Divide-and-Remember: Recursive Action-Relevant Memory for Long-Horizon VLA Policies

Xuehui Yu; Eason Yu; Meiyi Wang; Haozhe Du; Stefano V. Albrecht; Harold Soh

cs.ROOct 1, 2026

Divide-and-Remember: Recursive Action-Relevant Memory for Long-Horizon VLA Policies

Authors: Xuehui Yu, Eason Yu, Meiyi Wang, Haozhe Du, Stefano V. Albrecht, Harold Soh

Organizations: National University of Singapore · Nanyang Technological University, Singapore · LISTENAI

Abstract

Vision-language-action (VLA) models struggle on history-dependent manipulation tasks, where the current observation alone does not determine the action, and the policy needs a memory of the history. Existing memory methods decide what to remember by design, for example, keeping frames with large pixel changes, and show inconsistent gains across tasks. We view what to remember as an optimisation problem. From the POMDP formulation of imitation learning, we show that the optimal memory maximises the conditional mutual information $I(a_t; m_t \mid o_t)$ between the action and the memory given the current observation. Intuitively, this means preserving the action-relevant information in the history that is not already contained in the current observation. Based on our analysis, we propose Divide-and-Remember (D&R), a recursive memory method that learns a memory function $m_t = M(h_t)$ and scales to long contexts while staying compute-light. It involves two strategies: (1) the selection over the full history is divided recursively into subproblems of top- $K$ selection over $2K$ tokens, so that fixed-size, lightweight selectors learned end-to-end support an unbounded history; (2) all recursion blocks share one selector, which captures the selection rule common to every block and keeps the method efficient. On RoboMME, a benchmark of 16 long-horizon manipulation tasks that require remembering when, where, what, and how to act, D&R achieves a state-of-the-art average success rate with consistent gains across all four suites under a budget of only 64 tokens; real-robot experiments show the same gain. Code, checkpoints and more results are at https://dnr-memory.github.io/

Figures & tables

Figure 1: Memory is a question of what to remember. (a) A history-dependent task: “watch the video carefully, then move the cube to the target in the same manner as before”. Two videos show different manners (pick-and-place versus peg push), yet at time step $t$ the robot sees the same observation. A memoryless policy cannot recall how the cube was moved. (b) We view memory as an optimisation problem: find the memory $m_{t}=\mathcal{M}(h_{t})$ , selected from the full history, that carries the key fact the current observation lacks (here, the grasp pose at $t-3$ ), so that the policy knows how to move the cube.

Figure 2: Graphical models of (a) imitation learning in a POMDP, where

a_{t}

depends on the history

h_{t}

, and (b) a memory-augmented policy, where

m_{t}

makes

a_{t}

independent of

h_{t}

given

(o_{t},m_{t})

Figure 3: Overview of D&R. (a) At time step $t$ , the selection over the full history $\mathcal{H}_{t}$ is divided recursively into subproblems of top- $K$ selection over $2K$ tokens, each solved by the same lightweight selector $f_{\psi}$ ; the kept tokens are merged pairwise and re-selected level by level, and at the root the selected tokens are passed to the policy through the attention mask. Tokens keep their temporal order from left to right at every level. (b) D&R and all baselines modulate the memory into the action expert through an adaptive LayerNorm, where action features cross-attend to the memory tokens.

Figure 5: Memory at step $422$ of a MoveCube episode under a $64$ -token budget, with the task goal “watch the video carefully, then move the cube to the target in the same manner as before”. FrameSamp keeps $4$ evenly spaced frames. TokenDrop keeps the first frame and spends the rest of the budget on the patches whose pixels changed most. D&R selects $64$ patches recursively, $35$ of them from the video phase. Kept patches are shown, white ones are not in memory, and light blue borders mark the video phase. All tasks are in Appendix B.3.3 .

