BEAST: Efficient Tokenization of B-Splines Encoded Action Sequences for Imitation Learning
Authors: Hongyi Zhou
Weiran Liao
Xi Huang
Yucheng Tang
Fabian Otto
Xiaogang Jia
Xinkai Jiang
Simon Hilber
Ge Li
Qian Wang
Ömer Erdinç Yağmurlu
Nils Blank
Moritz Reuss
Rudolf Lioutikov
Venue: NeurIPS 2025 poster ()
Keywords: Action Tokenizer
Imitation Learning
Vision-Language-Action Models
Primary area: deep_learning
TL;DR: We present BEAST, a novel action tokenizer that encodes action sequences into compact discrete or continuous tokens using B-spline.

Abstract: We present the B-spline Encoded Action Sequence Tokenizer
(BEAST), a novel action tokenizer that encodes action sequences into compact discrete or continuous tokens using B-splines. In contrast to existing action tokenizers based on vector quantization or byte pair encoding,  BEAST requires no separate tokenizer training and consistently produces tokens of uniform length, enabling fast action sequence generation via parallel decoding. Leveraging our B-spline formulation, BEAST inherently ensures generating smooth trajectories without discontinuities between adjacent segments. We extensively evaluate BEAST by integrating it with three distinct model architectures: a Variational Autoencoder (VAE) with continuous tokens, a decoder-only Transformer with discrete tokens, and Florence-2, a pretrained Vision-Language Model with an encoder-decoder architecture, demonstrating BEAST's compatibility and scalability with large pretrained models.  We evaluate BEAST across three established benchmarks consisting of 166 simulated tasks and on three distinct robot settings with a total of 8 real-world tasks. Experimental results demonstrate that BEAST (i) significantly reduces both training and inference computational costs, and (ii) consistently generates smooth, high-frequency control signals suitable for continuous control tasks while (iii) reliably achieves competitive task success rates compared to state-of-the-art methods.

BEAST: Efficient Tokenization of B-Splines Encoded Action Sequences for Imitation Learning
Hongyi Zhout
Weiran Liao*
Xi Huang
Yucheng Tang
Fabian Otto
Xiaogang Jia
Xinkai Jiang"
Simon Hilber
Ge Li
Qian Wang'
Ömer Erding Yağmurlu Nils Blank Moritz Reuss
Rudolf Lioutikoy
Karlsruhe Institute of Technology
Robotics Institute Germany
8 Microsoft Research
Abstract
We present the B-spline Encoded Action Sequence Tokenizer (BEAST), novel action tokenizer that encodes action sequences into compact discrete or continuous tokens using B-spline. In contrast to existing action tokenizers based on vector quantization or byte pair encoding, BEAST requires no separate tokenizer training and consistently produces tokens of uniform length, enabling fast action sequence generation via parallel decoding. Leveraging our B-spline formulation, BEAST inherently ensures generating smooth trajectories without discontinuities between adjacent segments. We extensively evaluate BEAST by integrating it with three distinct model architectures: Variational Autoencoder (VAE) with continuous tokens, a decoder-only Transformer with discrete tokens, and Florence-2, a Vision- Language Model with encoder-decoder architecture, demonstrating BEAST's compatibility and scalability with large pretrained models. We evaluate BEAST across three established benchmarks consisting of 166 simulated tasks and on three distinct robot settings with a total of 8 real-world tasks. Experimental results demonstrate that BEAST (i) significantly reduces both training and inference computational costs, and (ii) consistently generates smooth, high-frequency control signals suitable for continuous control tasks while (iii) reliably achieves competitive task success rates compared to state-ot of-the-art art methods. Videos and code are available httpp::/intuutive-rooooo wwt...bbbbo.i.bbb.oob.obeaaaa website/
Introduction
Imitation learning has emerged powerful paradigm for training robots perform complex tasks by learning from human demonstrations 113]. Early works [4] [5) in this field primarily focused on predicting single-step actions based the current observation. However, recent research [6] highlights the importance of learning action sequences to capture the temporal coherence inherent in human demonstrations. Moreover, by modeling action sequences, we can reduce compounding errors 7 and create task demonstrations that more closely align with human methods [8] Given the success of autoregressive next-toke text-token prediction models in natural language processing and other domains [T] it is compelling to explore similar techniques for modeling action sequences, leveraging their ability predict and generate coherent sequences effectively.
In natural language processing, tokens typically represent words, which are inherently discrete elements. This discrete nature allows for effective next-token prediction, which extends well the generation and prediction of symbolic actions or in discrete action space. However, a significant challenge arises when attempting apply these approaches sub-symbolic, continuous actions, which are not inherently discrete. Discretization addresses this issue by compressing the continuous
"Correpondence to hongyi zhou@kit edu
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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action sequence while trying to retaining essential information. This process helps in balancing the expressivity of the action representation against computational efficiency.
Despite growing interest this area, effective strategies to create action sequences of discrete tokens remain underexplored. Existing approaches often focus single-step tokenization [2.14] vector quantization (15H77), or compression-based schemes [88.. However, they require training separate
encoder-decoder networks for the tokenizer [15] [9] or produce variable-length token sequences for inputs of the same duration [8], which complicates applying fast token generation techniques such as parallel decoding [20]. Furthermore, existing action tokenizers do not consider the smooth transitions between subsequent action chunks, which could result in undesired jumps at transition.
To address these challenges, propose the B-spline Encoded Action Sequence Tokenizer (BEAST), a novel tokenizer that represents continuous action sequences using B-splines [21]. BEAST offers versatility, allowing for effective integration with both discrete and continuous tokens. Different from tokenizers based on the vector quantization [15177), it does not require additional tokenizer training. BEAST compresses action trajectories into fixed-length token sequences enabling efficient parallel decoding for faster token generation, requiring 4 - 8x fewer tokens than binning-based tokenization. By using B-spline encoded control points as discrete tokens, BEAST ensures the generation of smooth action chunks, as well as the continuous connection between consecutive chunks.
Our contributions are: 1) We introduce BEAST, novel B-spline- tokenizer designed for modeling continuous action sequences. 2) We demonstrate the versatility of BEAST by integrating it into diverse model architectures that accommodate both continuous and discrete objectives. 3) We conduct extensive evaluations of simulated and real-world robotic tasks, showcasing its effectiveness. 4) We perform thorough ablation studies assess the impact of various design choices.
2 Related Work
Prior work has explored various action representations for policy learning. The most common approach is to directly predict low-level actions, such as joint positions or end-effector displacements, using a supervised learning objective [ (2). While simple, these approaches cannot tackle the
multimodality present in human behavior.
To address these limitations, ACT 6 introduces an Action Chunking Transformer trained as a conditional Variational Autoencoder (CVAE), which models multimodal behavior via a learned latent space. Instead of predicting single actions, ACT generates entire action chunks in single inference step. These chunks are short sequences of actions, which reduces covariate shift and improves performance. Another line of work focuses on generating action sequences with diffusion models. Diffusion Policies model complex, multimodal behaviors by iteratively denoising from Gaussian noise to generate action sequences 7 (6) While effective, these methods require multiple denoising steps per sequence, making inference comparatively expensive. In contrast, BEAST compresses full action sequences into compact control-point representations using B-spline approximation. This significantly reduces the number of predictions needed to model temporally extended behaviors. As result, it enables efficient action chunking with smooth transitions, combining the representational benefits of ACT and diffusion policies with the speed and simplicity of tokenized inference.
prominent research direction in action sequence representation is Movement Primitives (MPs) (27-30) Dynamic Movement Primitives (DMPs) [27 29) model trajectories using a second-order dynamical system with a goal attractor, ensuring smooth transitions in position and velocity. However, DMPs are limited in representing higher-order continuity. To address these limitations, recent works [30 31] have explored using B-splines as an alternative representation for MPs. Despite their effectiveness, existing works have primarily applied MPs within reinforcement learning (RL) frameworks 322 BII and only utilize them as continuous action representations.
Alternatively, robot actions can be represented as discrete values by discretizing them into a set of tokens. This discretization scheme is common many recent Vision- Larion-Lnnuugge-Cee Action models (VLAs) 12,37/400 [7]. These models, often based on Transformers, are well-suited to predicting discrete tokens due to their autoregressive pretraining on language. technique involves dividing the continuous action space into a fixed number of bins 41 [3). However, this strategy struggles to effectively model high-frequene robot data. Further it has very low inference
common discretization
speed. More sophisticated tokenization methods have been proposed. Behavior Transformers [19]

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1.0
9.75
0.50
100
9.75
3.5
325
3.00
0.50
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2.00
2.02
0.2 Narma Time.
0.73
Neraltad
Rermalless Time, Degres Pwl
Time Digras Pe
Tw3
Nermaleed Time,
Figure 1: From left to right: Clamped B-Spline Basis P 0, 1, 2, (top) and their generated trajectories (Bottom). Given the same control points, a higher degree will lead to smoother trajecto- ries. All generated trajectories start exactly from the first control point and end at the last control point. Notably, action chunk is conceptually equivalent to B-Splines of O-th degree, i.e. split-wise constants, as shown in the leftmost subplots. This relation is explained in details later in Section 4.1
use k-means clustering to form discrete action bins, combined with residual offsets via separate prediction heads. VQ-BeT [15] extends this idea by encoding action chunks into codebook vectors using a Residual VQ-VAE [1] While expressive, these methods require training encoder-decoder networks. which increases system complexity and introduces sensitivity hyperparameters and quantization loss. In contrast, BEAST requires no additional tokenizer training and avoids such instabilities through direct B-spline representation. BEAST does not require any additional tokenizer training and does not increase training complexity through its direct B-spline representation.
More recently, FAST [88] proposes compression-based tokenization strategy using discrete cosine transform and byte-pair encoding [42.. resulting in fewer tokens per action chunk. As a consequence, the resulting action sequences can have varying lengths. This complicate parallel decoding during inference. In comparison, BEAST produces fixed-length action representations. Fixed-length representations every inference step allow for parallel decoding, significantly speeding up inference. OLA [20] investigates how different tokenization strategies impact inference speed and policy performance in VLAs, showing that parallel decoding and action chunking can indeed lead to faster inference. However, OpenVLA-OFT does not compress the action tokens themselves, predicting an individual token for each action. BEAST compresses entire action chunks into a small set of B-spline control points. This enables both faster decoding and smooth, high-fidelity trajectories.
Preliminaries
Problem Formulation. Our goal train a policy :<( s) that capable of mapping given state to a corresponding sequence of actions a1:7 which has T' time steps and 1) Degrees of Freedom (DoF). To make this sequence prediction problem compatible with discrete generative models, we first transform the continuous action sequence into sequence of discrete tokens. The goal of action sequence tokenization is to obtain a discrete token sequence U1.J, where each token belongs to a vocabulary V/ with size [V/, by defining transformation tokenizer 01.7 t1:J,
B-Splines (Basis Splines)
43 are widely used in the field of computer graphics and computer-aided
design. A B-Spline curve y is formulated through linear basis function representation
N 1-DoF B-Spline: v(u)= Σ n=0
th ( e, (w)) PS u E [ko, kyr
(1)
where N control points and u is a continuous parameter, often interpreted as normalized time. The basis functions & (u) = [40 (u), & [ are polynomial basis functions of P-th degree. These basis functions are defined over M intervals determined by M + knots in vector [ko. kMl. and it satisfies M = N + [43]. Typically, the knot vector is normalized such that ku 0 and kM 1. The basis functions &." (u recursively computed using the Cex-de Boor recursion
We denote all recursive degree as = 0 P. For q = 0, the basis functions are
"The B- B-Spline degree P differs from the recursive degree Trajectories are represented by basis functions of degree P. while lower recursive degree serve as intermediate representations the recursive process.

