Controllable Pareto Trade-off (CPT)

• CPT is a multi-objective optimization framework that makes Pareto trade-offs explicitly controllable by mapping preference signals to Pareto-efficient solutions.
• It employs diverse mechanisms such as preference-conditioned hypernetworks, stabilized gradient steering, and threshold-based control to tailor trade-offs.
• Empirical evaluations reveal CPT’s improved alignment of objectives in areas like fairness, privacy, and multi-task learning compared to naive scalarization methods.

Controllable Pareto Trade-off (CPT) denotes a class of multi-objective optimization formulations in which the aim is not only to recover Pareto-optimal solutions, but to make the final compromise explicitly steerable. In recent work, the control signal is instantiated as a reference vector, preference vector, scalarization weight, cost threshold, privacy level, learned inverse map, or intrinsic scale. Across these formulations, the common objective is to expose feasible trade-offs, parameterize or rank Pareto-efficient solutions, and obtain outcomes that more closely match an expressed preference than conventional single-point optimization or ad hoc weighted sums (Du et al., 17 Sep 2025). A closely related line of work argues that raw objectives are not meaningfully comparable after arbitrary scaling, and instead maps them into a common score space on $[0,1]$ via the probability integral transform so that Pareto-efficient solutions can be ordered and desired trade-offs can be mapped back into preference space (Hönel et al., 2022).

1. Conceptual foundations

CPT arises from a standard difficulty in multi-objective optimization: Pareto-optimality identifies efficient solutions, but does not by itself provide a total order over them or a reliable mechanism for selecting the particular compromise that a decision maker wants. This issue is explicit in work on objective ordering, where Pareto-efficient solutions are described as not totally orderable in the usual sense, and in work on fairness–accuracy optimization, where a single “optimal” solution is treated as insufficient because different deployment scenarios may prefer different balances (Hönel et al., 2022).

The central conceptual move in CPT is to treat the trade-off itself as a controllable object. In some settings, this means defining a user-specified direction in objective space. In the fairness–accuracy setting for NLP, the control variable is a reference vector $\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$, which encodes the desired ratio between fairness and accuracy losses and is enforced during optimization rather than being used only for initialization (Du et al., 17 Sep 2025). In other settings, controllability is achieved by a preference vector that conditions a solution generator, so that different preferences directly produce different Pareto solutions. This is the formulation used in controllable Pareto multi-task learning, where a hypernetwork learns the map $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$ from preferences to task-network parameters (Lin et al., 2020).

A second foundational theme is that controllability is often motivated by the inadequacy of naive scalarization. The score-space approach to ordering objectives explicitly argues that the common practice of treating scaled objectives as directly comparable is a fallacy because objective distributions are typically non-uniform; equal changes in raw objective values do not correspond to equal changes in desirability (Hönel et al., 2022). Related concerns appear in differential privacy, where linear utilities are reported to select only endpoints on an S-shaped front, and in biological adaptation, where weighted-sum optimization reconstructs only the convex part of the Pareto front and can miss concave regions (Yang et al., 4 Sep 2025).

2. Mathematical forms of controllability

The mathematical structure of CPT is heterogeneous, but several recurrent forms appear.

One form is preference-conditioned multiobjective optimization. In controllable Pareto multi-task learning, multi-task training is written as

$$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$

and a preference vector $\vp$ selects a restricted Pareto solution through either linear scalarization or an angular-region constraint in loss space (Lin et al., 2020). A closely related construction appears in Pareto Continual Learning, where the continual-learning objective is

$$\min_{\theta} F(\theta)=\big(\mathcal{L}_{\text{replay}}(f_\theta),\mathcal{L}_{\text{new}}(f_\theta)\big)^\top,$$

and a preference-conditioned model $f_\theta(x;\alpha)$ learns a family of stability–plasticity trade-offs rather than a single replay weight (Lai et al., 30 Mar 2025).

