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Do Better Goal Representations Improve Goal-Conditioned Reinforcement Learning?

Syed Nazmus Sakib, Abdul Monaf Chowdhury, Nafiul Haque, Shifat E Arman, Md Mehedi Hasan

arXiv:2609.39901v1cs.LGcs.AI

TL;DR

Prior evaluations do not isolate how goal-representation quality affects downstream control. The paper tests exact and varied temporal-distance representations in offline navigation while holding the learner fixed, and compares these with interventions on the agent’s state. Goal-side changes produce little performance change, whereas state-side encoding improves success substantially.

  • Problem

    Prior evaluations vary representation-learning procedures and embeddings together, leaving unclear how goal-representation quality itself affects downstream control.

    Source: §1 · p. 1

  • Method

    The paper constructs exact temporal-distance goal representations in deterministic mazes, varies representation quality with the learner fixed, and tests state-side positional features.

    Source: §1 · p. 2, §3.2 · p. 4

  • Results

    Goal-representation changes do not substantially alter success across benchmarks, while state interventions more than double success and Fourier features raise GCIVL on antmaze-large from 32 to 84.

    Source: §1 · p. 2, §6 · p. 9

  • Takeaways & Limitations

    In the evaluated state-based offline navigation tasks, the state pathway is a more important performance bottleneck than goal representation.

    Source: §6 · p. 9

    The evaluation is restricted to state-based navigation; position features do not transfer to manipulation or pointmaze, and remain untested in pixel-based domains.

    Source: §7 · p. 9

Abstract

from arXiv · show

Goal-conditioned reinforcement learning (GCRL) relies heavily on how target goals are represented to the policy. While recent methods encode goals via temporal distance, occupancy, or controllability, it remains unclear how much downstream performance actually depends on representation quality. We study this in offline GCRL by constructing an exact temporal-distance goal representation in deterministic mazes. We then systematically corrupt its geometric quality while keeping the downstream learner fixed. Across OGBench navigation tasks and two algorithms, large changes in goal-representation quality produce almost no change in performance. However, applying the same interventions to the agent's current state more than doubles success, revealing the state pathway as the true bottleneck. Building on this insight, we show that simple random Fourier positional encodings substantially improve performance on the hardest navigation tasks without map information or objective modifications. Overall, our findings suggest that in state-based offline navigation, improving how the agent's current state is represented matters far more than refining the goal representation. Code will be released soon.

1 INTRODUCTION

The introduction asks whether better goal representations improve offline goal-conditioned control, a question prior evaluations could not isolate. It reports little gain from improving goal representations, while state-side interventions and Fourier position features substantially improve navigation performance.

1 INTRODUCTION

  • Prior evaluations changed both the representation-learning procedure and the resulting embedding, leaving the effect of goal-representation quality on downstream control unclear.The motivating question is how much better an agent would act if given a perfect goal representation.Source: §1 · p. 1
  • In deterministic mazes, exact shortest-path distances provide an ideal goal representation that can replace the learned approximation while the downstream algorithm and objective remain unchanged.This permits systematic variation of representation quality while holding the learner fixed.Source: §1 · p. 2
  • Learned goal representations perform comparably to exact distances, while state-side interventions more than double success and Fourier features raise GCIVL on antmaze-large from 32 to 84.The Fourier features roughly double success on humanoidmaze-medium as well.Source: §1 · pp. 2–3
  • The proposed random Fourier features of the agent’s coordinates require no maze layout or learning-objective modification and improve performance on the hardest navigation tasks studied.They are appended to the observation; the paper identifies the state pathway, rather than goal representation, as the more consequential input pathway.Source: §1 · pp. 2–3

2 PRELIMINARIES

The preliminaries define offline goal-conditioned learning and temporal-distance goal representations, then distinguish representation, interface, and state-pathway design choices. They describe the two downstream algorithms used to test these choices.