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Continuous Tokens
Discrete Tokenss
DoF2
Extract 4-Sppine
Gontrol Points as Continuous Tokens
Control
Quantooe
Points
DoFa
Reshape and Flatton
Dacl
2
Figure 2: Overview of the BEAST Encoding Pipeline: Given a normalized action sequence, BEAST pipeline first uses linear regression to extract continuous-valle valued control points, forming control point matrices that serve as intermediate continuous representations. These matrices are then quantized uniformly into discrete values within the range [o, 255] and subsequently flattened to produce discrete action tokens for auto-regressive next-token prediction or parallel prediction.
defined as piecewise constant and recursively using the (g 1)-th degree basis for q 0 with
(2) {1
if kn S < kn+1, and otherwise.
piecewise constant:
recursive: (2(
kx 40 (w) kg (1
where kg-1 -uu-kn)/(kn+t k,..
Clamped B-Spline. In this work, we employ the clamped uniform B-Spline, where the first and last P + knots are repeated to ensure that the resulting curve starts at the first control point and ends at the last control point. In Figure [ we demonstrate the resulting basis functions of degrees from P 0 to P 4, together with their generated trajectories, given the same five control points. Clamped uniform B B-splines are particularly suited for trajectory generation due to their smoothness, compact representation, and local support, where each control point only affects the curve locally.
Parallel Decoding. Unlike autoregressive generation, which predicts tokens sequentially and thus requires K forward passes for sequence of length K, parallel decoding (20) enables the prediction of the entire output sequence in a single forward pass. This achieved by feeding the model with K empty token embeddings and replacing the causal attention mask with bidirectional mask, allowing the decoder to infer the entire sequence simultaneously. OpenVLA-OFF (20 leverages this approach for action sequence generation. In this work, adopt the parallel decoding strategy to predict all BEAST tokens in a single pass, improving the inference efficiency without sacrificing accuracy.
B-Spline Encoded Action Sequence Tokenizer
In this section, we first describe how BEAST utilizes B-Spline to construct efficient action sequence tokenizer that converts action sequences into either continuous or discrete action tokens. We then explain how smooth transitions between consecutive action sequences achieved by enforcing the initial conditions of clamped B-splines. Finally, we discuss strategies for efficient integrating BEAST with various model architectures that predict discrete continuous tokens.
4.1 Action Sequence Tokenization with B-Spline Tokenizer
Following prior works in action tokenization
we first normalize the input actions such that the 1st and 99th quantile value of each action dimension in the dataset maps to the range of [-1, 1]. Using quantiles makes the normalization robust against outlier data points.
Figure 2 presents an overview of the tokenization process. We begin by considering the tokenization of 1-DoF trajectory. Given normalized action sequence 01:7 [01, .2,,... @r) of length T. goal is to determine set of N control points c, with N < T, that approximate the given action sequence at spline evaluations y(u) I.. The linear transformation u +/r maps from action timestep to the parametric coordinate of the B-Spline. The spline evaluations y(4)1: are approximated by minimizing the least-squares error
C = arg min I| a1.Tl arg min 1 p (u)c ai.

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B-Spline Trajectory
8-Splin
Florence-2 Encoder
B-Spire Trajetory
Reshape
B-Spline
Centenaou
Poinss
DaVIT
Tokenizer
Prompt
Top-K / Greedy Sampler
Da-quantize
eg
Florence-2 Decoder
Diccntte Tokers
AE
(0ptione)
Figure 3: BEAST-F is VLA model that combines BEAST encoding with Florence-2 45 lightweight VLM with 0.77B parameters. BEAST produces uniform-length tokens, which allows BEAST-F to perform parallel decoding via learnable action embeddings (AE), instead of autoregres- sive next-token prediction. These discrete tokens are fed into the B-Spline Decoder, which first maps them to real-valued control points and then transforms those control points into continuous action sequences. The Pr token denotes an optional proprioceptive state.
where (w) (u), 2 (u), (u) represents precomputed B-spline basis functions defined over interval u E [0, 1]. Ridge regression estimates the control points closed form, [Gg. .I,..., - (TT Ar)-1 a1.7, with À acting as a regularization parameter. This efficient computation typically introduces only small overhead, typically 3 to 5 milliseconds per batch. For a gigh-dimnssional action sequence, i.c. D > 1, each DoF is encoded independently into ca. resulting in matrix C' of shape D D, N, that stacks each DoF's control points, C²=(ci, C2 ..., cD]
To form the final token sequence, this matrix flattened by interleaving different action dimensions corresponding to the same basis functions, as illustrated in Figure 2 the temporal order inherent in the trajectory segments associated with each basis function.
This flattening strategy preserves
Remark 1: Connection to Action Chunking, Action chunking, defined as discrete sequence of actions ag, (1,..., (T, mathematically equivalent piecewise constant function generated by O-th degree B-splines. As demonstrated in Figure most, each action step a, can be identified as a control point C, of O-th degree B-Spline basis with n, T
4.2 Enforcing Smooth Transition with Clamped B-Spline
Executing long-horizon tasks typically requires producing multiple small action sequences that connect seamlessly (replanning). While predicting action sequences effectively improves consistency within individual action chunks, a significant challenge lies managing discontinuities at transitions between consecutive chunks, which often result in jerky motion during online execution. Common approaches to address this issue apply temporal ensembles of actions 66 46). calculating moving averages over multiple predictions. However, these temporal ensembles require high-frequency replanning (lypically every timestep) to generate sufficient chunks for effective ensemble averaging, which significantly constrains execution speed in online applications.
In contrast, BEAST employs clamped B-Spline to ensure smooth transitions between consecutive action chunks. As introduced in Section clamped B B-Spline is specialized variant of B B-Spline that guarantees to start from the first control point and end at the last control point, which is utilized generate seamlessly connected action sequences, illustrated in Figure [ To ensure smooth transitions, we directly set the first control point co to the last action of the previous sequence. We then compute the residual trajectory by subtracting the contribution of the first basis function: are determined by solving the
= a 20 The remaining control points è [C1, linear regression problem similar to equation 4 arg mina ( à f2||2- Through this approach, BEAST consistently generates action sequences with mathematically guaranteed smooth transitions between chunks. This will be further discussed in toy task experiment in Section 5.1
5

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CALVIN
LIBERO
ALOHA Sim
Franka Challenge
Real World Aloha
Franka Kitchen
Figure 4: Simulation 6
47
48
and real world (Franka Challenge, Aloha, Franka Kitchen) tasks.
0.4
0.4
0.4
Postion
Postton
Position
0.2
0.2
0.2
BEAST(Durs)
Binning
Binning +AC
0.0
0.0
0.0
2
Time
Time
Time
Figure 5: Comparison among BEAST, single-step binning tokenization and binning tokenization with action chunking (AC). The comparison is conducted through the same rego-regressive model with different tokenizers to fit the same ground truth cube splines given the same context points. BEAST is smooth within each sequence and continuous the transitions between sequences.
4.3 Combining BEAST tokens with different architectures
We first evaluate BEAST in simplified setting with decode-only transformer (see Figure I0) with CLIP 49 for language encoding and Film-conditioned ResNet-18 8 [50] [1] as image encoder. Rell-ResNets are used in many prior works given their high efficiency and strong
Discrete Tokens.
performance [52] 53 [3] The proprioceptive state of robot is projected to the embedding dimension with a two-layer MLP. We employ parallel decoding with bi-directional attention to accelerate the inference. To further demonstrate the scalability of BEAST with large-pretrained models, we combine BEAST with Florence-2, a small, pretrained VLM with Encoder- Decoder architecture (0.77B parameters). Following the previous works on autoregressive VLAs [12] [8], we overwrite the least used 256 tokens in the VLM vocabulary as our action tokens. We also employ a parallel
decoding technique for the Florence variant, which significantly improves the throughput and reduces the latency for action generation. We provide an in-depth overview in Figure 3
We also explore the performance of combining BEAST with ACT O ACT uses conditional VAE (CVAE) with Transformer We predict N BEAST continuous tokens (normalized B B-Spline control points without quantization), where each token has the dimension of D, instead of action sequences, this design choice keeps the temporal order inherent in the trajectory segments. Using BEAST, we reduce the length of predicted token sequence by 6.67 times (from 100 to 15) without sacrificing the task performance. addition, method enables smooth trajectories without requiring temporal aggregation.
Continuous Tokens.
Seeoddrr-eeeeerrteee to predict sequence of actions.
5 Experiments
We conducted extensive evaluations in both simulated and real-world settings, targeting answering five key research questions (RQs): 1) What advantages does BEAST offer commonly used binning-based tokenizers? 2) How does BEAST contribute to the performance on imitation learning benchmarks? 3) How does BEAST affect the training and inference efficiency? 4) Does BEAST generalize to real-world scenarios? 5) How do the design choices affect the performance of BEAST? BEAST is integrated into three different architectures: First, combine BEAST and Florence-2 45 and term this VLA variant as BEAST-F. Second, we integrate BEAST into a small decoder-only transformer trained from scratch, referred as BEAST-D. Finally, we employ continuous BEAST tokens on top of vanilla ACT(6). resulting in (BEAST-ACT), A detailed description of each architecture is provided in Appendix B In contrast many baselines, we test BEAST-F without second-stage pretraining on large-scale robot datasets.

