# Behavioral Refinement via Interpolant-based
Policy Diffusion

Kaiqi Chen1, Eugene Lim1, Kelvin Lin1, Yiyang Chen1, and Harold Soh1,2
1Dept. of Computer Science, National University of Singapore.
2Smart Systems Institute, NUS.
Contact Authors: {kaiqi<EMAIL_ADDRESS>
###### Abstract

Imitation learning empowers artificial agents to mimic behavior by learning
from demonstrations. Recently, diffusion models, which have the ability to
model high-dimensional and multimodal distributions, have shown impressive
performance on imitation learning tasks. These models learn to shape a policy
by diffusing actions (or states) from standard Gaussian noise. However, the
target policy to be learned is often significantly different from Gaussian and
this mismatch can result in poor performance when using a small number of
diffusion steps (to improve inference speed) and under limited data. The key
idea in this work is that initiating from a more informative source than
Gaussian enables diffusion methods to mitigate the above limitations. We
contribute both theoretical results, a new method, and empirical findings that
show the benefits of using an informative source policy. Our method, which we
call BRIDGeR, leverages the stochastic interpolants framework to bridge
arbitrary policies, thus enabling a flexible approach towards imitation
learning. It generalizes prior work in that standard Gaussians can still be
applied, but other source policies can be used if available. In experiments on
challenging benchmarks, BRIDGeR outperforms state-of-the-art diffusion
policies and we provide further analysis on design considerations when
applying BRIDGeR.

##  I Introduction

Imitation learning enables robots to learn policies from demonstrations and
has been applied to a variety of domains including manipulation [33, 56, 12],
autonomous driving [32, 4], and shared autonomy [38, 54]. Recently, there has
been a significant interest in the adaptation of diffusion models for
imitation learning [6, 36, 17]. These deep generative models, which
progressively transform Gaussian noise to a policy over a number of diffusion
steps, offer practical advantages over classical techniques [40, 41] — they
scale well with the number of dimensions in the action/state spaces (e.g., for
visuo-motor learning on a 7-DoF robot arm [6]) and are able to capture complex
multimodal distributions. However, current diffusion methods also require
large training datasets and typically have long inference times due to the
number of diffusion steps needed to obtain effective action distributions for
complex tasks [36].

Figure 1: (A) Overview of action generation with BRIDGeR. With trained
velocity $b$ and score $s$ functions, BRIDGeR transports the actions from
source distribution $\pi_{0}(a|x)$ to the target distribution $\pi_{1}(a|x)$
via the forward SDE (Eq. 9). (B) We tested BRIDGeR on challenging robot
benchmark tasks and show that using informative source policies enhances
performance. For example, in 6-DoF grasp generation, using heuristic or data-
driven source policies results in more successful grasps compared to the
conventional Gaussian.

An examination of existing diffusion-style imitation learning reveals a
fundamental issue: these models learn to shape a policy starting from standard
Gaussian noise, which is often starkly different from the intended policy or
action distribution. The key insight in our work is that initiating from
Gaussian noise isn’t a prerequisite. To explore this, we move beyond the
conventional diffusion framework and employ stochastic interpolants [2] for
bridging arbitrary densities within finite time (Fig. 1). This approach allows
us to leverage stochastic source policies, enabling the diffusion process to
begin from a more informative starting point. We find that this shift retains
the inherent advantages of diffusion-style imitation learning, but positively
impacts inference time and performance. Our approach also generalizes prior
work in diffusion-based imitation learning since if no source policy is
available, simple distributions such as the Gaussian can be used.

In this paper, we first contribute a theoretical analysis of the impact of
different source policies in diffusion. In brief, we find that under
reasonable assumptions, selecting a better source policy results in better
target policies. We then turn to a practical approach for incorporating source
policies into diffusion methods. Applying the stochastic interpolants
framework [2] to imitation learning, we derive a new method called BRIDGeR
(Behavioral Refinement through Interpolant-based Diffusion for Generative
Robotics). To our knowledge, our work is the first application of this
bridging methodology to imitation learning, contrasting with its previous use
in simple synthetic tasks [58] and image generation [58, 24]. In addition to
standard neural architecture design for the learnt forward model, the
stochastic interpolant framework relies on several critical design choices
including the source policy and interpolant. The interpolant dictates how a
sample transitions from the source to the target distribution, with the
transition modulated by noise introduced through time-dependent Gaussian
latent variables [2].

We contribute a systematic empirical study of the effects of using source
policies (and other design elements) on a diverse set of robot tasks,
including the Franka kitchen benchmark, grasp generation, and challenging
tasks using a robot hand. Overall, the experimental results coincide with our
theoretical findings; Gaussians were seldom the most effective source
distribution and surprisingly, even simple heuristic distributions resulted in
superior learnt policies compared to the Gaussian. We demonstrate that given a
good source policy, BRIDGeR surpasses existing state-of-the-art diffusion
policies. Additionally, we discuss the effects of the interpolant function
when learning highly multi-modal behaviors.

In summary, our work connects distribution bridging to imitation learning,
which results in improved performance and addresses inherent limitations of
standard diffusion, such as lengthy inference times. We contribute:

  * •

Theoretical results on the impact of diffusing from source policies of varying
quality;

  * •

A practical method that enables source distributions to be used in diffusion
methods for imitation learning, allowing for better trade-offs between
inference speed and performance;

  * •

A comprehensive empirical study demonstrating the impact of source
distributions and interpolant design on outcome quality across various robot
tasks.

From a broader perspective, our research demonstrates the potential of
bridging models in imitation learning. We hope that this work lays the
foundation for future imitation learning methods that leverage past policies
for lifelong robot learning.

##  II Preliminaries: Background and Related Work

###  II-A Problem Formulation

In imitation learning, we wish to learn a policy from expert demonstrations.
Let $\pi_{1}(a|x)$ be an expert policy; it captures the probability of an
expert selecting an action $a$ given an observation $x$. Suppose we have a
dataset $\mathcal{D}=\\{\smash{x^{(i)}},\smash{a^{(i)}}\\}_{i=1}^{N}$ drawn
from $\pi_{1}(a|x)$ and let $\pi_{0}(a|x)$ be a distribution that we can
easily sample $a$ from.

Our goal is to learn a model for transporting actions drawn from $\pi_{0}$ to
$\pi_{1}$. More concretely, given an observation $x$, let $\pi_{t}(a|x)$ over
time $t\in[0,1]$ be distributions on the _bridge_ that links $\pi_{0}(a|x)$
and $\pi_{1}(a|x)$. Let $\hat{\pi}_{t}$ be the density of the learned
distribution over time $t\in[0,1]$. We want the generated target density
$\hat{\pi}_{1}$ to match the ground truth density $\pi_{1}$.

Similar to recent research [6], our approach primarily generates _action
sequences_ rather than single-step actions. Upon completing (or partially
completing) an action sequence, the model assimilates a new observation to
generate the subsequent action sequence. In the following, $a$ denotes an
action sequence or a target pose, depending on the context.

###  II-B Diffusion-based Policy Learning

Diffusion-based policy methods [6, 36] are largely based on Denoising
Diffusion Probabilistic Models (DDPM) [16]. These methods operate by
progressively adding noise to an action $a_{1}$ during a forward process and
subsequently, employing a reverse process to learn how to denoise. This is
achieved by training a neural network $g_{\upsilon}(a_{t_{k}},t_{k})$ to
predict the noise $z\sim\mathcal{N}(0,I)$ added to the data sample, within a
defined temporal sequence $0=t_{0}<t_{1}<\dots<t_{K-1}<t_{K}=1$. The training
utilizes a regression loss,

$\mathcal{L}(\upsilon)=\mathbb{E}\left[\lVert
z-g_{\upsilon}(\sqrt{\bar{\alpha}_{k}}a_{1}+\sqrt{1-\bar{\alpha}_{k}}z,t_{k})\rVert^{2}\right].$
(1)

Upon completion of training, DDPM utilizes initial samples $a_{0}$ drawn from
a standard Gaussian distribution and applies a $K$-step denoising procedure to
synthesize the desired output $a_{1}$,

$a_{t_{k}}=\frac{1}{\sqrt{\alpha_{k}}}\left(a_{t_{k-1}}-\frac{1-\alpha_{k}}{\sqrt{1-\bar{\alpha}_{k}}}g_{\upsilon}(a_{t_{k}},t_{k})\right)+\sigma_{k}\mathcal{N}(0,I).$
(2)

One limitation the original DDPM is slow sampling; the number of steps $K$
required is typically large (e.g., hundreds to thousands). To improve sample
efficiency, diffusion-based policy methods [6, 36] employ Denoising Diffusion
Implicit Models (DDIM) [8], which apply a _deterministic_ non-Markovian
process to trade-off between sample quality and inference speed.

##  III Bridging Policies: Theoretical Considerations

The central premise of this work is the use of source policies for diffusion-
based imitation learning: we posit that starting with a more informative
source density facilitates the shaping of the target density. We examine this
hypothesis theoretically in this section. Under reasonable assumptions, we
show that a “good” source policy can enhance the resultant the target policy
up to an addictive factor. Note that these results apply to any diffusion-type
model whereby a source distribution is gradually adapted over time to match a
target.

Formally, we denote the “difference” between the action distribution
$\hat{\pi}$ at time $t$ (conditioned upon observation $x$) and the expert
policy $\pi_{1}$ as

$\phi_{F,\hat{\pi}}(t,x)=F({\hat{\pi}_{t}(\cdot|x)},{\pi_{1}(\cdot|x)}).$

where $F(\cdot,\cdot)$ is a measure of difference between two distributions
(e.g. KL divergence $\mathbb{KL}(\cdot,\cdot)$ and cross-entropy
$\mathbb{H}(\cdot,\cdot)$). In our setup, we diffuse from a source
distribution $\hat{\pi}$ from time $t=0$ to $t=1$, and would like
$\phi_{F,\hat{\pi}}(1,x)$ to be small.

######  Assumption 1.

There exist constants $\epsilon_{\max}>\epsilon_{\min}>0$ such that for all
$t,x$,

$0\geq-\epsilon_{\min}\geq\partial_{t}\phi_{F,\hat{\pi}}(t,x)\geq-\epsilon_{\max}.$

We can interpret $\epsilon_{\min}\,\mathrm{d}t$ and
$\epsilon_{\max}\,\mathrm{d}t$ as the minimum and maximum improvement in the
differences towards $\pi_{1}$ after diffusing for some infinitesimal
$\mathrm{d}t$ time. We believe is this reasonable given a trained model with
limited capacity.

######  Theorem 1.

Let $\hat{\pi}_{0}$ and $\hat{\rho}_{0}$ be two source distributions and given
that Assumption 1 holds. Then the improvement of the generated target
distribution is bounded by the improvement of the source distribution

$\displaystyle\,\phi_{F,\hat{\pi}}(1,x)-\phi_{F,\hat{\rho}}$
$\displaystyle(1,x)$ $\displaystyle\leq$
$\displaystyle\,\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x)+\epsilon_{\max}-\epsilon_{\min}.$

The proof can be found in the Appendix A. Intuitively, Theorem 1 states that
if $\hat{\pi}_{0}$ is a better source distribution than $\hat{\rho}_{0}$
(i.e., $\phi_{F,\hat{\pi}}(0,x)<\phi_{F,\hat{\rho}}(0,x)$), then after
diffusion, $\hat{\pi}_{1}$ is better than $\hat{\rho}_{1}$ up to an additive
factor of $\epsilon_{\max}-\epsilon_{\min}$. To elaborate, let us rewrite the
bound as

$\displaystyle\phi_{F,\hat{\pi}}(1,x)+d(\hat{\pi}_{0},\hat{\rho}_{0})-(\epsilon_{\max}-\epsilon_{\min})\leq\phi_{F,\hat{\rho}}(1,x)$

where
$d(\hat{\pi}_{0},\hat{\rho}_{0})=\phi_{F,\hat{\rho}}(0,x)-\phi_{F,\hat{\pi}}(0,x)$
is the difference in F between $\hat{\pi}$ and $\hat{\rho}$ at time $0$. If
$\phi_{F,\hat{\pi}}(0,x)<\phi_{F,\hat{\rho}}(0,x)$, then
$d(\hat{\pi}_{0},\hat{\rho}_{0})>0$. The positive factor
$\epsilon_{\max}-\epsilon_{\min}$ accounts for the variability in improvements
during the diffusion process; a greater disparity in the changes when starting
from $\hat{\pi}_{0}$ versus $\hat{\rho}_{0}$ can influence the quality of the
resulting target distributions. If we further assume that the improvements are
equal regardless of the initial source, then this factor disappears and we
obtain

$\displaystyle\phi_{F,\hat{\pi}}(1,x)+d(\hat{\pi}_{0},\hat{\rho}_{0})\leq\phi_{F,\hat{\rho}}(1,x).$

In practice, it is necessary to discretize time steps for sampling. Next, we
extend our theoretical results to discrete time. Suppose we split the domain
of time $[0,1]$ into $K+1$ discrete time steps
$0=t_{0}<t_{1}<\dots<t_{K-1}<t_{K}=1$. We make a similar assumption,

######  Assumption 2.

There exist constants $\epsilon_{\max}>\epsilon_{\min}>0$ such that for all
$t,x$,

$0\geq-\epsilon_{\min}\delta
t_{k}\geq\phi_{F,\hat{\pi}}(t_{k},x)-\phi_{F,\hat{\pi}}(t_{k-1},x)\geq-\epsilon_{\max}\delta
t_{k}$

where $k\in\\{1,\dots,K\\}$ and $\delta t_{k}=t_{k+1}-t_{k}$.

Here, $\epsilon_{\min}\,\delta t_{k}$ and $\epsilon_{\max}\,\delta t_{k}$
quantify the minimum and maximum improvement in $F$ towards $\pi_{1}$ after
diffusing with step size $\delta t_{k}$.

######  Theorem 2.

Let $\hat{\pi}_{0}$ and $\hat{\rho}_{0}$ be two source distributions and given
that Assumption 2 holds. Then the improvement of the generated target
distribution is bounded by the improvement of the source distribution

$\displaystyle\,\phi_{F,\hat{\pi}}(1,x)-\phi_{F,\hat{\rho}}$
$\displaystyle(1,x)$ $\displaystyle\leq$
$\displaystyle\,\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x)+\epsilon_{\max}-\epsilon_{\min}.$

Finally, we establish a similar result for the expected cost
$\mathbb{E}[c(a|x)]$ when $F$ is the cross-entropy and the expert is Boltzmann
rational.

######  Assumption 3.

The density of the expert policy

$\pi_{1}(a|x)=\frac{1}{Z}\exp(-c(a|x))$

where $Z$ is a normalizing constant.

######  Theorem 3.

Let $\hat{\pi}_{0}$ and $\hat{\rho}_{0}$ be two source distributions and given
that both Assumption 2 and 3 holds for $F=\mathbb{H}$ (cross-entropy). Then
for any observation $x$, we have

$\displaystyle\,\mathbb{E}_{a\sim\hat{\pi}_{1}}$
$\displaystyle[c(a|x)]-\mathbb{E}_{a\sim\hat{\rho}_{1}}[c(a|x)]$
$\displaystyle\leq$
$\displaystyle\,\mathbb{E}_{a\sim\hat{\pi}_{0}}[c(a|x)]-\mathbb{E}_{a\sim\hat{\rho}_{0}}[c(a|x)]+\epsilon_{\max}-\epsilon_{\min}.$

Similar to Theorem 1, if we let
$d_{c}(\hat{\pi}_{0},\hat{\rho}_{0})=\mathbb{E}_{a\sim\hat{\rho}_{0}}[c(a|x)]-\mathbb{E}_{a\sim\hat{\pi}_{0}}[c(a|x)])$
and $\hat{\pi}_{0}$ is a lower-cost source policy compared to
$\hat{\rho}_{0}$, i.e., $d_{c}(\hat{\pi}_{0},\hat{\rho}_{0})>0$, then under
equal improvement $(\epsilon_{\max}-\epsilon_{\min})=0$,

$\displaystyle\mathbb{E}_{\hat{\pi}_{1}}[c(a|x)]+d_{c}(\hat{\pi}_{0},\hat{\rho}_{0})<\mathbb{E}_{\hat{\rho}_{1}}[c(a|x)].$

In words, the policy derived from the better source policy achieves a lower
expected cost. In the next section, we discuss how we can practically bridge
distributions for imitation learning.

