A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity

Michinari MommaChaosheng DongJia Liu

article2022ICML61 citations

Develops a generic multi-objective learning framework based on Pareto stationarity that incorporates user preferences and extends weighted Chebyshev optimization to discover models outperforming existing baselines in a single training run.

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Modern machine learning systems, such as search ranking, recommendation engines, and multi-task deep neural networks, frequently require optimizing multiple competing objectives simultaneously. In commercial environments, engineering teams routinely face the costly challenge of retraining models on fresh data while ensuring the new model does not degrade performance on any key metric relative to an existing baseline. Traditional multi-objective optimization techniques either struggle to align models with explicit business preferences or require extensive, manual trial-and-error exploration across all objectives, leading to development cycles that typically consume several days.

The article develops a unified mathematical framework for multi-objective and multi-task learning that simultaneously achieves optimal trade-off efficiency—known as Pareto optimality—and alignment with specific user-defined preferences. Specifically, it introduces two gradient-based algorithms: weighted Chebyshev multi-gradient descent and an extended version that explicitly explores improvements relative to an existing baseline model.

To demonstrate and validate this framework, the authors conducted empirical evaluations across both synthetic benchmarks and multiple real-world tasks. The experimental suite included two-task image classification across three separate datasets (MultiMNIST, Multi-Fashion, and Multi-Fashion+MNIST with 120,000 training images each), eight-target river flow regression across the Mississippi River network, and multi-class emotion classification in music. The proposed method was evaluated against established industry benchmarks, including linear scalarization, Pareto multi-task learning, and Exact Pareto Optimal search, measuring relative loss profiles and overall hypervolume coverage.

The findings show that the extended algorithm successfully identifies models that strictly outperform an existing reference model while directly following specified trade-off preferences. In practice, this reduces the required tuning exploration from scaling linearly with the number of objectives down to a single optimization run. The proposed method achieved the highest hypervolume metric in five out of eight evaluation settings and ranked second in the remaining three, demonstrating superior ability to dominate competing approaches. Furthermore, the algorithm’s automatic parameter-tuning mechanism ensured smooth convergence within 100 iterations on synthetic tests, outperforming fixed-parameter setups that required over 130 iterations.

These results provide immediate practical value for production engineering. By reducing the tuning overhead to a single run, organizations can substantially reduce computational costs, eliminate days of manual engineering labor, and de-risk model updates by guaranteeing that baseline performance is maintained or exceeded. Organizations facing frequent model retraining should consider adopting this framework to automate multi-task model updates and replace manual hyperparameter searches. Confidence in these results is high across complex, competing objectives; however, practitioners should note that the method operates on gradient-based models and does not provide an advantage on simpler problems where basic linear combinations already suffice.

classification computational efficiency hypervolume Pareto stationarity multi-task learning framework multi-objective optimization

Momma et al (2022).pdf

Paper: Multi-Task Learning as Multi-Objective Optimization, Ozan Sener et al. (2018). It establishes the multi-task-as-multi-objective framing and Pareto-efficient gradient optimization that the source extends with preference-directed Chebyshev methods.

Paper: Rewards-in-Context: Multi-objective Alignment of Foundation Models with Dynamic Preference Adjustment, Rui Yang et al. (2024). It carries Pareto-optimal multi-objective learning into foundation-model alignment, using preference-conditioned trade-offs to extend the source’s optimization framework to deployment-time user preferences.

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Abstract

Multi-objective optimization (MOO) and multi-task learning (MTL) have gained much popularity with prevalent use cases such as production model development of regression / classification / ranking models with MOO, and training deep learning models with MTL. Despite the long history of research in MOO, its application to machine learning requires development of solution strategy, and algorithms have recently been developed to solve specific problems such as discovery of any Pareto optimal (PO) solution, and that with a particular form of preference. In this paper, we develop a novel and generic framework to discover a PO solution with multiple forms of preferences. It allows us to formulate a generic MOO / MTL problem to express a preference, which is solved to achieve both alignment with the preference and PO, at the same time. Specifically, we apply the framework to solve the weighted Chebyshev problem and an extension of that. The former is known as a method to discover the Pareto front, the latter helps to find a model that outperforms an existing model with only one run. Experimental results demonstrate not only the method achieves competitive performance with existing methods, but also it allows us to achieve the performance from different forms of preferences.