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Train
Method
Action Type VLM
chains)
Avg.
5
DEN-P-CNN
X
Dillusion
63.5%
35. 35.3%
19.4%
10.7%
6.4%
1.35
MDT 24
Diffusion
63.1%
42.9%
24.7%
15.1%
9.1%
1.55
Ope
Discrete
91.3%
77.
62.0%
52.1%
43.5%
3.27
3DDA 55
Diffusion
93.8%
80. 30
66.2%%
53.3%
41.2%
3.35
ABC-D
MoDE
[k]
Diffusion
81.1%
71.85
63.5%
4.01
VPP 35
Diffusion
95.7%
91 2%
36.3%%
81 .00%
75.0%
4.29
(ours)
99.8%
96.5%
82.7%
74.4%
Diff-P-CNN 7
Diffusion
86.3%
72. 7%
60, %
51.25%
41.7%
3.16
33
Diffusion
83.5%
77.9%
ABCD-D
MDT 24
Diffusion
98.6%
95. 899
91.6%%
86.2%
80.1%
4.52
BEAST-F (ours)
Discrete
98.1%
96.2%
93.0%
89.3%
84.8%
Table 1: CALVIN Benchmark results for ABC and ABCD. The table reports average success rates for individual tasks within instruction chains and the average rollout length (Avg. Len.) to complete 5 consecutive instructions, based 1000 chains. Zero standard deviation indicates methods without reported standard deviations. BEAST-F achieves SoT performance in both tasks.
Sputial
Object
Goal
Long
Average
SR (1)
Rank (1)
SR (7)
Rank (
SR (f)
Rank ())
SR (T)
Rank (1)
(f)
Rank (.)
DiN-P-CNN
78.3 = 1.1%
92.5 0.7%
4
68. 3 =: 1.2%
50.5 d: 1.3%
6
72.4
1::0::07 0.7%
6
Octo
78.9 + 1.0%
K5.7 0.99%
6
84.6 + 0.99
51,1 1 35%
5
75. 01.6%
OLanVLAA
84.7
0.8%
5
79.3 1.0%
53.7 1.3%
76.5 0.6%
No
RA-FAST
96.4%
96.8%
3
33.6%
60.25%
85.5%
BEAST-F
92.9 %
97.5 8
2
93.1 16
86.4 %
92.5%
Table Experimental Results for the LIBERO Benchmarks. SR: Success Rate. Best results in each column are shown in bold. BEAST-F achieves comparable performance state-of-the-art VLA, despite with a much smaller model and without robot data pretraining.
5.1 Comparing Against Binning-Bast Based Tokenization
To answer RQ1, we begin with ID toy task to investigate the advantages of BEAST over linning-based tokenization. We follow the autoregressive prediction pipeline used in previous works 12 18. Note that BEAST can be used for both autore- gressive prediction and parallel decoding. A small decoder-only transformer is trained to predict cubic splines from 3 control points. We compare against: 1) Single-step binning (denoted as Binning) [12]. which discretizes each action into one of 256 bins, and 2) Chunk-level binning (denoted as Binning+ACC), which discretizes entire action sequences of fixed length. We generate 2000 trajectories, each 100Hz resolution. Each model is trained for 8k steps and evaluated on 200 test se- quences. BEAST achieves the lowest MSE (0. 00.0004 + 0.0005),
Success Mate
54
30-
21
TransfeCube
Insertion
BIDP-CNN
EaTo BEACT IRBEAST-ACT
Figure ALOHA Benchmark results. The success rate is reported over 500 episodes of evaluation.
outperforming chunked binning (0.0009 + 0.0013) and single-step binning (0.0215 + 0.0216), with the latter performing two orders of magnitude worse. To simulate real-world action chunking [7]. repeat the rollout prediction three times. As visualized in Figure single-step binning fails to capture temporal structure and produces erratic outputs. Chunked binning captures some temporal coherence but results in jerky transitions due to discretization and lack of continuity across chunks. In contrast, BEAST generates smooth trajectories with minimal error and requires only tokens per 100-step sequence, resulting in approximately 20x reduction in inference steps.
5.2 Strong Performance on Established Simulation Benchmarks
To answer RQ2, we evaluate BEAST on established simulation benchmarks and compare with other SoTA imitation learning methods and VLAs.
Simulation Benchmarks. CALVIN
features 34 tabletop manipulation tasks with Franka
Panda robot using delta end-effector control across four scene configurations (splits A-D). The dataset contains 24, 000 language-annotated demonstrations. We evaluate two settings: CALVIN ABC (zero-shot generalization) and CALVIN ABCD (scaling with more data). Performance is measured by success rates on sequential tasks and mean sequence length completion. All evaluations require policies to follow free-form language instructions and complete 5 tasks in sequence across 1, 000
7

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different instruction chains. LIBERO 48
tests delta-EEF controlled Panda Robot across various scenes with 130 diverse tasks. We report results on four specialized benchmark settings with 10 tasks each (Long, Spatial, Object, and Goal). Success is measured as the percentage of successful task completions across 50 trials per task. AL.OHA [6] tests an absolute joint position controlled ALOHA Robot in two challenging bi-manual manipulation tasks that require high precision.
Baselines. We compare our Vision-Language-AAtiie Model (VLA) against SOTA VLA policies and specialized approaches, using results reported in prior publications for fair comparison. Our primary baselines are OpenVLA [2] (7.7B parameters), Ny [56] (3.3B parameters), To- F (3.3B parameters), and a standard Diffusion Policy using a CNN 77. For the bi-manual manipulation tasks, we compare the BEAST-AC variant with small action chunking models vanilla ACT ( To. and standard Diffusion Policy using CNN.
Results. Table T summarizes the performance of all policies the CALVIN benchmark, where BEAST-F outperforms a diverse set of baselines across two settings, establishing a new state of the art. Unlike the most competitive baselines, BEAST-F achieves these results without relying on additional pretraining. On the various LIBERO benchmarks, our tokenizer achieves strong performance, being surpassed only by To- -V-AA. However, To relies on large-scale pretraining to reach its performance, whereas BEAST-F remains competitive without it. In the most challenging long-horizon task setting, LIBERO-LONG, BEAST-F outperforms all baselines. See Table for detailed results. For the bi-manual tasks (Figure 6 BEAST-ACT and ACT demonstrate significantly better performance than To. BEAST-ACT achieved higher success rate than vanilla ACT in both tasks.
5.3 Advantages in Training and Inference Speed
Next, we verify the inference and training efficiency of BEAST to answer RQ3. Specifically, we consider the VLA variant BEAST-F and compare against
Method
Throughputs (H2)1
Latency (s)
DP (0.268)
130.67
0.341
OLe (7B)
6.09
0.164
Tu (3.3B)
288.11
0,104
56 18 as well as a stan- dard CNN-based Diffusion Policy 7 We measure the inference efficiency on an RTX 4090 GPU. As shown Table 3 BEAST-F demonstrates clear com- putational advantages. It achieves throughput of 617.3 Hz (e.g., generates approximately 617 actions per second), which 2.14x faster than To. 4.72x faster than Diffusion Policy, and 101.4x faster than Open VLA. In addition, BEAST-F achieves the lowest latency just 19 milliseconds, where latency refers to the time taken to generate one action chunk. These gains are due to the parallel decoding, which enables generating the action sequence in a single forward pass.
several recent VLAs 12
BEAST-F (0.77B)
617.3
0.019
Table 3: Mean inference efficiency (1000
steps in Bf16). All policies except Open use chunking length 50 (48 for DP).
REAST-F
FAST
We further evaluate the training efficiency by comparing BEAST- F against To and FAST. To exclude the bias introduced by the pretraining datasets, we trained all models without robot dataset pretraining. We report the success rate on LIBERO-LONG bench-
Success Rate
0.8
0.6
mark every 10k training steps in Figure 7 BEAST-F reaches a approximate 80% success rate at just 20k steps, whereas To reaches only around 20% at the same point. Notably, FAST shows no suc- cess till 30k steps. To- FAST's poor performance indicates a heavy reliance on robot dataset pretraining, which further underscores the training efficiency of our method.
0.4
0.2
0
10k
20k
30k
Training Steps
Figure 7: LIBERO-LONG.
5.4 Tokenizer Comparison under a Unified Backbone
100
To ensure fair comparison between different tokenizers with- out the confounding effects of varying backbone architectures
Success Rate (%)
$4.84 MD
and different pretraining robot dataset pretraining recipes. We integrate FAST tokenization and binning tokenization (used
50
30
by OpenVLA) into the Florence-2 model (45) without robot
dataset pretraining. This setup enables an unbiased assessment of tokenizer performance. Both FAST and binning tokeniz-
LIBERO LONG
CALVIN ABC D
IaBinning AR AR
IRBEAST-AR LIBEAST-PD
ers are trained with autoregressive objective (FAST-AR and
Figure Tokenizer Comparison.
8