##  IV Method: BRIDGeR for Imitation Learning

In this section, we present a method that can learn to adapt a source policy
to match observed demonstrations. The source policy could be a simple
Gaussian, or an action distribution hand-crafted using prior knowledge, or a
data-driven policy learned from data. We call our method Behavioral Refinement
via Interpolant-based Diffusion for Generative Robotics (BRIDGeR).

BRIDGeR is based on Stochastic Interpolants [2], a recently-proposed framework
to bridge densities in finite time. At a high-level, a stochastic interpolant
is a time-dependent random variable that “interpolates” between two arbitrary
densities. As a concrete example, consider two densities $p_{0}$ and $p_{1}$.
A simple linear stochastic interpolant is

$\displaystyle y_{t}=(1-t)y_{0}+ty_{1}+\sqrt{2t(1-t)}z$ (3)

where $y_{0}$ and $y_{1}$ are drawn from $p_{0}$ and $p_{1}$, respectively,
and $z$ is drawn from a standard Gaussian. By construction, the paths of
$y_{t}$ bridge samples from $p_{0}$ at time $t=0$ and from $p_{1}$ at $t=1$.
Our goal is to learn how to transport samples along the paths of this
interpolant.

In the following, we will present stochastic interpolants more formally and
show how they can be adapted to imitation learning. We will then detail the
key design elements explored in this work.

###  IV-A Stochastic Interpolants for Imitation Learning

Let $(C^{r}(\mathbb{R}^{n}))^{m}$ be the space of $r$ continuously
differentiable functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ and
$(C_{0}^{r}(\mathbb{R}^{n}))^{m}$ as the space of compactly supported and $r$
continuously differentiable functions from $\mathbb{R}^{m}$ to
$\mathbb{R}^{n}$. We extend the definition of stochastic interpolant in [2] to
condition on observation $x$, similar to the concurrent work [18].

######  Definition 1 (Stochastic Interpolant).

Denote $n_{x}$ and $n_{a}$ as the dimension of observation and action
respectively. Let $x\in\mathbb{R}^{n_{x}}$ be an observation. Given two
probability density functions $\pi_{0}(a|x)$ and $\pi_{1}(a|x)$, a stochastic
interpolant between $\pi_{0}(a|x)$ and $\pi_{1}(a|x)$ is a stochastic process
$\\{a_{t}\\}_{t\in[0,1]}$ satisfying

$\displaystyle a_{t}=I(t,a_{0},a_{1},x)+\gamma(t)z,\quad t\in\left[0,1\right]$
(4)

where

  * 1.

$I\in
C^{2}([0,1]\times\mathbb{R}^{n_{a}}\times\mathbb{R}^{n_{a}}\times\mathbb{R}^{n_{x}})^{n_{a}}$
satisfies the boundary conditions $I(0,a_{0},a_{1},x)=a_{0}$ and
$I(1,a_{0},a_{1},x)=a_{1}$

  * 2.

There exist some $C_{1}<\infty$ such that
$|\partial_{t}I(t,a_{0},a_{1},x)|\leq C_{1}|a_{0}-a_{1}|$ for $t\in[0,1]$.

  * 3.

$\gamma(t)$ satisfies $\gamma(0)=\gamma(1)=0$, $\gamma(t)>0$ for all
$t\in(0,1)$, and $\gamma^{2}\in C^{2}([0,1])$.

  * 4.

The pair $(a_{0},a_{1})$ is drawn from a probability measure $v$ that
marginalizes on $\pi_{0}(\cdot|x)$ and $\pi_{1}(\cdot|x)$.

  * 5.

$z$ is a standard Gaussian random variable independent of $(a_{0},a_{1})$.

From Definition 1 and Theorem 2.6 of [2], the transport equation between the
source and target distribution is

$\displaystyle\partial_{t}\pi+\nabla_{a}\cdot(b\pi)=0$ (5)

where $\pi(t,a,x)\coloneqq\pi_{t}(a|x)$ and the velocity $b$ is defined as

$\displaystyle
b(t,a,x)=\mathbb{E}\left[\partial_{t}I(t,a_{0},a_{1},x)+\dot{\gamma}(t)z\right].$
(6)

We can rewrite the transport equation (Eq. 5) as a forward Fokker-Planck
equation, along with the corresponding forward stochastic process [2]. For any
$\epsilon\in C_{0}([0,1])$ with $\epsilon(t)\geq 0$ for all $t\in[0,1]$, the
probability density $\pi$ in equation 4 satisfies the forward Fokker-Planck
equation

$\displaystyle\partial_{t}\pi+\nabla_{a}\cdot(b_{F}\pi)=\epsilon\nabla^{2}_{a}\pi,\quad\pi(0)=\pi_{0}$
(7)

where

$\displaystyle b_{F}(t,a,x)\coloneqq b(t,a,x)+\epsilon(t)s(t,a,x)$ (8)

and $s(t,a,x)\coloneqq\nabla_{a}\log\pi(t,a,x)$ is the score of
$\pi_{t}(\cdot|x)$. The solutions of the forward stochastic differentiable
equation (SDE) associated with the Fokker-Planck (Eqn. 7) satisfy

$\displaystyle\mathrm{d}a_{t}=b_{F}(t,a_{t},x)\,\mathrm{d}t+\sqrt{2\epsilon(t)}\,\mathrm{d}W_{t}$
(9)

solved forward in time from the initial action $a_{0}\sim\pi_{0}$.

Action Sampling. Using above properties, we can sample actions by transporting
samples from the source distribution $\pi_{0}$ to the target distribution
$\pi_{1}$, i.e., we solve the forward SDE in Eq. 9 (See Fig. 1). The time
interval $t\in[0,1]$ is discretized into points ${t_{0},\ldots,t_{K}}$, with
$K$ denoting the total number of diffusion steps and $\delta t$ representing
the uniform time step increment between $t_{k}$ and $t_{k+1}$. The specifics
of the sampling process are outlined in Algorithm 1.

Algorithm 1 BRIDGeR Sampling

Current observation $x$ and a sample from the source policy
$a_{0}\sim\pi_{0}(a_{0}|x)$

for $k\leftarrow 1$ to $K$ do

$z\sim\mathcal{N}(0,I)$

$b_{F}(t_{k},a_{t_{k}},x)=b(t_{k},x)+\epsilon(t_{k})s(t_{k},a_{t_{k}},x)$

$a_{t_{k+1}}=a_{t_{(}k)}+b_{F}(t,a_{t_{k}},x)\delta
t+\sqrt{2\epsilon(t_{k})}z$

end for

$a_{1}$

Model Training.  Sampling actions as above requires the velocity $b$ and score
$s$. Suppose that these two functions are parameterized by function
approximators (e.g., neural networks), denoted as $b_{\theta}$ and $s_{\eta}$.
The velocity $b$ defined in equation 6 can be trained by minimizing,

$\displaystyle
L_{b}(\theta)=\int_{0}^{1}\mathbb{E}\left[b_{\theta}(t,a_{t},x)-(\partial_{t}I(t,a_{0},a_{1},x)+\dot{\gamma}(t)z)\right]^{2}dt$
(10)

where $a_{t}$ is defined in Eqn. (4) and the expectation is taken
independently over $\pi_{0}(a_{0}|x)$, $\pi_{1}(a_{1}|x)$, and
$\mathcal{N}(z|0,I)$. Similarly, the score $s$ of the probability density
$\pi(t)$ can be trained by minimizing

$\displaystyle
L_{s}(\eta)=\int_{0}^{1}\mathbb{E}\left[s_{\eta}(t,a_{t},x)+\gamma^{-1}(t)z\right]^{2}dt.$
(11)

In practice, predicting $\gamma^{-1}(t)z$ can be difficult since
$\gamma^{-1}(t)$ approaches infinity as $t$ approaches $0$ or $1$. To address
this issue, we re-parameterize the score as
$s_{\eta}=\hat{s}_{\eta}\gamma^{-1}(t)$ in our model. To further improve
training stability, we decompose the velocity $b$ as suggested in [2],

$\displaystyle b(t,a,x)=v(t,a,x)-\dot{\gamma}(t)\gamma(t)s(t,a,x)$ (12)

where $v$ can be trained by minimizing the quadratic objective

$\displaystyle
L_{v}(\phi)=\int_{0}^{1}\mathbb{E}\left(v_{\phi}(t,a_{t},x)-\partial_{t}I(t,a_{0},a_{1},x))\right)^{2}dt$
(13)

Given samples in the dataset $D$ comprising tuples $(x,a_{1})$, the above
objectives can be approximated via Monte-Carlo sampling. Our training
algorithm is outlined in Algorithm 2.

Algorithm 2 BRIDGeR Training

$D$ and batch size $N$

while Not Converged do

$(x,a_{1})\sim D$ and $a_{0}\sim\pi_{0}(a|x)$ $\triangleright$ Sample data

$t\sim\mathrm{U}(0,1)$ $\triangleright$ Uniformly sample $t$

Sample $a_{t}\sim I(t,a_{0},a_{1},x)+\gamma(t)z$

Compute losses:

$L_{b}(\theta)=\frac{1}{N}\sum\left(b_{\theta}(t,a_{t},x)-(\partial_{t}I+\dot{\gamma}z)\right)$.

$L_{s}(\eta)=\frac{1}{N}\sum\left(s_{\eta}(t,a_{t},x)+\gamma^{-1}z\right)$.

$L_{v}(\phi)=\frac{1}{N}\sum\left(v_{\phi}(t,a_{t},x)-\partial_{t}I\right)$.

Gradient descent on $\theta$, $\eta$ and $\phi$

end while

$b_{\theta}$, $s_{\eta}$ and $v_{\phi}$

###  IV-B Design Decisions

To apply BRIDGeR in practice, we have to design several key components,
specifically the source distribution $\pi_{0}$, the interpolant
$I(t,a_{0},a_{1},x)$, and the noise schedule $\gamma(t)$ and the
$\epsilon(t)$. In this work, we will focus on comparing specific source
distributions and interpolants.

Figure 2: Intermediate distributions obtained from BRIDGeR and DDIM trained on
2D synthetic data. With a source distribution that is closer to the target
distribution (smaller Earth Mover’s Distance (EMD) values), BRIDGeR can better
recover the true target distribution.  Figure 3: Earth Mover’s Distance (EMD)
of the generated target distributions under different source distributions and
interpolant functions on our 2D synthetic dataset. Each point represents the
EMD between a source/target distribution and the true target distribution.

Source Distributions.  As shown by Theorem 1 in Sec. III, a source
distribution that closer to the target can yield better policies. Our
preliminary experiments with 2D synthetic samples supports this notion (Fig. 2
and 3) — we see that source distributions that are closer to the target (lower
EMD) are more similar to the true target distribution. Along with standard
Gaussians, our experiments will involve two kinds of source policies:

  * •

_Heuristic policies_ : hand-crafted policies (e.g., using rules) based on
prior knowledge. Our heuristic policies are task-dependent and detailed in
Appendix C.

  * •

_Data-driven policies_ : policies learned from a dataset. In our experiments,
we use lightweight Conditional Variational Autoencoders (CVAEs) [44] as a
representative policy. CVAEs tend to not fully capture complex target
distributions, but are cheap to sample from.

These policy types are commonly employed in robotics and by comparing them, we
aim to evaluate potential benefits that BRIDGeR may offer under different use-
cases.

Interpolant Function. The second major component is the interpolant function
$I(t,a_{0},a_{1},x)$. We will use spatially linear interpolants [2],
$a_{t}=\alpha(t)a_{0}+\beta(t)a_{1}+\gamma(t)z$, specifically,

  * •

_Linear Interpolant_ where $\alpha(t)=1-t$ and $\beta(t)=t$.

  * •

_Power3 Interpolant_ where $\alpha(t)=(1-t)^{m}$ and $\beta(t)=1-(1-t)^{m}$.
In our experiments, we set $m=3$, which worked well in preliminary tests.

The linear interpolant uniformly progresses samples from the source to the
target distribution, which we observed makes training more stable. Conversely,
the Power3 interpolant starts with larger steps that decelerate towards the
end. This introduces the target pattern sooner than the linear approach, as
shown in Fig. 4. Qualitatively, we find Power3 worked better in scenarios with
highly multi-modal demonstrations.

Figure 4: Intermediate distributions under different interpolant functions
(trained on 2D synthetic data).

Noise Schedule and Diffusion Coefficient.

Figure 5: Intermediate distributions with varying $\gamma(t)=d\sqrt{2t(1-t)}$.
When the support of the source distribution is narrower than the target
distribution, selecting a small $\gamma$ value ($d=0.03$) results in samples
clustering within the high-density areas of the source. Conversely, an
excessively large $\gamma$ ($d=3$) results in overdispersion. However, a well-
choosen $\gamma$ ($d=0.3$) facilitates coverage to ensure reasonable recovery
of the target.

The noise schedule governs the variance of Gaussian latent noise across the
bridge. In our experiments, we set $\gamma(t)=d\sqrt{2t(1-t)}$, with $d$
acting as a scalar to adjust $\gamma$’s magnitude. This configuration results
in minimal Gaussian noise at the onset of the transition from the source
distribution, increasing to a peak variance before tapering off to zero as
samples approach the target distribution. Selecting a larger $d$ facilitates
exploration of low-density areas but risks introducing excessive noise, as
illustrated in Figure 5. The diffusion coefficient $\epsilon(t)$ controls the
level of noise in the forward SDE (Eq. 9). In our experiments, we define
$\epsilon$ as $\epsilon(t)=c(1-t)$, where $c$ is a scalar to adjust its
magnitude. We set two choices for $c$ (1 and 3) and for $d$ (0.03 and 0.3) and
reported the best results. In general, our results were relatively robust to
these choices.

##  V Related Work

BRIDGeR builds upon a large body of work in imitation learning [19, 57, 49],
specifically behavior cloning.

Behavior cloning has had a long history, with early methods adopting
supervised learning techniques, principally regression [4, 33, 33, 56, 12, 37,
46]. Although efficient, regression-based methods were generally unable to
capture multi-modal behavior. Later methods used classification methods, by
discretizing the action space [40, 41, 3, 50], to address this limitation but
were sensitive to the hyperparameters (such as the level of discretization)
and struggled with high-precision tasks [11].

Modern imitation methods employ generative models, starting with Gaussian
Mixture Models [27], then transitioning to more powerful Energy-Based Models
(EBMs) [7, 11, 20, 43] and Diffusion Models [39, 6, 31, 30]. Compared to EBMs,
diffusion models have demonstrated better training stability [6] and
effectiveness in generating consistent, multi-modal action sequences for
visual-motor control [39, 6, 31, 30] and hierarchical, long-horizon planning
tasks [35, 15, 29, 5, 51].

Recent progress has been made in reducing the computational costs of diffusion
methods by adapting the number of diffusion steps [17] or leveraging DDIM
[36].

Unlike the above methods, BRIDGeR is the first to generalize diffusion-type
policy learning to exploit source stochastic policies. As we will see, this
leads to better performance compared to strong baselines (such as DDIM) when
informative policies are available, especially when using a small number of
diffusion steps.

Since BRIDGeR adapts existing distributions, it can be seen as a few-shot
learner, specifically for imitation learning [9, 49]. Existing work on few-
shot imitation learning requires either specific prior policies [10, 55, 23]
or a hierarchical structure [52]. In contrast, BRIDGeR only requires that we
are able to sample from the source policy. For fair comparison against few-
shot methods, we adopt residual learning which is used for few-shot
reinforcement learning [42, 1, 21], planning [28], and shared autonomy [38,
54].

##  VI Experiments

This section describes experiments designed to evaluate the performance of
BRIDGeR relative to recent methods, particularly diffusion-based imitation
learning. More importantly, we aimed to test our hypothesis that leveraging
better source distributions within a diffusion framework can lead to better
policies. We further hypothesized that BRIDGeR performs better than DDIM
policies given a small number of diffusion steps, and limited data. Finally,
we sought to examine the effect of decision decisions, particularly the
interpolant choice.

In the following, we first give an overview of our experimental setup (details
relegated to Appendix) followed by our main results.

###  VI-A Domains

To evaluate our hypotheses above, we chose six challenging robot benchmarks in
three domains [13, 34, 14, 47] (See Fig. 6). The tasks in these domains
feature multi-modal demonstrations, which comprise multiple stages with high-
dimensional and high-precision actions.