Topics

classification computational efficiency hypervolume image classification neural networks optimization pareto pareto optimal solutions pareto-optimal recommendation

Keywords

Pareto stationarity multi-task learning framework multi-objective optimization KKT conditions MGDA solution SOCP duality preference vector strong duality

1. Introduction
2. Related work
3. Methodology
3.1. An MOO / MTL problem
3.2. MGDA
3.3. Weighted Chebyshev - MGDA
3.3.1. WC-MGDA Formulation
3.3.2. Solution Strategy
3.3.3. Extended WC-MGDA
3.3.4. Computational Complexity
4. Experiments
4.1. Synthetic data
4.2. Real dataset
4.2.1. Image Classification
4.2.2. Multi-Target Regression
4.2.3. Multi-Class Classification
4.2.4. Hypervolumes for All Experiments
5. Conclusion
References
A. Proofs
A.1. Proof of Lemma 3.2
A.2. Proof of Lemma 3.3
A.3. Proof of Lemma 3.4

Knowls

Knowl 1 — WC-MGDA combines preference alignment with Pareto stationarity
model/method
Consider minimizing a vector of $m$ differentiable losses $l(x)\in\mathbb{R}^m$ over model parameters $x\in\mathbb{R}^n$ . Let $G(x)=[\nabla l_1(x),\ldots,\nabla l_m(x)]\in\mathbb{R}^{n\times m}$ , let $K(x)=(G(x)^\top G(x))^{1/2}\in\mathbb{R}^{m\times m}$ , and let $r\in\mathbb{R}_{++}^m$ be a componentwise-positive preference vector. Define $K_r=\operatorname{diag}(\sqrt r)K\operatorname{diag}(\sqrt r)$ , where the square root is componentwise. WC-MGDA formulates preference alignment and approximate Pareto stationarity in one optimization problem:
$\begin{aligned} \max_{\gamma,\,\alpha\in\mathbb{R}^m_+,\,x\in\mathbb{R}^n}\quad &\alpha^\top(r\odot l(x))-u\gamma\\ \text{subject to}\quad &e^\top\alpha=1,\qquad \|K_r\alpha\|_2\leq\gamma, \end{aligned}$
Here $e\in\mathbb{R}^m$ is the all-ones vector, $\odot$ denotes elementwise multiplication, $\alpha$ is a nonnegative simplex vector, $\gamma\geq0$ bounds the stationarity residual, and $u>0$ trades off preference alignment against that residual. The constraint $K_r\alpha=0$ implies Pareto stationarity: because every component of $r$ is positive, $K\widetilde\alpha=0$ for a nonzero nonnegative vector $\widetilde\alpha=\operatorname{diag}(\sqrt r)\alpha$ , which can be normalized to sum to one.

For fixed $x$ , the equivalent primal form is
$\begin{aligned} \min_{\rho\in\mathbb{R},\,d\in\mathbb{R}^m}\quad &\rho\\ \text{subject to}\quad &r\odot l(x)+K_rd\leq\rho e,\qquad \|d\|_2\leq u. \end{aligned}$
The primal minimizes the largest component of the preference-weighted losses after a correction that also encodes the stationarity trade-off. For fixed parameters this is a second-order cone problem; jointly optimizing over $x$ is nonconvex. The construction exemplifies the paper's approach of starting from a preference-bearing optimization problem and incorporating Pareto stationarity, rather than modifying a stationarity-only method after the fact.
Knowl 2 — XWC-MGDA pivots the preferred Pareto search around a reference model
model/method
Let $\widetilde b\in\mathbb{R}^m$ contain the losses of a reference model, such as an existing or pretrained model. Extended WC-MGDA (XWC-MGDA) applies the preference vector $r\in\mathbb{R}_{++}^m$ to the loss difference $l(x)-\widetilde b$ , so the search is centered on the reference rather than the origin. To discourage weak Pareto-optimal solutions, it also imposes a positive lower bound on the dual weight vector. For any componentwise-positive $v\in\mathbb{R}_{++}^m$ , define $w=v/(1+e^\top v)$ ; then $w>0$ and $e^\top w<1$ .