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Arg. Stocess Rate (1S
100
07
50
46
Towel Fold
Sweep
Mixer
Pour
Move Bonana
Open Oven
Move Pat
Cuhe Transfer
DACT ILFAST
IEBEATT-F
IEBASTTD
BEBESTT-ACT
Figure 9: Experimental Results on Real-World Robot Tasks. This figure shows the average task success rate across eight real-world tasks. Each task and method was evaluated over 10 runs (30 runs for Cube Transfer). Success rates are measured at the sub-task level. Detailed descriptions of all sub-tasks are provided in Appendix F BEAST variants achieve strong performance in real world.
Binning-AR),, following their original implementations. Figure & presents results on the challenging LIBERO-Long benchmark and the Calvin ABC-D generalization benchmark. The BEAST tokenizer with parallel decoding (BEAST-PD) achieves the highest task completion ratio in both benchmarks, followed by BEAST with autoregressive decoding (BEAST-AR). BEAST-AR matches FAST LIBERO-long and outperforms it on Calvin ABC-D. Overall, both compression-bads based tokenizers (BEAST and FAST) consistently outperform the binning tokenizer across all evaluations.
5.5
Real-World Evaluation with 3 Different Robot Setups
To answer RQ4, we assess the effectiveness of BEAST across diverse real-world scenarios with varying data collection frequencies. We evaluate BEAST on 8 challenging manipulation tasks across 3 different experimental setups: 1) Franka Challenge: Four tabletop manipulation tasks (Towel Fold, Sweep, Mixer, Pour) using a joint position-controlle
Franka robot with data collected at 20Hz, 2) Real Kitchen: Three manipulation tasks on a toy kitchen setup (Move Banana, Open Oven, Move Pot) with data collected at 35Hz, 3) Bi-manual ALOHA: A cube transfer task using bi-manual ALOHA robot with data recorded at 60Hz. For each task in the Franka Challenge and Franka Kitchen setups, we conduct 10 evaluation runs per method, while for the ALOHA cube transfer task, we performed 30 runs. The average success rate for each task is reported in Figure 9 For tasks comprising multiple stages, track intermediate milestones to better evaluate the completion of each sub-task. Appendix [FaanddxFFFppr provides a detailed description of all setups and tasks. We compare BEAST against To (56), -o-FAST (18, and ACT (6) We finetune mo and To-FAST from the official pretrained checkpoints for an additional 60k and 40k steps, respectively. For each method, we train multitask model for all four tasks, Real Franka tasks, and another for the Real Kitchen tasks. The results demonstrate that BEAST-F achieves 52.86% success rate and BEAST-D achieves 76.57%. In contrast To achieves 53.43% and FAST only 28.5%. Interestingly, the smaller model (BEAST-D) outperforms all the VLAs, including the Florence variant with BEAST. We attribute this effect to the relatively small real-world dataset of only 50 demonstrations for each task. For the Aloha Cube Transfer task, we compare BEAST-ACT against the base ACT that directly predicts action sequences in the joint space. BEAST-ACT achieves 70% success, which 21% higher than the base ACT.
5.6 Ablation Studies
To answer RQS, we conduct ablation studies to analyze the impact of various design choices of BEAST. All experiments in this section use the Florence variant BEAST and are evaluated on the CALVIN ABC benchmark. All results summarized in Table
Variant
Avg. Len.
BEAST-F (10)
4.43
BEAST-F (5)
3.88
BEAST-F
4.20
BEAST-F (20)
4.32
BEAST Binning- based Tokenizer. We first compare BEAST against commonly used binning- based tokenizer in [2) which discretizes single-step actions into one of 256 uniformly distributed bins. We imple- ment this baseline using the same Florence-2 backbone and denote it as Binning-F. F is trained to perform autoregressive token prediction. shown in Table 4 BEAST significantly outperforms the binning- based approach, improving the average sequence length from 1.41 4.43, underscoring the effectiveness of BEAST as action tokenizer.
BEAST-F (25)
4.23
BEAST-SF
3.98
BEAST-CT
3.93
Binning-F
1.41
Table 4: Average Se- quence Lengths for BEAST-F Ablation.

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Discrete Tokens vs. Continuous Tokens. Next, we study the choices between using discrete tokens or continuous tokens (denoted as BEAST-CT) the action representation. In the continuous variant, the final hidden states of the Florence decoder are directly mapped to continuous BEAST tokens via a linear layer, and the learning objective changed from cross-entropy to L.1 regression loss. Results show that discrete tokens yield 12.7% better performance. We attribute this to the greater expressiveness of discrete representations, which are better suited to model multi-modal distributions.
Choice of Number of Basis Functions. Next, we evaluate how the number of basis functions affects the policy performance. We evaluate using N [5, 10, 15, 20, 25] basis functions to model action chunks of 20 steps, denoted as BEAST-F [N] in Table 4 Fewer basis functions lead to fewer tokens for prediction, but it also reduces the expressiveness of the B- B-Spline representation. On the contrary, more basis functions increase representational power but reduce compression, which can also negatively influence the performance.
Scaling with Model Size. Finally, we assess the impact of model size on task performance. We com- pare BEAST-F, which uses Florence-2-large (0.77B parameters), with BEAST-SF, a smaller variant based on Florence-2-base (0.23B parameters). The larger model achieves an 11.3% improvement in average sequence length, demonstrating that BEAST benefits from increased model capacity. This result highlights its potential as a scalable building block for larger VLAs.
Conclusion
We present BEAST, a sesline-bbsed tokenizer for continuous robot actions that compresses arbitrary trajectories into fixed-length token sequences while preserving smooth transitions between segments. BEAST supports discrete and continuous outputs and integrates seamlessly with various model archi- tectures. By exploiting parallel decoding, it delivers fast inference and high compression rates without sacrificing performance. In extensive experiments--bthh simulation and on real robots-BEAs consistently achieves strong results, demonstrating the effectiveness of our tokenization strategy.
sensitive the choice of the number of B-spline basis functions. which can markedly affect task outcomes (Section (.66). The optimal count depends on the smoothness and sampling frequency of the trajectory; our expermeets indicate that using 5-10 bases works well for one-second robot trajectories. A heuristic for selecting the number of basis functions provided in Appendix A
Limitations: Although BEAST delivers strong performance,
We plan to extend BEAST to large-scale robot pretraining and to integrate contin- uous token representations with diffusion- and flow-matching objectives, aiming to further boost downstream task performance. Preliminary results on combining continuous BEAST tokens with flow-matching objective are provided in Appendix D
Future Work:
7 Acknowledgment
This work was supported by the pilot program Core Informatics of the Helmholtz Association (HGF), by the German Research Foundation (DFG, grant no. 448648559), and in part by the Helmholtz Association of German Research Centers. The authors also acknowledge support by the state of Baden- Baden-Württembeel Württemberg through the HoreKa supercomputer funded by the Ministry of Science, Research and the Arts Baden- BartenWWrrteebberg, and by the German Federal Ministry of Education and Research.
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A Choice of Number of Basis Functions
For finding the optimal N, we recommend starting with number of basis functions equal to half of the sequence length. In practice, we typically sample batch of action sequences from the dataset (e.g., 100 trajectories or more) and compute the reconstruction mean square error (MSE). An MSE below 1e-2 (for action sequences that are normalized into the range [-1, 1D usually indicates that the chosen number of basis functions allows BEAST to represent the original trajectories well. We then repeat this process to find the minimal number of basis functions that achieves this threshold, thereby maximizing compression. Since BEAST does not require separate tokenizer training, this hyperparameter tuning process be done in a few minutes across different action spaces and frequencies.
B Architectures
BEAST-F is new VLA model that integrates BEAST with the pretrained VLMM Florence-2 45 Florence-2 is compact vision-language model with 0.77B parameters, featuring an encoder-decoder language model architecture paired with DaViT image encoder[57] that efficiently compresses images into sequences into just 50 tokens. Florence-2 offers two unique advantages: First, Florence-2 was pretrained with a large-scale dataset focusing on image-grounding with tasks like bounding box prediction and image segmentation. These objectives are well aligned with robotic manipulation challenges. In addition, efficient and enable us to run VLA experiments on consumer-grad hardware. This combination of relk-relevant pretraining and computational efficiency made Florence-2 ideal for exploring our tokenizers within practical resource constraints.
small size and low image token count make it computational and memory
BEAST-D is compact Transformer model that integrates discrete BEAST tokens. The architecture shown in Figure
B-Spline Decoder
Condition Takens
Top Gready Sampler
Decoder-Only, Bi-Directional Attention
Camera #1
Camera #2
Comeraa
ReaNet
ReeNet
ResNit
Prompt: a.g-
CLIP
Prop. State (Optional)
Learnalle Action Embeddings
(Action Takens)
Figure 10: Overview of BEAST-D: BEAST-D is a small transformer model that integrates BEAST. It replaces the causal attention in the decoder-only transformer with bidirectional attention to enable fast parallel decoding. BEAST-D integrates ResNet as image encoder and CLIP as language encoder.
C Baselines Implementation
To: Tg is a generalist robot policy that combines a pre-trained VLM backbone with a lightweight action expert module trained from scratch generate continuous actions using flow matching. A key innovation of my is its cross-embodiment training strategy, which integrates over 900M timesteps of data from 7 distinct robot embodiments and 68 manipulation tasks, enabling generalization across heterogeneous hardware platforms. The model is trained using a two-phase pipeline: a large-scale pre-training stage leveraging Internet-scale semantic priors, followed by post-training on curated task-specific data to enhance performance on complex. dexterous tasks.
FAST: FAST introduces novel compression- basedd tokenization method, named Frequency-space Action Sequence Tokenization, for training autoregressive VLA models on high-frequency. dexterous robot control tasks. Unlike prior VLAs that struggle with discretizing continuous actions high frequencies, FAST leverages the Discrete Cosine Transform (DCT) and Byle-Puir Encoding (BPE) to produce compact, information-rich action tokens, marking significant advance in training efficiency.
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D Preliminary Results with Flow Matching
We conduct additional experiment on combining BEAST's continuous tokens with flow-matching |88s|58 [59] and mixture-of-expeee backbone 53 The resulting model, BEAST-Flow achieves performance comparable to the Flow policy with action chunking on LIBERO-Long. and outperforms it on Calvin-ABC, while predicting only half as many tokens. The full results are presented in Figure These results highlight the effectiveness of BEAST within a flow-matching framework.
100
91.5
91.5
Suceess Rate (5)
61.6
50
LIBERO LINGRO-LONG
CALVIN
Flow BEAST-Flow
Figure 11: Preliminary Results with Flow Matching.
E
Hyperparameters
Hyperparameter
LIBERO
CALVIN
SPATIAL
OBJECT
GOAL
LONG
ABCD D
ABC-DD
Action Sequence Length
20
20
20
20
20
20
Number of Basis
10
10
10
10
10
10
Vocabulary Size
256
256
256
256
Optimizer
AdamW
Adam W
Adam W
Adam W
Adam W
Adam W
Betas
[0.9, 0.95]
[0.9, 0.95]
[0.9, 0.95)
[0.9, 0.95]
[0.9, 0.95]
[0.9, 0.95]
Learning Rate
2e-5
2e-5
2e-5
2e-5
2e-5
2e-5
Batch Size
128
128
128
128
32
Train Steps (k)
35
35
50
30
Table 5: Summary of BEAST-F hyperparameters for all simulation experiments.
Hyperparameter
REAL KITCHEN
REAL FRANKA
Action Sequence Length
80
20
Number Basis
Vocabulary Size
256
256
Optimizer
Adam W
Adam W
Betas
[0.9, 0.95]
[0.9, 0.95]
Learning Rate
2e-5
2e-5
Batch Size
96
Train Steps (k)
60
60
Table 6: BEAST-F hyperparameters for real robot experiments.
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Hyperparameter
REAL KITCHEN
REAL FRANKA
Action Sequence Length
80
20
Number of Basis
10
5
Vocabulary Size
256
256
Transformer Layers
6
6
Attention Heads
8
8
Embedding Dim
256
256
Image Encoder
FiLM-ResNet18 CLIP ViT-B/32
FiLM-ResNet18
Goal Lang Encoder
CLIP ViT-B/32
Aun Dropout
0.1
0.1
Residual Dropout
0.1
0.1
MLP Dropout
0.1
0.1
Optimizer
Adam
AdamW
Betas
[0.9, 0.9991
[0.9, 0.9991
Learning Rate
3c-4
3e-4
Weight Decay (Trans/Other)
0.05 / 0.05
0.05 / 0.05
Batch Size
384
256
Train Steps (k)
60
60
EMA
False
False
Table 7: BEAST-D hyperparameters for real robot experiments.
F Real Robots Setup & Tasks
F1
Robot System Details
Real Kitchen. This setup consists of a single Franka Emika robot operating within a simulated kitchen environment. It is equipped with two OAK-D Lite cameras providing top-down and side perspectives, each delivering visual input
resolution of 250x250 pixels. The robot has an 8-dimensional configuration and action space, which includes seven joint and one gripper states.
Real Franka. This configuration features single Franka Emika robot situated in general-purpose tabletop environment designed for more challenging manipulation tasks. Visual observations are obtained from Orbbec Femto Bolt cameras, positioned to capture left and right perspectives. The input images are resized to a resolution of 180x320 pixels. The robot configuration and action space remain the same the Franka Kitchen setup.
ALOHA. Based on the ALOHA setup [6, the system incorporates two 6-DoF Trossen ViperX robotic arms. The environment includes wrist-mounted and an additional oo-mounted Logitech C920 camera. The combined system operates in a 14-dimensional configuration and action space, accounting for both arms' joint and gripper states.
F.2 Tasks Description and Evaluation Metrics
In the Real Kitchen setup, the robot performs pick-and-plan tasks, whereas in the Real Franka setup, the robot is required to execute more diverse manipulation behaviors, such as sweeping or pouring. For each task performed by the Franka Emika robot, a scoring rubric is defined to quantitatively evaluate task progression. The specific evaluation criteria for each task are detailed below.
Open the door: The task begins under one of two initial conditions: with without object placed on the stove. The objective is for the robot to open the oven door. Task completion is evaluated as success or failure. Although the task involves three key motion phases, as shown in Figure 12 (Open the door), all the policies under evaluation are capable of completing the task its entirety once the robot successfully grasps the handle. A trial is considered successful if the robot fully opens the oven door by first grasping the handle and then using fingers to push the opposite side of the door, ensuring that is completely open. We conducted four evaluation trials with no object on the stove and one with a randomly placed object.
Banana into the pot: In this task, the robot aims to grasp a banana and place it on or into pot. The initial conditions are categorized based on the relative positions of the pot and the

