FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
Qian He * 1 2 Zhenshuo Yang * 1 2 Wenqi Liang 3 Chunhui Hao 1 Nicu Sebe 3 Jiandong Tian 1
Abstract
et al., 2021; Gong et al., 2025). To mitigate compounding errors in imitation learning (Tu et al., 2021), action chunking has been adopted (Lai et al., 2022), allowing the policyto infer a multi-step action chunk per inference, rather than a single-step action (Zhao et al., 2023; Zhang et al., 2025b). Moreover, recent research emphasizes that accelerating the generation of action chunks is crucial for deploying visuomotor policies in the real world (Hu et al., 2024; Su et al., 2025c). Although effective, ensuring high-quality generation still necessitates a small number of iterative steps per inference, hindering real-world deployment. Consequently, efficiently generating action trajectories with minimal compounding errors remains a major bottleneck (Kim et al., 2024), which directly dictates the coherence and deployability of imitation learning policies (Park et al., 2025).
arXiv:2605.15944v1 [cs.RO] 15 May 2026
Visuomotor policies aim to learn complex manipulation tasks from expert demonstrations. However, generating smooth and coherent trajectories remains challenging, as it requires balancing proximal precision with distal foresight. Existing approaches typically focus on optimizing intrachunk action distributions, often neglecting the inter-chunk coherence. Consequently, inter-chunk discontinuities significantly impede the learning of coherent long-horizon actions. To overcome this limitation and achieve a synergetic balance between precision and foresight, we propose FocalPolicy, a foresight-aware visuomotor policy that combines Frequency-Optimized Chunking with Locally Anchored flow matching. We introduce a foresight composite objective that supervises time-domain alignment within the proximal actions while regularizing frequency-domain structure over multiple future action chunks to improve cross-chunk coherence. To efficiently learn complex action distributions, we design locally anchored campling to enhance target signal propagation efficiency during consistency flow matching training. Extensive experiments demonstrate that FocalPolicy outperforms existing approaches and confirm the generalizability of our modules to other baselines. Project website: https://focalpolicy.github.io/.
To address this issue, recent work (Song et al., 2023; Song & Dhariwal, 2024) aims to improve the accuracy of the trajectory in one-step generation. Prominent approaches achieve high-fidelity sampling through self-consistency constraints (Prasad et al., 2024; Lu et al., 2024) or consistency flow matching (Zhang et al., 2025a). Subsequent frameworks further improve performance by enhancing distributional fidelity (Wang et al., 2025c; Jia et al., 2024) or frequency-domain supervision (Su et al., 2025a; Zhong et al., 2025). As illustrated in Figure 1(a), these methods focus mainly on optimizing intra-chunk distributions, often neglecting inter-chunk discontinuities (Black et al., 2025), which leads to incoherence in subsequent action trajectories. To mitigate the discontinuities between chunks, an intuitive approach is to increase the inference horizon. Nevertheless, directly fitting the more complex action distribution introduced by longer chunks leads to policy degradation. Therefore, this necessitates a foresight-aware policy that can simultaneously plan fine-grained immediate actions and coarse subsequent trajectories. We argue that learning manipulation trajectories with a differentiated focus is crucial, as it enables the policy to not only finely align the fine-grained actions of proximal chunks but also attend to the coherence of subsequent chunks, thereby bridging the inter-chunk discontinuities. To this end, we propose FocalPolicy, a foresight-aware visuomotor policy that combines Frequency-Optimized Chunking with Locally An-
1. Introduction Recently, imitation learning-based visuomotor policies have made significant progress in robotic manipulation (Florence *
Equal contribution 1 State Key Laboratory of Robotics and Intelligent Systems, Shenyang Institute of Automation, Chinese Academy of Sciences 2 University of the Chinese Academy of Sciences 3 University of Trento. Correspondence to: Jiandong Tian <[email protected]>. Proceedings of the 43 rd International Conference on Machine Learning, Seoul, South Korea. PMLR 306, 2026. Copyright 2026 by the author(s).
1
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy (a) Previous Chunk-Based Policies
(b) FocalPolicy (Ours)
(c) Performance Comparison Adroit
Four-Stage Tasks 28％
Frequency-Domain
Intra-Chunk Refinement
Proximal Chunk
Intra-Chunk Refinement
Subsequent Chunks (Discontinuous)
Foresight-Aware Multi-Chunk Modeling
Proximal Chunk
Subsequent Chunks (Global Coherence)
77％
91％
Real World
82％
Three-Stage 37％ Tasks Two-Stage Tasks
MetaWorld Easy
65％
52％ 85％
MetaWorld Medium
Simulation
MetaWorld Hard
MetaWorld Very Hard FocalPolicy (Ours)
FlowPolicy
DP3
FreqPolicy
Figure 1. Comparison with chunk-based baselines. Unlike previous approaches (a) that prioritize intra-chunk refinement but overlook inter-chunk discontinuities, FocalPolicy (b) employs a Foresight Composite Objective (FCO) to synergize proximal precision with distal coherence across chunks. Extensive experiments (c) demonstrate that FocalPolicy outperforms state-of-the-art baselines.
chored flow matching for the efficient generation of longhorizon actions, as illustrated in Figure 1(b). We introduce a Foresight Composite Objective that supervises timedomain alignment within the proximal actions while regularizing frequency-domain structure over multiple future action chunks to improve cross-chunk coherence. To overcome the complex action distributions within multi-chunk trajectories, we design Locally Anchored Sampling to enhance target signal propagation efficiency during training.
time during training. This strategy reinforces direct supervision of the target distribution while preserving stochastic robustness, thereby significantly enhancing propagation efficiency and stabilizing training. Crucially, LAS acts as a general optimization booster that is broadly applicable to other consistency flow-based visuomotor policies. We summarize the contributions of our work as follows: • We propose FocalPolicy, a foresight-aware policy that explicitly addresses inter-chunk discontinuities in action generation via differentiated focus learning.
Foresight Composite Objective (FCO). While timedomain supervision enables precise local action alignment, frequency-domain features excel at capturing the global spatiotemporal relationships of action trajectories. Previous works (Su et al., 2025a; Zhong et al., 2025) have leveraged this synergy to significantly enhance the learning efficiency of single-chunk action distributions. However, the overlooked discontinuities in inter-chunk action distributions remain a major bottleneck for learning coherent manipulation trajectories. To bridge inter-chunk discontinuities, we propose the Foresight Composite Objective (FCO), which combines intra-chunk refinement in the proximal time-domain with foresight-aware multi-chunk modeling in frequency-domain, as illustrated in Figure 1. Unlike existing methods (Black et al., 2025; Park et al., 2025; Liu et al., 2025b) that regulate trajectory coherence by modifying the inference schemes of pre-trained models, FCO inherently strengthens the learning of action coherence at the fundamental level of policy training. As a generic and effective module, FCO can be seamlessly integrated into various action chunking-based policies.
• We introduce the Foresight Composite Objective (FCO) to synergize proximal precision with distal coherence, serving as a versatile module applicable to various action chunking-based policies. • We design Locally Anchored Sampling (LAS) to stabilize training for general consistency flow-based policies. Extensive experiments confirm the state-of-theart performance of FocalPolicy and the effective generalizability of both FCO and LAS.
2. Related Work 2.1. Spatiotemporal Awareness for Visuomotor Policies Understanding spatiotemporal relationships is crucial for robust robotic manipulation. Recent studies have enhanced this awareness through feature encoding and trajectory modeling (Liang et al., 2024; Ke et al., 2025; Wang et al., 2025a; Tian et al., 2025a; Lv et al., 2025; Su et al., 2025b; Liang et al., 2025; Wang et al., 2025b; Dong et al., 2026). At the representation level, DP4 (Liu et al., 2025c) incorporates a 3D Gaussian Splatting (3DGS) world model to guide the encoder in capturing spatiotemporal dynamics. Complementarily, recent work (Zhong et al., 2025) leverages frequency-domain analysis to structurally decouple global motion patterns from local details. Beyond feature-centric approaches, directly modeling object trajectories has proven
Locally Anchored Sampling (LAS). Consistency flow matching enforces consistency constraints on adjacent time points randomly sampled during training, which often weakens the propagation efficiency of target signals. This represents a key bottleneck in improving training efficiency when modeling the complex action distributions of multi-chunk trajectories. To address this, we design Locally Anchored Sampling (LAS), which anchors one sampling time point to a logit-normal distribution concentrated near the terminal 2
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Locally Anchored Sampling
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Figure 2. The pipeline of FocalPolicy. We propose Locally Anchored Sampling (LAS) to improve the training efficiency of consistency flow matching. The policy is optimized via a Foresight Composite Objective (FCO), which synergizes proximal precision (via time-domain loss) with distal coherence (via frequency-domain loss) to generate smooth multi-chunk trajectories.
3. Preliminaries
effective (Wen et al., 2023; Yuan et al., 2025; Bharadhwaj et al., 2024; Xu et al., 2025). For instance, ORCA (Huey et al., 2025) employs a dense per-timestep reward to enable sequence-level spatiotemporal matching. Furthermore, spatiotemporal modeling facilitates inference acceleration; Dense Policy (Su et al., 2025c) proposes keyframe-based trajectory completion, significantly improving efficiency while maintaining manipulation performance.
Flow Matching (FM) (Lipman et al., 2023) models a ddimensional data distribution by learning a time-dependent vector field vθ (x, τ ) : Rd × [0, 1] → Rd , which defines an ODE that transports samples from the noise distribution p0 (typically Gaussian) to the target distribution p1 at τ = 1. FM trains vθ to match the conditional target field uτ (xτ | x1 ) given data samples x1 . Under the Optimal Transport (OT) path (Amos et al., 2023) xτ = (1 − τ )x0 + τ x1 , the target field reduces to uτ (xτ | x1 ) = x1 − x0 . Accordingly, the FM objective is formulated as follows:
2.2. Action Chunking-based Manipulation Policies Action chunking (Zhao et al., 2023) mitigates compounding errors by predicting multi-step actions in a single pass, a paradigm now widely adopted in diffusion and flowbased policies. To ensure coherent real-time manipulation, RTC (Black et al., 2025) proposes an inference-time algorithm to overcome the discontinuities between action chunks. Recently, ManiCM (Lu et al., 2024) and FlowPolicy (Zhang et al., 2025a) have advanced one-step inference by utilizing consistency distillation and consistency flow matching (CFM) (Yang et al., 2024), respectively. Building on this, frequency consistency constraints are integrated into consistency flow policies (Su et al., 2025a), further strengthening the temporal correlation of action trajectories. While effective, the aforementioned approaches prioritize intra-chunk fidelity yet overlook inter-chunk discontinuities in longhorizon execution. To bridge this gap, we shift from singlechunk refinement to foresight-aware multi-chunk modeling, ensuring near-term precision and long-term coherence.
LFM (θ) = Ex,τ ∥vθ (xτ , τ ) − uτ (xτ |x1 )∥2 .
(1)
Although FM is effective, the iterative ODE integration required at inference time incurs a high number of function evaluations (NFE), hindering real-time deployment. To enable one-step generation, Consistency Flow Matching (CFM) (Yang et al., 2024) enforces straight flows (Lee et al., 2023) via velocity consistency, vθ (xτ , τ, c) ≡ vθ (x0 , 0, c), where c denotes the condition. Accordingly, CFM enforces this straight-line behavior by minimizing a self-consistency loss between predictions at neighboring flow times τ and τ + ∆τ , resulting in the following training objective: h LCFM (θ) = Ex,τ,c ∥fθ (xτ , τ, c) − fθ− (xτ +∆τ , τ + ∆τ, c)∥2 i + α∥vθ (xτ , τ, c) − vθ− (xτ +∆τ , τ + ∆τ, c)∥2 , (2) 3
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
where fθ (xτ , τ, c) = xτ + (1 − τ )vθ (xτ , τ, c), θ− denotes the network parameters updated via exponential moving average (EMA), and α > 0 balances the loss terms. Under this constraint, CFM realizes straight-line noise-to-data transport, enabling one-step generation at inference time.
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FlowPolicy (Zhang et al., 2025a) first introduces the consistency flow-based framework into robotic imitation learning to accelerate action inference. At each environment timestep t, the policy aims to generate an action chunk At = [at , at+1 , . . . , at+H−1 ] ∈ RH×d , representing H future d-dimensional actions. Accordingly, the flow state xτ in Eq. (2) is defined as the noisy action chunk Aτ at flow time τ , conditioned on the current observation ot . The training objective is adapted as: h LFP (θ) = EAτ ,τ,ot ∥fθ (Aτ , τ, ot )
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Locally Anchored Sampling
Figure 3. Time sampling comparison. Left: Standard uniform sampling draws (τ, r) uniformly along the flow trajectory, which can attenuate target-signal propagation for early τ . Right: Our locally anchored sampling biases r toward the terminal region to strengthen target-signal propagation.
while high-frequency components represent fine-grained details. These properties align well with our foresight requirement. Motivated by this observation, we apply the Discrete Cosine Transform (DCT) (Khayam, 2003) to project macrotrajectories into the frequency domain. For each action dimension d, the DCT coefficients Cd ∈ RL are computed as follows:
− fθ− (Aτ +∆τ , τ + ∆τ, ot )∥2 i + α∥vθ (Aτ , τ, ot ) − vθ− (Aτ +∆τ , τ + ∆τ, ot )∥2 . (3)
Cd [u] = cu
As shown in Eq. (3), it relies solely on self-consistency between adjacent time steps within an action chunk. This supervision lacks explicit guidance from the expert trajectory, making it difficult to perceive long-horizon motion trends and model complex multi-chunk action distributions.