Figure 6: Experimental Domains. (A)-(D) Adroit tasks that involve the control
of a 24-degree-of-freedom robot hand to accomplish four specific tasks: (A)
Door: opening a door, (B) Relocate: moving a ball to a target position, (C)
Hammer: driving a nail into a board, and (D) Pen: aligning a pen with a target
orientation [13]. (E) Franka Kitchen includes 7 objects available for
interaction and the aim is to accomplish 4 subtasks: opening the microwave,
relocating the kettle, flipping the light switch, and sliding open the cabinet
door, with arbitrary order. (F) The goal of 6-DoF Grasp-Pose generation is to
generate grasp poses capable of successfully picking up an object.

Franka Kitchen (State Observations). The goal in Franka Kitchen is to control
a 7 DoF robot arm to solve seven subtasks by reaching a desired state
configuration (Fig. 6). Franka Kitchen comprises three datasets and we chose
the mixed dataset, which is considered as the most challenging; it presents
various subtasks being executed with 4 target subtasks that are not completed
in sequence [13]. The model is trained on three varying data sizes: small (16k
sequences), medium (32k sequences), large (64k sequences).

Adroit (State Observations). The Adroit tasks require control of a 24-DoF
robot hand to accomplish four tasks and is considered one of the most
challenging task sets; it requires a model to generate intricate, high-
dimensional, and high-precision action sequences over long time horizons. We
use the dataset of human demonstrations provided in the DAPG repository [13].
Similar to Franka Kitchen, we train the model on three different data sizes;
small (1.25k sequences), medium (2.5k sequences), and large (5k sequences).

6-DoF Grasp Pose Generation (Point-Cloud Observations). The goal here is to
generate grasp poses capable of picking up an object given the object’s point
cloud. The target grasp-pose distribution is multi-modal [47], with high-
dimensional point clouds as conditioning observations. BRIDGeR is trained on
Acronym dataset [48] (552 objects with 200-2000 grasps per object) after a
LogMap transformation [47] of the grasp poses.

###  VI-B Compared Methods.

BRIDGeR was implemented in PyTorch and trained using the Algorithm 2.
Additional details regarding hyperparameters and network structure are given
in Appendix D. To evaluate performance, we compared BRIDGeR against strong
baselines, specifically:

  * •

_DDIM_ : Diffusion policy [6, 36], a recent diffusion-based imitation-learning
method which trains DDPM and applies DDIM [45] during test-time.

  * •

_Residual Policy_ : This baseline applies residual learning towards policy
learning [42, 1, 21] and is a representative method not based on diffusion.

  * •

_SE3_ : A state-of-the-art score-based diffusion model designed in SE(3) [47]
specifically designed for generating 6-Dof grasp poses. Note that this
baseline only applies to the grasp generation domain.

We also evaluated variants of BRIDGeR with different Interpolant functions
(Linear and Power3), along with different source policies. As stated in Sec.
IV, we investigate three different kinds of source policies:

  * •

_Gaussian_ : a standard Gaussian, similar to DDPM.

  * •

_CVAE_ : a data-driven policy. For fair comparison to the baselines, the CVAE
source policy was trained using the same dataset. Qualitatively, we find the
CVAE was generally not able to capture the diversity of demonstrations.

  * •

_Heuristic_ : hand-crafted policies for the Hammer, Door, and Grasp tasks
(described in Appendix C). Heuristic policies were not available for the other
tasks due to their complexity and the high-dimension of the action space.

We label each BRIDGeR variant as BRIDGeR / $[\cdots]$ / $[\cdots]$. For
example, BRIDGeR / CVAE / Linear represents BRIDGeR with a CVAE source policy
and the linear interpolant.

###  VI-C Test Methodology.

Following prior work, we compute success measures for the different tasks. We
report success rate (Adroit) and number of successful tasks (Franka Kitchen),
averaged over three different seeds. Under each seed, the models are tested
with 100 random initializations for each configuration (dataset sizes [small,
medium, large] $\times$ the number of diffusion steps).

For the Grasp task, we evaluated the models in Nvidia Isaac Gym [26] and we
report the rate of successful grasps; the robot was able to pick up the object
without dropping it [47]. Each object was evaluated using 100 grasps with
random initialization of the object pose. We tested our models on both on
unseen-objects belonging to the object categories seen in the dataset and
unseen-objects from unseen categories. In addition to success rates, we also
measured the Earth Mover’s Distance (EMD) between the generated grasps and the
training data distribution.

Table I: Average task performance of BRIDGeR against current state-of-the-art methods under a different number of diffusion steps when trained with the Large dataset. Overall, BRIDGeR outperforms all competing methods. Results are similar for the small and medium datasets with complete results in the Appendix. |  | BRIDGeR | Residual Policy | DDIM
---|---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic
Door | $k=5$ | $\mathbf{0.60\pm 0.15}$ | $0.00\pm 0.00$ | $0.08\pm 0.06$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.02\pm 0.02$
$k=20$ | $\mathbf{0.52\pm 0.23}$ | $0.45\pm 0.04$ | $0.10\pm 0.07$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.38\pm 0.11$
$k=80$ | $0.50\pm 0.14$ | $\mathbf{0.63\pm 0.08}$ | $0.12\pm 0.09$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.05\pm 0.02$
Relocate | $k=5$ | $\mathbf{0.75\pm 0.11}$ | - | $0.61\pm 0.05$ | $0.3\pm 0.04$ | - | $0.17\pm 0.04$
$k=20$ | $0.7\pm 0.11$ | - | $\mathbf{0.72\pm 0.09}$ | $0.34\pm 0.04$ | - | $0.37\pm 0.04$
$k=80$ | $0.79\pm 0.08$ | - | $\mathbf{0.81\pm 0.04}$ | $0.34\pm 0.04$ | - | $0.26\pm 0.03$
Hammer | $k=5$ | $\mathbf{0.44\pm 0.11}$ | $0.2\pm 0.24$ | $0.16\pm 0.04$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.01\pm 0.01$
$k=20$ | $0.63\pm 0.08$ | $\mathbf{0.74\pm 0.07}$ | $0.35\pm 0.02$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.17\pm 0.12$
$k=80$ | $\mathbf{0.72\pm 0.09}$ | $0.47\pm 0.25$ | $0.43\pm 0.09$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.03\pm 0.04$
Pen | $k=5$ | $0.45\pm 0.04$ | - | $0.43\pm 0.02$ | $0.33\pm 0.12$ | - | $\mathbf{0.54\pm 0.03}$
$k=20$ | $0.49\pm 0.09$ | - | $0.53\pm 0.02$ | $0.33\pm 0.12$ | - | $\mathbf{0.51\pm 0.03}$
$k=80$ | $\mathbf{0.55\pm 0.06}$ | - | $\mathbf{0.55\pm 0.03}$ | $0.33\pm 0.12$ | - | $0.52\pm 0.05$
Franka Kitchen | $k=5$ | $\mathbf{1.96\pm 0.03}$ | - | $1.18\pm 0.02$ | $1.55\pm 0.10$ | - | $1.84\pm 0.06$
$k=20$ | $\mathbf{2.09\pm 0.04}$ | - | $1.54\pm 0.03$ | $1.55\pm 0.10$ | - | $1.93\pm 0.07$
$k=80$ | $\mathbf{2.16\pm 0.03}$ | - | $1.70\pm 0.05$ | $1.55\pm 0.10$ | - | $1.92\pm 0.02$
Table II: Success rate (averaged over $100$ grasps on ten test objects). BRIDGeR significantly outperforms DDIM and Residual Policy across the number of diffusion steps. Compared to SE3, BRIDGeR achieve higher success rate when the number of diffusion steps is small. We show up to $k=160$ steps to be consistent with prior reported results [47]. |  | BRIDGeR | Residual Policy | DDIM | SE3
---|---|---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic
Seen Categories | $k=5$ | $\mathbf{0.73\pm 0.15}$ | $0.64\pm 0.17$ | $0.56\pm 0.17$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.52\pm 0.24$ | $0.38\pm 0.26$
$k=20$ | $\mathbf{0.93\pm 0.08}$ | $0.91\pm 0.07$ | $0.83\pm 0.28$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.64\pm 0.21$ | $0.78\pm 0.19$
$k=160$ | $0.88\pm 0.10$ | $\mathbf{0.91\pm 0.08}$ | $0.90\pm 0.06$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.64\pm 0.26$ | $\mathbf{0.91\pm 0.08}$
Unseen Categories | $k=5$ | $\mathbf{0.48\pm 0.12}$ | $0.43\pm 0.19$ | $0.45\pm 0.14$ | $0.10\pm 0.16$ | $0.12\pm 0.28$ | $0.33\pm 0.20$ | $0.20\pm 0.10$
$k=20$ | $\mathbf{0.67\pm 0.21}$ | $\mathbf{0.67\pm 0.26}$ | $0.65\pm 0.25$ | $0.10\pm 0.16$ | $0.12\pm 0.28$ | $0.41\pm 0.23$ | $0.55\pm 0.24$
$k=160$ | $\mathbf{0.71\pm 0.24}$ | $0.66\pm 0.23$ | $0.64\pm 0.23$ | $0.10\pm 0.16$ | $0.12\pm 0.28$ | $0.35\pm 0.19$ | $0.66\pm 0.25$

###  VI-D Main Results and Discussion

Figure 7: Fifteen sampled grasps across the diffusion steps for BRIDGeR, DDIM
and SE3. We visualize $15$ grasps samples every $2$ diffuse steps until
$k=10$.

In this section, we summarize our key findings, with full results and plots in
Appendix F. We focus on our main hypotheses and significant observations.

With a more informative source policy, BRIDGeR outperforms the baselines,
especially with a small of diffusion steps. We observed that BRIDGeR achieves
the best success rate across the diffusion steps for many of the tasks. The
differences in success rates for Adroit and Franka Kitchen tasks (Table I) are
significant when the number of diffusion steps was small $k=5$; the exception
was the Adroit pen task where DDIM performs slightly better. The grasp
generation results in Table II further supports this finding, where we see
BRIDGeR’s average success rates surpassing the state-of-the-art SE3 model with
$k=5$ and $k=20$. For higher diffusion steps, SE3 catches up and the methods
appear comparable in terms of success rate. However, SE3 achieves poorer EMD
scores compared to BRIDGeR (Fig. 9). Fig. 7 shows sample grasps at different
diffusion steps; we see that using an informative source policy enables
BRIDGeR to more quickly converge to a reasonable set of grasps.

Interestingly, we observed that BRIDGeR consistently achieved better scores
than DDIM and the Residual Policy regardless of the dataset size (See Fig. 8
for the Door task, with plots for other tasks in the Appendix). Potentially,
larger datasets may negate the benefit afforded by the source policy since
more data could enable DDIM to better generalize.

Figure 8: Average success rate under different training dataset on Adroit Door
task ($k=80$ for DDIM and BRIDGeR). BRIDGeR consistently surpasses baselines
across different training data size. Figure 9: Average Earth Mover’s Distance
between generated grasps pose and target grasp poses on Seen Categories in
Grasp Generation task ($k=160$ for SE3, DDIM and BRIDGeR). Lower EMD indicates
BRIDGeR is closer to the dataset distribution. Figure 10: Task performance of
BRIDGeR starting from different source policies using the Power3 interpolant.
In general, with better source distribution, BRIDGeR achieves higher task
performance.

BRIDGeR achieves better performance when using better source policies. Fig. 10
illustrates the difference in success rates between the source and final
action distributions. Overall, starting from a source policy with higher
success rates tended to result in a better action distributions. This
difference can persists even with $k=80$ steps, with potentially diminishing
marginal improvement as illustrated in the Relocate and Franka Kitchen tasks.
As suggested by our theoretical results, this could be due to variations in
the gradual improvements over the diffusion steps.

Figure 11: The success rate of BRIDGeR with the Power3 and Linear interpolants
for grasp generation. In general, Power3 achieves better scores when starting
from the same source policy, especially when the number of diffusion steps is
small. The error bars represent standard deviations over 10 test objects.

The Power3 interpolant is more appropriate than the Linear interpolant when
behavior distributions exhibit high multi-modality. We find that both
interpolants perform comparably in tasks like Adroit, where robot behaviors
are intricate, but largely uni-modal. In contrast, the Power3 interpolant
significantly outperforms the Linear interpolant in Grasp Generation (Fig. 11)
where the distribution of end-effector poses is highly multi-modal (Fig. 6.F).
This efficiency potentially stems from Power3’s rapid convergence to a variety
of high-density target areas, followed by fine-scale adaptation. This notion
is supported by Fig. 12, which illustrates the smoothness of the velocity and
score functions over time (as measured by approximate Lipschitz constants
[53]). We observe the Lipschitz constants for Power3 to be larger at the start
of the process (a “rougher” function) but gradually falls below that of
Linear. This suggests that the function is making more rapid changes at the
outset and smaller adaptations towards the middle and end of the diffusion
process.

Figure 12: Approximate Lipschitz constants of the velocity $b$ and score $s$
for the Grasp task.

##  VII Conclusions and Future Work

In this work, we investigate the potential of integrating informative source
distributions into diffusion-style imitation learning. We provide theoretical
results that support this idea and propose BRIDGeR, a stochastic interpolant
method for imitation learning. Our experiments results show that BRIDGeR
outperforms strong baselines, including state-of-the-art diffusion policies on
various benchmark tasks. We provide additional analyses on our experimental
results to elucidate the effect of various design decisions within BRIDGeR.

Limitations and Future Work. Here, we have shown that leveraging prior
knowledge (in the form of source policies) improves learned diffusion policy
performance. Here, the prior knowledge can be obtained either from human
experts or existing datasets. It would be interesting to explore other forms
of prior knowledge, e.g., policies designed for other tasks and those
constructed using foundation models. Specific to BRIDGeR, we explored a
salient but limited set of design considerations; future work can examine more
elaborate interpolant functions, along with more in-depth analysis of noise
schedules and diffusion coefficients. Moreover, our evaluations comprise
simulation benchmarks as our study involved tens of thousands of trials;
recent work [6] has shown that diffusion policies can work on real robots and
we plan to conduct real world trials using BRIDGeR in subsequent work.
Finally, we plan to extend BRIDGeR to enable classifier-guided sampling.