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banana, as well as the position of the banana relative to the corresponding stove. The pot is assumed be correctly positioned on the stove. The banana, however, may be placed directly on the stove in one of three orientations or slightly offset to the left or right. A total of 10 trials are conducted across these different initial configurations. Task performance scored on scale of O to 3, with one point awarded for each of the following criteria: (1) successfully positioning the banana between the robot's two fingers, (2) lifting the banana off the surface, and (3) placing the banana onto or into the pot. If the robot attempts to grasp more than three times, exhibits jerky hand movements, or significantly displaces the pot from its original position, the subtask is awarded 0.5 points to reflect partial completion.
Pot into the sink: This task is similar to the one described previously. The initial conditions in this task differ based on the relative position between the pot and the two stoves. Task performance is evaluated using three criteria: (1) successfully positioning the pot between the robot's two fingers note that directly grasping the pot with its handle is considered an unstable grasp and is awarded 0.5 points, (2) lifting the pot off the surface, and (3) placing the pot in the sink. In this task, penalties for jerky hand movements are still applied.
Towel folding: The objective of this task to neatly fold a towel that is randomly oriented at the beginning of each trial. One point is awarded for each of the following: lifting a corner of the towel, completing the fold, and achieving accurate alignment of the folded towel.
Sweep: In the Sweep task, the positions of the broom, dustpan, and trash vary across trials. Four pieces of trash are placed on the table for the robot to clean. A maximum of four points can be awarded, based on the following criteria: successfully grasping the broom, performing a single sweeping motion, demonstrating the ability to execute multiple sweeping motions, and sweeping all trash into the dustpan.
Mixer: In this task, cup and a mixer are placed on the table. The robot's objective is to sequentially (1) open the mixer, (2) grasp the cup, (3) place the cup on the mixer's platform, and (4) close the mixer. Task performance is evaluated based on the successful completion of these four subtasks, with one point awarded for each. Notably, unlike previous tasks, the language instructions provided to the robot consist of three separate sentences corresponding to the actions of opening/closing the mixer and placing the cup onto the platform.
Pour: In the Pour task, the source cup is initially placed on a platform and contains plastic
pellets that simulate liquid. The objective is to pour the pellets into designated target cup. Task performance is evaluated out of a maximum of 4 points, awarded based on the following criteria: (1) successfully grasping the source cup, (2) pouring the pellets into the target cup, (3) ensuring that all pellets are poured into the target cup, and (4) placing the source cup back on the platform.
ALOHA cube transfer: In the cube transfer task, the ALOHA robot is designed to pick up a randomly placed cube using its right arm and then transfer the cube to its left arm. The performance of the task is evaluated by assigning scores to three specific steps: (I) successfully picking up the cube, (2) successfully initiating the transfer with the left arm making contact with the cube, and (3) the left arm successfully taking possession of the cube while the right arm releases it.
G Compute Resources
We train and evaluate all the models based on our private clusters. Each node contains 4 NVIDIA A100, for BEAST-F we use 4 GPUs for training. For BEAST-D and BEAST-ACT, we use one GPU for training. We report the average training cost in Table [)
BEAST-F
BEAST-D
BEAST-ACT
vRAM
64GB 6000
8GB
15GB
steps/hour
10000
11000
Table 8: Training time for each variant.
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Open the Door
Pot into the sink
Banana into the pot
Fold the towel
Sweep
Arrange
Pour
Figure 12: Key frames for real world different tasks
26