L−1 X i=0
Mt [i, d] cos
π L
(i + 0.5)u ,
(5)
where u = 0, . . . , L − 1 is the frequency index, andp cu are p normalization constants with c0 = 1/L and cu = 2/L for u > 0. Eq. (5) further shows that each DCT coefficient aggregates information across all timesteps, providing a global view of the macro-trajectory.
4. Methodology
To model both distal motion trends and proximal execution details, FCO combines intra-chunk refinement in the time domain with foresight-aware multi-chunk modeling in the frequency domain. Concretely, we first define the multi-chunk frequency-domain loss Lfreq as the squared L2 distance between the DCT coefficients of the predicted trajectory M̂t and the expert trajectory Mt :
In this work, we propose FocalPolicy, a foresight-aware visuomotor policy as illustrated in Figure 2. Unlike prior works that focus on time-domain optimization within a single action chunk, FocalPolicy introduces a Foresight Composite Objective (FCO), an optimization target that integrates proximal time-domain alignment with multi-chunk frequency-domain structural regularization (Sec. 4.1). Furthermore, to enhance the target-signal propagation efficiency during training, we design a Locally Anchored Sampling (LAS) strategy (Sec. 4.2).
2
Lfreq = E DCT(M̂t ) − DCT(Mt )
, 2
(6)
where DCT(·) denotes the DCT operator defined in Eq. (5). The final Foresight Composite Objective, denoted as LFCO , is then formulated as a weighted combination of the proximal single-chunk time-domain consistency loss Ltime (Eq. (14)) and the multi-chunk frequency-domain loss Lfreq :
4.1. Foresight Composite Objective To achieve seamless long-horizon execution, we identify prioritized learning as a key necessity: the policy should ensure fine-grained fidelity for proximal actions while maintaining coarse structural coherence for distal trajectories. We therefore propose the Foresight Composite Objective (FCO). Specifically, we explicitly concatenate N consecutive expert action chunks into a macro-trajectory of horizon length L = N H, denoted as Mt ∈ RL×d : Mt = Concat At , At+H , . . . , At+(N −1)H . (4)
LFCO = Ltime + λ · Lfreq .
(7)
Here, λ = 10−4 is a balancing coefficient chosen to equilibrate the relative scales of the time-domain and frequencydomain loss components. This hybrid objective supervises time-domain alignment within proximal actions while regularizing frequency-domain structure over multiple future action chunks to improve cross-chunk coherence. Crucially, although Lfreq is defined in the frequency domain, Parseval’s theorem (Rao & Yip, 2014) ensures that
Moreover, prior work (Zhang et al., 2025a) indicates that low-frequency components encode global motion trends, 4
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
its error signal is consistent with the time-domain ℓ2 objective. In FCO, Ltime supervises only a single chunk of length H, while Lfreq supervises multiple chunks spanning a length of L, and the complete task trajectory is significantly longer than L. Under this formulation, the resulting network-parameter gradients remain non-conflicting: D E (H) ∇θ Lfreq , ∇θ Ltime > 0. (8)
Following the Optimal Transport (OT) construction, we define the interpolated states at times τ and r as follows:
Mr = (1 − r)M0 + rM1 ,
(12)
fθ (Mτ ′ , τ ′ , ot ) = Mτ ′ + (1 − τ ′ )vθ (Mτ ′ , τ ′ , ot ), (13)
4.2. Locally Anchored Sampling
where τ ′ ∈ {τ, r}. To enforce the straight-flow property, we impose a consistency constraint between the predictions at the uniform point τ and the anchored point r. The timedomain consistency loss is defined as: h i 2 Ltime = Eτ,r ∥[fθ (Mτ , τ, ot ) − fθ− (Mr , r, ot )]1:H ∥2 . (14) where the subscript 1 : H indicates that the consistency loss is computed exclusively on the first H timesteps (i.e., the proximal single chunk) of the predicted macro-trajectory. By assuming monotonically decreasing error near the terminal boundary, we mathematically demonstrate that our locally anchored sampling strategy achieves a strictly superior propagation efficiency E(τ ) compared to the standard random sampling strategy. Detailed theoretical derivations and proofs are provided in Appendix B.
While FCO effectively strengthens global coherence, it requires modeling the macro-trajectory, which greatly increases distributional complexity. As discussed in Sec. 3, FlowPolicy imposes consistency constraints between adjacent random time points. However, this often weakens the target-signal propagation efficiency, hindering the learning of complex distributions. Definition 4.1. To quantify the effectiveness of information flow during training, we define the Target-signal Propagation Efficiency E(τ ) as the fidelity of the consistency gradient gcons (τ, r) relative to the ideal supervised gradient gsup (τ ): h i 2 E(τ ) := −Er | τ ∥gcons (τ, r) − gsup (τ )∥ . (9) Remark 4.1. Under standard uniform sampling, the high variance of intermediate targets causes E(τ ) to decay rapidly as τ moves away from the terminal state, leading to suboptimal target distribution learning.
5. Experiments This section presents a comprehensive evaluation of FocalPolicy. We first detail the setup, baselines, and metric. Next, we demonstrate the superiority of FocalPolicy and validate the generalizability and distinct contributions of FCO and LAS. Finally, we conduct extensive ablation studies on 24 tasks to verify the effectiveness of our design choices.
To improve the target-signal propagation efficiency, we propose Locally Anchored Sampling (LAS). Figure 3 illustrates the difference between standard uniform sampling and our locally anchored strategy, which biases the anchor time toward the terminal region. Specifically, we sample a flow time τ ∼ U[0, 1] and an anchor time r from a logit-normal distribution. We formulate r as: ϵ ∼ N (0, 1),
(11)
where M0 is a Gaussian noise trajectory, and M1 corresponds to the target expert macro-trajectory Mt . Conditioned on observation ot , the model predicts velocity fields vθ (Mτ , τ, ot ) and vθ (Mr , r, ot ). We define the model predictions of the target action trajectory from any given flow state as follows:
Thus, optimizing Lfreq does not introduce conflicting optimization directions. A detailed derivation and further discussion on spectral sensitivity, along with the complete training process of FocalPolicy, are provided in the Appendix A.
r = sigmoid(µr + σr ϵ),
Mτ = (1 − τ )M0 + τ M1 ,
5.1. Experiments Setup Simulation experiments. To achieve a comprehensive evaluation across diverse robotic manipulation skills, we select 53 tasks from the Adroit (Rajeswaran et al., 2017) and MetaWorld (Yu et al., 2020) simulation platforms. The Adroit is primarily designed for evaluating dexterous hand manipulation tasks, while MetaWorld focuses on parallel gripper manipulation tasks. All tasks were constructed using simulators such as MuJoCo (Todorov et al., 2012) and SaPien (Xiang et al., 2020). For expert demonstration data, comprising 10 trajectories of 200 steps per task, we employed well-trained heuristic policies to ensure the generation of high-quality datasets, with Adroit using the VRL3 (Wang et al., 2022)
(10)
where µr , σr control the anchor bias. By biasing r toward the terminal boundary, we effectively strengthen the targetsignal propagation efficiency in consistency training. However, rather than fixing r to the terminal time, we keep it stochastic to stabilize training and preserve temporal continuity along the flow. In practice, we set µ = 4.0 and σ = 1.6, mapping to a median time of r ≈ 0.982 with a heavy left tail, which effectively provides the desired nearterminal supervision. 5
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 1. Main results in simulation. We report the number of function evaluations (NFE) and success rates (%) for each method across benchmarks. Averages computed over 53 tasks. * indicates results reproduced using the same expert demonstrations for fair comparison. Method \Task
NFE
Adroit (3)
MetaWorld Easy (28)
MetaWorld Medium (11)
MetaWorld Hard (6)
MetaWorld Very Hard (5)
Average
DP MainCM SDM
10 1 1
31.7 72.3 74.0
83.6 83.6 86.5
31.1 55.6 65.8
9.0 33.3 35.8
26.6 67.0 71.6
55.9 69.9 74.4
DP3* FlowPolicy* FreqPolicy* Ours
10 1 1
70.4 ± 3.2 69.0 ± 2.6 68.6 ± 0.8 76.9 ± 2.2
89.3 ± 0.1 91.1 ± 0.2 85.4 ± 0.2 91.4 ± 0.1
77.2 ± 2.7 71.9 ± 1.2 66.6 ± 2.0 81.9 ± 1.6
47.1 ± 2.3 45.2 ± 0.9 46.4 ± 2.1 51.8 ± 3.8
78.1 ± 0.9 77.9 ± 0.6 74.4 ± 2.4 85.1 ± 1.6
79.9 79.4 75.1 83.6
Table 2. Comparing FocalPolicy with more baselines on 7 simulation tasks. We compare Mamba Policy, DP3, FlowPolicy, and their variants, termed DP3 w. FCO and FlowPolicy w. LAS. Adroit
Method \Task
MetaWorld
Average
Hammer
Door
Pen
Sweep-Into
Hand-Insert
Pick-Place
Disassemble
Mamba Policy* FreqPolicy* Ours
99.7 ± 0.6 94.3 ± 2.1 100.0 ± 0.0
59.0 ± 2.0 58.3 ± 3.8 63.2 ± 5.3
58.3 ± 3.8 53.0 ± 3.5 67.5 ± 1.3
59.3 ± 17.0 19.3 ± 8.5 70.7 ± 22.0
15.3 ± 4.5 16.0 ± 3.0 29.3 ± 16.2
0.0 ± 0.0 57.7 ± 4.2 70.7 ± 4.2
79.7 ± 4.7 85.3 ± 1.5 86.7 ± 1.5
53.1 54.9 69.7
DP3* DP3 w. Ours
98.3 ± 2.9 98.3 ± 1.5
53.5 ± 5.1 58.3 ± 2.5
59.3 ± 3.5 59.7 ± 2.1
39.3 ± 25.3 47.0 ± 24.3
14.3 ± 2.9 15.3 ± 0.6
60.7 ± 11.9 61.7 ± 9.3
79.3 ± 7.0 84.0 ± 5.6
57.8 60.6
FlowPolicy* FlowPolicy w. Ours
94.0 ± 0.0 98.7 ± 2.3
53.0 ± 6.2 61.3 ± 3.2
61.0 ± 3.5 62.7 ± 4.5
37.3 ± 31.8 62.3 ± 17.6
16.0 ± 2.6 16.5 ± 0.7
55.3 ± 2.3 66.3 ± 4.6
71.0 ± 7.8 75.0 ± 6.0
55.4 63.3
Success Rate (％)
Adroit Hammer
MetaWorld Basketball
MetaWorld Disassemble
MetaWorld Push Wall
MetaWorld Sweep Into
MetaWorld Place Wall
FocalPolicy FlowPolicy 0
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Figure 4. Learning efficiency. We report learning curves of FocalPolicy and FlowPolicy across six tasks. Benefiting from higher target signal propagation efficiency during training, FocalPolicy converges faster to higher success rates than FlowPolicy.
algorithm and MetaWorld employing the PPO (Schulman et al., 2017) algorithm.
Figure 5 illustrates the setup and stages for representative tasks, with details in Appendix C.4.
Real-world experiments. Referencing the multi-stage task design in (Liu et al., 2025a; Tian et al., 2025b), we curate six long-horizon tasks to evaluate the real-world performance of FocalPolicy. We categorize these tasks into three levels, ranging from two to four stages: Two-stage tasks: 1) Water Pouring. The robot pours water from a kettle into a cup and relocates the kettle. 2) Drawer Loading. The robot places an object into an open drawer and pushes it closed. Three-stage tasks: 3) Pot Loading. The robot removes a pot lid, adds an ingredient, and recovers the pot. 4) Tower Stacking. The robot sequentially stacks three objects of varying shapes. Four-stage tasks: 5) Cup Matching. The robot relocates three cups onto corresponding saucers. 6) Object Sorting. The robot sorts four items into two bins.
Baselines and Implementation Details. We compare FocalPolicy against state-of-the-art visuomotor policies designed for 3D point-cloud observations and action chunking. Specifically, we re-evaluate FlowPolicy (Zhang et al., 2025a), DP3 (Ze et al., 2024), Mamba Policy (Cao et al., 2024) and FreqPolicy (Zhong et al., 2025) on our datasets as primary baselines. For DP (Chi et al., 2023), ManiCM (Lu et al., 2024), and SDM (Jia et al., 2024), we include their reported metrics for comparison. All baselines utilize original configurations, while for the parameters specific to FocalPolicy, we set the action chunk size H = 4 and the number of aggregated chunks N = 3. We adopt a consistent training and evaluation setup across all methods, with details provided in Appendix C.1. 6
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy RealSence D435i
Water Pouring Stage 1
UR-10e AG-95 Gripper
Stage 2
Pot Loading Stage 1
Stage 2
Stage 3
Stage 1
Stage 2
Stage 3
Tower Stacking
Cup Matching Stage 1
Stage 2
Stage 3
Stage 4
Figure 5. Real-world experimental setup and stage definitions. Left: Real-world experimental setup utilizing a UR-10e robotic arm equipped with an AG-95 gripper and a RealSense D435i camera. Right: Illustration of stage-wise definitions for four representative tasks.
