Dual Goal Representations in AI

• Dual goal representations are defined as structured objects encoding relationships (temporal, hierarchical, spatial) rather than atomic states, offering sufficient abstraction for optimal policy recovery.
• They leverage parameterized encoders and offline value learning to approximate temporal distances and achieve noise invariance, thereby enhancing sample efficiency and planning performance.
• Empirical studies demonstrate that dual goal representations boost performance in robotics, dialogue systems, and navigation by enabling hierarchical planning and robust multi-goal coordination.

Dual goal representations define a class of frameworks and computational methods that characterize goals not as atomic entities or single target states, but as structured objects—typically as sets, vectors, graphs, or functions—encoding their relations with other states, goals, or tasks. In contemporary artificial intelligence, particularly in reinforcement learning (RL), dialogue systems, robotics, and cognitive modeling, dual goal representations exploit this relational structure to improve planning, robustness, generalization, and efficiency. They encompass representations where a goal is defined by its relation (often via temporal distance, hierarchy, or functional transformation) to all states or subgoals, or where multiple goals (e.g., current and final, or spatial and temporal) are modeled in parallel. Under this paradigm, the representation of goals acquires several desirable theoretical and empirical properties: sufficiency and invariance for optimal policy recovery, improved sample efficiency, resilience to noise, and flexible coordination of multiple objectives.

1. Foundational Principles of Dual Goal Representations

Dual goal representations formalize the encoding of goals through relationships—either as transformations, distances, or hierarchical decompositions—rather than relying on raw goal observations. A key instantiation is the temporal-distance–based dual (Park et al., 8 Oct 2025), where any goal $g$ is represented by the functional

$\vee(g)(s) = d^{\star}(s, g),$

meaning the set of (optimal) temporal distances from every state $s$ to $g$. This relational encoding filters out exogenous noise (i.e., goal-irrelevant aspects of observation space), is defined purely by the system dynamics, and provides a provably sufficient representation for extracting optimal policies.

Two theoretical properties characterize the utility of this approach:

• Sufficiency: The dual representation $\vee(g)$ can recover the optimal goal-reaching policy via deterministic policy $\pi^\vee(s,\vee(g))$ so that $V^{\pi^\vee}(s,g)=V^{\star}(s,g)$.
• Noise Invariance: In environments with observation noise, if two goal observations correspond to the same latent state, their dual representations coincide ($\vee(g_1)=\vee(g_2)$).

Further, duality manifests in several forms:

• Hierarchical structures: Recursive decompositions where goals are partitioned into subgoals (cf. sub-goal trees (Jurgenson et al., 2019)).
• Multiplexed goal conditioning: Explicit parallel encoding for both current and future (or multi-perspective) goals (Serris et al., 27 Mar 2025, Lai et al., 2019).
• Contrastive and spatial/temporal combinations: Simultaneous learning of metrics satisfying temporal and structural constraints (Myers et al., 24 Sep 2025).

2. Methodologies for Learning Dual Goal Representations

Practical dual goal representations are typically parameterized for scalability to continuous or high-dimensional spaces:

• Parameterization via Encoders: The core technique is to learn encoders $\psi(s)$ for states and $\varphi(g)$ for goals, so that the inner product or other aggregation function approximates temporal distances:

$$d^\star(s, g) \approx f(\psi(s), \varphi(g)) = \psi(s)^\top \varphi(g).$$

• Offline Value Learning: The approach is often integrated with offline value-based RL, where implicit Q-learning or related objectives are used to fit $\psi$ and $\varphi$ to the optimal distance (or value-to-go) landscape.
• Generalized Functional Approximations: For tasks where direct functional representation is intractable, finite-dimensional embeddings serve as surrogates for dual representations (Park et al., 8 Oct 2025).
• Dynamic Programming and Non-BeLLMan Decompositions: In sub-goal tree frameworks (Jurgenson et al., 2019), trajectory optimization is reformulated dynamically via recursive, parallelizable partitioning of state space, departing from the classical BeLLMan setup.

Dual representations are modular and compatible with a variety of RL algorithms. They are commonly trained as a plug-in for downstream methods, including goal-conditioned value learning, contrastive RL, or behavior cloning, with the goal encoder replacing or augmenting the raw goal input.

3. Hierarchical, Sequential, and Multi-level Goal Structures

Recent research has illustrated the effectiveness of dual representations in hierarchically structured or sequential decision problems:

• Hierarchical Duality: Frameworks such as sub-goal trees (Jurgenson et al., 2019) recursively decompose trajectories, refining high-level plans into concurrent, parallelizable subgoals. This enables $O(\log T)$ step computation for trajectory prediction, a significant reduction from sequential $O(T)$ approaches, and permits dynamic programming solutions that explicitly encode both current and final goal structure.
• Multi-goal Conditioning: In multi-goal scenarios (cf. (Serris et al., 27 Mar 2025)), policies are conditioned on both current and next (or final) goals, encoded as $(s, g_1, g_2)$, to reduce myopic failure modes and optimize for future progress beyond immediate subgoal attainment. Empirical results indicate substantial improvements in both stability and sample efficiency compared with single-goal–conditioned policies.
• Representation Hierarchies in Dialogue and Recommender Systems: Multi-level duality extends to non-sequential domains; for instance, in conversational recommendation (Chen et al., 2023), cross-hierarchical structures are learned both in representation (goal type $\rightarrow$ entity) and optimization (bi-level loss weighting), leveraging cross-attention modules and dynamic soft labeling to enhance prediction robustness.