Reviews:
[{"review_id": "5GTREa1mjo", "reviewer": "Reviewer_Ca1e", "summary": "The paper proposes an action sequence modeling approach that is better suited for representing continuous actions and is more efficient in capturing temporal dynamics.", "questions": "When I read the introduction, I expect your work to address a concrete problem in VLA—specifically, the limitations of existing methods that use autoregressive models to predict discrete actions. However, when I get to Section 4.3 and the experiments, I no longer understand what problem your method is trying to solve. At one point, it is combined with ACT (which already predicts continuous actions rather than discrete ones), and at another point, it is compared with methods like Pi0, which use DIT to output continuous actions—even though the architectures are fundamentally different. I think you are trying to show your work is goot at lots of aspects, but such experiments design is weak to prove that.\n\nThat said, I think your idea is valuable. What I expect to see is whether your method can be plugged into existing frameworks—for example, replacing the action modeling strategy in Pi0 or OpenVLA—and bring consistent improvements. Instead, the current evaluation focuses on combining your method with DaViT and Florence to form Beast-VLA, but these vision-language models are rarely used in prior VLA work, which makes the comparison less convincing.\n\nThe following are detailed questions that may overlap with the overall points mentioned above:\n\n- Why do you choose DaViT and Florence to build your VLA model, instead of using more commonly adopted backbones such as DINO or SigLIP?\n\n- Why do you combine with ACT or replace ACT’s action modeling, given that ACT is unrelated to discrete action modeling?\n\n- Why not integrate your method into PI0 or OpenVLA to demonstrate that your action modeling approach can improve existing VLA models? While PI0 uses DiT to predict continuous actions and is not based on discrete action modeling, it still serves as a strong and relevant baseline to show your action modeling strategy can even outperform DiT's action modeling.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- Strengths：\n    - The idea is both interesting and intuitive. It adopts a more explainable approach to action modeling and demonstrates how it improves the smoothness of transitions between action chunks. The method proves to be effective and efficient in the context of robotic manipulation.\n-  Weaknesses:\n   - The main weakness lies in the experimental design used to demonstrate how the proposed method works and its advantages. I believe this is partly due to the experimental setup and partly due to the writing. Please refer to the questions for more details.", "was_revised": true, "final_justification": "After rebuttal, I keep my score as borderline accept", "rebuttal": "We would like to thank the reviewer for the positive evaluation of our work and the many helpful comments and suggestions. We hope the following replies sufficiently address the raised questions and concerns. We will update the paper accordingly.\n\n> Q1: Why do you choose DaViT and Florence to build your VLA model, instead of using more commonly adopted backbones such as DINO or SigLIP?\n> \n\nWe chose DaViT as vision encoder because Florence-2 was pre-trained with DaViT, we maintained DaViT to preserve the learned multimodal alignment between visual and language representations. Replacing DaViT with another vision encoder like DINO or SigLIP could break this carefully learned alignment. \n\nWe chose Florence-2 as our VLM backbone for two main reasons: \n\n1. Florence-2 was pre-trained on a large-scale dataset focusing on image-grounding tasks, such as bounding box prediction and image segmentation. These objectives are well aligned with robotic manipulation tasks. \n2. With only 0.77B trainable parameters, Florence-2 is significantly more compact and computationally  efficient than larger alternatives like PaliGemma (3B) used by pi0 [3] or Llama-2 (7B) by OpenVLA [5]. This makes it a practical backbone for our work, which emphasizes efficiency and tokenizer design rather than maximizing backbone performance. \n\n> Q2: Why do you combine with ACT or replace ACT’s action modeling, given that ACT is unrelated to discrete action modeling?\n> \n\nThe idea is to demonstrate that BEAST can also be used to generate continuous tokens. While ACT already generates continuous actions, it typically requires temporal ensemble [1,2] across predictions to smooth trajectories. In contrast, the BEAST formulation inherently ensures trajectory smoothness and continuity in transitions. Through empirical experiments, we demonstrate that the BEAST-enhanced ACT achieves these benefits without compromising performance. \n\n> Q3: Why not integrate your method into PI0 or OpenVLA to demonstrate that your action modeling approach can improve existing VLA models? While PI0 uses DiT to predict continuous actions and is not based on discrete action modeling, it still serves as a strong and relevant baseline to show your action modeling strategy can even outperform DiT's action modeling.\n> \n\nWe fully agree with the reviewer that integrating BEAST with PI0 and OpenVLA backbone would strengthen our argument. However, extremely high computational resources demand and closed-source, proprietary datasets make this kind of comparison extremely difficult. OpenVLA, for instance, employs Llama-2 7B as its backbone and was pretrained using **21,500 A100 GPU hours (equivalent to 14 days on a clusters of 64 A100s)** [5].  PI0 and PI0-FAST are pretrained on large closed-source,  proprietary dataset (they use a large pretrained dataset **~10000 hours with only 9.1% open-source data and 903M timesteps closed-source data** [3]). Furthermore,  the pretraining pipelines for PI0 and PI0-FAST are not publicly available, making integrating BEAST extremely challenge.\n\nTo still provide a meaningful and fair comparison, we instead combined FAST tokenization[4] and OpenVLA tokenization[5] with Florence-2 model. This enables a fair assessment of performance between different tokenizers. Here are the results on the most challenging LIBERO-Long benchmark and Calvin ABC→D generalization benchmark. BEAST tokenizer with parallel decoding (BEAST-F-PD) achieves the highest task completion in both benchmark, followed by BEAST with autoregressive (BEAST-F-AR), yielding equal or better performance than FAST and OpenVLA tokenizers.\n\n|  | **BEAST-F-PD** | BEAST-F-AR | FAST-F | OpenVLA-F |\n| --- | --- | --- | --- | --- |\n| LIBERO-Long | **0.864** | 0.837 | 0.837 | 0.304 |\n| Calvin-ABC | **4.42** | 3.57 | 2.3 | 1.41 |\n\nAdditionally, to simulate a minimal “PI0-style” alternative without requiring large-scale pretraining, we conduct an additional experiment on combining BEAST’s continuous tokens with flow-matching loss and a mixture-of-expert backbone [6]. The resulting model, **BEAST-Flow** achieves performance comparable to the Flow policy with action chunking on LIBERO-Long, and outperforms it on Calvin-ABC, while **predicting** **only half as many tokens**. The full results are presented in the table below. These results highlights the effectiveness of BEAST within a flow-matching framework.\n\n|  | **BEAST-Flow** | Flow |\n| --- | --- | --- |\n| LIBERO-Long | **0.915** | **0.915** |\n| Calvin-ABC | **3.18** | 3.08 |\n\n### References:\n\n**[1]** Zhao T. et al., \"Learning fine-grained bimanual manipulation with low-cost hardware,\" *RSS* 2023.\n\n**[2]** Li Q. et al., \"Cogact: A foundational vision-language-action model for synergizing cognition and action in robotic manipulation,\" *arXiv preprint arXiv:2411.19650*, Nov. 2024.\n\n**[3]** Black K. et al., \"$\\pi_0$: A Vision-Language-Action Flow Model for General Robot Control,\"  *RSS* 2025.\n\n**[4]** Pertsch K. et al., \"FAST: Efficient action tokenization for vision-language-action models,\"  *RSS* 2025.\n\n**[5]** Kim M. et al., \"OpenVLA: An open-source vision-language-action model,\" CoRL 2024.\n\n**[6]** Reuss M. et al., \"Efficient diffusion transformer policies with mixture of expert denoisers for multitask learning,\" *ICLR* 2025."}, {"review_id": "s1KChDFRDq", "reviewer": "Reviewer_JQnF", "summary": "The paper introduces BEAST (B-spline Encoded Action Sequence Tokenizer), a novel method for tokenizing continuous action sequences in imitation learning. Instead of relying on existing tokenization methods like vector quantization or binning, BEAST employs B-spline basis functions to compress continuous action trajectories into fixed-length, smooth representations. These can be used as discrete tokens (after quantization) or continuous control points, enabling compatibility with both autoregressive and parallel decoding architectures.\n\nKey contributions include:\n\n1. A tokenizer that does not require separate training and produces fixed-length, smooth action sequences.\n2. Compatibility with multiple model types (e.g., transformer, CVAE, Florence-2).\n3. Strong empirical performance across simulation benchmarks (CALVIN, LIBERO) and real-world robotic tasks.\n4. Substantial improvements in training/inference efficiency via parallel decoding and action compression.", "questions": "1. Section 5.5 shows that BEAST is sensitive to the number of basis functions. Can you provide heuristics or general rules for selecting this parameter across different robot platforms and frequencies?\n\n2. Are there scenarios where BEAST fails to outperform traditional tokenizers, such as in highly discontinuous or impulsive control settings?\n\n3. You highlight reduced token counts. Can you elaborate on any observed tradeoffs between compression and policy expressivity beyond MSE?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "# Quality\n\n**Strengths**: novel and practical method that improves both performance and efficiency; extensive experimental validation, including simulation and real-world benchmarks; strong ablations showing the value of B-spline representation and design choices; impressive improvements in inference speed (2-100x over baselines); smooth trajectory generation is demonstrably better than binning or quantized chunking.\n\n**Weaknesses**: some ablation results (e.g., choice of basis functions) suggest non-trivial sensitivity, but there's limited discussion on tuning strategies; paper could better detail failure cases or conditions where BEAST might underperform\n\n# Clarity\n\n**Strengths**: the methodology is well-explained, especially the B-spline encoding and its advantages; figures (e.g., tokenization pipeline, spline smoothness) are informative; experimental setup and benchmarks are clearly described.\n\n**Weaknesses**: the distinction between discrete and continuous token modes in some sections could be more explicit; descriptions of integration into large models like Florence-2 may assume familiarity with these architectures.\n\n\n# Significance\n\n**Strengths**: addresses a core bottleneck in modern VLAs-tokenization of continuous actions; broadly applicable to robotics, VLMs, and imitation learning; effective across multiple benchmarks and robots without large-scale pretraining; could lower the barrier to applying VLAs in real-world robotics.\n\n# Originality\n\n**Strengths**: B-spline-based tokenization is novel in this domain; clever exploitation of B-spline properties (local support, smooth transitions, fixed-length encoding); avoids training a separate tokenizer: conceptually elegant and practically useful.\n\n**Weaknesses**: some concepts (e.g., B-spline smoothing, ridge regression) are not new, though their application here is novel.", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the positive evaluation of our work and the many helpful comments and suggestions. We hope the following replies sufficiently address the raised questions and concerns. We will update the paper accordingly.\n\n> W1: some ablation results (e.g., choice of basis functions) suggest non-trivial sensitivity, but there's limited discussion on tuning strategies; paper could better detail failure cases or conditions where BEAST might underperform.\n> \n\nThank you for pointing this out, that’s indeed a valuable observation. We investigated the irregular performance drop from `N = 15` to `N = 20` in more detail. We found that it was due to overfitting in the ridge estimator (line 145~146) when too many basis functions are used. Specifically, the default regularization parameter was set to a very small value ($\\lambda=1e-9$), which we found insufficient.  After increasing it to $\\lambda=1e-4$, BEAST-F with `N = 20` also achieves 4.32 on CALVIN ABC. To further validate this, we re-evaluated the the model for `N = 15`, and added an additional ablation for `N = 25`. The complete ablation is attached below.