2 PRELIMINARIES

  • Offline goal-conditioned learning trains a policy to reach specified goals using a fixed trajectory dataset, with goals supplied by hindsight relabelling.The benchmark reward is 0 on reaching the goal and −1 otherwise, so maximizing return is equivalent to minimizing steps.Source: §2 · p. 3
  • Dual Goal Representations encode a goal by its temporal distances from states; in standard DGR, a learned bilinear value parameterization approximates those distances offline.The paper calls this estimate the learned representation and contrasts it with exact ground-truth distances, termed the ideal representation.Source: §2 · p. 3
  • The state–goal interface is commonly concatenation followed by an MLP, but the paper treats this inherited choice as a separate design axis from the representation and state pathway.Prior comparisons described here varied other aspects of representation learning or pixel fusion, not how a state-based goal representation is combined with state in the downstream policy.Source: §2 · p. 3
  • The study uses GCIVL, which learns a goal-conditioned value function by expectile regression, and CRL, which learns a bilinear contrastive critic.Their differing value-estimation and policy-extraction approaches allow effects specific to one algorithm family to be distinguished from those shared across both.Source: §2 · p. 3

3 REPRESENTATION INTERVENTIONS

The framework isolates the policy’s input interface by pairing exact goal-reachability representations with matched interventions on the agent’s current state.

3.1 GOAL-SIDE INTERVENTION

  • Exact goal codes use shortest-path distances from maze landmarks to reachable cells, with the downstream learner and other training components unchanged.The representation has the same landmark dimensionality as DGR: K = 256 for AntMaze and HumanoidMaze, and K = 64 for PointMaze.Source: §3.1 · p. 4
  • Min-plus decoding measures temporal-distance structure by Spearman correlation with true shortest-path distances, overall and within distance quartiles.The measure assesses retained geometry without fitting an auxiliary probe.Source: §3.1 · p. 4
  • A corruption ladder degrades exact distances with Gaussian noise across ranges or far-field noise that preserves nearby relationships; each perturbation is fixed during training.Varying σ changes the amount and type of distance information while leaving the downstream learner unchanged.Source: §3.1 · p. 4

3.2 STATE-SIDE INTERVENTION

  • Matched state-side interventions assign each maze cell a fixed code and append it to the observation, testing the same representational structure on the current state.Exact, corrupted, and independently sampled random codes distinguish geometric structure from making position more distinguishable; random Fourier features of coordinates provide a map-free alternative.Source: §3.2 · pp. 4–5

4 GOAL REPRESENTATIONS ARE SATURATED

Across the tested navigation settings, making goal representations exact or sharply degrading their distance information does not yield a consistent downstream advantage. The section contrasts this insensitivity with results for a code applied to the state pathway.

4.1 DOES AN EXACT GOAL REPRESENTATION IMPROVE DOWNSTREAM PERFORMANCE

  • Six of nine evaluations show no statistically distinguishable difference between exact and learned goals; the remaining advantages split across environments and algorithms.Exact goals gain 6.4 points on humanoidmaze-medium with GCIVL and 19.5 on pointmaze-medium with CRL, while learned goals gain 8.2 on pointmaze-medium with GCIVL. Across five GCIVL tasks, mean success is 48 for exact and 51 for learned goals; learning curves show no consistent advantage.Source: §4.1 · pp. 5–6, Figure 3 · p. 5

4.2 HOW SENSITIVE IS DOWNSTREAM PERFORMANCE TO GOAL REPRESENTATION QUALITY

  • Across 16 corruption settings, every performance estimate stays within 7.6 points of the uncorrupted representation and all confidence intervals include zero.Far-field corruption changes far-field correlation from 1.00 to −0.41 while near-field correlation remains 0.997; the authors do not claim exact flatness because several intervals extend beyond the five-point tolerance.Source: Figure 4 · p. 6, §4.2 · pp. 6–7

4.3 DOES THE GOAL REPRESENTATION NEED TO ENCODE DISTANCE AT ALL

  • An i.i.d. N(0, 1) goal table identifies the goal cell but contains no geometry, with rank correlation 0.004 to true distances.This tests whether the learner uses distance information at all, beyond the corruption ladder’s degraded representations.Source: §4.3 · p. 7

4.4 THE SAME CODE IS WORTH FORTY POINTS MORE ON THE STATE SIDE

  • 31.9 ± 2.4 for the random code versus 30.0 ± 8.0 for exact geodesics (+1.9 [−4.9, +8.8]) and 32.2 ± 8.9 for DGR’s learned representation (−0.2 [−7.9, +7.4]) on antmaze-large.The random, exact, and learned goal codes are indistinguishable in this comparison.Source: §4.4 · p. 7

5 THE STATE PATHWAY IS THE BOTTLENECK

Matched interventions show that changing the agent’s current-state input matters more than changing goal representations. Random Fourier position features improve success on the hardest navigation tasks without map information or objective changes.