		Motion-Centric					Time-Sensitive	Short-Horizon Video				Long-Horizon Video			Dynamic Scene-Change				Event-Salient
Type	Method	Pattern Lock	Route Stick	Swing Xtimes	Insert Peg	Avg	Stop Cube	Video Umsk	Video Repick	Video UmskS	Avg	Video PlcBtn	Video PlcOrd	Avg	Button Umsk	Button UmskS	Pick HighL	Avg	Pick Xtimes	Bin Fill	Move Cube	Avg
No memory	$\pi_{0.5}$	2.89	4.67	35.56	1.56	11.17	6.67	20.44	0.44	18.67	13.18	31.11	25.78	28.45	22.22	6.67	11.33	13.41	42.89	30.00	26.00	32.96
Perceptual	FrameSamp	15.56	17.78	88.45	2.22	31.00	27.34	30.00	14.22	20.89	21.70	26.89	25.56	26.23	6.67	5.11	18.67	10.15	60.44	34.44	52.66	49.18
Perceptual	TokenDrop	9.56	8.89	36.67	2.44	14.39	5.78	28.67	1.56	16.67	15.63	26.67	19.56	23.12	16.67	6.44	14.00	12.37	39.78	27.78	24.44	30.67
Latent	HAMLET	7.33	9.56	81.11	1.11	24.78	39.33	27.78	20.67	25.11	24.52	27.78	21.78	24.78	26.67	21.78	26.00	24.82	84.67	34.89	60.00	59.85
Latent	RB-VLA	0.67	0.20	86.00	3.33	22.55	10.67	30.00	26.00	23.33	26.44	24.67	24.00	24.34	28.00	20.00	26.67	24.89	79.33	36.67	36.67	50.89

Table 1: Per-task success rate (%) on RoboMME ( Dai et al., 2026 ) , tasks grouped by functional characteristic (the same results by suite are in Table 7 ). All memories are built on

\pi_{0.5}

with a budget of

64

tokens and differ in what they store: raw patch tokens kept by frame sampling (FrameSamp) or token dropping (TokenDrop); learned moment tokens (HAMLET) or a recursive belief (RB-VLA); or fast weights (TTT). Each number is the mean over the last three checkpoints and three seeds; standard deviations are in Table 7 . Avg is the mean over the tasks of a characteristic; bold is the highest success rate in each column.

Figure 6: (a) Average success rate against the TFLOPs of one policy forward pass; each line sweeps the memory budget

K

from

64

512

(Table 8 ). (b) Average success rates against

K

. (c) D&R per task characteristic ( Dai et al., 2026 ) as the candidate pool

|\mathcal{H}|

grows from

128

1024

tokens at

K=64

; mean

\pm

std over runs (Table 5 ).

Method	Put Bottles	Track Cube	Repick Cube	Draw Pattern	Total
$\pi_{0.5}$	0/10	2/10	0/10	1/10	3/40
FrameSamp	4/10	8/10	6/10	4/10	22/40
D&R	9/10	9/10	9/10	8/10	35/40

Table 2: Real-world results.

Appendix figures & tables27 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Graphical models of imitation learning in a POMDP . (a) A general policy for the POMDP inputs the current observation and the full history,

a_{t}\sim\pi(\cdot\mid o_{t},h_{t})

, but the history grows without bound. (b) A memoryless policy

a_{t}\sim\pi(\cdot\mid o_{t})

inputs only the current observation, which causes the memory problem in Figure 1 (a). (c) A belief policy

a_{t}\sim\pi(\cdot\mid b_{t})

inputs the belief

b_{t}(s)=P(s_{t}=s\mid h_{t},o_{t})

, the posterior over the current state and a sufficient statistic of the history. (d) A memory-augmented policy

a_{t}\sim\pi(\cdot\mid o_{t},m_{t})

inputs a memory

m_{t}=\mathcal{M}(h_{t})

and the current observation. Here

m_{t}

need only be sufficient to reproduce the expert’s action (Proposition 1 ), and the optimal memory carries the most action-relevant information

I\big(a_{t};\,m_{t}\mid o_{t}\big)

. Shaded circles are observed in the demonstrations, white circles are hidden.