Average Success Rate (％)
100
Water Pouring
Drawer Loading
Stage 1
Stage 1
Pot Loading
Tower Stacking
Cup Matching
Object Sorting
80 60 40 20 0
Stage 2
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FocalPolicy (Ours)
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DP3
Stage 3
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FlowPolicy
Figure 6. Real-world main results. We evaluate FocalPolicy, FlowPolicy, DP3, and FreqPolicy on six tasks. We report the average success score at each stage, and the horizontal segments indicate the policy performance across different stages of each task.
(63.3%) surpasses FlowPolicy by 7.9%. These improvements demonstrate that the benefits of FCO and LAS are not confined to FocalPolicy but are generalizable to similar model architectures.
Evaluation Metric. Following the evaluation protocol in DP3, we evaluate each task across three runs using seeds 0, 1, and 2. For each seed, we evaluate the policy 20 times every 200 epochs and report the average of the top five success rates, along with the mean and variance across seeds.
3) Mitigation of Compounding Errors and Discontinuities. As shown in Figure 7, we visualize the predicted spatial trajectories alongside the corresponding expert trajectories, and compute their Euclidean distances to the expert ground truth to evaluate the compounding errors. Under the same 12-step prediction horizon, FocalPolicy tracks the expert trajectories more closely and exhibits significantly lower compounding errors compared to FlowPolicy.
5.2. Simulation Results 1) Main Results. Table 1 summarizes the results on 53 tasks, where FocalPolicy reaches an average success rate of 83.6%, outperforming DP3 (79.9%), FlowPolicy (79.4%), and FreqPolicy (75.1%). Table 2 further compares FocalPolicy with more baselines on 7 representative tasks. FocalPolicy achieves the highest success rate on all tasks, with an average of 69.7%. These results demonstrate the effectiveness of foresight-aware multi-chunk modeling.
To rigorously quantify the inter-chunk discontinuities, we further adopt Action Total Variation (ATV) (Park et al., 2025) to measure the temporal coherence of the generated action sequences. The ATV is defined as:
2) Module Generalizability. To evaluate the generalizability of FCO and LAS, we directly integrate these modules into baseline models, denoted as DP3 w. Ours and FlowPolicy w. Ours. As shown in Table 2, DP3 w. Ours (60.6%) improves the average success rate by 2.8% over DP3, while FlowPolicy w. Ours
ATV =
L−1 d XX 1 |aj − ajt |, (L − 1) · d t=1 j=1 t+1
(15)
where t denotes the time step in a trajectory of length L, and j denotes the action dimension up to d. As reported 7
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy (a) Bin Picking
(b) Push Wall
Compounding Errors
Compounding Errors
Step
Step
Figure 7. Comparison of compounding errors. We visualize the 3D end-effector trajectories (left) inferred by FocalPolicy and FlowPolicy on two representative simulation tasks (Bin Picking and Push Wall), given the same initial state. The corresponding error curves (right) represent the Euclidean distance between the predicted and ground-truth coordinates (||ppred − pgt ||2 ) at each time step. Table 3. Quantitative comparison of Action Total Variation (ATV ↓). Lower values indicate higher temporal coherence and smoother action sequences across chunks.
Table 4. Ablation study of component effectiveness on 24 tasks. We report the average success rate (%) across tasks. A (3)
M (11)
H (5)
VH (5)
Average
Ours w/o LAS w/o FCO&LAS
76.9 ± 2.2 71.7 ± 1.8 46.2 ± 12.1
81.9 ± 1.6 65.6 ± 2.9 56.5 ± 3.2
62.2 ± 4.6 47.8 ± 1.8 43.1 ± 0.2
85.1 ± 1.6 73.3 ± 1.1 64.7 ± 2.4
77.9 64.3 54.1
Ltime -Only FCO Lfreq -Only FCO Fixed-r LAS
34.3 ± 1.2 74.0 ± 1.0 70.4 ± 2.2
65.9 ± 4.8 74.4 ± 1.9 77.4 ± 2.0
49.4 ± 2.1 60.1 ± 1.3 59.0 ± 3.7
78.1 ± 1.7 81.4 ± 1.8 80.2 ± 1.7
61.1 72.8 73.3
Designs
Task Bin Picking Disassemble Pick Out of Hole Pick Place Pick Place Wall Push Wall Shelf Place Sweep Into
FlowPolicy
Ours
0.99 0.35 0.94 0.03 0.42 0.39 0.48 1.66
0.87 0.21 0.92 0.03 0.39 0.33 0.46 1.45
Table 5. Horizon Ablation. We evaluate the impact of chunk size H and the number of aggregated chunks N across 24 tasks. We report the success rates under different settings of H and N . H
N
A (3)
M (11)
H (5)
VH (5)
Average
4
2 3 4 5
73.6 ± 3.2 76.9 ± 2.2 76.1 ± 3.4 70.4 ± 2.8
74.7 ± 1.1 81.9 ± 1.6 76.5 ± 3.1 65.9 ± 3.5
59.1 ± 5.0 62.2 ± 4.6 58.7 ± 1.8 49.5 ± 1.5
82.3 ± 2.1 85.1 ± 1.6 82.1 ± 3.1 73.4 ± 2.1
72.9 77.9 73.9 64.6
8
2 3 4 5
73.7 ± 2.3 74.4 ± 1.8 73.4 ± 3.7 72.6 ± 3.1
78.5 ± 2.1 71.2 ± 4.6 77.5 ± 0.8 75.4 ± 2.0
59.4 ± 1.7 57.1 ± 0.6 60.0 ± 1.0 59.7 ± 2.3
80.3 ± 1.0 78.9 ± 2.1 82.0 ± 1.4 80.0 ± 1.7
74.3 70.3 74.3 72.7
12
2 3 4 5
72.7 ± 2.1 73.6 ± 2.2 69.7 ± 2.9 69.4 ± 1.2
73.9 ± 1.3 76.7 ± 0.7 72.9 ± 1.6 71.8 ± 1.8
61.4 ± 0.3 61.6 ± 2.6 59.8 ± 2.2 61.3 ± 1.6
79.3 ± 2.2 79.3 ± 1.7 77.3 ± 2.0 76.3 ± 1.7
72.3 73.7 70.7 70.3
in Table 3, FocalPolicy achieves consistently lower ATV scores than FlowPolicy across the evaluated tasks. Together with the qualitative reduction in compounding errors, these quantitative results strongly demonstrate that FCO effectively mitigates the discontinuities between action chunks, thereby enabling FocalPolicy to maintain higher temporal coherence in long-horizon trajectory prediction. 4) Learning Efficiency. We track the success rates of FocalPolicy and FlowPolicy during training in Figure 4. The results show that FocalPolicy converges faster to a higher success rate than FlowPolicy, demonstrating superior learning efficiency. We attribute this improvement to LAS, which enhances target-signal propagation efficiency during consistency flow training. For detailed implementation specifics and more results, please refer to the Appendix C.3.
advantage becomes most pronounced in the final stage. We attribute this success to the foresight composite objective, which guides the policy to bridge inter-chunk discontinuities, thereby mitigating compounding errors in imitation learning. Furthermore, FlowPolicy suffers a sharp performance decline across task stages, likely due to limited long-horizon modeling, whereas FocalPolicy mitigates this degradation by accurately predicting longer action chunks.
5.3. Real World Results. Figure 6 illustrates the real-world performance comparison of four methods: FocalPolicy, DP3, FreqPolicy, and FlowPolicy. FocalPolicy consistently achieves superior average success rates compared to other baselines, demonstrating its exceptional capability in capturing the complex distributions of long-horizon actions. Notably, this performance
5.4. Ablation Study To provide a solid validation of our proposed component effectiveness, we conduct extensive ablation studies across 24 tasks. The evaluation benchmarks encompass Adroit (A) and MetaWorld tasks spanning three difficulty levels: 8
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
Acknowledgement
Medium (M), Hard (H), and Very Hard (VH). Our analysis focuses on: Component Effectiveness and the Impact of Horizon Hyperparameters. To ensure a fair comparison, we conduct all experiments across three random seeds (0, 1, 2) using the identical set of expert demonstrations.
We thank all reviewers for their valuable suggestions. This work is supported by he Nation Key R&D Program of China under Grant 2024YFE0115500, the LiaoNing Revitalization Talents Program under Grant XLYC 2502011, the FIS project GUIDANCE (No. FIS2023-03251) and the EU Horizon project ELLIOT (No. 101214398).
Component Effectiveness. Table 4 presents the ablation results for the individual components of FocalPolicy. We first conduct ablation studies on FCO and LAS. Specifically, compared to FocalPolicy, the average success rates of variants 1) w/o LAS and 2) w/o FCO&LAS decrease by 13.6% and 23.8%, respectively. To further validate specific design choices, we define additional variants: 3) Ltime -Only FCO: Replaces long-horizon frequency supervision with a long-horizon time-domain loss, validating the superiority of frequency-domain supervision in FCO; 4) Lfreq -Only FCO: Removes the time-domain loss from FCO to evaluate frequency-only supervision; and 5) Fixed-r LAS: Restricts the sampling time r in LAS strictly to the endpoint (r = 1). The superior performance of FocalPolicy demonstrates the benefits of FCO and LAS for visuomotor policies.
Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
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We introduce FocalPolicy, a foresight-aware visuomotor policy that mitigates inter-chunk discontinuities in action generation by employing coarse-to-fine prioritized planning. By leveraging a foresight composite objective and locally anchored sampling, it achieves higher success rates than recent methods. However, it does not incorporate foresight reasoning for high-dimensional decision-making. Furthermore, it does not explicitly address capability enhancements for handling suboptimal demonstrations, distribution shifts, or out-of-distribution generalization. Despite these challenges, we believe FocalPolicy serves as an important step towards achieving foresight-aware robotic manipulation.
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Appendix Appendix Contents • A. Gradient Consistency of Foresight Composite Objective • B. Theoretical Analysis of Propagation Efficiency • C. Experiment Details – C.1 Hyper-parameters Setup – C.2 Implementation details – C.3 Simulation experiment results – C.4 Real-World experiment details • D.Ablation study results
A. Theoretical Analysis: Gradient Consistency of Foresight Composite Objective In this appendix, we analyze the spectral component in the Foresight Composite Objective (FCO) of FocalPolicy. Unlike the standard setting where both time- and frequency-domain losses are defined on the same horizon, FCO uses a frequencydomain loss on a macro-trajectory of length L = N H and a time-domain loss on a prefix sub-trajectory of length H (steps 1:H). We establish two key results: (i) Non-conflicting gradients: the spectral and prefix time losses produce aligned (non-opposing) gradient signals, with strictly positive cosine similarity at the prediction level (and positive cosine similarity in parameter space under mild coupling conditions); (ii) Spectral advantage beyond Parseval: although Parseval’s theorem guarantees energy equivalence between time and frequency representations on the same length-L signal, we show that the spectral parameterization makes low-frequency deviations more salient via gradient concentration, improving the numerical detectability of global motion trends without requiring explicit low-frequency re-weighting. A.1. Definitions and Preliminaries For clarity, we consider a single action dimension and denote the expert and predicted macro-trajectories by m ∈ RL and m̂(θ) ∈ RL , respectively, where L = N H. Define the macro-trajectory error and Jacobian: e := m̂(θ) − m,
J :=
∂ m̂ ∈ RL×|θ| . ∂θ
(16)
Let D ∈ RL×L denote the length-L DCT-II matrix with orthonormal normalization (Eq. 5), so that D⊤ D = IL . The (length-L) full-band frequency-domain loss is (L)
Lfreq (θ) := ∥Dm̂(θ) − Dm∥22 = ∥De∥22 .
(17)
The time-domain loss is defined on the first H steps of the macro-trajectory (steps 1:H). Let S ∈ RH×L be a fixed selection matrix that extracts this supervised prefix, for example S = [IH 0H×(L−H) ]. Define eH := Se ∈ RH ,
(H)
Ltime (θ) := ∥eH ∥22 = ∥Se∥22 .
(18)
We assume S is a row-submatrix of IL (i.e., it selects H time indices without reweighting), hence P := S⊤ S ∈ RL×L is an orthogonal projector with P⊤ = P and P2 = P. Theorem A.1 (Parseval’s Theorem for Orthonormal DCT). For any x ∈ RL , ∥x∥22 = ∥Dx∥22 . Equivalently, D⊤ D = IL . A.2. Gradient Compatibility We first establish a strictly positive cosine similarity at the prediction level (w.r.t. m̂), and then provide a mild sufficient condition for positive cosine similarity in the parameter space (w.r.t. θ). 13
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
Gradients w.r.t. predictions m̂.
(L)
By Theorem A.1, Lfreq (θ) = ∥e∥22 . Thus, (H)
(L)
∇m̂ Ltime = 2Pe.
∇m̂ Lfreq = 2e,
(19)
Proposition A.2 (Strictly Positive Cosine Similarity in Prediction Space). Whenever the short-window error is non-zero (i.e., Pe ̸= 0), we have ⟨e, Pe⟩ ∥Pe∥2 (L) (H) cos ∇m̂ Lfreq , ∇m̂ Ltime = = > 0. (20) ∥e∥2 ∥Pe∥2 ∥e∥2 Therefore, the macro-scale spectral term never induces an opposing supervision direction for the prefix time objective at the prediction level. Proof. Since P is an orthogonal projector, ⟨e, Pe⟩ = e⊤ Pe = (Pe)⊤ (Pe) = ∥Pe∥22 . Substituting yields Eq. (20). Gradients w.r.t. parameters θ.
