability to generate unseen property combinations is unreported.The authors argue that advancing the Pareto front requires extrapolating beyond the training set.Source: Conditional generation · pp. 26–27

Hybrid approaches

  • Grantham et al. [45] encode molecules with a variational autoencoder, mutate latent vectors, decode new molecules, and use NSGA-II selection to retrain the autoencoder.Selection uses non-dominated sorting with a crowding-distance constraint.Source: Hybrid approaches · p. 27
  • Grantham et al. [45]’s hybrid method outperformed two Bayesian optimization baselines in final Pareto-front hypervolume for ClogP, QED, and synthesizability.The comparison reports no numerical hypervolume values.Source: Hybrid approaches · pp. 27–28
  • An iterative conditional graph generator was fine-tuned on molecules active against both JNK3 and GSK-3β, adding a bias toward favorable-property sampling [135].The workflow follows iterative retraining used in distribution learning.Source: Hybrid approaches · p. 28
  • Two methods [64,139] use RL to generate molecules predicted by pretrained surrogate models to be dual inhibitors of GSK3β and JNK3 receptors.Source: Hybrid approaches · p. 28
  • That RL result is qualified by dual inhibitors already present in the training data, making surrogate-classified active-space discovery easier than extrapolation with a continuous oracle.Success with Boolean-valued conditioning may be overoptimistic because optimization is harder to quantify with metrics such as hypervolume.Source: Hybrid approaches · p. 28

Discussion

The discussion identifies weaknesses in current evaluation and diversity practice, alongside Pareto scalability constraints, while outlining ways to bring model-guided optimization into de novo design.

Discussion

  • De novo design could adopt Bayesian principles by adding a retrainable surrogate to predict objectives and guide selection as new data arrive.The authors suggest such adjustments may bridge generation and model-guided optimization.Source: Discussion · p. 29
  • Individual property values and constraints reveal little about the combinations of properties that matter in multi-objective molecular optimization.Hypervolume improvement tracks Pareto-front shifts, but front density and average Pareto rank can also matter.Source: Discussion · p. 29
  • Because predicted high-performing molecules may fail experimental validation, dense Pareto fronts and structural diversity provide more candidates and a hedge against failed hits.Structural diversity explores a wider region of chemical space; neither it nor Pareto diversity has been shown to outperform the other in MMO tasks.Source: Discussion · p. 29
  • Whether and how to combine structural and Pareto diversity during acquisition remains unsettled, including whether diversity-aware acquisition benefits optimization.Both diversity measures can be used during optimization, but neither has been shown to outperform the other in MMO tasks.Source: Discussion · pp. 29–30
  • Pareto optimization faces scaling challenges beyond three objectives: non-dominated sorting differentiates solutions less, hypervolume estimation scales exponentially, and surrogate scoring adds cost.When some objectives matter more than others, scalarizing selected objectives or turning them into constraints may be practical.Source: Discussion · p. 30
  • Pareto fronts are easier to visualize with two or three objectives, and the best problem formulation depends on use case and subject-matter expertise.Expert collaboration can help keep formulations feasible without imposing unrealistic assumptions.Source: Discussion · p. 30
  • Molecular optimization needs realistic computational oracles and harder benchmarks; simple QED, ClogP, or classifier-output tasks poorly indicate robustness or generality.Generative models must also work with fewer oracle calls, often a bottleneck when scoring requires experiments or high-fidelity simulations.Source: Discussion · p. 31
  • Experimental applications must account for synthesizability, and prospective experimental validation is essential to demonstrate an algorithm’s viability.Synthesizability can be modeled as a continuous objective or a rigid constraint.Source: Discussion · p. 31

Conclusion

The conclusion contrasts relatively direct multi-objective extensions for library search with varied generative methods, and calls for uncertainty-aware workflows, stronger benchmarks, and attention to Pareto complexity.

Conclusion

  • Library-based multi-objective discovery is a relatively direct extension of Bayesian optimization, requiring changes such as adapting acquisition functions and adding surrogate models.De novo workflows are less directly analogous to Bayesian optimization and vary more in methodology.Source: Conclusion · p. 31
  • De novo methods use Pareto rank as an RL reward or non-dominated sorting to select molecules for distribution-learning retraining or genetic-algorithm propagation.These approaches replace greedy acquisition functions.Source: Conclusion · p. 31
  • New generative workflows could more directly incorporate model-guided optimization while accounting for model uncertainty.The conclusion identifies this as an opportunity for bridging generation and model-guided optimization.Source: Conclusion · p. 31
  • Diversity-aware batching can reward structural diversity, Pareto diversity, or both, but best practices are not established.The conclusion calls for challenging benchmarks and metrics such as dominated hypervolume or Pareto-front density.Source: Conclusion · pp. 31–32
  • Multi-objective methods must address algorithmic complexities introduced by Pareto optimization as molecular discovery tools increasingly emphasize multiple objectives.Source: Conclusion · p. 32
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