\n\n| Number of Basis | 5 | **10** | 15 | 20 | 25 |\n| --- | --- | --- | --- | --- | --- |\n| Calvin ABC | 3.88 | **4.43** | 4.20 | 4.32 | 4.23 |\n\n> W2.1: the distinction between discrete and continuous token modes in some sections could be more explicit;\n> \n\nWe apologize for the lack of clarity. To clarify: **continuous tokens** refer to the normalized B-spline control points used directly, without any quantization. In contrast, **discrete tokens** are obtained by discrete these control points into uniform bins ranging from 0 to 255. We will update the paper accordingly.\n\n> W2.2: Descriptions of integration into large models like Florence-2 may assume familiarity with these architectures.\n> \n\nWe agree with the reviewer that a short description of Florence-2 will enhance the clarity, therefore we will include the following description into the appendix for the final version:\n\nFlorence-2 [1] is a compact vision-language model with 0.77B parameters, featuring an encoder-decoder language model architecture paired with a DaViT image encoder that efficiently compresses images into sequences into just 50 tokens. Florence-2 offers two unique advantages: First, Florence-2 was pretrained with a large-scale dataset focusing on image-grounding with tasks like bounding box prediction and image segmentation. These objectives are well aligned with robotic manipulation challenges. In addition, its small size and low image token count make it computational and memory efficient and enable us to run VLA experiments on consumer-grade hardware. This combination of task-relevant pretraining and computational efficiency made Florence-2 ideal for exploring our tokenizers within practical resource constraints. \n\n> Q1: Section 5.5 shows that BEAST is sensitive to the number of basis functions. Can you provide heuristics or general rules for selecting this parameter across different robot platforms and frequencies?\n> \n\nAs we mentioned in our response to W1, the sensitivity was caused by an extremely small regularization parameter in the ridge regression, which has now been addressed. We agree with the reviewer that choosing an appropriate number of basis functions is crucial for the performance of BEAST. As a general heuristic, we recommend starting with a number of basis functions equal to half of the sequence length. In practice, we typically sample a batch of action sequences from the dataset (e.g., 100 trajectories or more) and compute the reconstruction mean square error (MSE). An MSE below 1e-2 usually indicates that the chosen number of basis functions allows BEAST to represent the original trajectories well. We then repeat this process to find the minimal number of basis functions that achieves this threshold, thereby maximizing compression. Since BEAST does not require separate tokenizer training, this hyperparameter tuning process can be done in a few minutes across different action spaces and frequencies.\n\n> Q2: Are there scenarios where BEAST fails to outperform traditional tokenizers, such as in highly discontinuous or impulsive control settings?\n> \n\nWe assume the reviewer refers to traditional tokenizers such as single-step binning or quantized chunking (i.e., Binning + AC). Action chunking can be interpreted as a special case of zero-degree B-splines, and we provide further discussion on this in Section 4.1. In highly discontinuous or impulsive control scenarios, where sharp changes or sudden actions dominate, we can configure BEAST to replicate traditional tokenizer’s behavior. Specifically, setting the spline degree `P = 0` and number of basis functions `N = 1` allows BEAST to act as a single-step binning tokenizer, while setting the `P = 0` and  `N = T` emulates quantized action chunking. We include a table below for clarity:\n\n|  | **Spline Degree P** | **Number of Basis Functions N** |\n| --- | --- | --- |\n| Binning | 0 | 1 |\n| Binning + AC | 0 | T (Action Sequence Length) |\n\nTherefore, even in highly discontinuous or impulsive control settings, BEAST does not inherently fail to outperform traditional methods, instead, it includes them as special cases within its formulation. This highlights one of BEAST’s key strengths: its representational flexibility to adapt to a wide range of control characteristics.\n\n> Q3: You highlight reduced token counts. Can you elaborate on any observed tradeoffs between compression and policy expressivity beyond MSE?\n> \n\nWe agree with the reviewer that there is a trade-off between compression and policy performance. Fewer basis functions yield smoother trajectories and shorter token sequences, improving compression. However, too few may underfit and miss essential trajectory features, resulting in sub-optimal policy performance, especially in tasks that require precise and dexterous control. This can be observed from the table in response to W1, where `N = 5` only achieves average sequence length of 3.88, which is noticeable lower than `N = {10, 15, 20, 25}`. On the other hand, more basis functions improve expressiveness but risks overfitting to noise present in the original action sequences and reduce compression gains. Notably, BEAST does not aim to solely minimize the MSE directly but prioritizes generating smooth trajectories with low MSE.\n\n> In line 320 and Table 4., the indices are incorrectly processed as hyperlinks to references.\n> \n\nThanks for pointing out the typo, we will update the paper accordingly\n\n## References:\n\n[1] Xiao, Bin, et al. \"Florence-2: Advancing a unified representation for a variety of vision tasks.\" *CVPR* 2024."}, {"review_id": "behGOshCdV", "reviewer": "Reviewer_s1wN", "summary": "The paper introduces BEAST (B-spline Encoded Action Sequence Tokenizer), a novel method for representing continuous robot action sequences as compact, fixed-length discrete or continuous tokens. A core aspect of BEAST is its use of B-splines, which intrinsically ensures smooth, continuous trajectories without requiring separate tokenizer training.\n\nThe key contributions of this work are:\n*   **Novel B-spline-based Tokenization:** BEAST encodes continuous action sequences using B-splines, providing a compact and efficient representation.\n*   **Versatility Across Architectures:** The paper demonstrates BEAST's compatibility by integrating it with various model architectures, including VAEs (BEAST-ACT), decoder-only Transformers (BEAST-D), and Vision-Language Models like Florence-2 (BEAST-F), supporting both continuous and discrete token objectives.\n*   **Enhanced Efficiency:** BEAST significantly reduces computational costs during both training and inference. Its fixed-length token sequences enable faster action generation through parallel decoding, requiring significantly fewer tokens compared to binning-based methods.\n*   **Guaranteed Smoothness:** By leveraging clamped B-splines and enforcing initial conditions, BEAST ensures mathematically guaranteed smooth transitions between consecutive action chunks, preventing jerky movements.\n*   **Strong Performance:** Extensive evaluations on 166 simulated tasks (CALVIN, LIBERO, ALOHA) and 8 real-world robotic tasks across three different setups demonstrate that BEAST achieves competitive or superior task success rates compared to state-of-the-art methods, often without relying on large-scale pretraining.\n*   **Ablation Studies:** The paper includes thorough ablation studies analyzing the impact of design choices such as discrete versus continuous tokens, the number of basis functions, and model size, providing insights into BEAST's performance characteristics.\n\nA noted limitation is BEAST's sensitivity to the chosen number of B-spline basis functions, which can impact performance.", "questions": "Here are some questions and suggestions for the authors, focusing on actionable points for a productive rebuttal:\n\n1.  **Robustness and Generalization of Basis Function Selection (Line 337-339):**\n    *   **Question:** The limitations section acknowledges sensitivity to the number of B-spline basis functions (`N`), stating that \"The optimal count depends on the smoothness and sampling frequency of the trajectory; our experiments indicate that using 5-10 bases works well for one-second robot trajectories.\" However, Table 4 shows a significant drop in average sequence length (performance) from `N=15` (4.14) to `N=20` (2.71) for a 20-step action chunk (Line 322). This suggests that `N=20` might be suboptimal or that higher `N` values *can* negatively influence performance, which contradicts the intuition that more basis functions provide more representational power. Could the authors elaborate on the trade-offs beyond \"compressing vs. representational power\"? Is there a specific threshold or relationship between `N` and `T` (action sequence length) beyond which performance degrades significantly, or a methodology for finding the optimal `N` given varying trajectory characteristics?\n    *   **Suggestion:** Expand on the reasons for the performance drop at `N=20` in Table 4. Discuss more deeply the practical considerations and a more systematic approach (e.g., a small search, heuristic, or adaptive method) for selecting `N` across different tasks and action sequence lengths, especially given the sensitivity mentioned.\n    *   **Criteria for Score Change:** Providing a more detailed analysis of the `N` selection, perhaps with a clear methodology for practitioners to determine `N` or a deeper insight into the observed performance drop, would strengthen the paper's claims about its practical applicability. If the authors can show a more robust way to handle this sensitivity, the score would increase.\n\n2.  **Comparison with Other Trajectory Parameterizations / Implicit Representations:**\n    *   **Question:** The paper primarily compares BEAST against \"binning-based tokenization\" (Section 5.1) and other VLA/Diffusion policies that might use simpler action chunking or direct prediction. While BEAST's B-spline approach offers advantages like smoothness and fixed-length tokens, there are other established methods for continuous trajectory parameterization in robotics (e.g., Gaussian Process Motion Priors, Dynamic Movement Primitives, or even learned implicit representations for trajectories). Could the authors discuss how BEAST compares conceptually and/or empirically (if relevant prior work exists) against these alternative trajectory representations, especially concerning their ability to ensure smoothness, achieve compression, and integrate with deep learning architectures?\n    *   **Suggestion:** Add a brief discussion in the Related Work or Limitations section acknowledging other continuous trajectory representations. Even a qualitative discussion of their pros and cons relative to BEAST's B-spline approach would add significant value by contextualizing BEAST within a broader spectrum of trajectory encoding methods.\n    *   **Criteria for Score Change:** Acknowledging and briefly discussing other continuous trajectory parameterization methods would demonstrate a more comprehensive understanding of the landscape and position BEAST more clearly within it, leading to a higher score.\n\n3.  **Implications of \"Without Pretraining\" (Line 247-248):**\n    *   **Question:** The paper frequently highlights that BEAST-F achieves strong results \"without relying on additional pretraining\" (Line 247) or \"without robot data pretraining\" (Line 248). While this is a significant advantage, Florence-2 itself is a large pre-trained Vision-Language Model. Could the authors clarify the distinction more explicitly? Is the claim specifically that BEAST-F *doesn't require additional robot-specific pretraining* (beyond what Florence-2 already has for general vision-language tasks), or that it performs well *from scratch* (which seems less likely given Florence-2's nature)? A nuanced explanation would prevent potential misinterpretations regarding the extent of \"pretraining\" involved.\n    *   **Suggestion:** Rephrase or elaborate on the \"without pretraining\" claim to clearly state that it refers to *robot-specific* or *second-stage* pretraining on large robot datasets, distinguishing it from the foundational pretraining of Florence-2. This would make the contribution clearer and more precise.\n    *   **Criteria for Score Change:** A precise clarification of what \"without pretraining\" entails in the context of BEAST-F, acknowledging the base VLM's pretraining, would increase the clarity and accuracy of the paper's claims, leading to a higher score.", "limitations": "Please check the weakness section.", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "### Strengths:\n\n*   **Originality:** The core idea of using B-splines for action sequence tokenization, specifically the direct encoding without separate training and the inherent smoothness guarantee, appears novel in the context of imitation learning and VLA models. This differentiates it from common binning, VQ-VAE, or compression-based methods.\n*   **Quality - Technical Soundness:** The mathematical formulation of B-splines and their application to action sequences (Equation 4, clamped B-splines for smooth transitions) is well-grounded. The claim of mathematically guaranteed smoothness is a strong technical advantage. The efficiency gains from fixed-length tokens and parallel decoding are clearly articulated and supported by empirical results.