5 THE STATE PATHWAY IS THE BOTTLENECK

  • CRL’s exact state code raised success from 80.8 ± 3.4 to 89.9 ± 1.5 (+9.2 [+6.1, +12.3]), while the goal-side intervention was worth −1.4 [−5.1, +2.2].The matched interventions differ only in which argument receives the code.Source: §5 · pp. 7–8

5.1 WHAT MUST THE STATE CODE CONTAIN?

  • A geometry-free state table exceeded the exact geodesic code by 6.8 points, while corrupting the table improved success by 19.8 points with Gaussian σ = 8 and 12.9 points with far-field σ = 8.The authors attribute the gains to a high-dimensional, distinguishable representation of the current position.Source: §5.1 · p. 8

5.2 POSITION FEATURES

  • On antmaze-large, position features raised GCIVL success from 32.2 to 83.9 with DGR’s learned representation; lower frequencies performed best across settings.Success roughly doubled on humanoidmaze-medium under both goal representations; the highest frequency performed worst.Source: Figure 5 · p. 7, §5.2 · p. 8
  • Position features raised a raw-goal baseline from 15.7 to 71.0, while DGR’s learned representation added +12.9 [+9.0, +16.9] beyond them.Standardising coordinates without adding a basis did not reproduce the gain and was severely harmful.Source: §5.2 · p. 8

5.3 RULING OUT CONFOUNDING MECHANISMS

  • Widening the actor to 1024 hidden units changed success by −0.4 [−8.8, +8.0], and compressing goal inputs produced no discernible change in either representation condition.Standardising coordinates dropped success from 32.2 to 7.1; cross-attention did not implement min-plus decoding.Source: §5.3 · p. 9
  • Cross-attention improved antmaze-large performance by +23.2 [+15.6, +30.9], but its gain was redundant when an explicit state code was present (+8.9 [−3.4, +21.2]).These benefits did not transfer to manipulation or low-dimensional pointmaze, where coordinates already dominate the state vector.Source: §5.3 · p. 9

6 CONCLUSION

Across offline goal-conditioned navigation benchmarks, changing goal representations had little effect on success, while interventions on the observation improved it substantially. Map-free Fourier position features raised GCIVL from 32 to 84 on antmaze-large and roughly doubled performance on humanoidmaze-medium.

6 CONCLUSION

  • Across benchmarks, exact, degraded, and random goal representations did not substantially change success, while observation interventions more than doubled antmaze-large success.A map-free Fourier positional encoding raised GCIVL from 32 to 84 and roughly doubled humanoidmaze-medium performance.Source: §6 · p. 9

7 LIMITATIONS

The paper leaves the precise mechanism behind the benefits of spatial distinguishability unresolved and limits its evaluation to state-based navigation. Related work addresses horizon difficulty through hierarchy, n-step returns, or replay-buffer search, while this paper changes how position enters the network.

7 LIMITATIONS

  • The precise mechanism by which spatial distinguishability aids policy learning remains unresolved, and position features remain untested in pixel-based domains.The evaluation is restricted to state-based navigation; benefits do not transfer to manipulation or low-dimensional pointmaze, and the work does not claim state-of-the-art results over hierarchical methods.Source: §7 · p. 9

A RELATED WORK

  • Prior work shortens the decision problem with hierarchy, n-step returns, or replay-buffer search; this paper instead changes how the agent’s position enters the network.Those methods remain stronger in absolute terms on the hardest mazes.Source: A RELATED WORK · p. 13
  • Fourier and random features are established coordinate encodings, but the paper’s frequency sweep does not support spectral bias as the usual justification.The paper also situates input combination methods such as attention and feature-wise modulation as a separate design axis.Source: A RELATED WORK · p. 13

B EXPERIMENTAL SETUP AND PROTOCOL

The experiments use OGBench state-based navigation tasks, two downstream algorithms, and controlled changes to goal and state inputs while holding the learner fixed. Representation quality is measured with a probe-free min-plus decoder, and exact goal representations generally tie the learned baseline across the navigation suite.

