(115.08, 501.21) (181.58, 539.61) (178.01, 545.79) (111.51, 507.40)        /F1 dot | backward | w/ negs	<|special_separator|>
(131.57, 499.36) (201.28, 539.61) (197.71, 545.79) (128.00, 505.54)        /F1 dot | backward | wo/ negs	<|special_separator|>
(152.57, 500.14) (220.95, 539.61) (217.38, 545.79) (149.00, 506.32)        /F1 dot | symmetric | w/ negs	<|special_separator|>
(169.06, 498.28) (240.66, 539.62) (237.09, 545.81) (165.49, 504.47)        /F1 dot | symmetric | wo/ negs	<|special_separator|>
(197.47, 503.33) (260.30, 539.61) (256.73, 545.79) (193.90, 509.52)        /F1 l2 | backward | w/ negs	<|special_separator|>
(213.95, 501.48) (279.98, 539.61) (276.41, 545.79) (210.38, 507.66)        /F1 l2 | backward | wo/ negs	<|special_separator|>
(234.96, 502.24) (299.67, 539.59) (296.10, 545.77) (231.39, 508.42)        /F1 l2 | symmetric | w/ negs	<|special_separator|>
(251.43, 500.40) (319.37, 539.62) (315.80, 545.81) (247.86, 506.59)        /F1 l2 | symmetric | wo/ negs	<|special_separator|>
(273.43, 501.07) (278.52, 504.00) (274.95, 510.19) (269.86, 507.25)        /F1 l2	<|special_separator|>
(277.15, 506.47) (279.52, 507.84) (277.02, 512.18) (274.65, 510.81)        /F2 2	<|special_separator|>
(281.07, 505.48) (338.80, 538.82) (335.23, 545.00) (277.50, 511.67)        /F1  | backward | w/ negs	<|special_separator|>
(289.72, 499.12) (294.81, 502.06) (291.24, 508.24) (286.15, 505.31)        /F1 l2	<|special_separator|>
(293.45, 504.51) (295.82, 505.88) (293.32, 510.22) (290.95, 508.85)        /F2 2	<|special_separator|>
(297.37, 503.54) (358.29, 538.72) (354.72, 544.90) (293.80, 509.72)        /F1  | backward | wo/ negs	<|special_separator|>
(310.85, 499.96) (315.94, 502.89) (312.37, 509.08) (307.28, 506.15)        /F1 l2	<|special_separator|>
(314.59, 505.35) (316.96, 506.72) (314.46, 511.05) (312.09, 509.69)        /F2 2	<|special_separator|>
(318.51, 504.38) (378.10, 538.78) (374.53, 544.97) (314.94, 510.56)        /F1  | symmetric | w/ negs	<|special_separator|>
(327.15, 498.00) (332.24, 500.93) (328.67, 507.12) (323.58, 504.18)        /F1 l2	<|special_separator|>
(330.87, 503.41) (333.24, 504.77) (330.74, 509.11) (328.37, 507.74)        /F2 2	<|special_separator|>
(334.79, 502.41) (397.61, 538.69) (394.04, 544.87) (331.22, 508.60)        /F1  | symmetric | wo/ negs	<|special_separator|>
(354.90, 503.33) (417.72, 539.61) (414.15, 545.79) (351.33, 509.52)        /F1 l1 | backward | w/ negs	<|special_separator|>
(371.39, 501.48) (437.42, 539.61) (433.85, 545.79) (367.82, 507.66)        /F1 l1 | backward | wo/ negs	<|special_separator|>
(392.39, 502.24) (457.09, 539.59) (453.52, 545.77) (388.82, 508.42)        /F1 l1 | symmetric | w/ negs	<|special_separator|>
(408.87, 500.40) (476.81, 539.62) (473.24, 545.81) (405.30, 506.59)        /F1 l1 | symmetric | wo/ negs	<|special_separator|>
(139.84, 543.26) (154.00, 543.26) (154.00, 553.64) (139.84, 553.64)        /F1 0.0	<|special_separator|>
(139.84, 585.76) (154.00, 585.76) (154.00, 596.14) (139.84, 596.14)        /F1 0.2	<|special_separator|>
(139.84, 628.26) (154.00, 628.26) (154.00, 638.64) (139.84, 638.64)        /F1 0.4	<|special_separator|>
(139.84, 670.75) (154.00, 670.75) (154.00, 681.13) (139.84, 681.13)        /F1 0.6	<|special_separator|>
(137.59, 605.04) (137.59, 672.87) (124.60, 672.87) (124.60, 605.04)        /F1 Solved Rate	<|special_separator|>
(203.45, 704.95) (458.57, 704.95) (458.57, 720.54) (203.45, 720.54)        /F1 Final Solved Rate (1M training steps)	<|special_separator|>
(108.00, 468.36) (504.00, 468.57) (504.00, 476.27) (108.00, 476.92)      /F116 Figure 10: Success rate on Rubik's Cube scrambled with 10 random moves, for models trained with different	<|special_separator|>