\n*   **Quality - Empirical Performance:** The paper presents extensive evaluations across numerous simulated benchmarks (CALVIN, LIBERO, ALOHA, 166 tasks) and real-world robot setups (8 tasks across 3 different robots). Consistently achieving competitive or state-of-the-art results, often with smaller models and without large-scale robot-specific pretraining, demonstrates the method's effectiveness. The ablation studies provide useful insights into design choices.\n*   **Clarity:** The paper is generally well-written and easy to follow. The introduction clearly outlines the problem and BEAST's proposed solution. Figure 1 and 2 effectively illustrate the core concepts of B-splines and the encoding pipeline. The experimental setup and results are presented clearly with appropriate tables and figures.\n*   **Significance:** BEAST offers a practical solution to several challenges in robot imitation learning, including action sequence discretization, ensuring smooth trajectories, and improving inference efficiency. Its compatibility with various architectures (VAE, Transformer, VLM) makes it a versatile tool. The efficiency gains (2.14x to 101.4x faster inference, faster training convergence) are highly significant for real-world robotic applications and research. The reduced token count is also a notable contribution to efficiency.\n\n### Weaknesses:\n\n*   **Quality - Sensitivity to Basis Function Selection (Actionability):** While the paper acknowledges the sensitivity to the number of B-spline basis functions (\\`N\\`) in the limitations, the explanation for why \\`N=20\\` performs significantly worse than \\`N=15\\` for a 20-step chunk in Table 4 is somewhat vague (\"reduce compression, which can also negatively influence the performance\"). This is a critical hyperparameter, and the guidance \"5-10 bases works well for one-second robot trajectories\" might not generalize well to different action chunk lengths or robot dynamics. A more principled or robust way to select \\`N\\`, or a deeper understanding of its impact beyond a simple trade-off, is missing. \n*   **Clarity - Notation for B-spline Parameters:** There's a slight potential for confusion between the B-spline degree \\`P\\` and the number of control points \\`N\\` (e.g., \"Remark 1\" stating \\`T=N\\` for \\`P=0\\` vs. \\`N <= T\\` for \\`P\\`-th degree, and how \\`N\\` is precisely chosen for \\`P > 0\\` degrees given a \\`T\\`-length action sequence). While explained, a concise clarification on the typical choices of \\`P\\` and \\`N\\` in relation to the action sequence length \\`T\\` in practice would improve understanding.", "was_revised": true, "final_justification": "Thank you for the detailed rebuttal, which has addressed most of my concerns. I will raise the scores.", "rebuttal": "We would like to thank the reviewer for taking the time to review our work and the many helpful comments and suggestions. We hope the following replies sufficiently address the raised questions and concerns. We will update the paper accordingly.\n\n> **Q1, W1:** …However, Table 4 shows a significant drop in average sequence length (performance) from `N=15` (4.14) to `N=20` (2.71) for a 20-step action chunk (Line 322). This suggests that `N=20` might be suboptimal or that higher `N` values *can* negatively influence performance, which contradicts the intuition that more basis functions provide more representational power. Could the authors elaborate on the trade-offs beyond \"compressing vs. representational power\"? Is there a specific threshold or relationship between `N` and `T` (action sequence length) beyond which performance degrades significantly, or a methodology for finding the optimal `N` given varying trajectory characteristics?\n> \n\nThank you for pointing this out, that’s indeed a valuable observation. We investigated the irregular performance drop from `N = 15` to `N = 20` in more detail. We found that it was due to overfitting in the ridge estimator (line 145~146) when too many basis functions are used. Specifically, the default regularization parameter was set to a very small value ($\\lambda=1e-9$), which we found insufficient.  After increasing it to $\\lambda=1e-4$, BEAST-F with `N = 20` also achieves 4.32 on CALVIN ABC. To further validate this, we re-evaluated the the model for `N = 15`, and added an additional ablation for `N = 25`. The complete ablation is attached below.\n\n| Number of Basis | 5 | **10** | 15 | 20 | 25 |\n| --- | --- | --- | --- | --- | --- |\n| Calvin ABC | 3.88 | **4.43** | 4.20 | 4.32 | 4.23 |\n\nFor finding the optimal `N`, we recommend starting with a number of basis functions equal to half of the sequence length. In practice, we typically sample a batch of action sequences from the dataset (e.g., 100 trajectories or more) and compute the reconstruction mean square error (MSE). An MSE below 1e-2 usually indicates that the chosen number of basis functions allows BEAST to represent the original trajectories well. We then repeat this process to find the minimal number of basis functions that achieves this threshold, thereby maximizing compression. Since BEAST does not require separate tokenizer training, this hyperparameter tuning process can be done in a few minutes across different action spaces and frequencies. We will include this practical tuning guide in the appendix.\n\n> **Q2:** The paper primarily compares BEAST against \"binning-based tokenization\" (Section 5.1) and other VLA/Diffusion policies that might use simpler action chunking or direct prediction. While BEAST's B-spline approach offers advantages like smoothness and fixed-length tokens, there are other established methods for continuous trajectory parameterization in robotics (e.g., Gaussian Process Motion Priors, Dynamic Movement Primitives, or even learned implicit representations for trajectories). Could the authors discuss how BEAST compares conceptually and/or empirically (if relevant prior work exists) against these alternative trajectory representations, especially concerning their ability to ensure smoothness, achieve compression, and integrate with deep learning architectures?\n> \n\nWe appreciate the reviewer’s constructive suggestion. In response, we will include the following discussion in the related works section:\n\nDynamic Movement Primitives (DMPs)[1] employ a second-order dynamic system with a goal attractor. When we use DMPs to encode action chunks, only trajectory position and velocity at the transitions are continuous, because of its second-order formulation. Additionally, their goal is only converged when time goes to infinity instead of finite time and its resulting position basis functions (after integrating the dynamic system) is asymmetric, as shown Fig. 1 in [2] , which potentially influences its representation capacity. In contrast, the B-Splines representation allows for arbitrary-order smoothness by increasing the spline degree and the basis functions are symmetric, as reported in Fig. 1 top row, in our paper. \n\nTo tackle the issues of DMPs, recent works [3, 4] have also explored using B-splines as a Movement Primitives representation. However, these methods mainly focus on reinforcement learning settings and only utilize it as continuous action abstraction. \n\n**To our best knowledge, BEAST is the first work that integrates B-splines into imitation learning architectures, generating discrete tokens for VLAs.** \n\nGaussian Process Motion Planner (GPMPs) [5] identifies waypoints as support states and uses Gaussian Process (GP) to interpolates the trajectories.  However, GP-based methods are usually constrained to conditioned on low-dimensional conditions, such as via-points. It is unclear how to employ them with high-dimensional conditions, such as images and language. To overcome this issue, Conditional Neural Movement Primitives (CNMPs) [7] was introduced to model trajectories using Conditional Neural Processes (CNPs) [6]. CNMPs need a separate training process and cannot guarantee continuous transitions (see Fig 10 (b) in [2]). **In contrast, BEAST requires no separate training for tokenization and guarantee smooth transitions between chunks.** \n\nWe summarize a comparison of these works in the table below: \n\n| Model | Need training | Chunk transition smoothness | High dimensional conditioning (Language, image) |\n| --- | --- | --- | --- |\n| DMP [1, 2] | No | Position, Velocity | Yes [2] |\n| GPMP [5] | No | No | No |\n| CNMP (CNP) [6, 7] | Yes | No | Yes [7] |\n| BEAST (ours) | No | Arbitrary orders | Yes |\n\n> **Q3:** The paper frequently highlights that BEAST-F achieves strong results \"without relying on additional pretraining\" (Line 247) or \"without robot data pretraining\" (Line 248). While this is a significant advantage, Florence-2 itself is a large pre-trained Vision-Language Model. Could the authors clarify the distinction more explicitly? Is the claim specifically that BEAST-F *doesn't require additional robot-specific pretraining* (beyond what Florence-2 already has for general vision-language tasks), or that it performs well *from scratch* (which seems less likely given Florence-2's nature)? A nuanced explanation would prevent potential misinterpretations regarding the extent of \"pretraining\" involved.\n> \n\nWe thank the reviewer for pointing this out, for the “without relying on additional pretraining” we indeed mean without additional robot-specific pretraining. We would highlight this even more in the final version. \n\n> **W2: Clarity - Notation for B-spline Parameters:** There's a slight potential for confusion between the B-spline degree `P` and the number of control points `N` (e.g., \"Remark 1\" stating `T=N` for `P=0` vs. `N <= T` for `P`-th degree, and how `N` is precisely chosen for `P > 0` degrees given a `T`-length action sequence). While explained, a concise clarification on the typical choices of `P` and `N` in relation to the action sequence length `T` in practice would improve understanding.\n> \n\nWe thank the reviewer for pointing out the confusion in the notation. We will update the paper accordingly. \n\nIn the BEAST tokenizer, there are three key hyperparameters:\n\n- `T`: the length of an action chunk (i.e., number of actions),\n- `P`: the degree of the B-spline, which determines the level of continuity of the trajectory,\n- `N`: the number of basis functions, i.e., control points per action dimension.\n\nHere we attach a table to demonstrate the relation between P and level of continuity:\n\n| P (Degree) | 0 | 1 | 2 | 3 | 4 |\n| --- | --- | --- | --- | --- | --- |\n| Continuity | None | Position | Pos+Vel | Pos+Vel+Acc | Pos+Vel+Acc+Jerk |\n\nIn BEAST, we typically choose `P = 4` to achieve continuity up to jerk trajectory.\n\nThe number of basis functions `N` determines how many control points are used for each action dimension within one action chunk. A higher `N` offers higher granularity in trajectory representation but risks overfitting to noise in action sequences. Smaller `N`, on the other hand, offers better compression but risks underfitting and dexterity loss. \n\nAccording to the recursive definition of B-splines, `N` must satisfy `N > P` [8].\n\nMathematically, there is no hard dependencies between T and N. In practice, using `N ≤ T` can compress token sequence length, resulting in more efficient training and inference. For details on how the optimal value of `N` is determined, please refer to our response to Q1.\n\nTo your question regarding Remark 1, when choosing `P = 0` and `N = T`, B-splines are equivalent to action chunking, where each control point correspond exact to an action point. \n\n### References:\n\n**[1]** Ijspeert A. et al., \"Dynamical movement primitives: learning attractor models for motor behaviors,\" *Neural Computation* 2013.\n\n**[2]** Li G. et al., \"Prodmp: A unified perspective on dynamic and probabilistic movement primitives,\" *IEEE RA-L* 2023.\n\n**[3]** Liao W. et al., \"BMP: Bridging the Gap between B-Spline and Movement Primitives,\" *2nd CoRL Workshop on Learning Effective Abstractions for Planning*, 2024.\n\n**[4]** Kicki P. et al., \"Bridging the gap between Learning-to-plan, Motion Primitives and Safe Reinforcement Learning,\" *CoRL* 2024.\n\n**[5]** Mukadam M. et al., \"Gaussian process motion planning,\" *IEEE ICRA* 2016.\n\n**[6]** Garnelo M. et al., \"Conditional neural processes,\" *ICML* 2018.\n\n**[7]** Seker M. Y. et al., \"Conditional Neural Movement Primitives,\" *RSS* 2019.\n\n**[8]** Prautzsch H. et al., *Bézier and B-spline Techniques*, Springer, Aug. 2002."}]

Author rebuttal:

Meta-review: The paper proposes an interesting idea for tokenizing continuous action sequences in imitation learning, which compresses continuous action trajectories into fixed-length, smooth representations. It addresses a core bottleneck in modern VLAs, and proves effective across multiple benchmarks and robots without large-scale pretraining.

All reviewers vote for acceptance, and the AC accordingly recommends accepting this submission.
Decision: Accept (poster)