With $K_r=\operatorname{diag}(\sqrt r)(G^\top G)^{1/2}\operatorname{diag}(\sqrt r)$ , XWC-MGDA's dual and primal forms are
$\begin{aligned} \max_{\gamma,\,\alpha\in\mathbb{R}^m,\,x\in\mathbb{R}^n}\quad &\alpha^\top\bigl(r\odot(l(x)-\widetilde b)\bigr)-u\gamma\\ \text{subject to}\quad &e^\top\alpha=1,\quad \alpha\geq w,\quad \|K_r\alpha\|_2\leq\gamma, \end{aligned}$ $\begin{aligned} \min_{\rho\in\mathbb{R},\,d\in\mathbb{R}^m,\,z\in\mathbb{R}^m_+}\quad &\rho-w^\top z\\ \text{subject to}\quad &r\odot(l(x)-\widetilde b)+K_rd+z\leq\rho e,\\ &\|d\|_2\leq u. \end{aligned}$
Here $z$ is a nonnegative slack vector, $u>0$ is the stationarity trade-off, and $\alpha$ is constrained componentwise. Setting $\widetilde b=0$ and $w=0$ recovers WC-MGDA. The reference-relative formulation lets a user seek solutions aligned with a preference from a chosen baseline; if the resulting model satisfies $l(x)-\widetilde b<0$ componentwise, it strictly improves every reference loss. The formulation seeks such improvements but does not guarantee that every run will produce a dominating model.
Knowl 3 — A primal-objective criterion automatically adjusts the stationarity trade-off
theoretical result
For WC-MGDA at fixed model parameters $x$ , let $\rho_0$ be the previous primal objective value and let $K_r$ be nonzero. The automatic adjustment chooses the smallest feasible trade-off $u$ subject to not worsening that objective:
$\begin{aligned} \min_{u\in\mathbb{R},\,\rho\in\mathbb{R},\,d\in\mathbb{R}^m}\quad &u\\ \text{subject to}\quad &r\odot l(x)+K_rd\leq\rho e,\\ &\|d\|_2\leq u,\qquad \rho\leq\rho_0. \end{aligned}$
The feasibility conditions distinguish whether the iteration can still improve preference alignment or needs to prioritize stationarity. If Pareto stationarity does not hold, or if the weighted Chebyshev constraints are feasible at $\rho=\rho_0$ , this adjustment problem is feasible. Conversely, if it is infeasible, Pareto stationarity holds and the weighted Chebyshev constraints cannot be satisfied at $\rho=\rho_0$ . When feasible, a solution with $\rho<\rho_0$ has only a trivial associated dual vector; the main WC-MGDA problem must then be solved to obtain a useful gradient. When $\rho=\rho_0$ , the minimum feasible $u$ yields a solution recoverable for the main problem.

The resulting strategy prioritizes reducing the primary weighted-loss objective by choosing small $u$ , and increases $u$ when needed to approach stationarity, while preventing the primary objective from worsening between iterations. For XWC-MGDA, the analogous adjustment uses the primary objective $\rho-w^\top z$ and its previous value $p_0$ as the non-increase criterion.
Knowl 4 — XWC-MGDA alternates trade-off adjustment and gradient updates
algorithm
The algorithm takes a preference vector $r\in\mathbb{R}_{++}^m$ , lower bound $w$ on the dual weights, reference loss vector $\widetilde b$ , learning rate $\eta$ , maximum iteration count $I_M$ , and residual tolerances $\tau_\alpha,\tau_d$ . It uses the gradient matrix $G(x)$ , the matrix $K_r$ , the XWC-MGDA primal-dual problem, and the automatic adjustment problem. The paper specifies a random initial parameter vector, a large initial primary-objective bound, and a small positive initial $u$ , but does not prescribe numerical values for these initializations. At each iteration, dual and primal variables from the adjustment or main problem supply the preference-weighted gradient and stationarity residuals.
Input: r, w, reference losses b_tilde, learning rate eta, maximum iterations I_M, tolerances tau_alpha and tau_d
Initialize x randomly, p0 to a large value, and u to a small positive value
For i = 1 to I_M:
    Compute G(x) and K_r(x)
    Solve the XWC-MGDA automatic adjustment problem with objective bound p0
    If the adjustment has a nontrivial solution:
        Update u and recover the associated primal-dual quantities alpha, d, and p
    Else:
        Solve the XWC-MGDA primal-dual problem at the current u to obtain alpha, d, and p
    If ||K_r(x) alpha||_2 <= tau_alpha and ||K_r(x) d||_2 <= tau_d:
        Return x
    Compute g = G(x)(r elementwise-multiplied by alpha)
    Update x = x - eta g
    Update p0 = p
Return the final x after I_M iterations
The parameter update is gradient descent on the model parameters using the preference-weighted combination of task gradients. The residual tests require both the weighted-gradient stationarity residual and the primal correction residual to be small.
Knowl 5 — Synthetic experiments show preference alignment and smoother automatic tuning
empirical result
The synthetic two-objective experiment used $n=20$ model parameters, a shared uniformly sampled initial point, and the highly nonconvex Pareto-front loss function
$l(x)=\begin{bmatrix}1-\exp\bigl(-\|x-e/\sqrt n\|_2^2\bigr)\\1-\exp\bigl(-\|x+e/\sqrt n\|_2^2\bigr)\end{bmatrix},$
where $e\in\mathbb{R}^n$ is the all-ones vector. Methods were evaluated with preference directions generated at equal angular intervals; XWC-MGDA additionally used a reference point. Linear scalarization found only the convex part of the front, while PMTL produced approximate but imperfect preference alignment. EPO and WC-MGDA aligned solutions with preferences better than PMTL. With a reference point, XWC-MGDA explored a preferred portion of the front that dominated the reference in the illustrated comparison.