Panel of Figure 8 — Figure 8: Different methods assume different sufficiency metrics on the history. Bisimulation measures reward and transition; policy similarity measures policy and transition; the belief takes all three. D&R keeps policy sufficiency alone, under a limited budget.

Hyperparameter	Value
Base model	$\pi_{0.5}$ base
Batch size	$64$ global
Action horizon	20
Action space	Joint-space
Proprioception	False
Total steps	80,000

Appendix

Table 3: Training hyperparameters.

Variant	Counting	Permanence	Reference	Imitation	AVG
$\|\mathcal{H}\|=128$ (8 frames)	58.25 $\pm$ 2.04	20.83 $\pm$ 4.79	27.00 $\pm$ 1.38	34.67 $\pm$ 1.70	35.19 $\pm$ 1.98
$\|\mathcal{H}\|=256$ (16 frames)	59.17 $\pm$ 2.84	25.17 $\pm$ 2.47	24.56 $\pm$ 2.13	38.67 $\pm$ 1.99	36.89 $\pm$ 0.81
$\|\mathcal{H}\|=512$ (32 frames)	61.17 $\pm$ 0.62	26.83 $\pm$ 1.03	28.50 $\pm$ 1.41	37.83 $\pm$ 2.78	38.58 $\pm$ 0.50
$\|\mathcal{H}\|=1024$ (64 frames)	61.89 $\pm$ 2.37	26.17 $\pm$ 2.32	20.94 $\pm$ 1.57	28.67 $\pm$ 2.58	34.42 $\pm$ 1.03

Appendix

Table 4: Candidate pool. Success rate (%) of D&R averaged over the four tasks of each suite, as mean

\pm

standard deviation over evaluation runs (checkpoints

\times

seeds

0/7/42

), as the size

|\mathcal{H}|

of the candidate pool the selector reduces grows from

8

64

frames. Every row uses a memory budget of

K=64

tokens. bold is the highest success rate in each column.

Variant	Motion- Centric	Time- Sensitive	Short-Horizon Video	Long-Horizon Video	Dynamic Scene-Change	Event- Salient	AVG
$\|\mathcal{H}\|=128$ (8 frames)	41.92 $\pm$ 1.62	20.67 $\pm$ 8.46	26.33 $\pm$ 1.58	33.17 $\pm$ 1.77	15.33 $\pm$ 5.84	61.11 $\pm$ 2.69	35.19 $\pm$ 1.98
$\|\mathcal{H}\|=256$ (16 frames)	45.94 $\pm$ 2.05	28.00 $\pm$ 7.06	26.52 $\pm$ 1.80	30.67 $\pm$ 3.06	19.33 $\pm$ 1.99	59.85 $\pm$ 2.77	36.89 $\pm$ 0.81
$\|\mathcal{H}\|=512$ (32 frames)	47.50 $\pm$ 1.08	31.33 $\pm$ 2.49	28.89 $\pm$ 1.37	31.67 $\pm$ 1.70	23.78 $\pm$ 1.26	58.22 $\pm$ 1.66	38.58 $\pm$ 0.50
$\|\mathcal{H}\|=1024$ (64 frames)	36.17 $\pm$ 1.41	32.44 $\pm$ 5.56	23.19 $\pm$ 2.06	25.78 $\pm$ 2.15	22.44 $\pm$ 2.95	61.70 $\pm$ 3.02	34.42 $\pm$ 1.03

Appendix

Table 5: Candidate pool by task characteristic. Success rate (%) of D&R averaged over the tasks of each characteristic of Dai et al. (2026) (the grouping of Table 1 ), as mean

\pm

standard deviation over evaluation runs (checkpoints

\times

seeds

0/7/42

), as the size

|\mathcal{H}|

of the candidate pool changes. Every row uses a memory budget of

K=64

tokens; the AVG column is that of Table 4 . bold is the highest success rate in each column.