Applying the chain rule, (L)
(H)
∇θ Lfreq = 2J⊤ e,
∇θ Ltime = 2J⊤ Pe.
(21)
Partition e = [eS ; eR ] and J = [JS ; JR ], where S corresponds to the prefix indices 1:H. Then (L) ⊤ ∇θ Lfreq = 2 J⊤ S eS + JR eR ,
(H)
∇θ Ltime = 2J⊤ S eS .
(22)
Assumption A.3 (Non-adversarial temporal coupling). The gradient contribution from the remaining (unobserved in time loss) steps is not adversarial to the supervised short-window contribution: ⊤ J⊤ R eR , JS eS ≥ 0.
(23) (H)
Proposition A.4 (Positive Cosine Similarity in Parameter Space). Under Assumption A.3, whenever ∇θ Ltime ̸= 0, D E 2 (L) (H) (H) ∇θ Lfreq , ∇θ Ltime ≥ ∇θ Ltime > 0, 2
(24)
(L) (H) and hence cos ∇θ Lfreq , ∇θ Ltime > 0. Proof. By decomposition, D
(L)
(H)
∇θ Lfreq , ∇θ Ltime
E
⊤ ⊤ = 4 J⊤ S eS + JR eR , JS eS 2
⊤ ⊤ = 4 J⊤ S eS 2 + 4 JR eR , JS eS .
(25)
Assumption A.3 makes the second term non-negative, yielding the claim. Remark A.5 (A sufficient structural condition). A sufficient structural condition for Assumption A.3 is JR J⊤ S = 0, i.e., the parameter sensitivities of the supervised prefix window and the remaining steps are orthogonal (e.g., temporally separated heads). Even when not exact, this orthogonality is often approximately satisfied, leading to empirically positive cosine similarity. Remark A.6 (Why Parseval does not contradict empirical gains). Parseval’s theorem implies that the macro-scale unweighted full-band spectral loss ∥De∥22 equals the macro-scale time-domain loss ∥e∥22 on the same length-L signal. In FCO, however, the time-domain term is defined only on a prefix window of length H, whereas the spectral term supervises the full macrotrajectory of length L = N H. Therefore, the two objectives are not equivalent, and the spectral term provides additional full-trajectory supervisory signal that can improve optimization and long-range consistency. 14
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
A.3. Spectral Sensitivity Analysis A core motivation for using frequency-domain supervision is its ability to capture overall motion trend. While Parseval’s theorem guarantees energy equivalence on the same length-L signal (i.e., ∥e∥2 = ∥De∥2 for orthonormal D), it does not imply equivalence in optimization dynamics in our setting: FCO pairs a full-trajectory spectral loss with a prefix-window time loss, and, moreover, the DCT representation can concentrate low-frequency (trend-level) errors into a small set of coefficients, making such errors more numerically salient under noisy stochastic optimization. Let E = De denote the DCT coefficients of the error. The unweighted full-band spectral loss can be written as L−1
(L)
Lfreq =
1 1X 2 ∥E∥22 = Ek , 2 2
(L)
∇E Lfreq = E.
(26)
k=0
This mode-wise decomposition makes explicit that each frequency component contributes independently, and that errors dominated by a few low-frequency modes yield sparse, concentrated gradients in coefficient space. Proposition A.7 (Gradient Concentration and Sensitivity Gain for Macroscopic Drift). Consider a macroscopic drift error modeled as a constant DC offset en = c for all n ∈ {1, . . . , L}, where |c| ≪ 1. Let ϕ0 = √1L 1 denote the DC DCT basis vector. Then the error concentrates entirely on the DC coefficient: √ E0 = ⟨e, ϕ0 ⟩ = L c, Ek = 0 for k > 0, (27) and hence the coefficient-space gradient is sparse: (√ (L) (∇E Lfreq )k =
0,
L c,
k = 0, k > 0.
(28)
Consequently, the peak gradient magnitude in coefficient space satisfies √ (L) ∥∇E Lfreq ∥∞ = L|c|.
(29) (L)
In contrast, in the time domain, the per-timestep gradient for the full-trajectory quadratic loss Ltime = 12 ∥e∥22 is uniformly distributed: (L) (L) ∇m̂n Ltime = en = c, ∀n, ∥∇m̂ Ltime ∥∞ = |c|. (30) √ Therefore, for coherent macroscopic drift, the DCT coefficient space exhibits a sensitivity gain of L in the peak gradient magnitude: (L) √ ∥∇E Lfreq ∥∞ Gain = = L. (31) (L) ∥∇m̂ Ltime ∥∞ Implication for numerical optimization (SNR viewpoint). In stochastic gradient optimization, gradient estimates are often corrupted by noise η (e.g., mini-batching noise or interference from other objective terms). When the macroscopic drift |c| is small, the uniform time-domain gradient magnitude |c| can be dominated by noise, leading to √ineffective updates. In the DCT coefficient space, however, the same drift produces a concentrated signal of magnitude L|c| on a small set of low-frequency coefficients (here, √ only the DC term), improving the effective signal-to-noise ratio. In particular, even when |c| ≲ ∥η∥∞ , the condition L|c| ≳ ∥η∥∞ can still hold for moderate L, enabling the optimizer to reliably detect and correct global trend errors. This provides a numerical explanation for why unweighted full-band spectral supervision can better capture macroscopic motion structure without requiring explicit low-frequency re-weighting. Why explicit low-frequency weighting is not superior. For completeness, consider a weighted spectral objective with W = diag(w0 , . . . , wL−1 ): L−1 1 1X 2 2 Lw-spec = ∥WE∥22 = wk Ek . (32) 2 2 k=0
Its gradients satisfy ∇E Lw-spec = W2 E,
∇m̂ Lw-spec = D⊤ W2 D e. 15
(33)
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
If wk decays with k (emphasizing low frequencies), then D⊤ W2 D shapes the time-domain gradient toward a low-pass update: it strengthens corrections to smooth global structure but suppresses higher-frequency corrections. While this can help when errors are purely trend-dominated, diverse manipulation behaviors often require fine-scale, high-frequency adjustments (e.g., contact transitions and rapid micro-corrections). Over-emphasizing low frequencies can therefore under-train these components, explaining why explicit band weighting is not better than the unweighted full-band spectral loss in practice.
B. Theoretical Analysis of Propagation Efficiency In this appendix, we analyze the proposed Locally Anchored Sampling (LAS) strategy in FocalPolicy. We formalize a metric of Target-signal Propagation Efficiency that measures how faithfully a consistency-based update at a student time τ matches an ideal supervised update, and prove that biasing the teacher (anchored) time r towards the terminal region improves this efficiency under a standard monotonicity assumption. Consequently, LAS promotes more accurate target-signal propagation during consistency training. B.1. Preliminaries and Definitions We use the same notation as Sec. 4.2. Let τ denote the student flow time and r denote the teacher (anchored) time. Along the OT path (Eq. (11)), the interpolated macro-trajectories are Mτ = (1 − τ )M0 + τ M1 ,
Mr = (1 − r)M0 + rM1 ,
(34)
where M0 is Gaussian noise and M1 is the expert macro-trajectory. Conditioned on the observation ot , the model predicts velocities vθ (Mτ , τ, ot ) and the EMA target network predicts vθ− (Mr , r, ot ). In consistency-based flow matching, for a given student timestep τ , we sample a teacher timestep r biased towards the terminal region (typically r ≥ τ ). The standard consistency loss (simplified for a single sample) is given by: 2
Lcons (θ; τ, r) = ∥vθ (Mτ , τ, ot ) − sg[vθ− (Mr , r, ot )]∥2 , where sg[·] denotes stop-gradient. The resulting (velocity-space) consistency gradient at time τ is gcons (τ, r) := ∇vθ (Mτ ,τ,ot ) Lcons = 2 vθ (Mτ , τ, ot ) − vθ− (Mr , r, ot ) .
(35)
(36)
Under the OT construction used in flow matching, the optimal target velocity is constant: u∗ (Mτ , τ, ot ) = M1 − M0 .
(37)
gsup (τ ) := ∇vθ (Mτ ,τ,ot ) ∥vθ (Mτ , τ, ot ) − u∗ (Mτ , τ, ot )∥2 = 2 vθ (Mτ , τ, ot ) − (M1 − M0 ) .
(38)
Thus the ideal supervised gradient at time τ is
Definition B.1 (Target Signal Propagation Efficiency). For a fixed student time τ , let p(r|τ ) denote the sampling distribution of the teacher time r. The propagation efficiency E(τ ) is defined as the negative expected mean squared error between the consistency gradient and the ideal supervised gradient: h i 2 E(τ ) := −Er∼p(r|τ ) ∥gcons (τ, r) − gsup (τ )∥2 . (39) Substituting the gradient definitions, we can simplify the efficiency term: 2 2 ∥gcons (τ, r) − gsup (τ )∥2 = 2 vθ (τ ) − vθ− (r) − 2 vθ (τ ) − u∗ 2 2
= 4 ∥u∗ − vθ− (Mr , r, ot )∥2 ,
(40)
where we abbreviate vθ (τ ) := vθ (Mτ , τ, ot ), vθ− (r) := vθ− (Mr , r, ot ), and u∗ := M1 − M0 . Remarkably, the student model’s prediction vθ (τ ) cancels out, indicating that the propagation efficiency depends solely on the quality of the teacher’s target estimation at time r. By construction of the OT path (Eq. (34)), the optimal velocity field implies a straight trajectory, thus u∗ (Mτ , τ ) = M1 − M0 is constant for all τ ∈ [0, 1]. Thus, the metric simplifies to the teacher’s prediction error: h i 2 E(τ ) ∝ −Er∼p(r|τ ) ∥vθ− (Mr , r, ot ) − (M1 − M0 )∥2 . (41) Eq. (41) shows that maximizing E(τ ) is equivalent to minimizing the expected teacher prediction error evaluated at the sampled teacher time r. 16
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
B.2. Proof of Superiority We now compare the proposed Locally Anchored Sampling against a standard baseline (e.g., uniform sampling or local neighborhood sampling). Assumption B.2 (Monotone Teacher Error Towards the Terminal). Let i h 2 ϵ(s) := E ∥vθ− (Ms , s, ot ) − (M1 − M0 )∥2 .
(42)
We assume ϵ(s) is strictly decreasing as s approaches the terminal boundary: ∀s1 , s2 ∈ [0, 1],
s1 < s2 ⇒ ϵ(s1 ) > ϵ(s2 ).
(43)
Definition B.3 (First-Order Stochastic Dominance). For a fixed τ , we say pLAS (r|τ ) first-order stochastically dominates pbase (r|τ ) on [τ, 1], denoted pLAS ⪰FSD pbase , if for all a ∈ [τ, 1], Pr (r ≥ a | τ ) ≥
r∼pLAS
Pr (r ≥ a | τ ),
r∼pbase
(44)
with strict inequality for some a. Proposition B.4. Under Assumption B.2, if pLAS (r|τ ) ⪰FSD pbase (r|τ ) for all τ , the Locally Anchored Sampling strategy yields strictly higher propagation efficiency than any sampling strategy that places probability mass further from the terminal state. Proof. From Eq. (41), for any sampling strategy π we can write the propagation efficiency (up to a positive constant factor) as Eπ (τ ) ∝ − Er∼pπ (r|τ ) ϵ(r) , (45) where ϵ(r) is defined in Assumption B.2 and is strictly decreasing on [0, 1]. We now compare π ∈ {LAS, base} for a fixed τ . By Definition B.3, the condition pLAS (·|τ ) ⪰FSD pbase (·|τ ) means that pLAS assigns (weakly) more probability mass to larger teacher times: Pr (r ≥ a | τ ) ≥
r∼pLAS
Pr (r ≥ a | τ ),
r∼pbase
∀a ∈ [τ, 1],
(46)
with strict inequality for at least one threshold a. Equivalently, letting FLAS (a|τ ) and Fbase (a|τ ) denote the conditional CDFs of r, this is the same as FLAS (a|τ ) ≤ Fbase (a|τ ), ∀a ∈ [τ, 1], (47) with strict inequality on a set of nonzero measure. Since ϵ(·) is strictly decreasing (Assumption B.2), the random variable ϵ(r) is therefore stochastically smaller when r is stochastically larger. Formally, a standard consequence of first-order stochastic dominance states that for any strictly decreasing measurable function ϕ, rLAS ⪰FSD rbase =⇒ E ϕ(rLAS ) < E ϕ(rbase ) , (48) whenever the dominance is strict. Applying this result with ϕ(·) = ϵ(·) yields Er∼pLAS (r|τ ) [ϵ(r)] < Er∼pbase (r|τ ) [ϵ(r)].