## References

  * Alakuijala et al. [2021] Minttu Alakuijala, Gabriel Dulac-Arnold, Julien Mairal, Jean Ponce, and Cordelia Schmid.  Residual reinforcement learning from demonstrations.  _arXiv preprint arXiv:2106.08050_ , 2021.
  * Albergo et al. [2023] Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden.  Stochastic interpolants: A unifying framework for flows and diffusions.  _arXiv preprint arXiv:2303.08797_ , 2023.
  * [3] Yahav Avigal, Lars Berscheid, Tamim Asfour, Torsten Kröger, and Ken Goldberg.  Speedfolding: Learning efficient bimanual folding of garments. in 2022 ieee.  In _RSJ International Conference on Intelligent Robots and Systems (IROS)_ , pages 1–8.
  * Bojarski et al. [2016] Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, et al.  End to end learning for self-driving cars.  _arXiv preprint arXiv:1604.07316_ , 2016.
  * Chen et al. [2023] Lili Chen, Shikhar Bahl, and Deepak Pathak.  Playfusion: Skill acquisition via diffusion from language-annotated play.  In _Conference on Robot Learning_ , pages 2012–2029. PMLR, 2023.
  * Chi et al. [2023] Cheng Chi, Siyuan Feng, Yilun Du, Zhenjia Xu, Eric Cousineau, Benjamin Burchfiel, and Shuran Song.  Diffusion policy: Visuomotor policy learning via action diffusion.  _arXiv preprint arXiv:2303.04137_ , 2023.
  * Datta et al. [2023] Gaurav Datta, Ryan Hoque, Anrui Gu, Eugen Solowjow, and Ken Goldberg.  Iifl: Implicit interactive fleet learning from heterogeneous human supervisors.  In _Conference on Robot Learning_ , pages 2340–2356. PMLR, 2023.
  * Du and Mordatch [2019] Yilun Du and Igor Mordatch.  Implicit generation and generalization in energy-based models.  _arXiv preprint arXiv:1903.08689_ , 2019.
  * Duan et al. [2017] Yan Duan, Marcin Andrychowicz, Bradly Stadie, OpenAI Jonathan Ho, Jonas Schneider, Ilya Sutskever, Pieter Abbeel, and Wojciech Zaremba.  One-shot imitation learning.  _Advances in neural information processing systems_ , 30, 2017.
  * Finn et al. [2017] Chelsea Finn, Tianhe Yu, Tianhao Zhang, Pieter Abbeel, and Sergey Levine.  One-shot visual imitation learning via meta-learning.  In _Conference on robot learning_ , pages 357–368. PMLR, 2017.
  * Florence et al. [2022] Pete Florence, Corey Lynch, Andy Zeng, Oscar A Ramirez, Ayzaan Wahid, Laura Downs, Adrian Wong, Johnny Lee, Igor Mordatch, and Jonathan Tompson.  Implicit behavioral cloning.  In _Conference on Robot Learning_ , pages 158–168. PMLR, 2022.
  * Florence et al. [2019] Peter Florence, Lucas Manuelli, and Russ Tedrake.  Self-supervised correspondence in visuomotor policy learning.  _IEEE Robotics and Automation Letters_ , 5(2):492–499, 2019.
  * Fu et al. [2020] Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine.  D4rl: Datasets for deep data-driven reinforcement learning.  _arXiv preprint arXiv:2004.07219_ , 2020.
  * Gupta et al. [2020] Abhishek Gupta, Vikash Kumar, Corey Lynch, Sergey Levine, and Karol Hausman.  Relay policy learning: Solving long-horizon tasks via imitation and reinforcement learning.  In _Conference on Robot Learning_ , pages 1025–1037. PMLR, 2020.
  * Ha et al. [2023] Huy Ha, Pete Florence, and Shuran Song.  Scaling up and distilling down: Language-guided robot skill acquisition.  In _Conference on Robot Learning_ , pages 3766–3777. PMLR, 2023.
  * Ho et al. [2020] Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models.  _Advances in neural information processing systems_ , 33:6840–6851, 2020.
  * Hu et al. [2023] Xixi Hu, Bo Liu, Xingchao Liu, et al.  Rf-policy: Rectified flows are computation-adaptive decision makers.  In _NeurIPS 2023 Foundation Models for Decision Making Workshop_ , 2023.
  * Huang et al. [2023] Ding Huang, Jian Huang, Ting Li, and Guohao Shen.  Conditional stochastic interpolation for generative learning.  _arXiv preprint arXiv:2312.05579_ , 2023.
  * Hussein et al. [2017] Ahmed Hussein, Mohamed Medhat Gaber, Eyad Elyan, and Chrisina Jayne.  Imitation learning: A survey of learning methods.  _ACM Computing Surveys (CSUR)_ , 50(2):1–35, 2017.
  * Jarrett et al. [2020] Daniel Jarrett, Ioana Bica, and Mihaela van der Schaar.  Strictly batch imitation learning by energy-based distribution matching.  _Advances in Neural Information Processing Systems_ , 33:7354–7365, 2020.
  * Johannink et al. [2019] Tobias Johannink, Shikhar Bahl, Ashvin Nair, Jianlan Luo, Avinash Kumar, Matthias Loskyll, Juan Aparicio Ojea, Eugen Solowjow, and Sergey Levine.  Residual reinforcement learning for robot control.  In _2019 International Conference on Robotics and Automation (ICRA)_ , pages 6023–6029. IEEE, 2019.
  * Kingma and Ba [2014] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  _arXiv preprint arXiv:1412.6980_ , 2014.
  * Li et al. [2021] Jiayi Li, Tao Lu, Xiaoge Cao, Yinghao Cai, and Shuo Wang.  Meta-imitation learning by watching video demonstrations.  In _International Conference on Learning Representations_ , 2021.
  * Liu et al. [2023] Guan-Horng Liu, Arash Vahdat, De-An Huang, Evangelos A Theodorou, Weili Nie, and Anima Anandkumar.  I2sb: Image-to-image schr$\backslash$” odinger bridge.  _arXiv preprint arXiv:2302.05872_ , 2023.
  * Loshchilov and Hutter [2017] Ilya Loshchilov and Frank Hutter.  Decoupled weight decay regularization.  _arXiv preprint arXiv:1711.05101_ , 2017.
  * Makoviychuk et al. [2021] Viktor Makoviychuk, Lukasz Wawrzyniak, Yunrong Guo, Michelle Lu, Kier Storey, Miles Macklin, David Hoeller, Nikita Rudin, Arthur Allshire, Ankur Handa, et al.  Isaac gym: High performance gpu based physics simulation for robot learning.  In _Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2)_ , 2021.
  * Mandlekar et al. [2022] Ajay Mandlekar, Danfei Xu, Josiah Wong, Soroush Nasiriany, Chen Wang, Rohun Kulkarni, Li Fei-Fei, Silvio Savarese, Yuke Zhu, and Roberto Martín-Martín.  What matters in learning from offline human demonstrations for robot manipulation.  In _Conference on Robot Learning_ , pages 1678–1690. PMLR, 2022.
  * Mandlekar et al. [2023] Ajay Mandlekar, Caelan Reed Garrett, Danfei Xu, and Dieter Fox.  Human-in-the-loop task and motion planning for imitation learning.  In _Conference on Robot Learning_ , pages 3030–3060. PMLR, 2023.
  * Mishra et al. [2023] Utkarsh Aashu Mishra, Shangjie Xue, Yongxin Chen, and Danfei Xu.  Generative skill chaining: Long-horizon skill planning with diffusion models.  In _Conference on Robot Learning_ , pages 2905–2925. PMLR, 2023.
  * Ng et al. [2023] Eley Ng, Ziang Liu, and Monroe Kennedy.  Diffusion co-policy for synergistic human-robot collaborative tasks.  _IEEE Robotics and Automation Letters_ , 2023.
  * Pearce et al. [2023] Tim Pearce, Tabish Rashid, Anssi Kanervisto, Dave Bignell, Mingfei Sun, Raluca Georgescu, Sergio Valcarcel Macua, Shan Zheng Tan, Ida Momennejad, Katja Hofmann, et al.  Imitating human behaviour with diffusion models.  In _The Eleventh International Conference on Learning Representations (ICLR 2023)_ , 2023.
  * Pomerleau [1988] Dean A Pomerleau.  Alvinn: An autonomous land vehicle in a neural network.  _Advances in neural information processing systems_ , 1, 1988.
  * Rahmatizadeh et al. [2018] Rouhollah Rahmatizadeh, Pooya Abolghasemi, Ladislau Bölöni, and Sergey Levine.  Vision-based multi-task manipulation for inexpensive robots using end-to-end learning from demonstration.  In _2018 IEEE international conference on robotics and automation (ICRA)_ , pages 3758–3765. IEEE, 2018.
  * Rajeswaran et al. [2018] Aravind Rajeswaran, Vikash Kumar, Abhishek Gupta, Giulia Vezzani, John Schulman, Emanuel Todorov, and Sergey Levine.  Learning complex dexterous manipulation with deep reinforcement learning and demonstrations.  _Robotics: Science and Systems XIV_ , 2018.
  * Reuss and Lioutikov [2023] Moritz Reuss and Rudolf Lioutikov.  Multimodal diffusion transformer for learning from play.  In _2nd Workshop on Language and Robot Learning: Language as Grounding_ , 2023.
  * Reuss et al. [2023] Moritz Reuss, Maximilian Li, Xiaogang Jia, and Rudolf Lioutikov.  Goal-conditioned imitation learning using score-based diffusion policies.  _arXiv preprint arXiv:2304.02532_ , 2023.
  * Ross et al. [2011] Stéphane Ross, Geoffrey Gordon, and Drew Bagnell.  A reduction of imitation learning and structured prediction to no-regret online learning.  In _Proceedings of the fourteenth international conference on artificial intelligence and statistics_ , pages 627–635. JMLR Workshop and Conference Proceedings, 2011.
  * Schaff and Walter [2020] Charles Schaff and Matthew R Walter.  Residual policy learning for shared autonomy.  _arXiv preprint arXiv:2004.05097_ , 2020.
  * Scheikl et al. [2023] Paul Maria Scheikl, Nicolas Schreiber, Christoph Haas, Niklas Freymuth, Gerhard Neumann, Rudolf Lioutikov, and Franziska Mathis-Ullrich.  Movement primitive diffusion: Learning gentle robotic manipulation of deformable objects.  _arXiv preprint arXiv:2312.10008_ , 2023.
  * Shafiullah et al. [2022] Nur Muhammad Shafiullah, Zichen Cui, Ariuntuya Arty Altanzaya, and Lerrel Pinto.  Behavior transformers: Cloning $k$ modes with one stone.  _Advances in neural information processing systems_ , 35:22955–22968, 2022.
  * Sharma et al. [2018] Pratyusha Sharma, Lekha Mohan, Lerrel Pinto, and Abhinav Gupta.  Multiple interactions made easy (mime): Large scale demonstrations data for imitation.  In _Conference on robot learning_ , pages 906–915. PMLR, 2018.
  * Silver et al. [2018] Tom Silver, Kelsey Allen, Josh Tenenbaum, and Leslie Kaelbling.  Residual policy learning.  _arXiv preprint arXiv:1812.06298_ , 2018.
  * Singh et al. [2023] Sumeet Singh, Stephen Tu, and Vikas Sindhwani.  Revisiting energy based models as policies: Ranking noise contrastive estimation and interpolating energy models.  _arXiv preprint arXiv:2309.05803_ , 2023.
  * Sohn et al. [2015] Kihyuk Sohn, Honglak Lee, and Xinchen Yan.  Learning structured output representation using deep conditional generative models.  _Advances in neural information processing systems_ , 28, 2015.
  * Song et al. [2020] Jiaming Song, Chenlin Meng, and Stefano Ermon.  Denoising diffusion implicit models.  In _International Conference on Learning Representations_ , 2020.
  * Toyer et al. [2020] Sam Toyer, Rohin Shah, Andrew Critch, and Stuart Russell.  The magical benchmark for robust imitation.  _Advances in Neural Information Processing Systems_ , 33:18284–18295, 2020.
  * Urain et al. [2023] Julen Urain, Niklas Funk, Jan Peters, and Georgia Chalvatzaki.  Se (3)-diffusionfields: Learning smooth cost functions for joint grasp and motion optimization through diffusion.  In _2023 IEEE International Conference on Robotics and Automation (ICRA)_ , pages 5923–5930. IEEE, 2023.
  * Vahrenkamp et al. [2010] Nikolaus Vahrenkamp, Martin Do, Tamim Asfour, and Rüdiger Dillmann.  Integrated grasp and motion planning.  In _2010 IEEE International Conference on Robotics and Automation_ , pages 2883–2888. IEEE, 2010.
  * Wang et al. [2020] Yaqing Wang, Quanming Yao, James T Kwok, and Lionel M Ni.  Generalizing from a few examples: A survey on few-shot learning.  _ACM computing surveys (csur)_ , 53(3):1–34, 2020.
  * Wu et al. [2020] Jimmy Wu, Xingyuan Sun, Andy Zeng, Shuran Song, Johnny Lee, Szymon Rusinkiewicz, and Thomas Funkhouser.  Spatial action maps for mobile manipulation.  _arXiv preprint arXiv:2004.09141_ , 2020.
  * Xian et al. [2023] Zhou Xian, Nikolaos Gkanatsios, Theophile Gervet, Tsung-Wei Ke, and Katerina Fragkiadaki.  Chaineddiffuser: Unifying trajectory diffusion and keypose prediction for robotic manipulation.  In _Conference on Robot Learning_ , pages 2323–2339. PMLR, 2023.
  * Xu et al. [2023] Mengda Xu, Zhenjia Xu, Cheng Chi, Manuela Veloso, and Shuran Song.  Xskill: Cross embodiment skill discovery.  In _Conference on Robot Learning_ , pages 3536–3555. PMLR, 2023.
  * Yang et al. [2023] Zhantao Yang, Ruili Feng, Han Zhang, Yujun Shen, Kai Zhu, Lianghua Huang, Yifei Zhang, Yu Liu, Deli Zhao, Jingren Zhou, et al.  Eliminating lipschitz singularities in diffusion models.  _arXiv preprint arXiv:2306.11251_ , 2023.
  * Yoneda et al. [2023] Takuma Yoneda, Luzhe Sun, Bradly Stadie, Ge Yang, and Matthew Walter.  To the noise and back: Diffusion for shared autonomy.  _arXiv preprint arXiv:2302.12244_ , 2023.
  * Yu et al. [2018] Tianhe Yu, Chelsea Finn, Annie Xie, Sudeep Dasari, Tianhao Zhang, Pieter Abbeel, and Sergey Levine.  One-shot imitation from observing humans via domain-adaptive meta-learning.  _arXiv preprint arXiv:1802.01557_ , 2018.
  * Zhang et al. [2018] Tianhao Zhang, Zoe McCarthy, Owen Jow, Dennis Lee, Xi Chen, Ken Goldberg, and Pieter Abbeel.  Deep imitation learning for complex manipulation tasks from virtual reality teleoperation.  In _2018 IEEE International Conference on Robotics and Automation (ICRA)_ , pages 5628–5635. IEEE, 2018.
  * Zheng et al. [2022] Boyuan Zheng, Sunny Verma, Jianlong Zhou, Ivor W Tsang, and Fang Chen.  Imitation learning: Progress, taxonomies and challenges.  _IEEE Transactions on Neural Networks and Learning Systems_ , (99):1–16, 2022.
  * Zhou et al. [2023] Linqi Zhou, Aaron Lou, Samar Khanna, and Stefano Ermon.  Denoising diffusion bridge models.  _arXiv preprint arXiv:2309.16948_ , 2023.

##  Appendix A Proofs

See 1

###### Proof.

By Lagrange’s Theorem, there exist $s\in[0,1]$ such that
$\partial_{t}\phi_{F,\hat{\pi}}(s,x)=\phi_{F,\hat{\pi}}(1,x)-\phi_{F,\hat{\pi}}(0,x)$.
Rearranging the terms and using the assumption that the derivative is upper
bounded by $-\epsilon_{\min}$, we have

$\displaystyle\phi_{F,\hat{\pi}}(1,x)$
$\displaystyle=\phi_{F,\hat{\pi}}(0,x)+\partial_{t}\phi_{F,\hat{\pi}}(s,x)$
$\displaystyle\leq\phi_{F,\hat{\pi}}(0,x)-\epsilon_{\min}.$ (14)

Similarly, there must exist $s^{\prime}$ such that
$\partial_{t}\phi_{F,\hat{\rho}}(s^{\prime},x)=\phi_{F,\hat{\rho}}(1,x)-\phi_{F,\hat{\rho}}(0,x)$.
Rearranging the terms and using the assumption that the derivative is lower
bounded by $-\epsilon_{\max}$,

$\displaystyle\phi_{F,\hat{\rho}}(1,x)$
$\displaystyle=\phi_{F,\hat{\rho}}(0,x)+\partial_{t}\phi_{F,\hat{\rho}}(s^{\prime},x)$
$\displaystyle\geq\phi_{F,\hat{\rho}}(0,x)-\epsilon_{\max}.$ (15)

Putting these results together, we have

$\displaystyle\,\phi_{F,\hat{\pi}}(1,x)-\phi_{F,\hat{\rho}}(1,x)$
$\displaystyle\leq$
$\displaystyle\,(\phi_{F,\hat{\pi}}(0,x)-\epsilon_{\min})-(\phi_{F,\hat{\rho}}(0,x)-\epsilon_{\max})$
$\displaystyle=$
$\displaystyle\,\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x)+\epsilon_{\max}-\epsilon_{\min}.$

where the inequality is due to Equation 14 and 15. ∎

See 2

###### Proof.

We prove by induction. In particular, we will prove that for all
$k\in\\{1,\dots,K\\}$, we have

$\displaystyle\,\phi_{F,\hat{\pi}}(t_{k},x)-\phi_{F,\hat{\rho}}(t_{k},x)$
$\displaystyle\leq$
$\displaystyle\,\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x)+(\epsilon_{\max}-\epsilon_{\min})\sum_{i=1}^{k}\delta
t_{i}.$ (16)

When $k=1$, we have

$\displaystyle\,\phi_{F,\hat{\pi}}(t_{1},x)-\phi_{F,\hat{\rho}}(t_{1},x)$
$\displaystyle\leq$
$\displaystyle\,(\phi_{F,\hat{\pi}}(t_{0},x)-\epsilon_{\min}\delta
t_{1})-(\phi_{F,\hat{\rho}}(t_{0},x)-\epsilon_{\max}\delta t_{1})$
$\displaystyle=$
$\displaystyle\,(\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x))+(\epsilon_{\max}-\epsilon_{\min})\delta
t_{1}.$

