Amortized Active Generation of Pareto Sets

TL;DR

This work presents Amortized Active Generation of Pareto Sets (A-GPS), a framework for online, discrete black-box multi-objective optimization that learns a conditional generative model of the Pareto set and supports a-posteriori user preferences through preference direction vectors. A-GPS uses a non-dominance class probability estimator to implicitly approximate the probability of hypervolume improvement (PHVI) and couples it with alignment cues to guide sampling toward preferred trade-offs, all within an amortized objective that updates online without retraining. The method avoids explicit hypervolume computations, scales to high-dimensional design spaces, and demonstrates strong sample efficiency across synthetic benchmarks and protein-design tasks, with additional ablations and discussions on practical considerations. The approach offers flexible, preference-conditioned exploration, enabling practitioners to tailor Pareto front sampling after model training and during design iterations. Code and experiments are provided to support reproducibility and extension to other generative backbones.

Abstract

We introduce active generation of Pareto sets (A-GPS), a new framework for online discrete black-box multi-objective optimization (MOO). A-GPS learns a generative model of the Pareto set that supports a-posteriori conditioning on user preferences. The method employs a class probability estimator (CPE) to predict non-dominance relations and to condition the generative model toward high-performing regions of the search space. We also show that this non-dominance CPE implicitly estimates the probability of hypervolume improvement (PHVI). To incorporate subjective trade-offs, A-GPS introduces preference direction vectors that encode user-specified preferences in objective space. At each iteration, the model is updated using both Pareto membership and alignment with these preference directions, producing an amortized generative model capable of sampling across the Pareto front without retraining. The result is a simple yet powerful approach that achieves high-quality Pareto set approximations, avoids explicit hypervolume computation, and flexibly captures user preferences. Empirical results on synthetic benchmarks and protein design tasks demonstrate strong sample efficiency and effective preference incorporation.

Figures (9)

• Figure 1: A visualization of a Pareto front, $\mathcal{F}_{\textrm{Pareto}}$, and the random variables used with A-GPS. $\mathbf{y}$ are noisy realizations of the objectives, ${\mathbf{f}\centerdot}$. When $z=1$ these observations lie on the Pareto front. Preference direction vectors, $\mathbf{u}$, are unit vectors pointing to a region of the Pareto front from a reference point, $\mathbf{r}$, \ref{['eq:pref_train']}. We derive aligned ($a=1$) training preference direction vectors, $\mathbf{u}_n$, from observation pairs $(\mathbf{y}_n, \mathbf{x}_n)$, and mis-aligned preference direction vectors from permuting these pairs, $(\mathbf{y}_{\rho(n)}, \mathbf{x}_n)$, \ref{['eq:cpe_pref']}. The aim is to learn the distribution of the Pareto set $q_{\phi}({\mathbf{x}}|{\mathbf{u}}) \approx p({\mathbf{x}}|{\mathbf{u}, z=1, a=1})$.
• Figure 2: Experimental results on three test functions commonly used in the literature. The top row reports per round, the bottom row demonstrates amortized preference conditioning by generating Pareto front samples (DTLZ7 is a PCA projection of the front).
• Figure 3: Ehrlich function vs. ProtBert naturalness score for different sequence lengths.
• Figure 4: Bi-grams and Stability vs. SASA results for relative hypervolume (HV) improvement and diversity.
• Figure 5: A visual depiction of \ref{['alg:optloop']} --- learns all the distributions involved by optimizing different components of a reverse loss. At time $t$, the optimized variational distribution $q_{\phi}({\mathbf{x}}|{\mathbf{u}})$ with parameters $\phi^*_t$ is used to generate new designs that can incorporate new user's preferences $\mathbf{u}^*$. We iterate until a convergence/user criterion is satisfied.

Theorems & Definitions (5)

• theorem 1: Equivalence of Indicators
• corollary 1: Non-Dominance cpe estimates phvi
• theorem 1: Equivalence of Indicators
• proof
• corollary 1: Non-Dominance cpe estimates phvi