(108.00, 458.61) (504.00, 458.61) (504.00, 466.30) (108.00, 466.30)      /F116 contrastive losses. Models using the backward loss consistently achieve better performance than those using	<|special_separator|>
(108.00, 448.64) (191.66, 448.64) (191.66, 456.34) (108.00, 456.34)      /F116 the symmetric variant.	<|special_separator|>
(197.33, 448.64) (493.03, 448.64) (493.03, 456.34) (197.33, 456.34)      /F116 Using the dot product without in-trajectory negatives performs similarly to the	<|special_separator|>
(496.23, 448.79) (500.03, 448.79) (500.03, 456.63) (496.23, 456.63)       /F65 ℓ	<|special_separator|>
(500.03, 448.37) (503.68, 448.37) (503.68, 453.59) (500.03, 453.59)       /F64 2	<|special_separator|>
(108.00, 438.68) (505.12, 438.68) (505.12, 446.38) (108.00, 446.38)      /F116 metric, while combining the dot product with in-trajectory negatives yields the highest success rate. In contrast,	<|special_separator|>
(108.00, 428.72) (505.12, 428.72) (505.12, 436.42) (108.00, 436.42)      /F116 combining in-trajectory negatives with symmetric loss results in a drop in performance, likely because, in CRTR,	<|special_separator|>
(108.00, 418.76) (364.44, 418.76) (364.44, 426.45) (108.00, 426.45)      /F116 such negatives are often closer to the correct solution in the state-space.	<|special_separator|>
(108.00, 386.27) (504.00, 386.27) (504.00, 394.82) (108.00, 394.82)      /F116 learning rate and batch size was guided by the performance of the contrastive baseline on Rubik's	<|special_separator|>
(108.00, 375.36) (503.99, 375.36) (503.99, 383.91) (108.00, 383.91)      /F116 Cube. Specifically, we evaluated solve rates on cubes shuffled 10 times, as shown in Figure 8. We	<|special_separator|>
(108.00, 364.45) (504.67, 364.45) (504.67, 373.00) (108.00, 373.00)      /F116 also conducted grid searches to find the optimal training parameters (learning rate and batch size)	<|special_separator|>
(108.00, 353.54) (404.95, 353.54) (404.95, 362.09) (108.00, 362.09)      /F116 for the supervised baseline on Sokoban, Lights Out, and Rubik's Cube .	<|special_separator|>
(409.69, 353.54) (504.00, 353.54) (504.00, 362.09) (409.69, 362.09)      /F116 We use the same batch	<|special_separator|>
(108.00, 342.63) (504.00, 342.63) (504.00, 351.18) (108.00, 351.18)      /F116 size and learning rate across all methods and environments, with the exception of Lights Out, where	<|special_separator|>
(108.00, 331.72) (492.30, 331.72) (492.30, 340.27) (108.00, 340.27)      /F116 increasing the batch size and learning rate in the supervised baseline led to a higher success rate.	<|special_separator|>
(108.00, 314.33) (199.75, 314.33) (199.75, 323.29) (108.00, 323.29)      /F107 Network architecture.	<|special_separator|>
(209.71, 314.34) (504.00, 314.34) (504.00, 322.90) (209.71, 322.90)      /F116 We adopt the network architecture proposed by Nauman et al. [44], using 8	<|special_separator|>
(108.00, 303.44) (504.00, 303.44) (504.00, 311.99) (108.00, 311.99)      /F116 layers with a hidden size of 512 and a representation dimension of 64. This configuration was found	<|special_separator|>
(108.00, 292.53) (505.74, 292.53) (505.74, 301.08) (108.00, 301.08)      /F116 to yield optimal performance for the contrastive baseline on Rubik's Cube, as illustrated in Figure 7.	<|special_separator|>
(107.53, 281.62) (465.90, 281.62) (465.90, 290.17) (107.53, 290.17)      /F116 We observed that this architecture performs well in all environments except for two cases:	<|special_separator|>
(135.40, 261.54) (138.88, 261.54) (138.88, 270.09) (135.40, 270.09)      /F116 •	<|special_separator|>
(143.87, 261.54) (484.08, 261.54) (484.08, 270.09) (143.87, 270.09)      /F116 In Sokoban, a convolutional architecture was required to achieve strong performance.	<|special_separator|>