A second form is reference-vector control. In the fairness–accuracy setting, CPT optimizes

$$\min_{\theta}\ \mathbf{L}(\theta)=\big(L_{\text{fair}}(\theta),L_{\text{acc}}(\theta)\big)^\top$$

and introduces a KL-divergence-style constraint

$$V(\mathbf{l},\mathbf{T})=D_{\mathrm{KL}}\!\left(\frac{\mathbf{l}}{\|\mathbf{l}\|_1}\,\middle\|\,\frac{\mathbf{T}}{\|\mathbf{T}\|_1}\right),$$

followed by a two-stage procedure consisting of a correction stage satisfying $V(\mathbf{l},\mathbf{T})<\tau$ and a multi-objective stage optimizing $\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$0 jointly (Du et al., 17 Sep 2025).

A third form is threshold-based control. In constrained Markov decision processes, Threshold UCT solves

$\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$1

with the threshold $\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$2 selecting a point on a cost–utility Pareto curve. When $\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$3 lies between two adjacent frontier vertices, the planner returns a randomized mixture whose coefficients interpolate between those vertices so that expected cost matches the available budget as closely as possible (Kurečka et al., 2024).

A fourth form is scalarized physical control. In thermal machines, controllability is encoded by weights in

$\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$4

where increasing one weight improves the associated quantity but moves the solution along the Pareto front by degrading the others (Almanza-Marrero et al., 20 Mar 2026). In intrinsically nonequilibrium control, the analogous scalarized functional is

$\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$5

and the weights combine with physical parameters into a single intrinsic scale that parametrizes the full front and defines control equivalence classes (Forão et al., 22 Jun 2026).

A fifth form replaces raw objectives by a common score representation. Objective-ordering work defines

$\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$6

so that all transformed objectives lie in $\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$7, can be aggregated, and can be used to learn a nonlinear map between preference space $\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$8 and score space $\mathbf{T}=(v_{\text{fair}},v_{\text{acc}})$9 (Hönel et al., 2022).

3. Algorithmic mechanisms

Several distinct algorithmic mechanisms realize CPT in practice.

One mechanism is preference-conditioned generation. Hypernetwork-based multi-task learning uses a hypernetwork to generate model parameters conditioned on a preference vector, enabling real-time trade-off control from a single stored model (Lin et al., 2020). Expensive black-box optimization extends the same idea through Co-PSL, where a hypernetwork learns $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$0 after a warm-start Bayesian optimization phase has constructed higher-quality Gaussian process priors; the resulting preference-to-solution map supports real-time control over expensive Pareto fronts (Nguyen et al., 2023). Pareto Continual Learning uses a hypernetwork only for the final classifier layer, sampling preference vectors during training and selecting, at inference, the lowest-entropy output among multiple sampled preferences (Lai et al., 30 Mar 2025).

A second mechanism is stabilized gradient steering. In fair NLP, CPT augments MGDA-style common descent with moving-average gradients,

$\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$1

to smooth noisy fairness updates, and gradient pruning,

$\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$2

to retain only critical parameters before computing the common descent direction (Du et al., 17 Sep 2025). In machine unlearning, CUP constructs two anchors—an efficacy anchor and a fidelity anchor—and defines a pivoting direction $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$3 controlled by a single unlearning-intensity parameter $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$4, so that the update moves smoothly between fidelity-preserving and forgetting-oriented directions (Hwang et al., 22 Oct 2025).

A third mechanism is explicit frontier estimation and interaction. In differential privacy, the Pareto front is modeled directly as a one-dimensional curve

$\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$5

rather than as two generic surrogates over the full hyperparameter space. Preference learning is then conducted by presenting hypothetical Pareto curves to the user and asking them to select their preferred point, with Knowledge Gradient selecting both the next hypothetical curve and the next privacy level to evaluate (Yang et al., 4 Sep 2025). Threshold UCT likewise estimates Pareto curves throughout a search tree and uses thresholded mixing plus a sound threshold update to maintain feasibility during online planning (Kurečka et al., 2024).

