Self-Supervised Distance-to-Goal Predictor

• Self-supervised distance-to-goal predictors are functions that estimate the number of steps an agent needs to reach a goal using only trajectory-derived temporal cues.
• They employ techniques like metric regression, temporal classification, and Bellman regression to infer distances without relying on explicit reward signals.
• This approach enhances goal-conditioned planning and curriculum generation across diverse domains, from low-dimensional MDPs to high-dimensional visual and language-guided tasks.

A self-supervised distance-to-goal predictor is a learned function estimating the expected steps or effort required, under an agent’s dynamics, to reach a specified goal state given the current state, without access to privileged supervision. This approach is integral to goal-conditioned reinforcement learning (RL) and model-based planning, where a robust, dynamics-aware measure of progress toward a goal is necessary but often unavailable in closed-form or as a direct supervision signal. Self-supervised predictors leverage trajectories generated by the agent (on- or off-policy), mining pairs of states and using their temporal separation or reachability as labels, thereby sidestepping the need for ground-truth reward functions or access to geometric distances. These predictors enable efficient and scalable learning in high-dimensional, visual, or partially defined environments.

1. Mathematical Foundations

Self-supervised distance-to-goal estimation in goal-conditioned RL is most formally stated via the Markov Decision Process (MDP) tuple $(\mathcal{S}, \mathcal{A}, P, G, r, \gamma)$, where $\mathcal{S}$ is the state space, $\mathcal{A}$ the action space, $P$ the transition dynamics, $G$ the goal space, $r$ a sparse indicator reward, and $\gamma$ a discount factor. The central object is a distance function $d(s, g)$ that approximates the expected number of steps under an optimal (or fixed) policy needed to reach goal $g$ from state $s$: $$d(s, g) \approx \min_\pi \mathbb{E}_\pi[\text{\# steps from } s \text{ to } g]$$ In practice, empirical first-passage times sampled from trajectories are used: for any $(s_i, s_j)$, their temporal separation $|t - t'|$ when appearing within the same episode is a sample of the action distance. For a symmetric, dynamics-aware distance, the commute time is often adopted: $$d^\pi(s_i, s_j) = \frac{\rho(s_i)m^\pi(s_j|s_i) + \rho(s_j)m^\pi(s_i|s_j)}{\rho(s_i) + \rho(s_j)}$$ where $m^\pi(s_j|s_i)$ is the expected first-passage time and $\rho$ is the stationary distribution under $\pi$ (Venkattaramanujam et al., 2019).

Alternative frameworks define the distance as the value corresponding to the optimal policy’s goal-conditioned Q-function,

$Q^\star(o, a, g) = \gamma^{d^\star(o, a, g)}$

where $d^\star$ is the minimal step count; conversely,

$$d^\star(o,a,g) = \frac{\log Q^\star(o,a,g)}{\log \gamma}$$

This formulation is central for visual domains using Q-learning with “hindsight” relabeling (Tian et al., 2020).

Classification-based alternatives discretize the temporal distance into logarithmic bins, converting the problem into a self-supervised classification task (Prakash et al., 2021). This approach improves early learning stability and robustness to scale variance.

Recent architectures designed for navigation from visual and language inputs extend these distance predictors to take image or text-based goals, predicting both distance and model confidence (Milikic et al., 8 Dec 2025).

2. Self-Supervised Learning Objectives and Training Protocols

Training objectives uniformly exploit the temporal information within unlabeled trajectories, in both on-policy and off-policy settings:

• Metric regression: Embeddings $e_\theta(\cdot)$ are learned so that the $p$-norm (often $\ell_1$ or $\ell_2$) between the embeddings of $s$ and $g$ regresses to empirical or estimated distances,

$$\min_\theta \sum_{i<j} w_{ij}\left(\|e_\theta(s_i) - e_\theta(s_j)\|_p^q - d^\pi(s_i, s_j)\right)^2$$

where $w_{ij}$ are visitation-based weights (Venkattaramanujam et al., 2019).

• Temporal classification: Predict discrete bins of temporal separation, using cross-entropy loss,

$$\mathcal{L}_{\rm CE} = -\mathbb{E}_{(s_i, s_j)} \log p_\phi(b(k) | s_i, s_j)$$

with geometric binning for label balance (Prakash et al., 2021).

• Bellman regression: For model-free Q-distance, the Bellman loss is used,

$$\mathcal{L}(\phi) = \mathbb{E}_{(o_t,a_t,o_{t+1}),g}\left[ Q_\phi(o_t, a_t, g) - y_t \right]^2$$

with

$$y_t = 1_{\{o_{t+1}=g\}} + \gamma 1_{\{o_{t+1}\neq g\}} \max_{a'} Q_{\bar\phi}(o_{t+1}, a', g)$$

incorporating target networks and conservative Q-learning (CQL) penalties (Tian et al., 2020).

• Gaussian mixture NLL: For visual-language navigation, models output both a distance and a confidence, trained with a negative log-likelihood loss under a mixture of Gaussians (inlier/outlier variance) (Milikic et al., 8 Dec 2025).

All methods emphasize exclusively self-supervised training, typically using only the temporal structure of the trajectory buffer, randomized tail explorations, or synthetic/hard negative sampling for robust representation of unreachable or out-of-distribution pairs.