The experiment also compared fixed $u=0.001$ with automatic adjustment for WC-MGDA. Fixed small $u$ produced nonsmooth changes in the dual weights and primary objective and took more than 130 iterations to converge. Automatic adjustment kept the primary objective non-increasing, produced smoother weight changes, and converged in fewer than 100 iterations.
Knowl 6 — Image-classification experiments test origin-based and reference-based preferences
experimental setup
The image experiments used Multi-MNIST, Multi-Fashion, and Multi-Fashion+MNIST, each with 120,000 training samples and 20,000 test samples. Each dataset had two classification tasks: classify the top-left image and classify the bottom-right image. All methods used the same LeNet architecture and shared random seeds; individual-task training served as a baseline. The comparison included XWC-MGDA, EPO, PMTL, and linear scalarization. In comparisons without a reference point, XWC-MGDA used $w=10^{-6}e$ .

With preference rays from the origin, XWC-MGDA produced test-loss solutions aligned with the requested preferences, especially for two of the three plotted directions. EPO was closer to one other direction, but XWC-MGDA generally matched or dominated competing gradient-based methods in the displayed loss and accuracy comparisons. The authors interpret the accuracy results as consistent with XWC-MGDA seeking Pareto-optimal rather than merely weakly Pareto-optimal points. In reference-based runs, XWC-MGDA used reference losses $(0.5,0.5)$ , $(0.7,0.7)$ , and $(0.55,0.55)$ for the three datasets, respectively; the resulting models followed preference rays from those references. This demonstrates the intended one-run search from a baseline, rather than repeated exploration to find a model that matches it on every task.
Knowl 7 — Regression and music-label experiments reveal dataset-dependent method behavior
empirical result
For multi-target regression, the River Flow dataset predicts river flows 48 hours ahead at eight sites in the Mississippi River network. The experiment used a four-layer fully connected network, mean squared error per task, and 20 preference vectors. On test data, XWC-MGDA and EPO had similarly distributed relative loss profiles across the eight tasks and performed better than PMTL and linear scalarization; their task losses were also reported as similar to the single-task baseline and much lower than those of the other multi-task methods, particularly PMTL.

For multi-label music classification, the Emotions dataset contains 593 songs with six emotion labels. The experiment used a four-layer fully connected network, sigmoid binary cross-entropy per task, and 50 preference vectors. All methods produced nearly uniform relative loss profiles, and none dominated the others. The authors take linear scalarization's strong performance as evidence that the Pareto front in this dataset may be convex.

Knowl 8 — Reported hypervolumes favor XWC-MGDA in five of eight comparisons

data/table

The table reports mean hypervolumes for test-loss and test-accuracy fronts across three image datasets, plus test-loss fronts for River Flow and Emotions. Larger hypervolume was interpreted by the authors as stronger Pareto-front performance. For image classification, the reported means are over five trials with random seeds shared across methods; nadir points from single-task baselines were used as hypervolume reference points. The authors report that XWC-MGDA has the highest hypervolume in five of the eight comparisons and the second-highest in the other three.

Dataset	Metric	XWC-MGDA	EPO	PMTL	LinScalar
MultiMNIST	Loss	0.0142	0.0135	0.0141	0.0168
MultiFashion	Loss	0.0677	0.0602	0.0590	0.0675
MultiFashion+MNIST	Loss	0.0389	0.0376	0.0303	0.0363
MultiMNIST	Accuracy	0.0016	0.0015	0.0015	0.0019
MultiFashion	Accuracy	0.0099	0.0087	0.0087	0.0099
MultiFashion+MNIST	Accuracy	0.0052	0.0050	0.0040	0.0048
River	Loss	6.83E+28	5.93E+28	1.08E+28	6.64E+28
Emotion	Loss	0.000348	0.000366	0.000230	0.000258

The comparisons show that the best method varies by dataset and metric: XWC-MGDA leads on several image and River Flow results, while linear scalarization has the largest reported hypervolume for MultiMNIST accuracy and ties XWC-MGDA on MultiFashion accuracy; EPO has the largest value for Emotions loss.