Now assume that Equation 16 is true for $k$. Then

$\displaystyle\,\phi_{F,\hat{\pi}}(t_{k+1},x)-\phi_{F,\hat{\rho}}(t_{k+1},x)$
$\displaystyle\leq$
$\displaystyle\,(\phi_{F,\hat{\pi}}(t_{k},x)-\epsilon_{\min}\delta
t_{k+1})-(\phi_{F,\hat{\rho}}(t_{k},x)-\epsilon_{\max}\delta t_{k+1})$
$\displaystyle=$
$\displaystyle\,(\phi_{F,\hat{\pi}}(t_{k},x)-\phi_{F,\hat{\rho}}(t_{k},x))+(\epsilon_{\max}-\epsilon_{\min})\delta
t_{k+1}$ $\displaystyle\leq$
$\displaystyle\,\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x)+(\epsilon_{\max}-\epsilon_{\min})\sum_{i=1}^{k+1}\delta
t_{i}.$

In particular, when $k=K$, we have

$\displaystyle\,\phi_{F,\hat{\pi}}(1,x)-\phi_{F,\hat{\rho}}(1,x)$
$\displaystyle=$
$\displaystyle\,\phi_{F,\hat{\pi}}(t_{K},x)-\phi_{F,\hat{\rho}}(t_{K},x)$
$\displaystyle\leq$
$\displaystyle\,\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x)+(\epsilon_{\max}-\epsilon_{\min})\sum_{i=1}^{K}\delta
t_{i}$ $\displaystyle\leq$
$\displaystyle\,\phi_{F,\hat{\pi}}(0,x)-\phi_{F,\hat{\rho}}(0,x)+\epsilon_{\max}-\epsilon_{\min}$

where the sum telescopes to $t_{K}-t_{0}=1$. ∎

See 3

###### Proof.

We prove by induction. In particular, we will prove that for all
$k\in\\{1,\dots,K\\}$, we have

$\displaystyle\,\mathbb{E}_{\hat{\pi}_{[k]}}[c(a|x)]-\mathbb{E}_{\hat{\rho}_{[k]}}[c(a|x)]$
$\displaystyle\leq$
$\displaystyle\,\mathbb{E}_{\hat{\pi}_{[k-1]}}[c(a|x)]-\mathbb{E}_{\hat{\rho}_{[k-1]}}[c(a|x)]+(\epsilon_{\max}-\epsilon_{\min})\sum_{i=1}^{k}\delta
t_{i}.$

where for brevity, we denote $\hat{\pi}_{t_{k}}$ as $\hat{\pi}_{[k]}$. Let
$\epsilon=\epsilon_{\max}-\epsilon_{\min}$. When $k=1$, we have

$\displaystyle\,\mathbb{E}_{\hat{\pi}_{[1]}}[c(a|x)]-\mathbb{E}_{\hat{\rho}_{[1]}}[c(a|x)]$
$\displaystyle=$
$\displaystyle\,\int(\hat{\pi}_{[1]}(a|x)-\hat{\rho}_{[1]}(a|x))\,c(a|x)\,\mathrm{d}a$
$\displaystyle=$
$\displaystyle\,-\int(\hat{\pi}_{[1]}(a|x)-\hat{\rho}_{[1]}(a|x))(\ln
Z-\ln\pi_{1}(a|x))\,\mathrm{d}a$ $\displaystyle=$
$\displaystyle\,-\int\hat{\pi}_{[1]}(a|x)\ln\pi_{1}(a|x)-\hat{\rho}_{[1]}(a|x)\ln\pi_{1}(a|x)\,\mathrm{d}a$
$\displaystyle=$
$\displaystyle\,\mathbb{H}(\hat{\pi}_{[1]}(\cdot|x),\pi_{1}(\cdot|x))-\mathbb{H}(\hat{\rho}_{[1]}(\cdot|x),\pi_{1}(\cdot|x))$
$\displaystyle\leq$
$\displaystyle\,\mathbb{H}(\hat{\pi}_{[0]}(\cdot|x),\pi_{1}(\cdot|x))-\mathbb{H}(\hat{\rho}_{[0]}(\cdot|x),\pi_{1}(\cdot|x))+\epsilon\delta
t_{1}$ $\displaystyle=$
$\displaystyle\,-\int\hat{\pi}_{[0]}(a|x)\ln\pi_{1}(a|x)-\hat{\rho}_{[0]}(a|x)\ln\pi_{1}(a|x)\,\mathrm{d}a+\epsilon\delta
t_{1}$ $\displaystyle=$
$\displaystyle\,-\int(\hat{\pi}_{[0]}(a|x)-\hat{\rho}_{[0]}(a|x))(\ln
Z-\ln\pi_{1}(a|x))\,\mathrm{d}a+\epsilon\delta t_{1}$ $\displaystyle=$
$\displaystyle\,\int(\hat{\pi}_{[0]}(a|x)-\hat{\rho}_{[0]}(a|x))\,c(a|x)\,\mathrm{d}a+\epsilon\delta
t_{1}$ $\displaystyle=$
$\displaystyle\,\mathbb{E}_{\hat{\pi}_{[0]}}[c(a|x)]-\mathbb{E}_{\hat{\rho}_{[0]}}[c(a|x)]+(\epsilon_{\max}-\epsilon_{\min})\,\delta
t_{1}.$

The inductive case can be proven similarly. At $k=K$, the sum telescopes to 1,
thus yielding our result. ∎

##  Appendix B Experiment Domains

Franka Kitchen (State). The goal of Franka Kitchen is to control a 7 DoF robot
arm to interact with the various objects to reach a desired state
configuration (Fig. 6). The environment includes 7 objects available for
interaction [13]. Franka Kitchen comprises three datasets: 1) a complete
dataset that includes demonstrations of all 4 target subtasks performed in a
specific order, 2) a partial dataset that includes sub-trajectories where the
4 subtasks are completed sequentially, and 3) a mixed dataset, presenting
various subtasks being executed, but the 4 subtasks are not completed in
sequence and the demonstrated behaviors are multi-modal [13]. We train the
models on the mixed dataset, which is considered the most challenging among
the three. The model is trained on varying data sizes: 1) small size: $16000$
sequences, 2) medium size: $32000$ sequences, and 3) large size: $64000$
sequences. We evaluate CVAE [44] source distributions trained on the same
dataset (using state observations). Similar to [6], the models predict action
sequences with receding-horizon control ($16$ steps of actions, of which $8$
steps of actions are executed on the robot without re-planning).

Adroit (State). Adroit encompasses the control of a 24-degree-of-freedom
robotic hand to accomplish four specific tasks (Fig. 6). Adroit is one of the
most challenging task sets, as it requires a model to generate intricate,
high-dimensional, and high-precision action sequences over long time horizons.
We use the dataset of human demonstrations provided in the DAPG repository
[13]. Similar to Franka Kitchen, we train the model on different data sizes:
1) small size: $1250$ sequences, 2) medium size: $2500$ sequences, and 3)
large size: $5000$ sequences. We evaluate two source distributions: 1) CVAE
[44] trained on the same dataset (on $4$ tasks) and 2) hand-craft heuristic
source distribution (on door and hammer). The models are trained on state
observation. Similar to Franka Kitchen, we apply receding-horizon control; the
models predict $64$ steps of actions, of which $48$ steps of actions are
executed on the robot without re-planning.

6-DoF Grasp Pose Generation (Point-Cloud). The objective is to generate grasp
poses capable of picking up an object based on the object’s point cloud (Fig.
6). The demonstrated behavior’s distribution is intricate and highly multi-
modal [47], with high-dimensional point clouds as observations. Our model is
trained on $11$ object categories sourced from the Acronym dataset [48],
encompassing $552$ objects with approximately $200$-$2000$ grasps for each
object. Similar to Adroit, we evaluate two different source distributions: 1)
CVAE [44] trained on the same dataset and 2) hand-crafted heuristic source
distributions. Both source distributions use point-cloud observations.

##  Appendix C Source Policies

As mentioned above, our experiments involve heuristic policies and data-driven
CVAE policies:

Heuristic policies. We used hand-crafted policies for the following tasks:

  * •

Adroit Hammer: the policy involves moving the hand towards the hammer, closing
the hand, and subsequently rotating the wrist to swing the hammer.

  * •

Adroit Door: the policy involves moving the hand towards the handle of the
door, closing the hand, rotating the wrist, and pulling the hand back.

  * •

Grasp Generation: We sample grasp poses directed at the object, distributing
them uniformly on a sphere with its center at the object. The sphere’s radius
is determined by adding $0.5$ to the average distance from the normalized
point cloud to its center.

Data-driven policies. We use lightweight Conditional Variational Autoencoders
(CVAEs) as a representative policy. The CVAEs are trained to generate action
sequences conditioned on observations specific to each domain (as described
above). In Franka Kitchen and Adroit tasks, the CVAEs are parameterized by
multi-layer perceptrons with $\approx 3\times 10^{6}$ parameters ($10$ times
fewer parameters compared to the diffusion models). In Grasp Generation task,
CVAE is parameterized using the Grasp SE(3)-DiffusionFields neural network
[47] with $\approx 3.5\times 10^{6}$ parameters (approximately half of the
number of parameters of the diffusion models).

Table III: Values of $\gamma$ scale $d$ and $\epsilon$ scale $c$ in BRIDGeR under different source policies. | CVAE | Heuristic | Gaussian
---|---|---|---
|  $\gamma$ scale $d$ |  $\epsilon$ scale $c$ |  $\gamma$ scale $d$ |  $\epsilon$ scale $c$ |  $\gamma$ scale $d$ |  $\epsilon$ scale $c$
Door | $0.03$ | $1.0$ | $0.03$ | $1.0$ | $0.03$ | $1.0$
Relocate | $0.3$ | $1.0$ | - | - | $0.3$ | $1.0$
Hammer | $0.03$ | $3.0$ | $0.3$ | $1.0$ | $0.03$ | $3.0$
Pen | $0.03$ | $1.0$ | - | - | $0.03$ | $1.0$
Franka Kitchen | $0.03$ | $3.0$ | - | - | $0.03$ | $3.0$
Grasp Generation | $0.3$ | $1.0$ | $0.3$ | $1.0$ | $0.3$ | $1.0$

##  Appendix D Baselines

We compare BRIDGeR Policy with the current state-of-the-art diffusion-style
behavior cloning. Since BRIDGeR can be viewed as one kind of residual model,
we also compare it against a residual policy. For ablation studies, we
evaluate BRIDGeR under different configurations, focusing on the source
distributions and interpolants. In the following, we present an overview of
each methods.

DDIM. Diffusion Policy [6, 36] trains a Denoising Diffusion Probabilistic
Model [16] and applied Denoising Diffusion Implicit Model (DDIM) [45] during
testing. In Adroit and Franka Kitchen tasks, the model is parameterized by the
CNN-based U-net [6]. In Grasp Generation tasks, the model is parameterized by
Grasp SE(3)-DiffusionFields neural network [47].

SE3 for Grasping. Score-based diffusion model designed in SE(3) space [47].
The model is specifically designed for generating 6-DoF grasp poses and
parameterized by Grasp SE(3)-DiffusionFields neural network. It represents a
state-of-the-art generative model for grasps.

Residual Policy. Residual Policy [42, 1, 21] takes in the sample from the
source distribution as input and predicts the difference between the source
sample and the target sample. The model is parameterized by the CNN-based
U-net [6] similar to the network used in DDIM and is trained by Mean Square
Error (MSE). For Grasp Generation tasks, it is parameterized by the Grasp
SE(3)-DiffusionFields neural network [47].

BRIDGeR. Our stochastic interpolant policy. The model is parameterized by the
CNN-based U-net [6] for the Adroit and Franka Kitchen tasks and parameterized
by Grasp SE(3)-DiffusionFields neural network [47] for Grasp Generation. When
compared with other baselines, we set $\gamma$ scale $d$ and $\epsilon$ scale
$c$ as in Table. III. We conduct an ablation study of BRIDGeR with different
source distribution, interpolant functions, $\gamma$ and $\epsilon$ scale. We
name each ablation as BRIDGeR / $[\cdots]$ / $[\cdots]$. For example, BRIDGeR
/ Linear / CVAE represents BRIDGeR with CVAE as the source distribution and
linear function as the interpolant.

For fair comparison, each method has approximately the same number of
parameters. In Adroit and Franka Kitchen, methods have $\approx 6.7\times
10^{7}$ parameters each. In Grasp Generation tasks, the parameter count for
each method is $\approx 6.2\times 10^{6}$.

##  Appendix E Model Training

In both Adroit and Franka Kitchen, we utilized AdamW [25] as the optimizer.
The model was trained for $3000$ epochs for Adroit and $5000$ epochs for
Franka Kitchen. The learning rate was set to $5\times 10^{-6}$ with a learning
rate decay of $0.5$ every $500$ epochs. The batch size was fixed at $256$.

For Grasp Generation, we employed Adam [22] as the optimizer. The model was
trained for 4000 epochs with the learning rate $5\times 10^{-4}$. Similar to
Adroit and Franka Kitchen, the learning rate decay was set to $0.5$ every
$500$ epochs. The batch contains $100$ objects with $80$ grasps per object.

##  Appendix F Additional Results and Plots

In this section, we present additional results, complementing the main paper.

BRIDGeR against baselines. Performance comparisons between BRIDGeR v.s. the
baselines on different training data sizes and diffusion steps for Adroit and
Franka Kitchen are shown in Table. IV (large training data size), V (medium
training data size) and VI (small training data size). Results for Grasp
Generation with varying number of steps are presented in Table. VII and VIII.
We report the results of BRIDGeR with Power3 interpolant function and
decomposed velocity $b$. The values of $\gamma$ scale and $\epsilon$ scale are
listed in Table. III.

BRIDGeR with Power3 and Linear interpolant function. The performance metrics
comparing the different interpolant functions on the Adroit and Franka Kitchen
tasks are shown in Table. IX (large training data size), X (medium training
data size) and XI (small training data size). The results for grasp generation
presented in Table. XII and XIII.

###  F-A Other Findings

Figure 13: Relative improvement is the ratio of the source distribution’s
performance to the best performance achieved by BRIDGeR. Increasing the
$\gamma$ scale parameter from $d=0.03$ to $d=0.3$ yields relative improvements
in scenarios where the source distribution significantly deviates from the
target distribution, as observed in the Hammer and Relocate tasks. This
adjustment promotes exploration, enhancing performance. Conversely, in cases
where the source distribution closely matches the target, such as in the
Franka Kitchen task, increasing the $\gamma$ scale parameter does not
contribute to performance gains.  Figure 14: Success rate of BRIDGeR / Power3
in the Adroit tasks with standard and decomposed velocity functions. Error
bars represent standard deviation over three seeds. Figure 15: Success rate of
BRIDGeR / Linear in Adroit tasks with standard and decomposed velocity
functions. Error bars represent standard deviation over three seeds.

Larger $\gamma$ value encourages exploration. Our findings indicate that
increasing the $\gamma$ scale $d$ is beneficial when there is a significant
discrepancy between the source and target distributions. Specifically, in
experiments, the heuristic source policy in Hammer and the CVAE source policy
in Relocate exhibited suboptimal performance, with relative ratios of source
policy performance to the best performance of our method being $0.41$ and
$0.21$, respectively. Comparing $\gamma$ scale values ($0.03$ vs. $0.3$), we
observed that an increased $\gamma$ scale led to improved outcomes, as
depicted in Figure 13. We believe this is due to larger exploration in the
sampling process during training. However, when the source distribution
closely resembles the target distribution, increasing $\gamma$ does not
further enhance performance. This is exemplified in the Franka Kitchen
scenario, where the relative ratio between source-only and optimal performance
of our method stands at $0.73$, suggesting a relatively minor gap. In such
contexts, our method yields comparable results regardless of the increased
$\gamma$ scale, as illustrated in Figure 13.