A fourth mechanism is exact or closed-form parameterization. In thermal machines, endoreversible systems admit an exact analytical parameterization of the full four-objective Pareto front, and lower-dimensional trade-offs follow universal formulae independent of machine parameters once the relevant quantities are normalized (Almanza-Marrero et al., 20 Mar 2026). In beyond diagonal reconfigurable intelligent surfaces, the control variable is the circuit complexity $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$6, which determines the number of connected components $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$7 and hence an exact mapping from complexity to the optimal architecture and maximal average received power (Nerini et al., 2023).

4. Representative domains and control variables

The term CPT is used across a wide range of domains, but the underlying pattern is stable: a controllable parameter selects a region or point on a Pareto front.

Domain	Objectives	Control variable
Fair NLP classification	fairness and accuracy	reference vector $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$8
Differential privacy	privacy and accuracy	utility weights $\vtheta_{\vp}=g(\vp\mid \vphi^\ast)$9 and privacy level $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$0
Multi-task learning	multiple task losses	preference vector $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$1
Continual learning	stability and plasticity	preference vector $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$2
Machine unlearning	unlearning efficacy and model fidelity	unlearning intensity $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$3
Safe sequential planning	payoff and cost	threshold $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$4
Thermal machines	$$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$5	weights $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$6
BD-RIS design	received power and circuit complexity	complexity $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$7

In NLP fairness, CPT is presented as a method that can train models to perform different fairness–accuracy trade-offs according to human-defined reference vectors on Jigsaw toxic comment classification with race bias and BiasBios occupation classification with gender bias (Du et al., 17 Sep 2025). In differential privacy, the same general idea appears as interactive discovery of the preferred privacy–accuracy compromise for differentially private logistic regression and deep transfer learning, where the front is S-shaped and is modeled with sigmoid or Gompertz priors (Yang et al., 4 Sep 2025).

In learning systems, controllability often appears as preference-conditioned adaptation. Multi-task learning uses a single hypernetwork-based model to synthesize task parameters for different trade-off preferences in real time (Lin et al., 2020). Continual learning replaces a fixed replay weight by a family of preference-conditioned solutions, enabling both manual control and automatic inference-time adaptation (Lai et al., 30 Mar 2025). Machine unlearning reframes forgetting versus fidelity as a Pareto problem and uses a single intuitive hyperparameter to traverse the frontier (Hwang et al., 22 Oct 2025).

Outside machine learning, CPT also appears in exact or structural forms. In thermal machines, scalar weights traverse universal Pareto fronts and generate two-, three-, and four-objective trade-off relations (Almanza-Marrero et al., 20 Mar 2026). In intrinsically nonequilibrium thermodynamic control, the relative weights combine with physical parameters into a single intrinsic scale that governs the full trade-off and yields control equivalence classes (Forão et al., 22 Jun 2026). In wireless communications, complexity itself becomes the control knob: varying $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$8 moves continuously in the discrete sense between low-complexity single-connected RIS and high-performance tree-connected RIS (Nerini et al., 2023).

5. Empirical behavior and evaluation

Empirical evaluation of CPT methods typically measures two distinct properties: the quality of the attainable Pareto set and the fidelity with which a requested trade-off is realized.

In fairness-aware NLP, controllability is assessed through alignment with reference vectors and through hypervolume. CPT is reported to “precisely follow each reference vector,” while MGDA tends to collapse to essentially one solution, PMTL produces limited diversity, and EPO can generalize worse in fairness. Table 2 reports hypervolume values of $$\min_{\theta}\ \mathcal{L}(\theta)=\big(\mathcal{L}_1(\theta),\ldots,\mathcal{L}_m(\theta)\big),$$9 on Jigsaw and $\vp$0 on BiasBios for CPT, with ablations showing lower values for variants without gradient averaging or without pruning (Du et al., 17 Sep 2025).

