Dual-Objective Representation Learning

• Dual-objective representation learning is a framework that jointly optimizes two nontrivially interacting objectives, balancing discriminative, generative, and regularizing losses.
• It spans various settings—from supervised and self-supervised to multi-task learning—using techniques like scalarization, gradient balancing, and decoupled architectures to mitigate interference.
• Empirical studies show improved accuracy, robustness, and out-of-distribution detection, highlighting the importance of adaptive weighting and modular design in practical applications.

Dual-objective representation learning is a paradigm in which two distinct, nontrivially interacting objectives are jointly optimized to produce representations that integrate the benefits of both criteria. Typical instances balance objectives such as discriminative fitness (e.g., classification loss), structural or statistical regularization (e.g., contrastive, information-theoretic, or generative reconstruction loss), or domain/task specificity (e.g., expert-feature alignment vs. task loss). This framework subsumes a range of influential models across supervised, self-supervised, multi-task, multi-view, and privacy-sensitive learning, and is supported by a mature mathematical, algorithmic, and empirical foundation.

1. Formal Definition and Optimization Landscape

Let $\theta$ denote model parameters and $f_1(\theta), f_2(\theta)$ two objectives, the canonical dual-objective formulation is: $\min_{\theta} \; [f_1(\theta), f_2(\theta)].$ When $f_1$ and $f_2$ are conflicting, there is generally no global minimizer; instead, the solution is characterized by the Pareto set: $$P = \{\theta^* \mid \nexists \theta : \forall i, f_i(\theta) \leq f_i(\theta^*) \text{ and } \exists j, f_j(\theta) < f_j(\theta^*)\}.$$ This Pareto set defines the frontier of achievable trade-offs between the two objectives. Common scalarization methods reduce this to a weighted sum: $$\mathcal{L}(\theta) = \lambda f_1(\theta) + (1-\lambda) f_2(\theta), \quad \lambda \in [0,1],$$ but recent advances incorporate adaptive weighting, multi-gradient methods, or explicit Pareto-stationarity conditions to address issues such as gradient interference and non-convex trade-offs (Peitz et al., 2 Dec 2024, Nguyen et al., 12 Feb 2024).

2. Representative Algorithmic Classes

Dual-objective representation learning encompasses several methodological archetypes:

• Scalarization: Fixed or learned $\lambda$ parameters combine objectives additively, as in $\beta$-VAE for disentanglement (reconstruction vs. KL divergence).
• Multi-objective gradient methods: MGDA and variants solve, at each update, for a convex combination of task gradients whose sum yields a common descent direction, satisfying the Karush-Kuhn-Tucker (KKT) first-order condition:

$$\exists \alpha \in [0,1]:\ \alpha \nabla f_1(\theta^*) + (1 - \alpha) \nabla f_2(\theta^*) = 0$$

as applied in contrastive topic modeling (Nguyen et al., 12 Feb 2024).

• Dynamic gradient recombination or balancing: PCGrad, CAGrad, and GradNorm project or rebalance gradients to circumvent interference—often critical when objectives operate over different aspects of the representation space (Peitz et al., 2 Dec 2024).
• Architectural dualism: Explicit branches or decoupled heads for each objective (e.g., DCCAE's pairwise autoencoders and correlation constraints (Wang et al., 2016), dual-encoder PiRO for object/category separation (Sarkar et al., 1 Mar 2024), or parallel classification/variational modules in PM+MO (Armitage et al., 2020)).

3. Exemplary Models and Domains

3.1 Classification and Contrastive Integration

The ESupCon loss (Aljundi et al., 2022) unifies cross-entropy (likelihood-based discrimination) and supervised contrastive loss (robust pairwise structure regularization) in a single objective: $$\ell_{\rm ESupCon} = \frac{1}{2N + K} \Biggl[ \sum_{k=1}^K \frac{1}{|C_k|} \sum_{i:y_i=k} \ell_{\rm pt}(z_i, w_k) + \sum_{i=1}^{2N} \ell_{\rm SupCon}(z_i, P_i) \Biggr]$$ where $\ell_{\rm pt}$ interpolates between prototypical discrimination and pairwise contrastive tightness. Empirically, ESupCon yields both superior accuracy and robustness to class imbalance and label corruption compared to single-objective or two-stage baselines.

3.2 Multi-view and Correlation-based Learning

DCCAE (Wang et al., 2016) jointly minimizes per-view reconstruction error and negative canonical correlation: $$\min_{\theta_x, \theta_y, \phi_x, \phi_y} -\mathrm{corr}(f_{\theta_x}, g_{\theta_y}) + \alpha (L_{\mathrm{recon}}^X + L_{\mathrm{recon}}^Y)$$ This model achieves top-performing clustering and predictive accuracy on noisy vision, speech, and language tasks; the correlation constraint eliminates nuisance factors that simple autoencoders retain.