Table IV: Full results of average task performance of BRIDGeR against current state-of-the-art methods under varying number of diffusion steps when trained with the Large dataset. |  | BRIDGeR | Residual Policy | DDIM
---|---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic
Door | $k=0$ | $0.13\pm 0.04$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$ | $0.13\pm 0.04$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.60\pm 0.15}$ | $0.00\pm 0.00$ | $0.08\pm 0.06$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.02\pm 0.02$
$k=10$ | $\mathbf{0.54\pm 0.18}$ | $0.27\pm 0.01$ | $0.07\pm 0.05$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.31\pm 0.08$
$k=20$ | $\mathbf{0.52\pm 0.23}$ | $0.45\pm 0.04$ | $0.10\pm 0.07$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.38\pm 0.11$
$k=40$ | $0.48\pm 0.19$ | $\mathbf{0.72\pm 0.02}$ | $0.07\pm 0.05$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.03\pm 0.00$
$k=80$ | $0.50\pm 0.14$ | $\mathbf{0.63\pm 0.08}$ | $0.12\pm 0.09$ | $0.04\pm 0.06$ | $0.00\pm 0.00$ | $0.05\pm 0.02$
Relocate | $k=0$ | $\mathbf{0.31\pm 0.15}$ | - | $0.00\pm 0.00$ | $\mathbf{0.31\pm 0.15}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.75\pm 0.11}$ | - | $0.61\pm 0.05$ | $0.30\pm 0.04$ | - | $0.17\pm 0.04$
$k=10$ | $\mathbf{0.75\pm 0.08}$ | - | $0.73\pm 0.11$ | $0.30\pm 0.04$ | - | $0.38\pm 0.11$
$k=20$ | $0.70\pm 0.11$ | - | $\mathbf{0.72\pm 0.09}$ | $0.30\pm 0.04$ | - | $0.37\pm 0.04$
$k=40$ | $0.77\pm 0.11$ | - | $\mathbf{0.78\pm 0.05}$ | $0.30\pm 0.04$ | - | $0.38\pm 0.11$
$k=80$ | $0.79\pm 0.08$ | - | $\mathbf{0.81\pm 0.04}$ | $0.30\pm 0.04$ | - | $0.26\pm 0.03$
Hammer | $k=0$ | $\mathbf{0.16\pm 0.03}$ | $0.11\pm 0.08$ | $0.00\pm 0.00$ | $\mathbf{0.16\pm 0.03}$ | $0.11\pm 0.08$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.44\pm 0.11}$ | $0.20\pm 0.24$ | $0.16\pm 0.04$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.01\pm 0.01$
$k=10$ | $0.54\pm 0.17$ | $\mathbf{0.62\pm 0.11}$ | $0.30\pm 0.04$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.11\pm 0.08$
$k=20$ | $0.63\pm 0.08$ | $\mathbf{0.74\pm 0.07}$ | $0.35\pm 0.02$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.17\pm 0.12$
$k=40$ | $\mathbf{0.58\pm 0.12}$ | $\mathbf{0.58\pm 0.21}$ | $0.36\pm 0.05$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.01\pm 0.01$
$k=80$ | $\mathbf{0.72\pm 0.09}$ | $0.47\pm 0.25$ | $0.43\pm 0.09$ | $0.13\pm 0.06$ | $0.24\pm 0.07$ | $0.03\pm 0.04$
Pen | $k=0$ | $\mathbf{0.29\pm 0.05}$ | - | $0.00\pm 0.00$ | $\mathbf{0.29\pm 0.05}$ | - | $0.00\pm 0.00$
$k=5$ | $0.45\pm 0.04$ | - | $0.43\pm 0.02$ | $0.33\pm 0.12$ | - | $\mathbf{0.54\pm 0.03}$
$k=10$ | $0.49\pm 0.02$ | - | $0.46\pm 0.04$ | $0.33\pm 0.12$ | - | $\mathbf{0.54\pm 0.04}$
$k=20$ | $0.49\pm 0.09$ | - | $\mathbf{0.53\pm 0.02}$ | $0.33\pm 0.12$ | - | $0.51\pm 0.03$
$k=40$ | $0.52\pm 0.07$ | - | $\mathbf{0.54\pm 0.05}$ | $0.33\pm 0.12$ | - | $0.53\pm 0.02$
$k=80$ | $\mathbf{0.55\pm 0.06}$ | - | $\mathbf{0.55\pm 0.03}$ | $0.33\pm 0.12$ | - | $0.52\pm 0.05$
Franka Kitchen | $k=0$ | $\mathbf{1.53\pm 0.09}$ | - | $0.00\pm 0.00$ | $\mathbf{1.53\pm 0.09}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{1.96\pm 0.03}$ | - | $1.18\pm 0.02$ | $1.55\pm 0.10$ | - | $1.84\pm 0.06$
$k=10$ | $\mathbf{2.13\pm 0.01}$ | - | $1.28\pm 0.24$ | $1.55\pm 0.10$ | - | $1.91\pm 0.01$
$k=20$ | $\mathbf{2.09\pm 0.04}$ | - | $1.54\pm 0.03$ | $1.55\pm 0.10$ | - | $1.93\pm 0.07$
$k=40$ | $\mathbf{2.13\pm 0.06}$ | - | $1.67\pm 0.01$ | $1.55\pm 0.10$ | - | $1.99\pm 0.04$
$k=80$ | $\mathbf{2.16\pm 0.03}$ | - | $1.70\pm 0.05$ | $1.55\pm 0.10$ | - | $1.92\pm 0.02$
Table V: Full results of average task performance of BRIDGeR against current state-of-the-art methods under varying number of diffusion steps when trained with the Medium dataset. |  | BRIDGeR | Residual Policy | DDIM
---|---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic
Door | $k=0$ | $0.12\pm 0.08$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$ | $0.12\pm 0.08$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.20\pm 0.11}$ | $0.07\pm 0.00$ | $0.08\pm 0.00$ | $0.16\pm 0.11$ | $0.02\pm 0.02$ | $0.00\pm 0.00$
$k=10$ | $0.15\pm 0.07$ | $0.00\pm 0.00$ | $0.02\pm 0.02$ | $\mathbf{0.16\pm 0.11}$ | $0.02\pm 0.02$ | $0.02\pm 0.01$
$k=20$ | $\mathbf{0.13\pm 0.06}$ | $0.00\pm 0.00$ | $0.08\pm 0.10$ | $0.16\pm 0.11$ | $0.02\pm 0.02$ | $0.03\pm 0.02$
$k=40$ | $0.13\pm 0.08$ | $0.01\pm 0.02$ | $0.12\pm 0.11$ | $\mathbf{0.16\pm 0.11}$ | $0.02\pm 0.02$ | $0.00\pm 0.00$
$k=80$ | $\mathbf{0.18\pm 0.13}$ | $0.08\pm 0.06$ | $0.13\pm 0.11$ | $0.16\pm 0.11$ | $0.02\pm 0.02$ | $0.00\pm 0.00$
Relocate | $k=0$ | $\mathbf{0.24\pm 0.15}$ | - | $0.00\pm 0.00$ | $\mathbf{0.24\pm 0.12}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.55\pm 0.02}$ | - | $0.52\pm 0.08$ | $0.29\pm 0.07$ | - | $0.01\pm 0.01$
$k=10$ | $\mathbf{0.56\pm 0.04}$ | - | $0.61\pm 0.09$ | $0.29\pm 0.07$ | - | $0.09\pm 0.01$
$k=20$ | $0.62\pm 0.16$ | - | $\mathbf{0.63\pm 0.12}$ | $0.29\pm 0.07$ | - | $0.12\pm 0.04$
$k=40$ | $0.55\pm 0.12$ | - | $\mathbf{0.58\pm 0.15}$ | $0.29\pm 0.07$ | - | $0.020\pm 0.01$
$k=80$ | $\mathbf{0.64\pm 0.11}$ | - | $0.54\pm 0.10$ | $0.29\pm 0.07$ | - | $0.03\pm 0.01$
Hammer | $k=0$ | $0.09\pm 0.15$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$ | $0.09\pm 0.15$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$
$k=5$ | $0.04\pm 0.05$ | $\mathbf{0.17\pm 0.20}$ | $0.04\pm 0.01$ | $0.11\pm 0.02$ | $0.12\pm 0.05$ | $0.00\pm 0.00$
$k=10$ | $0.14\pm 0.14$ | $\mathbf{0.43\pm 0.18}$ | $0.11\pm 0.03$ | $0.11\pm 0.02$ | $0.12\pm 0.05$ | $0.03\pm 0.02$
$k=20$ | $0.25\pm 0.11$ | $\mathbf{0.51\pm 0.18}$ | $0.24\pm 0.03$ | $0.11\pm 0.02$ | $0.12\pm 0.05$ | $0.04\pm 0.04$
$k=40$ | $0.42\pm 0.10$ | $\mathbf{0.48\pm 0.27}$ | $0.29\pm 0.02$ | $0.11\pm 0.02$ | $0.12\pm 0.05$ | $0.00\pm 0.00$
$k=80$ | $\mathbf{0.37\pm 0.06}$ | $0.32\pm 0.24$ | $0.31\pm 0.07$ | $0.11\pm 0.02$ | $0.12\pm 0.05$ | $0.00\pm 0.00$
Pen | $k=0$ | $\mathbf{0.21\pm 0.05}$ | - | $0.00\pm 0.00$ | $\mathbf{0.21\pm 0.05}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.48\pm 0.06}$ | - | $\mathbf{0.48\pm 0.02}$ | $0.19\pm 0.05$ | - | $0.42\pm 0.02$
$k=10$ | $\mathbf{0.50\pm 0.04}$ | - | $0.46\pm 0.07$ | $0.19\pm 0.05$ | - | $\mathbf{0.50\pm 0.07}$
$k=20$ | $\mathbf{0.57\pm 0.05}$ | - | $0.46\pm 0.05$ | $0.19\pm 0.05$ | - | $0.49\pm 0.03$
$k=40$ | $\mathbf{0.58\pm 0.04}$ | - | $0.45\pm 0.07$ | $0.19\pm 0.05$ | - | $0.52\pm 0.03$
$k=80$ | $0.56\pm 0.05$ | - | $0.49\pm 0.06$ | $0.19\pm 0.05$ | - | $\mathbf{0.57\pm 0.04}$
Franka Kitchen | $k=0$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{1.88\pm 0.04}$ | - | $1.57\pm 0.14$ | $1.4\pm 0.06$ | - | $1.82\pm 0.08$
$k=10$ | $\mathbf{2.04\pm 0.02}$ | - | $1.62\pm 0.22$ | $1.4\pm 0.06$ | - | $1.74\pm 0.04$
$k=20$ | $\mathbf{2.12\pm 0.05}$ | - | $1.48\pm 0.23$ | $1.4\pm 0.06$ | - | $1.81\pm 0.04$
$k=40$ | $\mathbf{2.09\pm 0.05}$ | - | $1.51\pm 0.36$ | $1.4\pm 0.06$ | - | $1.79\pm 0.07$
$k=80$ | $\mathbf{2.11\pm 0.05}$ | - | $1.44\pm 0.28$ | $1.4\pm 0.06$ | - | $1.79\pm 0.06$
Table VI: Full results of average task performance of BRIDGeR against current state-of-the-art methods under varying number of diffusion steps when trained with the Small dataset. |  | BRIDGeR | Residual Policy | DDIM
---|---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic
Door | $k=0$ | $0.08\pm 0.12$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$ | $0.08\pm 0.12$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.10\pm 0.06}$ | $0.00\pm 0.00$ | $0.00\pm 0.00$ | $0.00\pm 0.00$ | $0.01\pm 0.02$ | $0.00\pm 0.00$
$k=10$ | $\mathbf{0.10\pm 0.07}$ | $0.00\pm 0.00$ | $0.01\pm 0.01$ | $0.00\pm 0.00$ | $0.01\pm 0.02$ | $0.00\pm 0.00$
$k=20$ | $\mathbf{0.12\pm 0.08}$ | $0.00\pm 0.00$ | $0.05\pm 0.02$ | $0.00\pm 0.00$ | $0.01\pm 0.02$ | $0.00\pm 0.00$
$k=40$ | $\mathbf{0.14\pm 0.14}$ | $0.00\pm 0.00$ | $0.06\pm 0.01$ | $0.00\pm 0.00$ | $0.01\pm 0.02$ | $0.00\pm 0.00$
$k=80$ | $\mathbf{0.14\pm 0.14}$ | $0.00\pm 0.00$ | $0.06\pm 0.01$ | $0.00\pm 0.00$ | $0.01\pm 0.02$ | $0.00\pm 0.00$
Relocate | $k=0$ | $\mathbf{0.18\pm 0.13}$ | - | $0.00\pm 0.00$ | $\mathbf{0.18\pm 0.13}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.56\pm 0.06}$ | - | $0.23\pm 0.05$ | $0.20\pm 0.07$ | - | $0.00\pm 0.00$
$k=10$ | $\mathbf{0.55\pm 0.06}$ | - | $0.31\pm 0.05$ | $0.20\pm 0.07$ | - | $0.00\pm 0.00$
$k=20$ | $\mathbf{0.51\pm 0.03}$ | - | $0.28\pm 0.04$ | $0.20\pm 0.07$ | - | $0.00\pm 0.00$
$k=40$ | $\mathbf{0.49\pm 0.04}$ | - | $0.37\pm 0.06$ | $0.20\pm 0.07$ | - | $0.00\pm 0.00$
$k=80$ | $\mathbf{0.56\pm 0.14}$ | - | $0.31\pm 0.04$ | $0.20\pm 0.07$ | - | $0.01\pm 0.01$
Hammer | $k=0$ | $0.09\pm 0.15$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$ | $0.07\pm 0.12$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$
$k=5$ | $0.04\pm 0.04$ | $\mathbf{0.16\pm 0.08}$ | $0.03\pm 0.03$ | $0.07\pm 0.04$ | $0.08\pm 0.04$ | $0.00\pm 0.00$
$k=10$ | $\mathbf{0.09\pm 0.08}$ | $0.04\pm 0.01$ | $0.06\pm 0.02$ | $0.07\pm 0.04$ | $0.08\pm 0.04$ | $0.01\pm 0.01$
$k=20$ | $\mathbf{0.21\pm 0.12}$ | $0.14\pm 0.09$ | $0.18\pm 0.07$ | $0.07\pm 0.04$ | $0.08\pm 0.04$ | $0.01\pm 0.01$
$k=40$ | $\mathbf{0.28\pm 0.21}$ | $0.21\pm 0.18$ | $0.18\pm 0.04$ | $0.07\pm 0.04$ | $0.08\pm 0.04$ | $0.01\pm 0.01$
$k=80$ | $0.21\pm 0.13$ | $\mathbf{0.24\pm 0.26}$ | $0.17\pm 0.01$ | $0.07\pm 0.04$ | $0.08\pm 0.04$ | $0.01\pm 0.00$
Pen | $k=0$ | $\mathbf{0.24\pm 0.11}$ | - | $0.00\pm 0.00$ | $\mathbf{0.24\pm 0.11}$ | - | $0.00\pm 0.00$
$k=5$ | $0.38\pm 0.07$ | - | $0.34\pm 0.03$ | $\mathbf{0.48\pm 0.10}$ | - | $0.41\pm 0.05$
$k=10$ | $0.44\pm 0.05$ | - | $0.43\pm 0.03$ | $\mathbf{0.48\pm 0.10}$ | - | $0.45\pm 0.03$
$k=20$ | $\mathbf{0.50\pm 0.08}$ | - | $0.40\pm 0.03$ | $0.48\pm 0.10$ | - | $0.47\pm 0.01$
$k=40$ | $\mathbf{0.53\pm 0.09}$ | - | $0.48\pm 0.01$ | $0.48\pm 0.10$ | - | $0.47\pm 0.04$
$k=80$ | $\mathbf{0.55\pm 0.05}$ | - | $0.38\pm 0.08$ | $0.48\pm 0.10$ | - | $0.47\pm 0.03$