Knowl 9 — Per-iteration computational cost is dominated by gradient Gram-matrix formation
theoretical result
For $m$ objectives and $n$ model parameters with $n\gg m$ , the paper gives XWC-MGDA a computational complexity of $O(m^2n)$ per iteration. The dominant operation is forming $G(x)^\top G(x)$ from the $n\times m$ gradient matrix. Computing the matrix square root required for $K_r$ costs $O(m^3)$ , and solving the second-order cone program costs $O(m^{2.87})$ under the cited solver complexity. The stated total is therefore dominated by the gradient-matrix computation in the usual machine-learning regime $n\gg m$ .
Knowl 10 — The parameter update omits derivatives of the stationarity matrix
limitation
When updating model parameters, the method uses the gradient of the preference-weighted loss combination and does not differentiate through the parameter dependence of $K_r(x)$ . The authors state that including this dependence would require Hessian computation with respect to the model parameters, which they regard as prohibitive for the large-dimensional regime $n\gg1$ . The proposed update therefore does not include those second-order terms.

Coverage note — Formal proofs and proof-only intermediate steps are omitted; the main feasibility and recovery conclusions they establish are included. Detailed dataset preprocessing is not reproduced because the paper delegates it to prior work.

References

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Citation

MLA

Momma, M., et al. “A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity”. International Conference on Machine Learning, vol. 162, 2022, pp. 15895–907, https://proceedings.mlr.press/v162/momma22a.html.

APA

Momma, M., Dong, C., & Liu, J. (2022). A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity. International Conference on Machine Learning, 162, 15895–15907. https://proceedings.mlr.press/v162/momma22a.html

Chicago

Momma, M., C. Dong, and J. Liu. 2022. “A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity”. International Conference on Machine Learning 162: 15895–907. https://proceedings.mlr.press/v162/momma22a.html.

Harvard

Momma, M., Dong, C. and Liu, J. (2022) “A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity”, International Conference on Machine Learning. PMLR, pp. 15895–15907. Available at: https://proceedings.mlr.press/v162/momma22a.html.

Vancouver

1. Momma M, Dong C, Liu J (2022) A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity. In: International Conference on Machine Learning. PMLR, pp 15895–15907

BibTeX

@InProceedings{pmlr-v162-momma22a,
  title = 	 {A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity},
  author =       {Momma, Michinari and Dong, Chaosheng and Liu, Jia},
  booktitle = 	 {Proceedings of the 39th International Conference on Machine Learning},
  pages = 	 {15895--15907},
  year = 	 {2022},
  editor = 	 {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
  volume = 	 {162},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {17--23 Jul},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v162/momma22a/momma22a.pdf},
  url = 	 {https://proceedings.mlr.press/v162/momma22a.html},
  abstract = 	 {Multi-objective optimization (MOO) and multi-task learning (MTL) have gained much popularity with prevalent use cases such as production model development of regression / classification / ranking models with MOO, and training deep learning models with MTL. Despite the long history of research in MOO, its application to machine learning requires development of solution strategy, and algorithms have recently been developed to solve specific problems such as discovery of any Pareto optimal (PO) solution, and that with a particular form of preference. In this paper, we develop a novel and generic framework to discover a PO solution with multiple forms of preferences. It allows us to formulate a generic MOO / MTL problem to express a preference, which is solved to achieve both alignment with the preference and PO, at the same time. Specifically, we apply the framework to solve the weighted Chebyshev problem and an extension of that. The former is known as a method to discover the Pareto front, the latter helps to find a model that outperforms an existing model with only one run. Experimental results demonstrate not only the method achieves competitive performance with existing methods, but also it allows us to achieve the performance from different forms of preferences.}
}

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A Multi-objective / Multi-task Learning Framework Induced by Pareto Stationarity

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — WC-MGDA combines preference alignment with Pareto stationarity

Knowl 2 — XWC-MGDA pivots the preferred Pareto search around a reference model

Knowl 3 — A primal-objective criterion automatically adjusts the stationarity trade-off

Knowl 4 — XWC-MGDA alternates trade-off adjustment and gradient updates

Knowl 5 — Synthetic experiments show preference alignment and smoother automatic tuning

Knowl 6 — Image-classification experiments test origin-based and reference-based preferences

Knowl 7 — Regression and music-label experiments reveal dataset-dependent method behavior

Knowl 8 — Reported hypervolumes favor XWC-MGDA in five of eight comparisons

Knowl 9 — Per-iteration computational cost is dominated by gradient Gram-matrix formation

Knowl 10 — The parameter update omits derivatives of the stationarity matrix

References

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