(49)
Finally, substituting Eq. (49) into Eq. (45) and noting the leading negative sign, we obtain ELAS (τ ) > Ebase (τ ),
(50)
In summary, by shifting the sampling distribution towards the terminal region where the teacher is more reliable, Locally Anchored Sampling effectively reduces the variance of the supervision signal, leading to higher propagation efficiency. 17
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 6. Ablation of anchor time r configurations in LAS. We compare the sampling distributions of r with different µr and σr settings. The configuration (µr = 4.0, σr = 1.6, biased toward 1.0) represents our optimal setting, which yields the best performance across most tasks. Adroit
Config. of r
Meta-World
Average
Hammer
Door
Pen
Sweep-Into
14.0 ± 4.4 80.7 ± 3.5
17.7 ± 6.7 59.7 ± 5.9
35.0 ± 7.5 63.0 ± 3.6
12.3 ± 3.1 32.0 ± 22.3
8.3 ± 1.5 16.7 ± 2.5
1.0 ± 1.0 29.0 ± 3.6
9.0 ± 1.0 52.3 ± 9.3
13.9 47.6
Ours (µr = 4.0, σr = 1.6) 100.0 ± 0.0 63.2 ± 5.3 67.5 ± 1.3 70.7 ± 22.0
29.3 ± 16.2
70.7 ± 4.2
86.7 ± 1.5
69.7
µr = 0.4, σr = 0.5 (≈ 0.6) µr = 1.4, σr = 0.5 (≈ 0.8)
H AND -I NSERT P ICK -P LACE D ISASSEMBLE
C. Experiment Details C.1. Hyperparameters Setup This section details the hyper-parameter configurations used for training and evaluating the proposed FocalPolicy. Table. 7 summarizes the complete set of hyper-parameters used in our experiments. For all baseline methods, including DP3, FlowPolicy, and FreqPolicy, we use the default hyper-parameter settings to avoid introducing unintended advantages. For both the Adroit and Meta-World benchmarks, all models are trained for 3000 training iterations to ensure convergence and a fair comparison of learning dynamics. Hyperparameters for FocalPolicy in LAS (µ = 4.0, σ = 1.6). Our theoretical analysis (Proposition B.4) indicates that the sampling distribution should be strictly skewed towards the terminal time r = 1 to maximize propagation efficiency while maintaining non-zero variance for robustness. We employ a Logit-Normal distribution r = σ(x), x ∼ N (µ, σ 2 ). • Choice of µ = 4.0: This maps to a median time of r ≈ 0.982, ensuring that the majority of consistency supervision comes from the near-terminal region where the teacher’s prediction error ϵ(r) is minimal. • Choice of σ = 1.6: This standard deviation provides a heavy left tail in the probability density. Specifically, it ensures that approximately 95% of the sampling mass is concentrated in the high-fidelity region r ∈ [0.5, 1.0], while retaining a small probability for r < 0.5 to enforce global consistency constraints preventing overfitting to the terminal state. To further investigate the impact of different anchoring strategies, we compare three representative parameter configurations for the time r: µr = 1.4, σr = 0.5 (anchored near 0.8), µr = 0.4, σr = 0.5 (anchored near 0.6), and our optimal setting µr = 4.0, σr = 1.6 (biased toward 1.0). The resulting sampling distributions for the flow timesteps τ and r, derived from 50,000 random samples, are visualized in Figure 8. We compared the performance of our model under three different sampling parameter configurations across three Adroit tasks and four MetaWorld tasks, as summarized in Table 6. The results demonstrate a significant performance degradation as the anchor time r shifts away from the terminal region; specifically, the average success rate plummets from 69.7% to 13.9% when the anchor center moves from approximately 1.0 to 0.6. These observations directly validate Assumption B.2, confirming that the teacher model’s predictive accuracy diminishes significantly as the anchor time r moves toward earlier, more stochastic flow stages. As elucidated in our theoretical analysis in Appendix B, this degradation in teacher reliability leads to a rapid decay of the target-signal propagation efficiency E(τ ), because the consistency gradient gcons fails to faithfully approximate the ideal supervised gradient gsup . Such empirical findings provide strong evidence that local anchoring within the high-fidelity terminal region is an indispensable condition for providing stable, low-variance supervision. This mechanism plays a decisive role in enabling FocalPolicy to effectively capture the complex distributions inherent in the longer macro-trajectories. C.2. Implementation details Network Architecture and Hardware. Following prior work (Chi et al., 2023; Ze et al., 2024; Jia et al., 2024; Lu et al., 2024; Zhang et al., 2025a; Su et al., 2025a), we adopt a standard 1D CNN-based U-Net architecture as the backbone of FocalPolicy to enable fair and well-controlled comparisons with existing visuomotor policies. FocalPolicy is primarily designed for 3D visuomotor manipulation. We encode the input 3D point clouds with a lightweight MLP to obtain compact feature representations, which are then fused with the action-generation backbone. This design provides a favorable trade-off between representational capacity and computational efficiency, and is consistent with the point-cloud encoders commonly 18
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 7. Hyperparameters setup for FOCAL. Hyperparameter
Value
Horizon (L) Action steps (H) Observation steps (nobs ) Pointcloud feature dim State mlp size Frequency weight (λf req ) Anchor r mean (µr ) Anchor r std (σr ) Condition type U-Net down dims Kernel size Number of groups (ngroups ) Pointnet type Batch size Number of epochs Optimizer Betas (β1 , β2 ) Learning rate Weight decay LR scheduler EMA max value
12 4 2 64 64 1.0e-4 4.0 1.6 FiLM [512, 1024, 2048] 5 8 MLP 128 3000 AdamW [0.95, 0.999] 1.0e-4 1.0e-6 Cosine (500 steps warmup) 0.9999
LAS Time Distribution Uniform Timestep r ( =4.0 =1.6, biased toward 1.0) r ( =1.4 =0.5, centered ~0.8) r ( =0.4 =0.5, centered ~0.6)
Sampling Density
8 6 4 2 0
0.0
0.2
0.4
0.6
Timestep ( , r)
0.8
1.0
Figure 8. Visualization of the sampling distributions in LAS. We compare standard uniform sampling (dashed blue curve) against our Locally Anchored Time Sampling (solid red curve) for flow timepoints. By biasing the anchor time r towards the terminal boundary (τ → 1), FOCAL significantly enhances the target signal propagation efficiency compared to uniform sampling, thereby stabilizing training for complex multi-chunk action distributions.
used in prior 3D manipulation policies (Ze et al., 2024; Zhang et al., 2025a; Su et al., 2025a). All experiments are conducted on NVIDIA RTX 4090, RTX 3090, and A6000 GPUs. The complete training process is summarized in Algorithm 1. Training Time and Computational Costs. We benchmark the computational efficiency of FocalPolicy against DP3 and FlowPolicy using a workstation equipped with an Intel Core i9-14900KF CPU and an NVIDIA GeForce RTX 4090 GPU. The evaluation is conducted across four MetaWorld tasks spanning a spectrum of difficulty levels: Reach Wall (Easy), Sweep Into (Medium), Pick Place (Hard), and Disassemble (Very Hard). Detailed metrics are summarized in Table 8. The results indicate that FocalPolicy achieves the lowest evaluation latency (26.22s). This efficiency stems from our hierarchical 19
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
Algorithm 1 FocalPolicy Training Algorithm Require: Dataset D, Train Horizon L, Execution Horizon H, Anchor params µr , σr , Weight λ. Initialize: Policy network θ, Target network θ− ← θ. ▷ Sampling Step: (ot , Mt ) ∼ D; M0 ∼ N (0, I); ϵ ∼ N (0, 1); τ ∼ U[0, 1]; r = sigmoid(µr + σr · ϵ); ▷ Construct OT interpolants Mτ = (1 − τ )M0 + τ Mt ; Mr = (1 − r)M0 + rMt ; ▷ Flow Prediction: vθτ ← vθ (Mτ , τ, ot ); vθr− ← vθ− (Mr , r, ot ); M̂τ ← fθ (Mτ , τ, ot ) = Mτ + (1 − τ ) vθτ ; M̂r ← fθ− (Mr , r, ot ) = Mr + (1 − r) vθr− ; ▷ Dual-Horizon Loss: Ltime = ∥(M̂τ )1:H − (M̂r )1:H ∥2 ; Lfreq = ∥DCT(M̂τ ) − DCT(Mt )∥2 ; Ltotal = Ltime + λLfreq ; ▷ Optimization: θ ← θ − η∇θ Ltotal ; θ− ← βθ− + (1 − β)θ; Return: Optimized Policy Parameters θ. Table 8. Computational Costs and Performance Comparison. Metrics are measured on a single workstation equipped with an Intel Core i9-14900KF CPU and an NVIDIA GeForce RTX 4090 GPU. Method
GPU Memory
Train Time
Eval Time (s) ↓
Success Rate (%) ↑
DP3 FlowPolicy
7769 MB 7377 MB
1.3 h 1.4 h
42.49 28.92
61.3 59.6
FocalPolicy
8035 MB
1.5 h
26.22
78.6
design: compared to FlowPolicy (28.92s), FocalPolicy utilizes a longer prediction horizon per inference; compared to DP3 (42.49s), it requires significantly fewer Number of Function Evaluations (NFE). Although the training time and GPU memory footprint of FocalPolicy (8035MB) are slightly higher than those of FlowPolicy (7377MB) and DP3 (7769MB) due to additional frequency-domain transformations and the modeling of extended macro-trajectories, this marginal increase in resource consumption is well-justified by a substantial gain of approximately 20% in average success rate. Module Generalizability setups. To evaluate the plug-and-play potential of our modules, we integrate Foresight Composite Objective (FCO) and Locally Anchored Sampling (LAS) into representative baselines. First, since DP3 relies on diffusion sampling steps, it is not directly compatible with our flow-based time sampling (LAS). We therefore only integrate FCO into DP3, denoted as DP3 w. Ours. Concretely, we apply the frequency-domain loss to DP3’s 16-step predicted trajectory, while applying the time-domain loss on the 8-step execution chunk. Second, for FlowPolicy, we integrate our LAS module, denoted as FP w. Ours. In this variant, we replace FlowPolicy’s original time sampling with our LAS. Since our sampling strategy differs from standard consistency flow matching, we remove FlowPolicy’s original segmented time-domain loss to align with the LAS formulation. Note that we do not integrate FCO into FlowPolicy in this plug-and-play test. This is because FlowPolicy, in its default configuration, predicts and executes only 4 steps. This short prediction horizon lacks the multi-chunk macro-trajectory structure necessary for frequency-domain structural regularization. Integrating FCO would require altering FlowPolicy’s horizon settings, which is beyond the scope of this evaluation of module compatibility. C.3. Simulation experiment results To comprehensively evaluate the robustness and generalization of the FocalPolicy, we report the performance across 53 challenging manipulation tasks from the Adroit and Meta-World benchmarks. For each task, we conduct evaluations over 3 independent random seeds (0, 1, 2). We report the mean success rate (%) along with the sample standard deviation to rigorously account for stochastic variability during training and rollout. To ensure a principled and fair comparison, all evaluated methods—including our FocalPolicy and all baselines—are trained on the identical set of expert demonstrations and evaluated under a unified protocol. The exhaustive per-task results are documented in Table. 15. 20
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 9. Supplementary evaluation on RoboMimic and LIBERO benchmarks. We report the mean success rate (0–1) across diverse and modern tasks, alongside the average inference latency (Cost). RoboMimic
Method \Task
LIBERO
Average ↑
Cost (ms) ↓
0.54 0.64 0.22
0.73 0.76 0.70
5.40 18.90 9.50
0.45
0.81
5.40
Lift
Can
Square
Transport
Open-Drawer
Put-Bowl
Stack-Bowl
FlowPolicy FreqPolicy ACG
0.99 0.98 0.99
0.98 0.90 0.99
0.74 0.74 0.78
0.44 0.38 0.50
0.88 1.00 0.97
0.55 0.70 0.46
Ours
1.00
1.00
0.82
0.67
1.00
0.73
Generalization to Modern and Diverse Benchmarks. To further validate the robustness and generalization of FocalPolicy beyond scripted heuristics, we extend our evaluation to more diverse and modern benchmarks, including RoboMimic (Mandlekar et al., 2021) and LIBERO (Liu et al., 2023). Unlike Adroit and Meta-World, RoboMimic provides expert datasets collected from human demonstrators, which introduce significant behavioral diversity and noise, posing a greater challenge for action chunking. LIBERO (2023) represents a more recent and complex suite of tasks designed to test the limits of imitation learning policies. As summarized in Table 9, we compare FocalPolicy against several strong baselines, including ACG (Park et al., 2025) (integrated into FlowPolicy for a fair comparison), which is explicitly designed to address cross-chunk coherence. FocalPolicy consistently outperforms all baselines across both benchmarks. Notably, on the high-dimensional human-demonstrated RoboMimic tasks (e.g., Square and Transport), FocalPolicy achieves the highest success rates, demonstrating its superior ability to model complex, multi-modal expert distributions. Furthermore, we evaluate the computational efficiency in terms of mean inference latency. Despite its enhanced foresight and coherence, FocalPolicy maintains a low inference cost of 5.40 ms, matching the vanilla FlowPolicy and significantly outperforming FreqPolicy (18.90 ms) and ACG (9.50 ms). This efficiency stems from our FCO formulation, which regularizes the policy during training without requiring auxiliary modules or iterative optimization during inference. These results confirm that FocalPolicy advances the state-of-the-art by providing a more effective and efficient solution for coherent long-horizon manipulation. Mitigation of Compounding Errors. To evaluate the robustness of our model against compounding errors in longtrajectory prediction, we conduct a comparative analysis between FocalPolicy and the baseline FlowPolicy. Both models are trained for 3,000 epochs on the same expert training set using identical configurations: a macro-trajectory length of L = 36 and an execution chunk of H = 12. To isolate the accumulated drift inherent in temporal predictions, we perform an open-loop rollout on a held-out test set. Specifically, both policies are initialized using only the first two frames of observations from the test sequences. From this initial state, the models predict the full subsequent trajectory of 36 steps without further environmental feedback. Given that MetaWorld data is recorded in incremental coordinates, we normalize the starting position of all trajectories to the origin (0, 0, 0) to facilitate a direct comparison of spatial deviation. We then visualize the predicted trajectories against the ground-truth expert demonstrations and quantify the compounding error at each timestep using the Euclidean distance of absolute positions. This setup rigorously tests the policies’ ability to maintain structural coherence over long trajectories without the corrective benefit of closed-loop observations. As shown in Figure 9, to provide a more comprehensive demonstration of FocalPolicy, we compare the predicted trajectories and compounding errors against FlowPolicy on additional MetaWorld tasks, including Pick Place, Pick Out of Hole, Sweep Into, Pick Place Wall Disassemble and Shelf Place. Table 10. Quantitative comparison of Trajectory Smoothness score (TSscore ↓). A lower score indicates that the policy’s trajectory smoothness is closer to the expert’s. Method FlowPolicy Ours
Bin Picking
Disassemble
Pick Out of Hole
Pick Place
Pick Place Wall
Push Wall
Shelf Place
Sweep Into
0.27 0.13
0.09 0.01
0.04 0.02
0.01 0.01
0.03 0.00
0.09 0.00
0.03 0.01
0.50 0.40
Quantitative Evaluation of Trajectory Smoothness. Furthermore, we calculated the Trajectory Smoothness (TS) metric introduced in (Songwei et al., 2026) to rigorously evaluate the temporal smoothness of the predicted trajectories. In this 21
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy (a) Pick Place
(b) Pick Out of Hole Compounding Errors
Compounding Errors
Step
Step
(d) Pick Place Wall
(c) Sweep Into Compounding Errors
Compounding Errors
Step
Step
(f) Shelf Place
(e) Disassemble Compounding Errors
Compounding Errors
Step
Step
Figure 9. Comparison of compounding errors. We visualize the 3D end-effector trajectories (left) inferred by FocalPolicy and FlowPolicy on more representative simulation tasks, given the same initial state. The corresponding error curves (right) represent the Euclidean distance between the predicted and ground-truth coordinates (||ppred − pgt ||2 ) at each time step.