Franka Kitchen | $k=0$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{1.88\pm 0.04}$ | - | $1.41\pm 0.06$ | $1.33\pm 0.07$ | - | $1.81\pm 0.03$
$k=10$ | $\mathbf{2.02\pm 0.11}$ | - | $1.66\pm 0.07$ | $1.33\pm 0.07$ | - | $1.94\pm 0.12$
$k=20$ | $\mathbf{1.99\pm 0.06}$ | - | $1.60\pm 0.09$ | $1.33\pm 0.07$ | - | $1.92\pm 0.04$
$k=40$ | $\mathbf{2.01\pm 0.03}$ | - | $1.70\pm 0.07$ | $1.33\pm 0.07$ | - | $1.71\pm 0.05$
$k=80$ | $\mathbf{2.00\pm 0.01}$ | - | $1.60\pm 0.03$ | $1.33\pm 0.07$ | - | $1.78\pm 0.09$
Table VII: Success rate (averaged over $100$ grasps on ten test objects). BRIDGeR significantly outperforms DDIM and Residual Policy across the number of diffusion steps. Compared to SE3, BRIDGeR achieve higher success rate when the number of diffusion steps is small. We show up to $k=160$ steps to be consistent with prior reported results [47]. |  | BRIDGeR | Residual Policy | DDIM | SE3
---|---|---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic
Seen Categories | $k=0$ | $\mathbf{0.26\pm 0.02}$ | $0.06\pm 0.00$ | $0.00\pm 0.00$ | $\mathbf{0.26\pm 0.02}$ | $0.06\pm 0.00$ | $0.00\pm 0.00$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.73\pm 0.15}$ | $0.64\pm 0.17$ | $0.56\pm 0.17$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.52\pm 0.24$ | $0.38\pm 0.26$
$k=10$ | $0.85\pm 0.17$ | $\mathbf{0.87\pm 0.12}$ | $0.72\pm 0.27$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.61\pm 0.27$ | $0.56\pm 0.20$
$k=20$ | $\mathbf{0.93\pm 0.08}$ | $0.91\pm 0.07$ | $0.83\pm 0.28$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.64\pm 0.21$ | $0.78\pm 0.19$
$k=40$ | $\mathbf{0.91\pm 0.08}$ | $0.90\pm 0.08$ | $0.81\pm 0.27$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.61\pm 0.26$ | $0.89\pm 0.04$
$k=80$ | $\mathbf{0.88\pm 0.18}$ | $\mathbf{0.88\pm 0.08}$ | $0.82\pm 0.05$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.64\pm 0.24$ | $0.83\pm 0.20$
$k=160$ | $0.88\pm 0.10$ | $\mathbf{0.91\pm 0.08}$ | $0.90\pm 0.06$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.64\pm 0.26$ | $\mathbf{0.91\pm 0.08}$
Unseen Categories | $k=0$ | $\mathbf{0.23\pm 0.04}$ | $0.00\pm 0.00$ | $0.00\pm 0.00$ | $\mathbf{0.23\pm 0.04}$ | $0.00\pm 0.00$ | $0.00\pm 0.00$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.48\pm 0.12}$ | $0.43\pm 0.19$ | $0.45\pm 0.14$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.33\pm 0.20$ | $0.20\pm 0.10$
$k=10$ | $0.59\pm 0.31$ | $\mathbf{0.61\pm 0.26}$ | $0.60\pm 0.21$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.38\pm 0.21$ | $0.39\pm 0.15$
$k=20$ | $\mathbf{0.67\pm 0.21}$ | $\mathbf{0.67\pm 0.26}$ | $0.65\pm 0.25$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.41\pm 0.23$ | $0.55\pm 0.24$
$k=40$ | $0.63\pm 0.29$ | $\mathbf{0.69\pm 0.22}$ | $\mathbf{0.69\pm 0.20}$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.33\pm 0.19$ | $0.61\pm 0.21$
$k=80$ | $\mathbf{0.71\pm 0.20}$ | $0.68\pm 0.25$ | $0.69\pm 0.21$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.38\pm 0.23$ | $0.63\pm 0.29$
$k=160$ | $\mathbf{0.71\pm 0.24}$ | $0.66\pm 0.23$ | $0.64\pm 0.23$ | $0.09\pm 0.15$ | $0.01\pm 0.03$ | $0.35\pm 0.19$ | $0.66\pm 0.25$
Table VIII: Average Earth Mover’s Distance between generated grasp poses and target grasp poses. BRIDGeR significantly outperforms SE3, DDIM and Residual Policy across the number of diffusion steps.. |  | BRIDGeR | Residual Policy | DDIM | SE3
---|---|---|---|---|---
|  | CVAE | Heuristicx | Gaussian | CVAE | Heuristic
Seen Categories | $k=0$ | $\mathbf{0.86\pm 0.09}$ | $1.11\pm 0.06$ | $1.34\pm 0.08$ | $\mathbf{0.66\pm 0.09}$ | $1.21\pm 0.06$ | $1.34\pm 0.08$ | $1.45\pm 0.08$
$k=5$ | $0.38\pm 0.03$ | $\mathbf{0.36\pm 0.03}$ | $0.39\pm 0.05$ | $0.69\pm 0.02$ | $1.55\pm 0.03$ | $1.41\pm 0.14$ | $0.44\pm 0.03$
$k=10$ | $0.38\pm 0.03$ | $\mathbf{0.36\pm 0.03}$ | $0.37\pm 0.04$ | $0.69\pm 0.02$ | $1.55\pm 0.03$ | $1.41\pm 0.14$ | $0.41\pm 0.03$
$k=20$ | $0.39\pm 0.05$ | $\mathbf{0.35\pm 0.03}$ | $0.39\pm 0.05$ | $0.79\pm 0.03$ | $1.55\pm 0.03$ | $0.64\pm 0.21$ | $0.40\pm 0.02$
$k=40$ | $0.37\pm 0.02$ | $\mathbf{0.35\pm 0.02}$ | $0.40\pm 0.06$ | $0.73\pm 0.03$ | $1.55\pm 0.03$ | $0.52\pm 0.24$ | $0.40\pm 0.01$
$k=80$ | $0.38\pm 0.02$ | $\mathbf{0.35\pm 0.02}$ | $0.40\pm 0.03$ | $0.74\pm 0.02$ | $1.55\pm 0.03$ | $0.52\pm 0.24$ | $0.45\pm 0.02$
$k=160$ | $0.38\pm 0.02$ | $\mathbf{0.35\pm 0.02}$ | $0.43\pm 0.08$ | $0.70\pm 0.02$ | $1.55\pm 0.03$ | $0.64\pm 0.26$ | $0.50\pm 0.03$
Unseen Categories | $k=0$ | $\mathbf{0.83\pm 0.15}$ | $1.21\pm 0.12$ | $1.42\pm 0.04$ | $\mathbf{0.83\pm 0.15}$ | $1.21\pm 0.12$ | $1.42\pm 0.04$ | $1.38\pm 0.02$
$k=5$ | $0.48\pm 0.13$ | $\mathbf{0.44\pm 0.09}$ | $0.45\pm 0.09$ | $1.48\pm 0.06$ | $1.57\pm 0.06$ | $0.73\pm 0.08$ | $0.49\pm 0.07$
$k=10$ | $0.49\pm 0.15$ | $\mathbf{0.45\pm 0.11}$ | $0.44\pm 0.09$ | $1.48\pm 0.06$ | $1.57\pm 0.06$ | $0.80\pm 0.10$ | $0.45\pm 0.06$
$k=20$ | $0.50\pm 0.15$ | $\mathbf{0.42\pm 0.10}$ | $0.48\pm 0.11$ | $1.48\pm 0.06$ | $1.57\pm 0.06$ | $0.76\pm 0.09$ | $0.45\pm 0.09$
$k=40$ | $0.49\pm 0.13$ | $\mathbf{0.43\pm 0.08}$ | $0.48\pm 0.12$ | $1.48\pm 0.06$ | $1.57\pm 0.06$ | $0.80\pm 0.13$ | $0.47\pm 0.06$
$k=80$ | $0.50\pm 0.15$ | $\mathbf{0.45\pm 0.11}$ | $0.49\pm 0.12$ | $1.48\pm 0.06$ | $1.57\pm 0.06$ | $0.81\pm 0.12$ | $0.56\pm 0.10$
$k=160$ | $0.49\pm 0.141$ | $\mathbf{0.44\pm 0.11}$ | $0.52\pm 0.15$ | $1.48\pm 0.06$ | $1.57\pm 0.06$ | $0.78\pm 0.14$ | $0.58\pm 0.10$
Table IX: Full results of average task performance of BRIDGeR with Power3 and Linear interpolant functions under varying number of diffusion steps when trained with the Large dataset. |  |  BRIDGeR / Power3 |  BRIDGeR / Linear
---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic | Gaussian
Door | $k=0$ | $0.13\pm 0.04$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$ | $0.13\pm 0.04$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.60\pm 0.15}$ | $0.00\pm 0.00$ | $0.08\pm 0.06$ | $0.44\pm 0.26$ | $0.16\pm 0.01$ | $0.04\pm 0.06$
$k=10$ | $\mathbf{0.54\pm 0.18}$ | $0.27\pm 0.01$ | $0.07\pm 0.05$ | $0.44\pm 0.32$ | $0.16\pm 0.02$ | $0.21\pm 0.15$
$k=20$ | $\mathbf{0.52\pm 0.23}$ | $0.45\pm 0.04$ | $0.10\pm 0.07$ | $0.45\pm 0.29$ | $0.18\pm 0.03$ | $0.31\pm 0.20$
$k=40$ | $0.48\pm 0.19$ | $\mathbf{0.72\pm 0.02}$ | $0.07\pm 0.05$ | $0.46\pm 0.30$ | $0.11\pm 0.02$ | $0.33\pm 0.18$
$k=80$ | $0.50\pm 0.14$ | $\mathbf{0.63\pm 0.08}$ | $0.12\pm 0.09$ | $0.42\pm 0.29$ | $0.14\pm 0.06$ | $0.20\pm 0.09$
Relocate | $k=0$ | $\mathbf{0.31\pm 0.15}$ | - | $0.00\pm 0.00$ | $\mathbf{0.31\pm 0.15}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.75\pm 0.11}$ | - | $0.61\pm 0.05$ | $0.64\pm 0.12$ | - | $0.29\pm 0.12$
$k=10$ | $\mathbf{0.75\pm 0.08}$ | - | $0.73\pm 0.11$ | $0.68\pm 0.16$ | - | $0.74\pm 0.04$
$k=20$ | $0.70\pm 0.11$ | - | $0.72\pm 0.09$ | $0.64\pm 0.12$ | - | $\mathbf{0.76\pm 0.07}$
$k=40$ | $0.77\pm 0.11$ | - | $\mathbf{0.78\pm 0.05}$ | $0.63\pm 0.19$ | - | $0.71\pm 0.04$
$k=80$ | $0.79\pm 0.08$ | - | $\mathbf{0.81\pm 0.04}$ | $0.68\pm 0.19$ | - | $0.74\pm 0.01$
Hammer | $k=0$ | $\mathbf{0.16\pm 0.03}$ | $0.11\pm 0.08$ | $0.00\pm 0.00$ | $\mathbf{0.16\pm 0.03}$ | $0.11\pm 0.08$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.44\pm 0.11}$ | $0.20\pm 0.24$ | $0.16\pm 0.04$ | $0.31\pm 0.18$ | $0.06\pm 0.04$ | $0.04\pm 0.02$
$k=10$ | $0.54\pm 0.17$ | $\mathbf{0.62\pm 0.11}$ | $0.30\pm 0.04$ | $0.54\pm 0.13$ | $0.22\pm 0.07$ | $0.08\pm 0.05$
$k=20$ | $0.63\pm 0.08$ | $\mathbf{0.74\pm 0.07}$ | $0.35\pm 0.02$ | $0.56\pm 0.05$ | $0.50\pm 0.10$ | $0.14\pm 0.04$
$k=40$ | $\mathbf{0.58\pm 0.12}$ | $\mathbf{0.58\pm 0.21}$ | $0.36\pm 0.05$ | $0.48\pm 0.06$ | $0.50\pm 0.12$ | $0.24\pm 0.06$
$k=80$ | $\mathbf{0.72\pm 0.09}$ | $0.47\pm 0.25$ | $0.43\pm 0.09$ | $0.48\pm 0.09$ | $0.50\pm 0.17$ | $0.29\pm 0.10$
Pen | $k=0$ | $\mathbf{0.29\pm 0.05}$ | - | $0.00\pm 0.00$ | $\mathbf{0.29\pm 0.05}$ | - | $0.00\pm 0.00$
$k=5$ | $0.45\pm 0.04$ | - | $0.43\pm 0.02$ | $\mathbf{0.48\pm 0.05}$ | - | $0.42\pm 0.03$
$k=10$ | $\mathbf{0.49\pm 0.02}$ | - | $0.46\pm 0.04$ | $\mathbf{0.49\pm 0.03}$ | - | $\mathbf{0.49\pm 0.05}$
$k=20$ | $0.49\pm 0.09$ | - | $\mathbf{0.53\pm 0.02}$ | $0.51\pm 0.03$ | - | $0.52\pm 0.04$
$k=40$ | $0.52\pm 0.07$ | - | $0.54\pm 0.05$ | $0.49\pm 0.04$ | - | $\mathbf{0.51\pm 0.04}$
$k=80$ | $0.55\pm 0.06$ | - | $0.55\pm 0.03$ | $\mathbf{0.59\pm 0.03}$ | - | $0.57\pm 0.05$
Franka Kitchen | $k=0$ | $\mathbf{1.53\pm 0.09}$ | - | $0.00\pm 0.00$ | $\mathbf{1.53\pm 0.09}$ | - | $0.00\pm 0.00$
$k=5$ | $1.96\pm 0.03$ | - | $1.18\pm 0.02$ | $\mathbf{2.03\pm 0.11}$ | - | $1.12\pm 0.09$
$k=10$ | $\mathbf{2.13\pm 0.01}$ | - | $1.28\pm 0.24$ | $2.10\pm 0.01$ | - | $1.55\pm 0.09$
$k=20$ | $2.09\pm 0.04$ | - | $1.54\pm 0.03$ | $\mathbf{2.11\pm 0.05}$ | - | $1.70\pm 0.04$
$k=40$ | $\mathbf{2.13\pm 0.06}$ | - | $1.67\pm 0.01$ | $2.08\pm 0.06$ | - | $1.79\pm 0.04$
$k=80$ | $\mathbf{2.16\pm 0.03}$ | - | $1.70\pm 0.05$ | $2.06\pm 0.06$ | - | $1.79\pm 0.01$
Table X: Full results of average task performance of BRIDGeR with Power3 and Linear interpolant functions under varying number of diffusion steps when trained with the Medium dataset. |  |  BRIDGeR / Power3 |  BRIDGeR / Linear
---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic | Gaussian
Door | $k=0$ | $0.12\pm 0.08$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$ | $0.12\pm 0.08$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$
$k=5$ | $0.20\pm 0.11$ | $0.07\pm 0.00$ | $0.08\pm 0.00$ | $\mathbf{0.32\pm 0.11}$ | $0.02\pm 0.02$ | $0.04\pm 0.05$
$k=10$ | $0.15\pm 0.07$ | $0.00\pm 0.00$ | $0.02\pm 0.02$ | $\mathbf{0.25\pm 0.11}$ | $0.00\pm 0.00$ | $0.22\pm 0.12$
$k=20$ | $0.13\pm 0.06$ | $0.00\pm 0.00$ | $0.08\pm 0.10$ | $\mathbf{0.25\pm 0.09}$ | $0.00\pm 0.00$ | $\mathbf{0.25\pm 0.16}$
$k=40$ | $0.13\pm 0.08$ | $0.01\pm 0.02$ | $0.12\pm 0.11$ | $\mathbf{0.28\pm 0.12}$ | $0.00\pm 0.00$ | $0.22\pm 0.12$
$k=80$ | $0.18\pm 0.13$ | $0.08\pm 0.06$ | $0.13\pm 0.11$ | $\mathbf{0.28\pm 0.12}$ | $0.02\pm 0.03$ | $0.26\pm 0.16$
Relocate | $k=0$ | $\mathbf{0.24\pm 0.15}$ | - | $0.00\pm 0.00$ | $\mathbf{0.24\pm 0.12}$ | - | $0.00\pm 0.00$