Variant	Counting	Permanence	Reference	Imitation	AVG
D&R (reference)	59.17 $\pm$ 2.84	25.17 $\pm$ 2.47	24.56 $\pm$ 2.13	38.67 $\pm$ 1.99	36.89 $\pm$ 0.81
$+$ pred	58.62 $\pm$ 1.85	22.69 $\pm$ 1.52	22.12 $\pm$ 1.43	30.75 $\pm$ 2.96	33.55 $\pm$ 0.70
$+$ text	52.00 $\pm$ 2.31	21.94 $\pm$ 3.51	23.11 $\pm$ 2.02	24.22 $\pm$ 3.15	30.32 $\pm$ 1.54

Appendix

Table 6: Ablations. Success rate (%) of D&R and its variants at a candidate pool of

|\mathcal{H}|=256

, as mean

\pm

standard deviation over evaluation runs (checkpoints

\times

seeds

0/7/42

), averaged over the four tasks of each suite (top) and per task (bottom).

+

pred adds RB-VLA’s world-model loss to the action loss;

+

text adds the language instruction to the candidate pool the selector scores. Every row uses a memory budget of

K=64

tokens. bold is the highest success rate in each column.

Type	Method	Counting	Permanence	Reference	Imitation	AVG
No memory	$\pi_{0.5}$	28.78	17.00	17.17	8.78	17.93
Perceptual	FrameSamp	52.67	15.67	21.34	22.06	27.93
Perceptual	TokenDrop	27.50	17.11	15.45	11.33	17.85
Latent	HAMLET	60.00	25.34	24.06	19.50	32.22
Latent	RB-VLA	53.17	25.33	25.34	10.22	28.52
Parametric	TTT	28.84	16.72	13.61	9.34	17.13

Appendix

Table 7: Full results on RoboMME. Success rate (%) of every method under the

64

-token budget, grouped by what the memory stores. Top: mean over the four tasks of each suite and over all

16

tasks (AVG). Bottom: every task, as mean

\pm

standard deviation over seeds. The

\pi_{0.5}

row is the number reported by Dai et al. (2026) . bold is the highest success rate in each column.

Overall
Budget	FrameSamp	TokenDrop	HAMLET	D&R
64	27.93	17.85	32.22	38.58
128	36.54	32.76	33.00	49.01
256	42.90	36.36	32.50	50.11
512	44.51	36.27	36.06	46.40

Appendix

Table 8: Effect of the memory budget. Average success rate (%) on RoboMME as the memory budget grows from

64

512

tokens, overall and per suite. The FrameSamp entries from

128

512

tokens and the TokenDrop entries at

128

and

256

tokens are the numbers reported by Dai et al. (2026) ; the remaining entries are our runs.

		Counting				Permanence
Type	Method	BinFill	PickXtimes	SwingXtimes	StopCube	VideoUnmask	ButtonUnmask	VideoUnmaskSwap	ButtonUnmaskSwap
No memory	$\pi_{0.5}$	30.00 $\pm$ 6.48	42.89 $\pm$ 6.41	35.56 $\pm$ 5.27	6.67 $\pm$ 5.20	20.44 $\pm$ 2.96	22.22 $\pm$ 10.22	18.67 $\pm$ 4.24	6.67 $\pm$ 4.12
Perceptual	FrameSamp	39.56 $\pm$ 5.27	87.33 $\pm$ 2.45	92.00 $\pm$ 2.24	42.00 $\pm$ 11.36	32.67 $\pm$ 3.16	25.11 $\pm$ 3.18	24.44 $\pm$ 6.06	18.22 $\pm$ 3.80
Perceptual	TokenDrop	31.33 $\pm$ 6.02	84.67 $\pm$ 4.84	86.00 $\pm$ 4.56	11.00 $\pm$ 4.69	30.67 $\pm$ 2.07	23.67 $\pm$ 4.46	22.67 $\pm$ 5.01	21.00 $\pm$ 6.03
Latent	HAMLET	31.78 $\pm$ 2.91	88.89 $\pm$ 3.48	89.33 $\pm$ 3.74	52.67 $\pm$ 7.42	29.78 $\pm$ 3.23	25.11 $\pm$ 4.26	23.33 $\pm$ 3.32	22.67 $\pm$ 3.61
Latent	RB-VLA	39.67 $\pm$ 3.13	78.31 $\pm$ 5.01	81.00 $\pm$ 8.50	12.02 $\pm$ 1.15	30.01 $\pm$ 3.21	24.20 $\pm$ 2.66	23.65 $\pm$ 0.13	19.10 $\pm$ 3.00