context, we utilize TS as a relative measure of imitation fidelity rather than an absolute performance metric. Higher trajectory quality is defined by how closely a policy’s TS aligns with the expert’s TS, rather than its absolute magnitude. For a more intuitive representation, we define the TS score as TSscore (↓) = |TSexpert − TSpolicy | and report these scores in Table 10. As shown in the table, FocalPolicy achieves lower or equal TSscore across all 8 evaluated tasks, indicating that its predicted trajectories better match the smoothness of the expert demonstrations. This demonstrates that FocalPolicy significantly improves the modeling of multi-chunk trajectories compared to the baseline FlowPolicy. Through the comparison of these quantitative metrics, we demonstrate that FocalPolicy produces more coherent action trajectories that closely resemble expert actions, effectively addressing the challenge of discontinuity across action chunks in imitation learning. We attribute this improvement to the Foresight Composite Objective (FCO), which enhances the model’s foresight in action prediction, and the Locally Anchored Sampling (LAS) strategy, which improves the learning efficiency of the policy. C.4. Real-World experiment details Hardware Specifications. Our real-world robotic platform consists of a UR-10e 6-DOF industrial arm and an AG-95 two-finger gripper. Visual input is captured by a fixed RealSense D435i RGB-D camera at a resolution of 840 × 480. All computations, including policy inference and vision processing, are performed on a workstation equipped with an Intel Core i9-14900KF CPU and an NVIDIA GeForce RTX 4090 GPU. Expert Demonstration and Training. For each task, we collected 30-50 expert demonstrations using a Logitech F710 Wireless Gamepad, recording end-effector Cartesian poses and gripper states. During training, we applied standard data augmentations, including random cropping and color jittering, to enhance the policy’s visual robustness. Task Definitions and Multi-stage Decomposition. As illustrated in Figure 10, we designed six challenging tasks to evaluate 22
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
Water Pouring Stage 1
Stage 2
Stage 1
Stage 2
Drawer Loading
Pot Loading Stage 1
Stage 2
Stage 3
Tower Stacking Stage 1
Stage 2
Stage 3
Cup Matching Stage 1
Stage 2
Stage 3
Stage 4
Object Sorting Stage 1
Stage 2
Stage 3
Stage 4
Figure 10. Illustration of real-world task phase decomposition. The figure displays the phase segmentation of six tasks on the left, alongside the corresponding variant types for each task on the right. Table 11. Real-world results across staged manipulation tasks. We report Success Rates (%) ↑ / Score ↑ for six real-world tasks—Water Pouring, Pot Loading, Cup Matching, Drawer Loading, Tower Stacking, and Object Sorting—each decomposed into sequential stages (Stage 1, Stage 2, etc.).
Alg \ Task
Water Pouring Stage 1 Stage 2
Stage 1
Pot Loading Stage 2
Stage 3
Stage 1
Cup Matching Stage 2 Stage 3
Stage 4
DP3 FreqPolicy FlowPolicy FocalPolicy
80.0 / 12.0 73.3 / 11.0 83.3 / 12.5 86.7 / 13.0
60.0 / 9.0 50.0 / 7.5 53.3 / 8.0 63.3 / 9.5
76.7 / 11.5 63.3 / 9.5 66.7 / 10.0 76.7 / 11.5
53.3 / 8.0 46.7 / 7.0 50.0 / 7.5 53.3 / 8.0
30.0 / 4.5 23.3 / 3.5 20.0 / 3.0 33.3 / 5.0
70.0 / 10.5 63.3 / 9.5 60.0 / 9.0 73.3 / 11.0
53.3 / 8.0 46.7 / 7.0 50.0 / 7.5 56.7 / 8.5
43.3 / 6.5 36.7 / 5.5 33.3 / 5.0 50.0 / 7.5
23.3 / 3.5 10.0 / 1.5 13.3 / 2.0 30.0 / 4.5
Alg \ Task
Drawer Loading Stage 1 Stage 2
Stage 1
Tower Stacking Stage 2 Stage 3
Stage 1
Object Sorting Stage 2 Stage 3
Stage 4
DP3 FreqPolicy FlowPolicy FocalPolicy
90.0 / 13.5 86.7 / 13.0 80.0 / 12.0 93.3 / 14.0
80.0 / 12.0 76.7 / 11.5 83.3 / 12.5 86.7 / 13.0
83.3 / 12.5 86.7 / 13.0 80.0 / 12.0 86.7 / 13.0
66.7 / 10.0 70.0 / 10.5 63.3 / 9.5 73.3 / 11.0
20.0 / 3.0 16.7 / 2.5 6.7 / 1.0 26.7 / 4.0
60.0 / 9.0 53.3 / 8.0 56.7 / 8.5 66.7 / 10.0
56.7 / 8.5 53.3 / 8.0 46.7 / 7.0 53.3 / 8.0
33.3 / 5.0 23.3 / 3.5 20.0 / 3.0 40.0 / 6.0
36.7 / 5.5 33.3 / 5.5 26.7 / 4.0 46.7 / 7.0
FocalPolicy’s ability to handle long-horizon, multi-stage manipulation: • 1) Water Pouring (2 stages, 4 variants): The robot grasps a kettle and maneuvers it above a cup. Subsequently, it pours water into the cup and returns the kettle to a designated resting position. This task challenges the smoothness and accuracy of the manipulation trajectory. • 2) Drawer Loading (2 stages, 6 variants): The robot deposits an object into an open drawer (on either the top or bottom tier) and subsequently closes the drawer using its end-effector. This task evaluates the coordination between free-space object manipulation and articulated mechanism actuation. 23
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy
• 3) Pot Loading (3 stages, 3 variants): The robot lifts and removes the pot lid, deposits an ingredient into the pot, and places the lid back to cover it. This task evaluates the policy’s capability to manipulate unattached constraints (the lid) and coordinate sequential interactions within a confined space. • 4) Tower Stacking (3 stages, 3 variants): The robot sequentially stacks three objects of varying shapes and decreasing sizes. This task requires high vertical alignment accuracy and stable object release. • 5) Cup Matching (4 stages, 4 variants): The robot relocates three colored cups onto their corresponding saucers. This task requires strict adherence to color matching, focusing on ordered repetitive operations and precise placement. • 6) Object Sorting (4 stages, 3 variants): The robot sequentially sorts four items into two bins based on attributes. This task involves repeated pick-and-place operations, serving as a benchmark for testing long-horizon execution stability. Evaluation Metric. To quantify performance, we employ a graded scoring system for each stage: 0.5 points for contacting the target (demonstrating intent) and 1.0 point for successful completion. We compute the average completion rates over 15 independent trials per task. Detailed results are presented in Table 11.
D. Ablation Study Results This section presents a comprehensive ablation analysis of FocalPolicy, accompanied by detailed experimental results. Consistent with the experimental setup in the main manuscript, our comprehensive evaluation spans a total of 24 simulation tasks. The benchmarks encompass Adroit (denoted as A) and MetaWorld tasks, where the latter are categorized into three difficulty levels based on their complexity: Medium (M), Hard (H), and Very Hard (VH). The analysis is organized into three parts: First, Sec. D.1 elaborates on the Component Ablations, corresponding to the Component Effectiveness analysis in the main text. Notably, to gain deeper insights into the contributions of various design choices, we extend our analysis beyond the main manuscript by investigating two supplementary variants: FCO-FullMacro and FCO-Prox-Freq. Subsequently, Sec. D.2 presents the Horizon Ablation, providing specific configurations and detailed results corresponding to the Impact of Horizon Hyperparameters analysis. Third, Appendix D.3 conducts a Sensitivity Analysis of λ, empirically justifying the choice of the balancing coefficient within the Foresight Composite Objective. Finally, Appendix D.4 presents the Ablation on Spectral Components, investigating the individual impacts of low-frequency and high-frequency components on modeling long-horizon manipulation trajectories. D.1. Component Ablations To isolate the contributions of the proposed Foresight Composite Objective (FCO) and Locally Anchored Sampling (LAS), and to examine the sensitivity of specific design choices, we evaluate a comprehensive set of variants. Unless otherwise specified, all variants maintain the same macro-trajectory horizons (L, H) and training hyperparameters as the original FocalPolicy to ensure a controlled comparison. We report success rates (mean ± std) over three random seeds. Ablation Configurations.
We define the specific implementation details of the compared variants as follows:
• w/o LAS: Replaces the locally anchored sampling with standard uniform sampling (as in FlowPolicy), testing the benefit of the biased sampling strategy. • w/o FCO & LAS: Removes both LAS and FCO, yielding a baseline similar to FlowPolicy but using our horizon configuration (L, H). • Ltime -Only FCO: Replaces the frequency-domain structural regularization in FCO with a standard time-domain MSE loss acting on the entire macro-trajectory. • Lfreq -Only FCO: Trains the policy using solely the frequency-domain loss, removing the temporal alignment constraint on the proximal chunk. • Fixed-r LAS: A deterministic version of LAS where the anchor time r is fixed to the terminal boundary (r = 1). This variant tests the necessity of stochasticity in anchor sampling. • FCO-Full-Macro: Extends the time-domain loss Ltime to the entire macro-trajectory, aligning its horizon with Lfreq . 24
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 12. Ablation study of component effectiveness on 24 tasks. We report the average success rate (%) across tasks.