$k=5$ | $0.55\pm 0.02$ | - | $0.52\pm 0.08$ | $\mathbf{0.63\pm 0.09}$ | - | $0.52\pm 0.08$
$k=10$ | $0.56\pm 0.04$ | - | $0.61\pm 0.09$ | $\mathbf{0.65\pm 0.06}$ | - | $\mathbf{0.65\pm 0.02}$
$k=20$ | $0.62\pm 0.16$ | - | $0.63\pm 0.12$ | $0.61\pm 0.05$ | - | $\mathbf{0.65\pm 0.02}$
$k=40$ | $0.55\pm 0.12$ | - | $0.58\pm 0.15$ | $0.57\pm 0.01$ | - | $\mathbf{0.60\pm 0.08}$
$k=80$ | $0.64\pm 0.11$ | - | $0.54\pm 0.10$ | $0.65\pm 0.05$ | - | $\mathbf{0.66\pm 0.06}$
Hammer | $k=0$ | $0.09\pm 0.15$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$ | $0.09\pm 0.15$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$
$k=5$ | $0.04\pm 0.05$ | $\mathbf{0.17\pm 0.20}$ | $0.04\pm 0.01$ | $0.10\pm 0.07$ | $0.09\pm 0.01$ | $0.03\pm 0.01$
$k=10$ | $0.14\pm 0.14$ | $\mathbf{0.43\pm 0.18}$ | $0.11\pm 0.03$ | $0.22\pm 0.09$ | $0.24\pm 0.07$ | $0.20\pm 0.02$
$k=20$ | $0.25\pm 0.11$ | $\mathbf{0.51\pm 0.18}$ | $0.24\pm 0.03$ | $0.28\pm 0.08$ | $0.33\pm 0.07$ | $0.22\pm 0.07$
$k=40$ | $0.42\pm 0.10$ | $\mathbf{0.48\pm 0.27}$ | $0.29\pm 0.02$ | $0.43\pm 0.15$ | $0.25\pm 0.03$ | $0.36\pm 0.05$
$k=80$ | $0.37\pm 0.06$ | $0.32\pm 0.24$ | $0.31\pm 0.07$ | $\mathbf{0.40\pm 0.08}$ | $0.35\pm 0.01$ | $0.28\pm 0.10$
Pen | $k=0$ | $\mathbf{0.21\pm 0.05}$ | - | $0.00\pm 0.00$ | $\mathbf{0.21\pm 0.05}$ | - | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.48\pm 0.06}$ | - | $\mathbf{0.48\pm 0.02}$ | $\mathbf{0.48\pm 0.05}$ | - | $0.42\pm 0.03$
$k=10$ | $\mathbf{0.50\pm 0.04}$ | - | $0.46\pm 0.07$ | $0.49\pm 0.03$ | - | $0.49\pm 0.05$
$k=20$ | $\mathbf{0.57\pm 0.05}$ | - | $0.46\pm 0.05$ | $0.51\pm 0.03$ | - | $0.52\pm 0.04$
$k=40$ | $\mathbf{0.58\pm 0.04}$ | - | $0.45\pm 0.07$ | $0.49\pm 0.04$ | - | $0.51\pm 0.04$
$k=80$ | $0.56\pm 0.05$ | - | $0.49\pm 0.06$ | $\mathbf{0.59\pm 0.03}$ | - | $0.56\pm 0.05$
Franka Kitchen | $k=0$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$
$k=5$ | $1.88\pm 0.04$ | - | $1.57\pm 0.14$ | $\mathbf{2.06\pm 0.02}$ | - | $1.02\pm 0.04$
$k=10$ | $2.04\pm 0.02$ | - | $1.62\pm 0.22$ | $\mathbf{2.09\pm 0.01}$ | - | $1.41\pm 0.09$
$k=20$ | $\mathbf{2.12\pm 0.05}$ | - | $1.48\pm 0.23$ | $2.04\pm 0.02$ | - | $1.75\pm 0.02$
$k=40$ | $2.09\pm 0.05$ | - | $1.51\pm 0.36$ | $\mathbf{2.11\pm 0.05}$ | - | $1.81\pm 0.16$
$k=80$ | $\mathbf{2.11\pm 0.05}$ | - | $1.44\pm 0.28$ | $2.03\pm 0.07$ | - | $1.82\pm 0.06$
Table XI: Full results of average task performance of BRIDGeR with Power3 and Linear interpolant functions under varying number of diffusion steps when trained with the Small dataset. |  |  BRIDGeR / Power3 |  BRIDGeR / Linear
---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic | Gaussian
Door | $k=0$ | $0.08\pm 0.12$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$ | $0.08\pm 0.12$ | $\mathbf{0.16\pm 0.06}$ | $0.00\pm 0.00$
$k=5$ | $0.10\pm 0.06$ | $0.00\pm 0.00$ | $0.00\pm 0.00$ | $\mathbf{0.13\pm 0.04}$ | $0.00\pm 0.00$ | $0.03\pm 0.03$
$k=10$ | $0.10\pm 0.07$ | $0.00\pm 0.00$ | $0.01\pm 0.01$ | $\mathbf{0.11\pm 0.06}$ | $0.00\pm 0.00$ | $0.06\pm 0.07$
$k=20$ | $\mathbf{0.12\pm 0.08}$ | $0.00\pm 0.00$ | $0.05\pm 0.02$ | $0.09\pm 0.03$ | $0.00\pm 0.00$ | $0.08\pm 0.06$
$k=40$ | $\mathbf{0.14\pm 0.14}$ | $0.00\pm 0.00$ | $0.06\pm 0.01$ | $0.11\pm 0.05$ | $0.00\pm 0.00$ | $0.12\pm 0.02$
$k=80$ | $\mathbf{0.14\pm 0.14}$ | $0.00\pm 0.00$ | $0.06\pm 0.01$ | $0.12\pm 0.05$ | $0.00\pm 0.00$ | $0.13\pm 0.03$
Relocate | $k=0$ | $\mathbf{0.18\pm 0.13}$ | - | $0.00\pm 0.00$ | $\mathbf{0.18\pm 0.13}$ | - | $0.00\pm 0.00$
$k=5$ | $0.56\pm 0.06$ | - | $0.23\pm 0.05$ | $\mathbf{0.59\pm 0.14}$ | - | $0.11\pm 0.10$
$k=10$ | $0.55\pm 0.06$ | - | $0.31\pm 0.05$ | $\mathbf{0.60\pm 0.05}$ | - | $0.59\pm 0.16$
$k=20$ | $0.51\pm 0.03$ | - | $0.28\pm 0.04$ | $0.59\pm 0.02$ | - | $\mathbf{0.63\pm 0.06}$
$k=40$ | $0.49\pm 0.04$ | - | $0.37\pm 0.06$ | $0.57\pm 0.11$ | - | $\mathbf{0.58\pm 0.04}$
$k=80$ | $0.56\pm 0.14$ | - | $0.31\pm 0.04$ | $\mathbf{0.61\pm 0.02}$ | - | $0.55\pm 0.02$
Hammer | $k=0$ | $0.09\pm 0.15$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$ | $0.07\pm 0.12$ | $\mathbf{0.11\pm 0.08}$ | $0.00\pm 0.00$
$k=5$ | $0.04\pm 0.04$ | $0.16\pm 0.08$ | $0.03\pm 0.03$ | $0.05\pm 0.01$ | $\mathbf{0.17\pm 0.01}$ | $0.04\pm 0.06$
$k=10$ | $0.09\pm 0.08$ | $0.04\pm 0.01$ | $0.06\pm 0.02$ | $\mathbf{0.22\pm 0.01}$ | $0.20\pm 0.07$ | $0.12\pm 0.10$
$k=20$ | $0.21\pm 0.12$ | $0.14\pm 0.09$ | $0.18\pm 0.07$ | $0.20\pm 0.08$ | $\mathbf{0.28\pm 0.07}$ | $0.19\pm 0.11$
$k=40$ | $0.28\pm 0.21$ | $0.21\pm 0.18$ | $0.18\pm 0.04$ | $0.24\pm 0.09$ | $\mathbf{0.36\pm 0.18}$ | $0.17\pm 0.11$
$k=80$ | $0.21\pm 0.13$ | $0.24\pm 0.26$ | $0.17\pm 0.01$ | $0.29\pm 0.00$ | $\mathbf{0.37\pm 0.21}$ | $0.21\pm 0.08$
Pen | $k=0$ | $\mathbf{0.24\pm 0.11}$ | - | $0.00\pm 0.00$ | $\mathbf{0.24\pm 0.11}$ | - | $0.00\pm 0.00$
$k=5$ | $0.38\pm 0.07$ | - | $\mathbf{0.34\pm 0.03}$ | $\mathbf{0.42\pm 0.09}$ | - | $0.35\pm 0.03$
$k=10$ | $0.44\pm 0.05$ | - | $0.43\pm 0.03$ | $0.42\pm 0.10$ | - | $\mathbf{0.45\pm 0.03}$
$k=20$ | $\mathbf{0.50\pm 0.08}$ | - | $0.40\pm 0.03$ | $0.45\pm 0.11$ | - | $0.47\pm 0.01$
$k=40$ | $\mathbf{0.53\pm 0.09}$ | - | $0.48\pm 0.01$ | $0.46\pm 0.08$ | - | $0.46\pm 0.09$
$k=80$ | $\mathbf{0.55\pm 0.05}$ | - | $0.38\pm 0.08$ | $0.49\pm 0.07$ | - | $0.52\pm 0.05$
Franka Kitchen | $k=0$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$ | $\mathbf{1.43\pm 0.17}$ | - | $0.00\pm 0.00$
$k=5$ | $1.88\pm 0.04$ | - | $1.41\pm 0.06$ | $\mathbf{2.02\pm 0.04}$ | - | $1.04\pm 0.07$
$k=10$ | $\mathbf{2.02\pm 0.11}$ | - | $1.66\pm 0.07$ | $\mathbf{2.02\pm 0.04}$ | - | $1.42\pm 0.03$
$k=20$ | $1.99\pm 0.06$ | - | $1.60\pm 0.09$ | $\mathbf{2.07\pm 0.04}$ | - | $1.69\pm 0.05$
$k=40$ | $\mathbf{2.01\pm 0.03}$ | - | $1.70\pm 0.07$ | $2.00\pm 0.02$ | - | $1.78\pm 0.07$
$k=80$ | $\mathbf{2.00\pm 0.01}$ | - | $1.60\pm 0.03$ | $1.93\pm 0.04$ | - | $1.57\pm 0.11$
Table XII: Success rate (averaged over $100$ grasps on ten test objects). BRIDGeR with Power3 interpolant function performs better than Linear interpolant function. We show up to $k=160$ steps to be consistent with prior reported results [47]. |  |  BRIDGeR / Power3 |  BRIDGeR / Linear
---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic | Gaussian
Seen Categories | $k=0$ | $\mathbf{0.26\pm 0.02}$ | $0.06\pm 0.00$ | $0.00\pm 0.00$ | $\mathbf{0.26\pm 0.02}$ | $0.06\pm 0.00$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.73\pm 0.15}$ | $0.64\pm 0.17$ | $0.56\pm 0.17$ | $0.52\pm 0.29$ | $0.31\pm 0.28$ | $0.30\pm 0.30$
$k=10$ | $0.85\pm 0.17$ | $\mathbf{0.87\pm 0.12}$ | $0.72\pm 0.27$ | $0.71\pm 0.30$ | $0.53\pm 0.27$ | $0.68\pm 0.24$
$k=20$ | $\mathbf{0.93\pm 0.08}$ | $0.91\pm 0.07$ | $0.83\pm 0.28$ | $0.73\pm 0.33$ | $0.86\pm 0.10$ | $0.77\pm 0.14$
$k=40$ | $\mathbf{0.91\pm 0.08}$ | $0.90\pm 0.08$ | $0.81\pm 0.27$ | $0.72\pm 0.33$ | $0.77\pm 0.28$ | $0.76\pm 0.28$
$k=80$ | $\mathbf{0.88\pm 0.18}$ | $\mathbf{0.88\pm 0.08}$ | $0.82\pm 0.05$ | $0.76\pm 0.25$ | $0.79\pm 0.27$ | $0.78\pm 0.29$
$k=160$ | $0.88\pm 0.10$ | $\mathbf{0.91\pm 0.08}$ | $0.90\pm 0.06$ | $0.85\pm 0.16$ | $0.86\pm 0.14$ | $0.78\pm 0.28$
Unseen Categories | $k=0$ | $\mathbf{0.23\pm 0.04}$ | $0.00\pm 0.00$ | $0.00\pm 0.00$ | $\mathbf{0.23\pm 0.04}$ | $0.00\pm 0.00$ | $0.00\pm 0.00$
$k=5$ | $\mathbf{0.48\pm 0.12}$ | $0.43\pm 0.19$ | $0.45\pm 0.14$ | $0.20\pm 0.14$ | $0.19\pm 0.18$ | $0.20\pm 0.14$
$k=10$ | $0.59\pm 0.31$ | $\mathbf{0.61\pm 0.26}$ | $0.60\pm 0.21$ | $0.52\pm 0.25$ | $0.50\pm 0.22$ | $0.52\pm 0.25$
$k=20$ | $\mathbf{0.67\pm 0.21}$ | $\mathbf{0.67\pm 0.26}$ | $0.65\pm 0.25$ | $0.63\pm 0.26$ | $0.57\pm 0.24$ | $0.63\pm 0.26$
$k=40$ | $0.63\pm 0.29$ | $\mathbf{0.69\pm 0.22}$ | $\mathbf{0.69\pm 0.20}$ | $0.63\pm 0.27$ | $0.62\pm 0.25$ | $0.63\pm 0.27$
$k=80$ | $\mathbf{0.71\pm 0.20}$ | $0.68\pm 0.25$ | $0.69\pm 0.21$ | $0.60\pm 0.28$ | $0.57\pm 0.25$ | $0.60\pm 0.28$
$k=160$ | $\mathbf{0.71\pm 0.24}$ | $0.66\pm 0.23$ | $0.64\pm 0.23$ | $0.60\pm 0.31$ | $0.66\pm 0.23$ | $0.60\pm 0.31$
Table XIII: Average Earth Mover’s Distance between generated grasp poses and target grasp poses. BRIDGeR with Power3 interpolant function performs better than Linear interpolant function. |  |  BRIDGeR / Power3 |  BRIDGeR / Linear
---|---|---|---
|  | CVAE | Heuristic | Gaussian | CVAE | Heuristic | Gaussian
Seen Categories | $k=0$ | $\mathbf{0.86\pm 0.09}$ | $1.11\pm 0.06$ | $1.34\pm 0.08$ | $\mathbf{0.66\pm 0.09}$ | $1.21\pm 0.06$ | $1.34\pm 0.08$
$k=5$ | $0.38\pm 0.03$ | $\mathbf{0.36\pm 0.03}$ | $0.39\pm 0.05$ | $0.56\pm 0.12$ | $0.42\pm 0.04$ | $0.42\pm 0.04$
$k=10$ | $0.38\pm 0.03$ | $\mathbf{0.36\pm 0.03}$ | $0.37\pm 0.04$ | $0.49\pm 0.09$ | $0.42\pm 0.06$ | $0.43\pm 0.05$
$k=20$ | $0.39\pm 0.05$ | $\mathbf{0.35\pm 0.03}$ | $0.39\pm 0.05$ | $0.57\pm 0.15$ | $0.39\pm 0.04$ | $0.51\pm 0.06$
$k=40$ | $0.37\pm 0.02$ | $\mathbf{0.35\pm 0.02}$ | $0.40\pm 0.06$ | $0.54\pm 0.12$ | $0.39\pm 0.05$ | $0.51\pm 0.12$
$k=80$ | $0.38\pm 0.02$ | $\mathbf{0.35\pm 0.02}$ | $0.40\pm 0.03$ | $0.49\pm 0.07$ | $0.39\pm 0.05$ | $0.54\pm 0.13$
$k=160$ | $0.38\pm 0.02$ | $\mathbf{0.35\pm 0.02}$ | $0.43\pm 0.08$ | $0.50\pm 0.07$ | $0.38\pm 0.04$ | $0.51\pm 0.08$
Unseen Categories | $k=0$ | $\mathbf{0.83\pm 0.15}$ | $1.21\pm 0.12$ | $1.42\pm 0.04$ | $\mathbf{0.83\pm 0.15}$ | $1.21\pm 0.12$ | $1.42\pm 0.04$
$k=5$ | $0.48\pm 0.13$ | $\mathbf{0.44\pm 0.09}$ | $0.45\pm 0.09$ | $0.56\pm 0.08$ | $0.51\pm 0.10$ | $0.54\pm 0.05$
$k=10$ | $0.49\pm 0.15$ | $\mathbf{0.45\pm 0.11}$ | $0.44\pm 0.09$ | $0.62\pm 0.18$ | $0.48\pm 0.10$ | $0.54\pm 0.10$
$k=20$ | $0.50\pm 0.15$ | $\mathbf{0.42\pm 0.10}$ | $0.48\pm 0.11$ | $0.64\pm 0.18$ | $0.45\pm 0.06$ | $0.53\pm 0.13$
$k=40$ | $0.49\pm 0.13$ | $\mathbf{0.43\pm 0.08}$ | $0.48\pm 0.12$ | $0.59\pm 0.15$ | $0.48\pm 0.08$ | $0.57\pm 0.11$
$k=80$ | $0.50\pm 0.15$ | $\mathbf{0.45\pm 0.11}$ | $0.49\pm 0.12$ | $0.66\pm 0.18$ | $0.45\pm 0.08$ | $0.64\pm 0.19$
$k=160$ | $0.49\pm 0.14$ | $\mathbf{0.44\pm 0.11}$ | $0.52\pm 0.15$ | $0.58\pm 0.10$ | $0.46\pm 0.08$ | $0.69\pm 0.11$