B EXPERIMENTAL SETUP AND PROTOCOL

  • The five OGBench navigation tasks range from 4- to 69-dimensional observations, with two position dimensions in every task.The tasks are pointmaze-medium/large, antmaze-medium/large, and humanoidmaze-medium; temporal distance is a graph geodesic in these settings.Source: B EXPERIMENTAL SETUP AND PROTOCOL · p. 13
  • Runs train for one million gradient steps and report mean success over the final three evaluations, never the best evaluation during training.Each evaluation uses five held-out goals and fifty episodes per goal; comparisons use Welch’s unequal-variance t-test and 95% confidence intervals.Source: B EXPERIMENTAL SETUP AND PROTOCOL · pp. 13–14

C.1 NETWORKS

C.2 DOWNSTREAM ALGORITHMS

  • GCIVL uses expectile value regression and advantage-weighted policy regression, whereas CRL trains a bilinear critic to distinguish reached goals from batch-sampled goals.The two algorithms differ in value estimation and policy extraction, allowing comparison across algorithm families.Source: §C.2 · pp. 14–15

C.3 THE INPUTS WE INTERVENE ON

  • The input interventions include learned and exact goal representations, corrupted goal representations, per-cell state codes, and random Fourier position features.Position features append fixed sine and cosine projections of the agent’s two coordinates and require neither a maze map nor an objective change.Source: §C.3 · p. 15

D MEASURING REPRESENTATION QUALITY

  • Representation quality is measured by correlating min-plus-decoded distances with true shortest-path distances, including within distance quartiles, without fitting a probe.With every cell used as a landmark, the decoder reproduced the true geodesic to 9.5 × 10−7; a learned probe was uninformative because goal identity itself determines position in a maze.Source: D MEASURING REPRESENTATION QUALITY · pp. 15–16

E.1 THE FULL NAVIGATION SUITE

  • Across five environments and both algorithms, exact goal representations tied the learned baseline in most cases; observed differences were small relative to seed variance and switched signs between algorithms.The full-suite figure reports mean ± 1 SD over 5 to 8 seeds; CRL was not run on humanoidmaze-medium.Source: §E.1 · p. 16, Figure 8 · p. 16

E.2 CORRUPTION BEYOND ANTMAZE-LARGE

Far-field corruption does not produce a monotone performance decline across the additional settings, despite sharply damaging long-range distance structure. Normalisation affects control more than exact-versus-learned representation, while changing landmark count leaves control unchanged.

E.2 CORRUPTION BEYOND ANTMAZE-LARGE

  • Across antmaze-medium and pointmaze-large under GCIVL, corruption produced no monotone decline; pointmaze-large under CRL was a high-variance exception with only two seeds per corrupted setting.The exception scored 43.0 ± 17.6 uncorrupted, versus 14.7 to 31.5 for corrupted settings; the authors do not interpret it as evidence of representation-quality dependence.Source: §E.2 · p. 16, Table 3 · p. 16

E.3 WHAT THE TWO CORRUPTIONS DO TO THE GEOMETRY

  • Gaussian corruption degrades distance structure across all ranges, whereas far-field corruption preserves the nearest quartile and reverses the furthest, yet both lead to similar downstream outcomes.At the strongest far-field corruption, near-field correlation remains 0.997 while far-field correlation falls from 1.00 to −0.41.Source: §E.3 · p. 17, Figure 9 · p. 17

E.4 NORMALISATION OF THE EXACT REPRESENTATION

  • Length normalisation benefits both algorithms more than the difference between exact and learned representations, showing that vector presentation matters alongside its encoded geometry.Table 4 compares the paper’s standard scheme with unit-length and raw-step-count alternatives.Source: §E.4 · p. 17, Table 4 · p. 17

E.5 NUMBER OF LANDMARKS

  • Halving or quadrupling the landmark count leaves control unchanged in every tested case, ruling out landmark count as the source of state-side or interface effects.The comparison uses the ideal representation with concatenation; Table 5 reports differences between the larger and smaller counts.Source: §E.5 · p. 17, Table 5 · p. 17

F.1 RESULTS THE MAIN TEXT REFERS TO

The results show that policy interfaces and state inputs can matter substantially, while capacity, coordinate rescaling, and goal-vector width do not explain the gains. Longer-wavelength positional features perform best in the tested sweep, and cross-attention’s performance varies across controlled comparisons.