3. Model Architectures and Input Modalities

Architectures are adapted to the state representation:

• Low-dimensional MDPs: Multilayer perceptrons (MLPs) with a single hidden layer of moderate width (e.g., 64 ReLU units; output dim 20) suffice for joint angle or coordinate spaces (Venkattaramanujam et al., 2019), and siamese MLP encoders for embedding both state and goal (Prakash et al., 2021).
• Visual inputs:
  • Stacked image pairs (current and goal) processed via a convolutional encoder (e.g., $4 \times [8,16,32,64]$ 2D-convs with $4\times4$ kernels and strides 2, LeakyReLU and BatchNorm, culminating in a fully connected “head”) (Tian et al., 2020).
  • For navigation, a transformer fusion model: DINOv2 image encoder (384d), CLIP-text encoder (projected to 384d), cross-attended by a multi-layer decoder with observation queries. Output heads regress distance (ReLU) and confidence (sigmoid) (Milikic et al., 8 Dec 2025).
• Integration of Action Inputs: In model-based settings, action sequences are concatenated with embeddings, often in deep Q or actor network heads (Tian et al., 2020).
• Temporal and Multi-View Fusion: For video-based navigation, self-attention across recent frames is used before fusion, yielding marginal improvements (Milikic et al., 8 Dec 2025).

4. Goal Generation and Curriculum Learning

Self-supervised distance-to-goal functions provide a mechanism for adaptive goal generation:

• Online Buffering: Off-policy “exploration tails” (K random actions post-episode) accumulate a buffer of achieved states, regularly refreshed with the most recent trajectory ends, ensuring that goals are feasible and appropriately challenging relative to the agent’s policy (Venkattaramanujam et al., 2019).
• Curriculum Construction: Predicted distances (or binned class labels) support selection of goals at intermediate difficulty—typically targets in the 80–90th percentile of predicted distances from current agent state, resulting in a dynamic curriculum that adapts as exploration covers new regions (Prakash et al., 2021).
• Qualitative Dynamics: Early in training, sampled goals populate “easy” regions; as agent competence increases, more distant or complex goals are introduced automatically.
• Visual and Language Goals: For visual navigation, goals span images, text, or both, supporting open-ended and multimodal curriculum construction from large-scale internet-mined video or corpus data (Milikic et al., 8 Dec 2025).

5. Reinforcement Learning Integration and Planning

Distance-to-goal predictors are foundational for both model-free RL and model-based planning:

• Reward Function Augmentation: Learned distances replace or augment sparse indicator rewards, e.g., $r(s, g) = 1\{ \hat{d}(s, g) < \epsilon \}$, progressively broadening the region considered “reachable” as training proceeds (Venkattaramanujam et al., 2019).
• Automated Curriculum in RL: Integrating dynamical distance-based goal selection into off-policy HER + DDPG or similar frameworks increases sample efficiency by 1.5–2× across several robotic manipulation and navigation domains (Prakash et al., 2021).
• Model-Based Visual Planning: Functional distances act as the cost-to-go estimate in action-conditioned video prediction frameworks, enabling model-predictive control (CEM optimization in latent space) with learned visual dynamics (Tian et al., 2020).
• Noisy Distance Injection: For sim-to-real navigation transfer, a noise model calibrated to the learned distance predictor’s error is used to ensure RL policies remain robust when swapped from ground-truth geometric distance rewards to self-supervised visual or language distances (Milikic et al., 8 Dec 2025).

6. Applications, Empirical Validation, and Comparisons

Empirical studies validate the effectiveness of self-supervised distance-to-goal predictors across diverse domains:

Task Domain	Architecture/Method	Key Performance Indicator	Reference
2D/robotic continuous RL	Spectral embedding	Coverage, success rates, curriculum adaptivity	(Venkattaramanujam et al., 2019)
Fetch/Antenna navigation	Bin-classifier DDF	~1.5–2× sample efficiency gain over random-goal HER baseline	(Prakash et al., 2021)
Visual manipulation/planning	Q-function/convnet	40–55% success, outperforms pixel-MSE, direct regression	(Tian et al., 2020)
Visual-language navigation	Transformer fusion	Ordinal consistency (Kendall’s τ: .82 at horizon-20), SR/SPL gains vs. baselines	(Milikic et al., 8 Dec 2025)

Ablations confirm that:

• Off-policy training with random exploration tails stabilizes the distance estimate and avoids nonstationarity (Venkattaramanujam et al., 2019).
• Log-scale binning and classification objectives yield more stable learning (Prakash et al., 2021).
• Mining both positive and “hard” negative goals in Q-distance training is crucial for realistic distance prediction and generalization (Tian et al., 2020).
• Visual-language predictors, decoupled from RL, achieve superior ordinal consistency over contrastive or metric-learning baselines, especially for ambiguous or in-the-wild video navigation (Milikic et al., 8 Dec 2025).

7. Challenges, Limitations, and Future Directions

Despite their generality and effectiveness, self-supervised distance-to-goal predictors are subject to several bottlenecks:

• Scalar bottleneck: Collapsing high-dimensional sensory and semantic information to a single distance (plus confidence) may discard key topological or directional cues, particularly in ambiguous or multi-path scenarios (Milikic et al., 8 Dec 2025).
• Memoryless agents: Policies that lack episodic memory or topological maps may exhibit looping behavior or premature stopping in spatially ambiguous environments (Milikic et al., 8 Dec 2025).
• Hard negatives and “unreachable” goals: Ensuring robust detection of unreachable or deceptive goals requires mining negative samples, careful confidence calibration, and, in some settings, explicit curriculum adjustment (Tian et al., 2020).
• Generalization: Transferring to highly novel scenes or goals without exploration or explicit semantic alignment remains challenging.

Proposed directions include incorporating richer output modalities (e.g., distributions or direction vectors), hierarchical/hybrid predictors (object-level, room-level), temporal context fusion, end-to-end encoder fine-tuning post RL, and explicit topological mapping on top of learned distances (Milikic et al., 8 Dec 2025).

Self-supervised distance-to-goal predictors have established themselves as a general, domain-agnostic technique for providing actionable progress signals in complex RL and planning problems, scaling from low-dimensional MDPs to visual-linguistic navigation and hybrid model-based control.