A(3)
M(11)
H(5)
VH(5)
Average
Ours w/o LAS w/o FCO&LAS
76.9 ± 2.2 71.7 ± 1.8 46.2 ± 12.1
81.9 ± 1.6 65.6 ± 2.9 56.5 ± 3.2
62.2 ± 4.6 47.8 ± 1.8 43.1 ± 0.2
85.1 ± 1.6 73.3 ± 1.1 64.7 ± 2.4
77.9 64.3 54.1
Ltime -Only FCO Lfreq -Only FCO Fixed-r LAS FCO-Full-Macro FCO-Prox-Freq
34.3 ± 1.2 74.0 ± 1.0 70.4 ± 2.2 73.8 ± 0.5 72.8 ± 2.5
65.9 ± 4.8 74.4 ± 1.9 77.4 ± 2.0 75.4 ± 1.0 75.4 ± 2.1
49.4 ± 2.1 60.1 ± 1.3 59.0 ± 3.7 60.2 ± 1.2 60.7 ± 1.0
78.1 ± 1.7 81.4 ± 1.8 80.2 ± 1.7 81.9 ± 1.6 83.0 ± 2.7
61.1 72.8 73.3 73.4 73.6
Designs
• FCO-Prox-Freq: Reverses our primary design by constraining the frequency-domain loss Lfreq to the proximal chunk (H), while applying the time-domain loss Ltime to the full macro-trajectory (L). Effectiveness of Key Modules. As shown in Table 12, FocalPolicy achieves the strongest and most consistent performance across the Adroit and Meta-World task groups. Removing LAS (w/o LAS) consistently degrades performance, indicating that locally anchored sampling provides a more effective training signal than uniform time sampling for learning the target distribution. When both modules are removed (w/o FCO&LAS), performance drops further, suggesting that FCO and LAS are complementary: FCO improves trajectory-level coherence via multi-chunk structural regularization, while LAS stabilizes and accelerates optimization by strengthening target-signal propagation. Analysis of Objective Designs. We further analyze the impact of different loss formulations based on the results in Table 12: • Time vs. Frequency Supervision: Ltime -Only FCO exhibits the most significant performance degradation, particularly on Adroit where the average success rate plummets from 76.9% to 34.3%. This highlights that vanilla temporal alignment is insufficient for capturing the overall motion trend, whereas spectral-domain supervision provides a more structured signal for global trajectory modeling. Conversely, Lfreq -Only FCO also leads to suboptimal results (e.g., 74.0% on Adroit and 74.4% on Meta-World), indicating that frequency alignment alone lacks the precision required for accurate proximal execution. • Horizon Alignment Strategy: FCO-Full-Macro performs generally weaker than FocalPolicy, supporting our design choice of applying the time-domain loss only to the proximal chunk for precision, while using the frequency-domain loss for global trends. Furthermore, the performance drop in FCO-Prox-Freq confirms that the primary strength of our DCT-based loss lies in enforcing global structural consistency over the long horizon, rather than serving as a tool for local proximal alignment. • Stochasticity in LAS: The performance of Fixed-r LAS is typically inferior to the stochastic version, suggesting that retaining randomness in the anchor time r (while keeping it biased towards the terminal) improves training stability and robustness. Detailed experimental results for all individual tasks are provided in Table 16. D.2. Horizon Ablation. Table 17 studies the effect of chunk size H and macro multiplier N (i.e., macro-horizon L = N H) on Adroit and MetaWorld tasks (excluding Easy tasks). Across configurations, moderate macro-horizons (e.g., H=4, N =3 in our default setting) tend to offer the best trade-off: increasing N enlarges the foresight window and can improve global structure, but overly large N makes the macro-trajectory distribution harder to model and may reduce success rates. Similarly, larger H increases per-chunk prediction difficulty and can hurt performance on tasks that require fine-grained control. These results motivate our default choice of H=4 and N =3, which provides robust performance across benchmarks while keeping inference efficient. 25
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 13. Ablation study on the balancing coefficient λ across six validation tasks. Results are reported as success rates (0–1).
λ −3
1 · 10 5 · 10−4 1 · 10−4 5 · 10−5 1 · 10−5
Bin-Picking
Peg-Insert-Side
Push-Wall
Sweep-Into
Hand-Insert
Shelf-Place
Disassemble
Average ↑
0.21 0.24 0.41 0.39 0.15
0.76 0.81 0.84 0.82 0.74
0.75 0.80 0.84 0.81 0.38
0.83 0.87 0.88 0.85 0.24
0.38 0.46 0.48 0.37 0.10
0.70 0.69 0.75 0.65 0.54
0.78 0.81 0.85 0.85 0.76
0.63 0.67 0.72 0.68 0.42
Table 14. Ablation study on the impact of low-frequency (LF) and high-frequency (HF) components across four MetaWorld tasks. Results are reported as success rates (0–1).
Variant Ours w. Only LF Ours w. Only HF Ours
Sweep-Into
Hand-Insert
Pick-Place
Disassemble
Average ↑
0.56 0.41 0.71
0.25 0.23 0.29
0.70 0.66 0.71
0.78 0.82 0.87
0.57 0.53 0.65
D.3. Sensitivity Analysis of λ. The hyperparameter λ in Eq. (7) balances the proximal time-domain consistency Ltime and the foresight frequency-domain regularization Lfreq . As shown in Table 13, we evaluate the sensitivity of FocalPolicy to λ by sweeping across five candidates: {1 · 10−3 , 5 · 10−4 , 1 · 10−4 , 5 · 10−5 , 1 · 10−5 } on seven representative validation tasks. We observe that λ = 10−4 consistently achieves the highest average success rate of 0.72. The relatively small magnitude of λ is primarily due to the scale difference between the two objectives: Lfreq aggregates spectral information over the entire macro-trajectory L, leading to naturally larger numerical gradients compared to the single-chunk Ltime . Increasing λ to 10−3 tends to over-regularize the policy, potentially sacrificing the fine-grained precision of proximal actions, while decreasing it to 10−5 diminishes the foresight-aware structural constraint, leading to a drop in cross-chunk coherence. These empirical results justify our choice of λ = 10−4 as a robust balancing coefficient. D.4. Ablation on Spectral Components. As discussed in Section 4.1, we posit that low-frequency (LF) components encode the global motion trend of the trajectory, while high-frequency (HF) components represent fine-grained execution details. To empirically validate the necessity of utilizing the full frequency spectrum, we conduct an ablation study isolating the impact of these components. Table 14 compares the performance of the full FocalPolicy against two degraded variants: one supervised exclusively by the low-frequency component (Ours w. Only LF) and another exclusively by the high-frequency component (Ours w. Only HF). We evaluate these variants across four representative MetaWorld tasks. As shown in the results, omitting either spectral component leads to noticeable performance degradation. Specifically, relying solely on LF or HF components results in an average success rate drop of 0.08 and 0.12, respectively, compared to our full model (0.65). These findings support our hypothesis that the frequency-domain supervision in FocalPolicy effectively captures holistic trajectory patterns by balancing common structural trends (LF) and specific local behaviors (HF). By integrating both, our method provides a comprehensive characterization of global trajectory features across multiple chunks. Exploring more sophisticated mechanisms to dynamically weight or combine these two spectral components remains an interesting direction for future work.
26
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 15. Complete simulation results across all tasks in Adroit and MetaWorld. Values represent the success rate (mean ± standard deviation) over multiple seeds.
Alg \ Task
Adroit Door
Pen
53.5 ± 5.1 53.0 ± 6.2 58.3 ± 3.8 63.2 ± 5.3
59.3 ± 3.5 61.0 ± 3.5 53.0 ± 3.5 67.5 ± 1.3
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
Dial Turn
Door Close
Door Lock
MetaWorld (Easy) Door Open Door Unlock Drawer Close Drawer Open Faucet Close
65.7 ± 2.3 87.0 ± 1.0 65.7 ± 4.2 86.0 ± 1.7
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
97.7 ± 0.6 100.0 ± 0.0 92.0 ± 6.2 100.0 ± 0.0
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
100.0 ± 0.0 100.0 ± 0.0 98.0 ± 1.7 100.0 ± 0.0
96.0 ± 2.6 100.0 ± 0.0 92.7 ± 1.2 100.0 ± 0.0
Faucet Open Handle Press
Handle Pull
MetaWorld (Easy) Handle Press Handle Pull Side Side
Lever Pull
Plate Slide
Plate Slide Back
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
41.0 ± 9.5 48.3 ± 21.1 40.7 ± 10.0 41.3 ± 14.4
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
76.0 ± 5.0 79.7 ± 2.3 69.0 ± 5.3 83.3 ± 3.1
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
Hammer
DP3 98.3 ± 2.9 FlowPolicy 94.0 ± 0.0 FreqPolicy 94.3 ± 2.1 FocalPolicy 100.0 ± 0.0 Alg \ Task DP3 FlowPolicy FreqPolicy FocalPolicy Alg \ Task
DP3 FlowPolicy FreqPolicy FocalPolicy Alg \ Task
DP3 FlowPolicy FreqPolicy FocalPolicy Alg \ Task DP3 FlowPolicy FreqPolicy FocalPolicy Alg \ Task
DP3 FlowPolicy FreqPolicy FocalPolicy Alg \ Task
82.7 ± 1.5 100.0 ± 0.0 85.0 ± 1.0 100.0 ± 0.0
MetaWorld (Easy) Button Press Button Press Button Press Button Press Topdown Topdown Wall Wall 99.7 ± 0.6 100.0 ± 0.0 69.3 ± 7.6 100.0 ± 0.0
89.3 ± 2.1 58.0 ± 5.0 68.3 ± 4.7 56.0 ± 6.2
85.7 ± 8.3 100.0 ± 0.0 58.3 ± 2.1 100.0 ± 0.0
MetaWorld (Easy) Reach Wall Window Close
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
Coffee Button 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
Plate Slide Back Side
Plate Slide Side
Reach
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
22.7 ± 2.1 27.0 ± 6.1 22.3 ± 3.5 24.3 ± 2.9
Basketball
Bin Picking
Box Close
100.0 ± 0.0 88.3 ± 9.8 99.0 ± 1.7 100.0 ± 0.0
65.3 ± 14.2 49.7 ± 24.8 28.7 ± 12.4 64.3 ± 20.3
59.7 ± 3.2 54.3 ± 6.0 48.0 ± 6.1 61.3 ± 1.5
Push Wall
MetaWorld (Medium) Soccer Sweep
Sweep Into
Assembly
MetaWorld (hard) Hand Insert Pick Out of Hole
Pick Place
95.3 ± 4.7 49.0 ± 3.6 89.7 ± 5.5 81.0 ± 6.6
31.7 ± 8.4 40.3 ± 8.1 24.3 ± 1.5 41.0 ± 7.5
99.3 ± 1.2 100.0 ± 0.0 90.3 ± 3.1 100.0 ± 0.0
39.3 ± 25.3 37.3 ± 31.8 19.3 ± 8.5 70.7 ± 22.0
98.0 ± 1.7 98.3 ± 1.5 98.0 ± 1.0 99.7 ± 0.6
14.3 ± 2.9 16.0 ± 2.6 16.0 ± 3.0 29.3 ± 16.2
60.7 ± 11.9 55.3 ± 2.3 57.7 ± 4.2 70.7 ± 4.2
MetaWorld (Hard) Push Push Back
Shelf Place
MetaWorld (Very Hard) Disassemble Stick Pull Stick Push
58.7 ± 2.1 62.3 ± 6.1 39.3 ± 11.6 69.3 ± 4.9
79.3 ± 7.0 71.0 ± 7.8 85.3 ± 1.5 86.7 ± 1.5
DP3 71.3 ± 5.0 FlowPolicy 80.3 ± 2.5 FreqPolicy 66.7 ± 10.7 FocalPolicy 76.3 ± 4.7
0.0 ± 0.0 0.0 ± 0.0 0.0 ± 0.0 0.0 ± 0.0
66.0 ± 5.3 74.7 ± 5.0 74.3 ± 2.5 86.3 ± 2.5
Window Open
Peg Unplug Side
100.0 ± 0.0 100.0 ± 0.0 93.3 ± 0.6 100.0 ± 0.0
79.0 ± 3.0 77.3 ± 2.1 62.0 ± 3.6 82.3 ± 3.1
Hammer
Peg Insert Side
100.0 ± 0.0 99.0 ± 1.7 100.0 ± 0.0 99.7 ± 0.6
78.7 ± 9.5 83.0 ± 6.1 63.7 ± 9.6 88.3 ± 4.5
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
MetaWorld (Medium) Coffee Pull Coffee Push 88.3 ± 2.9 95.3 ± 3.2 82.0 ± 7.0 98.0 ± 1.0
27
91.3 ± 6.1 93.0 ± 2.6 87.3 ± 6.7 96.7 ± 1.5
67.7 ± 5.0 78.3 ± 2.1 72.7 ± 3.8 80.7 ± 1.5
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
38.3 ± 0.6 21.0 ± 4.6 40.0 ± 2.0 35.0 ± 7.2
Pick Place Wall 85.0 ± 4.4 77.7 ± 3.2 74.7 ± 2.3 89.0 ± 4.4
Overall Avg
79.9 79.4 75.1 83.6
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 16. Ablation results on Adroit and MetaWorld (excluding Easy tasks). Values are success rate (%, mean ± std) over multiple seeds.