F.1 RESULTS THE MAIN TEXT REFERS TO

  • Cross-attention adds +33.2 points with the ideal goal representation but only +3.9 with the learned one; its gain is +8.9 with an explicit state code versus +23.2 without one.State codes also improve CRL by +9.2 points on the task where goal-side intervention achieved nothing.Source: §F.1 · p. 18, Table 6 · p. 18

F.2 WHICH WAVELENGTHS WORK

  • Longer wavelengths perform better, peaking at roughly eight maze cells; they preserve local similarity while separating distant positions, unlike short wavelengths that act as random codes.The sweep stops at eight cells, so where the trend eventually peaks remains open.Source: §F.2 · p. 18, Figure 10 · p. 18

G INTERFACE DEFINITIONS

  • Cross-attention treats each goal-representation coordinate as a token, forms a query from the state, and appends the attended context before the policy trunk.Concatenation is the baseline; the comparison also includes FiLM and a bottleneck interface.Source: G INTERFACE DEFINITIONS · pp. 18–19

H FULL INTERFACE SWEEP

  • Across fifteen controlled comparisons, replacing concatenation with cross-attention shifts performance from −8.4 to +26.9 points, with six runs separated from zero and five favoring cross-attention.The comparison holds representations, algorithms, and hyperparameters fixed; the authors emphasize interface choice rather than cross-attention as a universal prescription.Source: H FULL INTERFACE SWEEP · p. 19

I.1 CAPACITY, SCALE AND INPUT WIDTH

  • Doubling actor width changes performance by −0.4 points, while coordinate standardisation drops success from 32.2 to 7.1 and goal-vector bottlenecks show no measurable change.The reported bottleneck differences are −2.8 [−13.5, +7.8] for the ideal representation and −0.1 [−6.5, +6.3] for raw goal observations.Source: §I.1 · p. 19

I.2 THE POLICY DOES NOT ROUTE THROUGH LANDMARKS

The attention analysis finds that the policy attends to goal regions rather than routing through landmarks to compute shortest paths. Cross-attention and explicit state codes provide largely overlapping benefits.

I.2 THE POLICY DOES NOT ROUTE THROUGH LANDMARKS

  • Attention assigns only 1.59× and 1.43× chance mass to the true min-plus landmark, below a goal-nearest baseline’s 2.52× and 1.85×; its peak path lengths also exceed the true argmin.The attention-peak path lengths are 9.0 and 11.8 cells, versus 5.4 cells at the true argmin.Source: §I.2 · pp. 19–20

I.3 CROSS-ATTENTION AND THE STATE CODE OVERLAP

  • Cross-attention is worth +23.2 points alone and +33.2 when extended to the critic, but only +8.9 with an explicit state code, consistent with overlapping routes to the same quantity.The +8.9 interval, [−3.4, +21.2], includes zero.Source: §I.3 · p. 20

J WHERE THE INTERVENTIONS DO NOT HELP

Cross-attention does not improve the tested pointmaze settings or transfer its interface benefit to the two manipulation tasks. The authors relate these boundaries to how much positional information the observations already contain and whether the task has maze-like geometry.

J WHERE THE INTERVENTIONS DO NOT HELP

  • On pointmaze-medium and pointmaze-large, cross-attention with the ideal representation yields near-zero changes, and increasing landmark count to 256 does not alter that pattern.The reported changes at 64 landmarks are −0.2 [−12.1, +11.8] and +1.0 [−9.7, +11.8]; at 256 they are +1.8 [−7.2, +10.8] and −2.2 [−9.9, +5.5].Source: J WHERE THE INTERVENTIONS DO NOT HELP · p. 20
  • On manipulation tasks, cross-attention is worse on cube-single-play (−3.6 [−6.5, −0.7]) and no better on scene-play (−6.5 [−13.2, +0.2]); neither task has maze geometry or a single agent-position coordinate pair.The comparisons use GCIVL with the learned representation.Source: J WHERE THE INTERVENTIONS DO NOT HELP · pp. 20–21, Figure 12 · p. 21

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