Appendix

Table 9: Results with a memory budget of

512

tokens. Per-task success rate (%) on RoboMME ( Dai et al., 2026 ) as mean

\pm

standard deviation over seeds, with

\pi_{0.5}

as the backbone and methods grouped by what the memory stores. The

\pi_{0.5}

, FrameSamp, and TTT rows are the numbers reported by Dai et al. (2026) . bold is the highest success rate in each column.

Figure 9: BinFill (Counting): “put one red cube into the bin, then press the button to stop”.

Figure 10: PickXtimes (Counting): “pick up the green cube and place it on the target, then press the button to stop”.

Figure 11: SwingXtimes (Counting): “pick up the green cube, move it to the top of the right-side target, then move it to the top of the left-side target, repeating this back-and-forth motion two times, finally press the button to stop”.

Figure 12: StopCube (Counting): “press the button to stop the cube just as it reaches the target for the fourth time”.

Figure 13: VideoUnmask (Permanence): “watch the video carefully, then pick up the container hiding the green cube”.

Figure 14: ButtonUnmask (Permanence): “first press the button, then pick up the container hiding the green cube”.

Figure 15: VideoUnmaskSwap (Permanence): “watch the video carefully, then pick up the container hiding the green cube”.

Figure 16: ButtonUnmaskSwap (Permanence): “first press both buttons on the table, then pick up the container hiding the green cube”.

Figure 17: PickHighlight (Reference): “first press the button, then pick up all cubes that have been highlighted with white areas on the table”.

Figure 18: VideoRepick (Reference): “watch the video carefully, then pick up the same block that was previously picked up again, finally put it down and press the button to stop”.

Figure 19: VideoPlaceButton (Reference): “watch the video carefully, then place the green cube on the target right before the button was pressed”.

Figure 20: VideoPlaceOrder (Reference): “watch the video carefully, then place the green cube on the second target it was previously placed on”.

Figure 21: MoveCube (Imitation): “watch the video carefully, then move the cube to the target in the same manner as before”.

Figure 22: InsertPeg (Imitation): “watch the video carefully, then grasp the same end of the same peg you’ve picked before and insert it into the same side of the box”.

Figure 23: PatternLock (Imitation): “watch the video carefully, then use the stick attached to the robot to retrace the same pattern”.

Figure 24: RouteStick (Imitation): “watch the video carefully, then use the stick attached to the robot to navigate around the sticks on the table, following the same path”.

Figure 25: The four real-world tasks, recorded on our Franka Panda: PutBottles (Counting), TrackCube (Permanence), RepickCube (Reference) and DrawPattern (Imitation). For each task, the upper dashed box shows frames from the earlier history (the video the policy watched; for PutBottles, earlier frames of the episode, including a human intervention), and the lower box shows the front-camera observation during execution.

Figure 26: Real-robot platform. A 7-DoF Franka Emika Panda with a Robotiq 2F-140 gripper and two RealSense cameras: a wrist camera mounted on the gripper and a front camera on a tripod facing the workspace. The objects of the four tasks (baskets, cubes, cups, and bottles) are laid out on the table.

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