In score-based Pareto control, ordering and preference correction are evaluated by the discrepancy between desired and obtained trade-offs. The corrected preference method is reported to reduce the median mean absolute error to about $\vp$1, with all deviations below about $\vp$2, while online score-space optimization also improves substantially over raw-objective optimization (Hönel et al., 2022). This suggests that controllability depends not only on locating the Pareto front, but also on compensating for the nonlinear map from expressed preference to achieved score vector.

In differential privacy, evaluation uses preference inference error and regret. The interactive curve-based method is reported to achieve lower preference inference error, lower regret, faster convergence, lower user interaction cost, and lower computational cost than baselines, and ablations indicate that direct Pareto-front modeling and hypothetical-front queries are both important (Yang et al., 4 Sep 2025). In machine unlearning, solution-set quality is measured by the hypervolume indicator, with CUP reported to achieve $\vp$3 and $\vp$4 on CIFAR-10, and $\vp$5 and $\vp$6 on SVHN (Hwang et al., 22 Oct 2025).

In safe planning, T-UCT is evaluated by feasibility and payoff under cost constraints rather than by static front quality. It is reported to be much more reliable than RAMCP in finding feasible policies, less conservative than CC-POMCP, and the only method on the Manhattan benchmark to achieve both good safety and substantial reward at scale (Kurečka et al., 2024). In continual learning, ParetoCL is reported to trace a well-distributed Pareto front, with dynamic preference adaptation outperforming a fixed inference preference (Lai et al., 30 Mar 2025).

6. Misconceptions, limitations, and open directions

A recurrent misconception in CPT-related work is that arbitrary scaling renders objectives comparable. The score-space approach rejects this directly, arguing that objective distributions are inhomogeneous and that linear rescaling of raw values is deceptive because it conceals the density of attainable outcomes (Hönel et al., 2022). A related misconception is that one “best” compromise is sufficient. Fairness–accuracy work, continual learning, and controllable multi-task learning all treat this as too restrictive when preferences differ across users, deployment scenarios, training stages, or inputs (Du et al., 17 Sep 2025).

A second limitation concerns scalarization itself. Linear scalarization is repeatedly described as incomplete: in differential privacy it often selects only endpoints on an S-shaped front; in controllable multi-task learning it cannot recover non-convex parts of the Pareto front; and in biological adaptation weighted-sum reconstruction can miss concave front regions and induce jumps between disconnected branches (Yang et al., 4 Sep 2025). The biological literature makes the implication explicit: concave regions indicate coexistence of distinct optimization mechanisms and multiple nearby compromise-optimal solutions rather than a single smooth continuum (Tabanera-Bravo et al., 2024).

A third limitation is algorithmic sensitivity and computational cost. Fair NLP CPT notes that moving-average weights are sensitive, that tuning $\vp$7 can be tedious, that weights may not transfer across datasets, and that per-class trade-offs may not follow the global preference even when the overall fairness–accuracy balance does (Du et al., 17 Sep 2025). Threshold UCT identifies Pareto-curve maintenance and Bellman-style propagation as more expensive than scalar-value UCT, and explicitly points to function approximation for Pareto sets as a scaling direction (Kurečka et al., 2024). Hypernetwork-based controllable MTL states that it cannot guarantee the true Pareto front in general and that coverage depends on model capacity and preference sampling (Lin et al., 2020).

A final open direction concerns the geometry of control. Exact thermodynamic work shows that the trade-off may be parameterized by a universal front or by a single intrinsic scale (Almanza-Marrero et al., 20 Mar 2026). This suggests a stronger notion of controllability than merely choosing among stored solutions: in some systems, a low-dimensional control law organizes the entire set of optimal compromises. By contrast, learned mappings in score-space correction, expensive black-box Pareto set learning, and preference-conditioned neural models indicate that in many high-dimensional settings the preference-to-solution relation must be learned or approximated rather than derived analytically (Nguyen et al., 2023).