(135.40, 246.79) (138.88, 246.79) (138.88, 255.35) (135.40, 255.35)      /F116 •	<|special_separator|>
(143.87, 246.79) (474.20, 246.79) (474.20, 255.35) (143.87, 255.35)      /F116 In Lights Out, the convolutional network was necessary to ensure training stability.	<|special_separator|>
(108.00, 226.70) (142.20, 226.70) (142.20, 235.66) (108.00, 235.66)      /F107 Test set.	<|special_separator|>
(152.17, 226.72) (504.00, 226.72) (504.00, 235.27) (152.17, 235.27)      /F116 For Sokoban, we construct a separate test set comprising 100 trajectories, which is used	<|special_separator|>
(108.00, 215.81) (504.17, 215.81) (504.17, 224.36) (108.00, 224.36)      /F116 to compute evaluation metrics such as accuracy, correlation, and t-SNE visualizations. For all other	<|special_separator|>
(108.00, 204.90) (505.24, 204.90) (505.24, 213.45) (108.00, 213.45)      /F116 environments, a separate test set is unnecessary, as we train for only a single epoch. In this setting,	<|special_separator|>
(108.00, 193.99) (396.93, 193.99) (396.93, 202.54) (108.00, 202.54)      /F116 evaluation is performed directly on unseen data sampled during training.	<|special_separator|>
(108.00, 176.60) (178.49, 176.60) (178.49, 185.55) (108.00, 185.55)      /F107 Contrastive loss.	<|special_separator|>
(188.44, 176.61) (505.65, 176.61) (505.65, 185.17) (188.44, 185.17)      /F116 We use the backward version of the contrastive loss, which we found to consis-	<|special_separator|>
(108.00, 165.70) (504.00, 165.70) (504.00, 174.26) (108.00, 174.26)      /F116 tently outperform the symmetrized variant on Rubik's Cube as shown in Figure 10. We also found	<|special_separator|>
(108.00, 154.79) (505.61, 154.79) (505.61, 163.35) (108.00, 163.35)      /F116 the backward version to work better on 15-Puzzle and slightly better in the remaining environments.	<|special_separator|>
(108.00, 138.41) (381.21, 138.41) (381.21, 146.96) (108.00, 146.96)      /F116 For Rubik's Cube, we use the dot product as the similarity metric.	<|special_separator|>
(385.91, 138.41) (504.00, 138.41) (504.00, 146.96) (385.91, 146.96)      /F116 Performance across different	<|special_separator|>
(108.00, 127.50) (493.08, 127.50) (493.08, 136.05) (108.00, 136.05)      /F116 metrics is presented in Figure 10. While the contrastive baseline performs comparably under the	<|special_separator|>
(495.58, 127.66) (499.73, 127.66) (499.73, 136.37) (495.58, 136.37)       /F27 ℓ	<|special_separator|>
(499.73, 126.74) (503.70, 126.74) (503.70, 132.83) (499.73, 132.83)       /F23 2	<|special_separator|>
(108.00, 116.59) (504.00, 116.59) (504.00, 125.14) (108.00, 125.14)      /F116 metric, CRTR achieves significantly better results with the dot product. Based on similar empirical	<|special_separator|>
(108.00, 105.68) (368.89, 105.68) (368.89, 114.23) (108.00, 114.23)      /F116 evaluations, we use the following metrics for other environments:	<|special_separator|>
(135.40, 084.68) (138.88, 084.68) (138.88, 093.23) (135.40, 093.23)      /F116 •	<|special_separator|>
(143.87, 084.68) (189.53, 084.68) (189.53, 093.23) (143.87, 093.23)      /F116 Lights Out:	<|special_separator|>
(192.62, 084.84) (196.77, 084.84) (196.77, 093.55) (192.62, 093.55)       /F27 ℓ	<|special_separator|>
(196.77, 083.92) (200.74, 083.92) (200.74, 090.02) (196.77, 090.02)       /F23 2	<|special_separator|>
(203.73, 084.68) (238.87, 084.68) (238.87, 093.23) (203.73, 093.23)      /F116 distance,	<|special_separator|>
(135.40, 069.93) (138.88, 069.93) (138.88, 078.49) (135.40, 078.49)      /F116 •	<|special_separator|>
(143.87, 069.93) (309.38, 069.93) (309.38, 078.49) (143.87, 078.49)      /F116 Digit Jumper and 15-Puzzle: dot product,	<|special_separator|>
(301.02, 040.05) (310.98, 040.05) (310.98, 048.60) (301.02, 048.60)      /F116 17