GCSL from Pixels: Geometric Regularization for Navigable Visual Goal Spaces

Goal-Conditioned Supervised Learning (GCSL) learns goal-reaching behavior through hindsight relabeling and self-imitation, without a reward function, value function, or expert demonstrations. We propose that its success in physical state spaces may depend on an assumption it never states, that the state/goal space carries the environment's factors in linearly decodable form. While low-dimensional states provide this for free, we find that a naively trained pixel encoder often fails to obtain linear decodability, and that this gap is fixable. Pretraining the image encoder with a reconstruction loss plus a geometric regularizer combining variance-covariance (VICReg) and an eigenvalue-based collapse penalty (EigenReg) restores linear decodability and reliable goal-reaching across four control environments. GCSL's relabeling objective stays unchanged, though reliable pixel performance also needs frequency-balanced hindsight sampling and horizon weighting in the actor loop. Two controlled comparisons isolate where the improvement comes from. Against end-to-end pixel GCSL, which fails outright with 0% success across all four environments despite working reliably from physical state, our method reaches 94 to 100% success. Against GCBC, a one-shot cloning variant without iterative self-improvement, we isolate the contribution of iteration alone. A further comparison against vanilla-VAE, Beta-VAE, VICReg-only, and EigenReg-only isolates the regularizer's own contribution. Finally, comparing linear against nonlinear decodability of the same representations tests whether linear recoverability is associated with goal-reaching in these environments, or recoverability of any kind is. These results give an empirical characterization of when GCSL's reward-free formulation extends from physical state spaces to pixels.