4. Empirical Benefits and Practical Applications

Dual goal representations deliver several concrete advantages demonstrated across state- and pixel-based benchmarks (e.g., OGBench, Fetch robotics, PandaGym, dialogue corpora, and complex navigation tasks):

• Offline RL and Generalization: Experiments on OGBench (Park et al., 8 Oct 2025) show that dual goal representations improve mean performance across 20 tasks (navigation, manipulation, and visual domains), outperforming metric-based, contrastive, variational, and self-supervised baselines. Gains are particularly marked in manipulation settings and under pixel-based observation noise.
• Trajectory Planning and Multi-modal Prediction: In motion planning, sub-goal tree approaches lead to substantial increases in success rates for hard problems (e.g., 1–1.5% to 24–26% in imitation learning tasks) and drastic reduction in prediction time due to concurrent computation (Jurgenson et al., 2019).
• Hierarchical and Multi-Goal RL: Empirical studies in navigation and pole-balancing (Serris et al., 27 Mar 2025) show that dual conditioning on sequential goals yields near-100% success rates and more informative value propagation, as opposed to the flat profiles seen with standard myopic critics.
• Dialogue, Recommendation, and LLMs: Dual-space hierarchies (Chen et al., 2023) and plan-grounded dual objectives (Glória-Silva et al., 1 Feb 2024) enable context-sensitive recommendation, improved accuracy in multi-turn conversation, and robustness to dialogue context shifts—marked by significant F1 and precision gains over flat baselines or single-goal decoders.

5. Theoretical Insights and Invariance Properties

The theoretical appeal of dual goal representations is grounded in two principal results:

1. Policy Extraction Sufficiency: Given the dual representation $\vee(g)$ (e.g., as the vector of temporal distances to all states), it is possible to construct deterministic policies that, when conditioned on this representation, realize the optimal value function for any $(s, g)$ pair. Formally:

$$\tilde{\pi}(s,g) = \pi^\vee(s, \vee(g)), \quad V^{\tilde{\pi}}(s,g) = V^*(s,g).$$

The policy can be written explicitly as

$$\pi^\vee(s, f) = \arg\max_{a\in A} \mathbb{E}_{s'\sim p(\cdot|s,a)}[\gamma^{f(s')}],$$

with $f = \vee(g)$ (Park et al., 8 Oct 2025).

1. Exogenous Noise Invariance: Dual representations, depending solely on temporal or dynamical relationships, are provably robust to observation-level noise. If two observations $g_1$ and $g_2$ correspond to the same underlying latent state, then $\vee(g_1) = \vee(g_2)$ (Park et al., 8 Oct 2025).

Further, by integrating contrastive learning objectives with temporal and structural quasimetric constraints (triangle inequality, action invariance) (Myers et al., 24 Sep 2025), dual representations ensure that even with off-policy or stochastic data, the learned representation provides optimal goal-reaching information and supports trajectory stitching—outperforming both pure contrastive and traditional value learning approaches.

6. Extensions and Connections to Broader Learning Paradigms

Dual goal representations extend beyond classic RL or planning frameworks:

• Dialogue and Multi-Objective LLMs: In multi-turn conversational systems and plan-grounded instruction following, dual representations allow models to jointly optimize for both procedural plan navigation and reactive (user instruction–driven) objectives, with multi-objective losses and explicit separation of plan and open-domain goals (Glória-Silva et al., 1 Feb 2024).
• Telic States and Cognitive Frameworks: The notion of telic states (Amir et al., 20 Aug 2025, Amir et al., 20 Jun 2024) offers an overview of descriptive (state modeling) and prescriptive (value) aspects, positing that state representations and goal preference relations co-emerge from the agent’s experience. Algorithms in this setting adaptively cluster experience into telic equivalence classes by optimizing the statistical divergence (KL) between policy-induced distributions and goal-directed telic states, balancing complexity and flexibility through deliberate abstraction.
• Graph-Based and Structural Extensions: Graph-structured dual representations (e.g., via equivalence mappings or reachability graphs (Netanyahu et al., 2022, Zadem et al., 2023)) extend the duality paradigm by encoding spatial, relational, or reachability equivalences. These approaches support robust reward modeling, data augmentation, and efficient abstraction in planning and manipulation domains.
• Successor Features and Analogy Metrics: Dual representations unify with successor feature methods and bisimulation metrics. Embedding state–goal pairs via bisimulation (Hansen-Estruch et al., 2022) or learning analogy–preserving transformations between state representations further enables skill reuse, analogy-based reasoning, and compositionality.

7. Limitations, Trade-Offs, and Future Directions

While dual goal representations provide robust theoretical and empirical advantages, certain challenges and trade-offs are documented:

• Computational Overhead: Encoding temporal relationships to all states, or learning full quasimetric spaces, can be computationally demanding in large or continuous environments.
• Representation Granularity: There is a trade-off between precision (fine partitions or embeddings) and the complexity/capacity needed for policy updates or refinement (e.g., in telic-controllable representations (Amir et al., 20 Jun 2024)).
• Dependency on Dynamics: The meaningfulness of the dual representation depends critically on accurately modeled dynamics; approximation errors or function class misspecification can diminish optimality or invariance guarantees.
• Integration with Multi-Modal, Language, or Graph-Based Goals: Extending the formal duality principle to multi-modal (visual, linguistic, spatial) representations and integrating these with relational or symbolic structures remains an active area of research.

As evidenced by advances in reinforcement learning, robotics, dialogue systems, and cognitive modeling, dual goal representations provide a unified foundation for robust, interpretable, and efficient learning in goal-directed agents, with ongoing research focusing on scalability, abstraction, and integration with complex, structured domain knowledge.