3.3 Generative–Contrastive Hybrids

Hybrid generative-contrastive representation learning (Kim et al., 2021) leverages autoregressive likelihood for generative robustness and InfoNCE for semantic clustering: $$L_{\rm total} = \alpha L_{\rm gen} + \beta L_{\rm con}$$ with a staged schedule for stability. This yields state-of-the-art accuracy and OOD detection, especially effective in low-label regimes.

3.4 Domain and Task Decoupling

The "Universal Representations" framework (Li et al., 2022) frames multi-task/domain learning as dual objectives—task loss alignment and distillation from frozen expert networks via adapters—demonstrating that balancing feature-space alignment with task-specific loss yields a universal backbone capable of matching or exceeding many-task SOTA with no dynamic loss reweighting.

3.5 Out-of-Distribution and Disentanglement

Dual Representation Learning for OOD detection (Zhao et al., 2022) decomposes the representation into strongly label-discriminative and complementary, weakly label-related features. The two representations are trained with cross-entropy losses but with explicit mutual diversity construction, and OOD detection is by confidence disagreement. This approach yields robust calibration and strong AUROC improvements over prior methods.

3.6 Self-supervised Speech and 3D Learning

Speech SimCLR (Jiang et al., 2020) and AsymDSD (Leijenaar et al., 26 Jun 2025) use distinct but simultaneously optimized objectives—contrastive alignment and reconstruction or masked latent prediction. In both cases, empirical results confirm that the dual objective leverages the complementary strengths: discriminativity and signal fidelity in speech, semantic abstraction and spatial context in 3D point clouds.

4. Empirical Benefits and Trade-offs

A broad survey (Peitz et al., 2 Dec 2024) and individual empirical studies report that dual-objective frameworks regularly outperform single-objective counterparts along several axes:

Setting	Dual-Objective Advantage	Source
Supervised image classification	+0.5–2.5% accuracy, improved calibration, stability	(Aljundi et al., 2022)
OOD detection	AUROC +2–4% via dual-representation scoring	(Zhao et al., 2022)
Few-shot and multi-domain classification	SOTA on 11/13 domains, high Recall@k, no extra params	(Li et al., 2022)
Pose-invariant object recognition	+20% accuracy, +33.7% retrieval mAP (over single-objective)	(Sarkar et al., 1 Mar 2024)
Topic modeling	NPMI and topic diversity gain via Pareto-stationary updates	(Nguyen et al., 12 Feb 2024)
Privacy-preserving ML	Maintained utility with privacy via encoder-only exposure	(Ouaari et al., 2023)
Speech and speaker recognition	Phonetic WER drop (–4.2pp), EER reduction (–1.5%)	(Xie et al., 2022)
3D self-supervised learning	+5.3% accuracy (ScanObjectNN), robust to mask/crop variation	(Leijenaar et al., 26 Jun 2025)

Ablation studies establish that both objectives are necessary: removal of either leads to reduced accuracy, stability, or calibration; combining them in a naive shared embedding ("single-space") often results in gradient conflict and marginal gains, emphasizing the importance of decoupled architectures or Pareto-optimizing updates.

5. Practical Design Considerations and Optimization Techniques

Effective dual-objective representation learning depends on architectural and optimization design:

• Objective weighting requires careful selection or adaptation; fixed $\lambda$ works in some cases, but Pareto-stationary solvers (Nguyen et al., 12 Feb 2024) or gradient-balancing methods (Peitz et al., 2 Dec 2024) address evolving scales and trade-offs.
• Gradient interference between objectives can be mitigated with methods such as PCGrad, CAGrad, or architectural separation (dual-branch or multi-head designs).
• Efficient scaling is achieved by modular networks, staged schedules (e.g., warm-up phases (Kim et al., 2021)), and memory-efficient adapters (Li et al., 2022).
• Regularization and stabilization are routine in dual-objective setups. Probabilistic or variational inference in fusion layers (Armitage et al., 2020) controls variance in multimodal systems; attention to centering and sharpening prevents collapse in momentum student-teacher frameworks (Leijenaar et al., 26 Jun 2025).

6. Theoretical Insights and Limitations

Dual-objective models frequently instantiate core principles from information bottleneck theory, Pareto optimality, and domain adaptation. By optimizing for complementary mutual information criteria—retaining essential content while enforcing invariance or diversity—they often reach solutions unattainable by single objectives.

Nevertheless, limitations remain: naive scalarization may miss non-convex Pareto regions (Peitz et al., 2 Dec 2024); dynamic weight selection demands robust metaparameter schedules; architectural decoupling can add moderate parameter or compute overhead. Formal privacy guarantees in privacy-aware dual-objective encodings may be lacking absent adversarial or cryptographic objectives (Ouaari et al., 2023).

Ongoing research explores extending these paradigms to higher-resolution, multi-modality, federated, and continual learning settings, and improved theoretical analysis of convergence and generalization under joint optimization.