Hammer
Adroit Door
Pen
FocalPolicy w/o LAS w/o FCO&LAS
100.0 ± 0.0 95.0 ± 2.6 58.0 ± 8.7
63.2 ± 5.3 55.0 ± 2.6 39.0 ± 3.6
67.5 ± 1.3 65.0 ± 5.6 41.7 ± 29.4
Ltime -Only FCO Lfreq -Only FCO LAS → r = 1 FCO-Full-Macro FCO-Prox-Freq
55.0 ± 4.0 97.0 ± 3.0 97.3 ± 4.6 97.7 ± 1.5 99.0 ± 0.0
7.0 ± 1.7 57.0 ± 1.0 54.7 ± 3.5 54.7 ± 2.5 56.3 ± 3.5
41.0 ± 3.5 68.0 ± 3.6 59.3 ± 4.0 69.0 ± 2.0 63.0 ± 6.9
Variant \ Task
Basketball
Bin Picking
MetaWorld (Medium) Box Close Coffee Pull Coffee Push
Hammer
Peg Insert Side
FocalPolicy w/o LAS w/o FCO&LAS
100.0 ± 0.0 70.0 ± 9.5 56.3 ± 20.6
64.3 ± 20.3 37.0 ± 18.0 20.0 ± 7.2
61.3 ± 1.5 49.7 ± 10.4 38.0 ± 10.0
98.0 ± 1.0 91.7 ± 2.9 86.7 ± 5.0
96.7 ± 1.5 94.7 ± 3.2 86.0 ± 2.6
99.7 ± 0.6 99.7 ± 0.6 99.0 ± 1.7
88.3 ± 4.5 70.3 ± 11.7 54.3 ± 7.5
Ltime -Only FCO Lfreq -Only FCO LAS → r = 1 FCO-Full-Macro FCO-Prox-Freq
84.3 ± 12.3 94.3 ± 9.8 100.0 ± 0.0 99.0 ± 1.7 99.0 ± 1.7
50.0 ± 20.8 51.7 ± 16.4 51.7 ± 29.2 56.7 ± 14.5 54.7 ± 16.4
34.0 ± 7.0 49.0 ± 3.0 56.7 ± 7.5 47.3 ± 3.8 49.0 ± 9.0
93.3 ± 5.7 98.3 ± 0.6 97.3 ± 2.3 97.3 ± 2.3 96.7 ± 0.6
85.3 ± 2.5 96.0 ± 3.5 94.0 ± 5.0 94.7 ± 2.9 93.0 ± 5.6
98.7 ± 1.5 100.0 ± 0.0 99.0 ± 1.7 100.0 ± 0.0 100.0 ± 0.0
76.0 ± 2.6 76.3 ± 3.5 82.3 ± 6.8 73.3 ± 6.4 74.0 ± 10.1
Sweep Into
Assembly
Variant \ Task
Variant \ Task
Push Wall
MetaWorld (Medium) Soccer Sweep
MetaWorld (Hard) Hand Insert Pick Out Hole
FocalPolicy w/o LAS w/o FCO&LAS
81.0 ± 6.6 56.7 ± 4.7 43.3 ± 10.1
41.0 ± 7.5 44.7 ± 12.7 35.7 ± 7.1
100.0 ± 0.0 87.0 ± 9.6 82.7 ± 4.7
70.7 ± 22.0 20.3 ± 1.5 19.3 ± 5.1
99.7 ± 0.6 96.7 ± 2.5 94.3 ± 3.5
29.3 ± 16.2 11.0 ± 1.0 10.3 ± 2.1
35.0 ± 7.2 13.3 ± 6.1 13.3 ± 4.5
Ltime -Only FCO Lfreq -Only FCO LAS → r = 1 FCO-Full-Macro FCO-Prox-Freq
46.7 ± 25.4 51.7 ± 12.3 68.0 ± 4.4 66.3 ± 4.0 67.3 ± 12.1
38.0 ± 8.7 35.0 ± 9.5 37.0 ± 8.7 33.7 ± 7.8 35.7 ± 13.7
77.3 ± 4.0 95.3 ± 3.1 98.3 ± 2.9 96.7 ± 2.9 98.3 ± 1.5
41.7 ± 30.9 70.3 ± 11.7 67.3 ± 18.6 64.0 ± 17.1 61.3 ± 21.0
93.0 ± 4.4 100.0 ± 0.0 99.0 ± 1.7 99.7 ± 0.6 99.7 ± 0.6
23.0 ± 13.9 22.0 ± 10.6 26.3 ± 15.2 21.3 ± 8.4 22.0 ± 8.7
6.3 ± 5.9 33.3 ± 2.1 28.7 ± 3.1 34.0 ± 7.0 34.7 ± 3.1
Variant \ Task
MetaWorld (Hard) Pick Place Push
Shelf Place
MetaWorld (Very Hard) Disassemble Stick Pull Stick Push
FocalPolicy w/o LAS w/o FCO&LAS
70.7 ± 4.2 63.0 ± 5.3 56.0 ± 3.5
76.3 ± 4.7 55.0 ± 4.6 41.7 ± 3.5
69.3 ± 4.9 52.3 ± 5.5 41.7 ± 2.1
86.7 ± 1.5 67.3 ± 2.9 60.7 ± 4.5
80.7 ± 1.5 68.3 ± 4.0 53.3 ± 5.1
100.0 ± 0.0 100.0 ± 0.0 99.7 ± 0.6
89.0 ± 4.4 78.7 ± 4.5 68.0 ± 7.9
Ltime -Only FCO Lfreq -Only FCO LAS → r = 1 FCO-Full-Macro FCO-Prox-Freq
67.3 ± 7.6 67.0 ± 9.5 67.0 ± 6.6 68.3 ± 7.0 68.0 ± 6.2
57.3 ± 1.5 78.0 ± 7.0 74.0 ± 6.1 77.7 ± 6.4 79.0 ± 1.7
57.0 ± 3.0 67.7 ± 6.1 56.3 ± 4.5 69.0 ± 7.2 70.0 ± 3.0
83.0 ± 1.7 79.7 ± 2.1 82.0 ± 5.6 80.3 ± 3.8 80.3 ± 2.5
79.7 ± 2.5 77.7 ± 2.5 76.7 ± 3.2 76.7 ± 4.7 80.7 ± 5.5
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
70.7 ± 10.0 82.0 ± 9.0 86.0 ± 10.4 83.7 ± 6.0 84.0 ± 5.3
28
Pick Place Wall
FocalPolicy: Frequency-Optimized Chunking and Locally Anchored Flow Matching for Coherent Visuomotor Policy Table 17. Horizon ablation results on Adroit and MetaWorld (excluding Easy tasks). Values are success rate (%, mean ± std) over multiple seeds.
Hammer
Adroit Door
Pen
2 3 4 5
97.7 ± 2.1 100.0 ± 0.0 98.3 ± 2.1 95.0 ± 4.4
59.7 ± 6.7 63.2 ± 5.3 61.3 ± 5.9 56.7 ± 4.2
63.3 ± 2.5 67.5 ± 1.3 68.7 ± 4.5 59.7 ± 9.1
8
2 3 4 5
97.7 ± 1.5 97.3 ± 2.5 98.0 ± 1.7 93.7 ± 1.5
61.3 ± 0.6 61.7 ± 4.7 63.0 ± 6.0 55.7 ± 8.6
62.0 ± 7.0 64.3 ± 4.5 59.7 ± 5.6 68.3 ± 6.1
12
2 3 4 5
97.3 ± 3.1 94.7 ± 4.0 93.3 ± 4.7 91.0 ± 4.4
54.3 ± 2.1 56.7 ± 2.3 48.0 ± 3.6 52.0 ± 0.0
66.3 ± 1.5 69.3 ± 2.1 67.7 ± 2.1 65.3 ± 2.3
H
N
4
Basketball
Bin Picking
Box Close
MetaWorld (Medium) Coffee Pull
Coffee Push
Hammer
Peg Insert Side
4
2 3 4 5
99.0 ± 1.7 100.0 ± 0.0 99.7 ± 0.6 83.3 ± 11.9
55.0 ± 26.0 64.3 ± 20.3 55.0 ± 25.0 15.3 ± 7.5
57.7 ± 11.9 61.3 ± 1.5 57.0 ± 2.6 48.0 ± 1.0
95.3 ± 2.5 98.0 ± 1.0 96.3 ± 1.2 90.0 ± 3.5
92.3 ± 3.2 96.7 ± 1.5 93.3 ± 3.2 93.0 ± 1.7
99.0 ± 1.7 99.7 ± 0.6 99.0 ± 1.7 99.0 ± 1.0
84.0 ± 7.0 88.3 ± 4.5 85.0 ± 4.4 78.0 ± 1.0
8
2 3 4 5
99.0 ± 1.7 92.7 ± 4.9 100.0 ± 0.0 99.7 ± 0.6
69.7 ± 13.8 40.7 ± 36.7 66.3 ± 11.6 57.7 ± 23.2
53.7 ± 9.5 51.0 ± 7.8 49.7 ± 7.4 44.3 ± 4.2
95.7 ± 2.1 95.3 ± 1.5 90.7 ± 3.1 92.7 ± 0.6
92.7 ± 2.5 94.0 ± 3.6 93.3 ± 5.5 94.3 ± 4.0
100.0 ± 0.0 99.0 ± 1.7 99.3 ± 1.2 100.0 ± 0.0
80.7 ± 11.0 78.7 ± 3.5 80.7 ± 9.5 82.0 ± 7.5
12
2 3 4 5
96.3 ± 2.5 99.7 ± 0.6 97.7 ± 2.5 96.0 ± 4.0
63.7 ± 14.6 68.7 ± 12.5 60.7 ± 14.3 61.3 ± 12.9
46.3 ± 5.5 49.7 ± 2.1 42.3 ± 2.5 38.7 ± 7.6
91.3 ± 1.2 90.7 ± 0.6 88.7 ± 3.5 88.7 ± 2.5
90.3 ± 4.0 91.3 ± 6.4 91.0 ± 5.2 90.0 ± 4.6
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
72.0 ± 9.8 74.0 ± 7.2 69.7 ± 10.1 70.3 ± 10.6
H
N
Sweep Into
Assembly
MetaWorld (Hard) Hand Insert
Pick Out Hole
4
2 3 4 5
48.3 ± 12.6 81.0 ± 6.6 74.0 ± 9.8 44.7 ± 8.1
37.7 ± 8.5 41.0 ± 7.5 37.7 ± 9.7 44.0 ± 11.5
98.7 ± 1.2 100.0 ± 0.0 99.7 ± 0.6 96.7 ± 3.2
54.7 ± 21.8 70.7 ± 22.0 45.0 ± 36.8 33.0 ± 36.6
98.0 ± 3.5 99.7 ± 0.6 98.3 ± 2.1 93.7 ± 2.1
25.3 ± 14.5 29.3 ± 16.2 24.3 ± 17.1 12.7 ± 2.1
37.7 ± 3.8 35.0 ± 7.2 29.7 ± 4.5 16.7 ± 6.5
8
2 3 4 5
74.7 ± 5.9 54.0 ± 7.9 79.3 ± 4.5 82.0 ± 1.0
40.7 ± 9.0 42.0 ± 9.2 37.3 ± 11.0 37.3 ± 7.8
97.3 ± 3.8 94.7 ± 0.6 98.0 ± 2.6 97.7 ± 4.0
60.0 ± 13.2 41.7 ± 23.7 58.0 ± 24.2 41.7 ± 38.2
97.7 ± 2.5 99.3 ± 1.2 98.3 ± 1.5 93.7 ± 1.2
21.7 ± 9.0 15.3 ± 2.1 23.0 ± 7.0 24.0 ± 10.6
29.3 ± 2.1 20.0 ± 4.6 33.3 ± 3.1 32.0 ± 6.0
12
2 3 4 5
76.0 ± 6.2 79.3 ± 2.9 77.0 ± 11.3 72.7 ± 15.0
35.7 ± 11.1 36.0 ± 5.6 34.7 ± 9.5 35.7 ± 8.1
91.7 ± 3.5 93.3 ± 2.3 91.0 ± 3.6 91.0 ± 5.6
49.3 ± 21.4 61.3 ± 25.3 49.3 ± 24.8 45.3 ± 30.0
99.3 ± 1.2 98.7 ± 1.5 99.0 ± 1.7 99.3 ± 1.2
16.7 ± 5.5 16.7 ± 9.0 14.3 ± 2.1 18.0 ± 7.9
35.0 ± 2.6 38.0 ± 5.2 33.7 ± 6.4 38.7 ± 1.2
H
N
MetaWorld (Hard) Pick Place Push
Shelf Place
Disassemble
MetaWorld (Very Hard) Stick Pull
Stick Push
Pick Place Wall
4
2 3 4 5
60.0 ± 1.0 70.7 ± 4.2 64.3 ± 5.0 56.0 ± 1.0
74.3 ± 8.1 76.3 ± 4.7 76.7 ± 3.1 68.7 ± 4.5
68.3 ± 4.5 69.3 ± 4.9 67.3 ± 6.5 49.0 ± 2.6
80.3 ± 3.8 86.7 ± 1.5 82.7 ± 3.1 71.3 ± 5.1
78.7 ± 4.2 80.7 ± 1.5 78.0 ± 6.6 72.3 ± 2.1
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
84.3 ± 3.1 89.0 ± 4.4 82.3 ± 8.6 74.3 ± 2.5
8
2 3 4 5
72.3 ± 3.8 70.3 ± 3.8 68.0 ± 5.0 65.7 ± 5.9
76.0 ± 1.7 80.3 ± 1.5 77.3 ± 4.0 77.7 ± 2.9
60.7 ± 5.1 58.3 ± 4.9 61.3 ± 4.5 60.3 ± 8.5
79.7 ± 3.8 78.7 ± 5.7 85.0 ± 3.6 81.7 ± 2.5
75.7 ± 3.1 73.3 ± 4.7 75.7 ± 2.1 73.0 ± 4.6
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
85.7 ± 4.7 84.3 ± 3.2 88.0 ± 1.7 85.0 ± 2.6
12
2 3 4 5
75.3 ± 6.4 73.0 ± 6.2 71.0 ± 5.0 71.7 ± 5.8
80.7 ± 4.2 81.7 ± 5.8 81.0 ± 5.3 79.0 ± 5.2
58.3 ± 4.9 57.7 ± 3.8 56.0 ± 7.9 52.3 ± 11.6
82.7 ± 1.5 84.0 ± 3.6 81.7 ± 4.0 76.7 ± 4.5
72.0 ± 6.1 72.7 ± 3.1 67.3 ± 3.5 68.3 ± 1.5
100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0 100.0 ± 0.0
83.3 ± 4.9 82.3 ± 4.5 81.3 ± 3.2 84.0 ± 3.5
H
N
Push Wall
MetaWorld (Medium) Soccer Sweep
29