# LEO: Learning Energy-based Models in
Factor Graph Optimization

Paloma Sodhi1,2, Eric Dexheimer1, Mustafa Mukadam2, Stuart Anderson2, Michael
Kaess1
1Carnegie Mellon University, 2 Facebook AI Research

###### Abstract

We address the problem of learning observation models end-to-end for
estimation. Robots operating in partially observable environments must infer
latent states from multiple sensory inputs using observation models that
capture the joint distribution between latent states and observations. This
inference problem can be formulated as an objective over a graph that
optimizes for the most likely sequence of states using all previous
measurements. Prior work uses observation models that are either known
a-priori or trained on surrogate losses independent of the graph optimizer. In
this paper, we propose a method to directly optimize end-to-end tracking
performance by learning observation models with the graph optimizer in the
loop. This direct approach may appear, however, to require the inference
algorithm to be fully differentiable, which many state-of-the-art graph
optimizers are not. Our key insight is to instead formulate the problem as
that of energy-based learning. We propose a novel approach, LEO, for learning
observation models end-to-end with graph optimizers that may be non-
differentiable. LEO alternates between sampling trajectories from the graph
posterior and updating the model to match these samples to ground truth
trajectories. We propose a way to generate such samples efficiently using
incremental Gauss-Newton solvers. We compare LEO against baselines on datasets
drawn from two distinct tasks: navigation and real-world planar pushing. We
show that LEO is able to learn complex observation models with lower errors
and fewer samples.

> Keywords: factor graphs, energy-based learning, observation models

##  1 Introduction

00footnotetext: Code and supplementary material can be found on
https://psodhi.github.io/leo

We focus on the problem of learning observation models end-to-end for
estimation. Consider a robot hand manipulating an object with only tactile
feedback. It must reason over _a sequence of_ touch observations over time to
collapse uncertainty about the latent object pose. A common way to solve this
is as an inference over a factor graph which relies on observation models that
can map between states and observations [1, 2, 3]. However, in many domains,
sensors that produce observations are complex and difficult to model. _Can we
instead learn observation models from data?_

Given a batch of ground truth trajectories and corresponding measurements, how
should we learn observations models? One approach would be to learn a direct
mapping from measurement to state, for example as a regression or
classification problem [4, 5, 6, 7]. However, while easy to optimize given a
direct supervised loss, this only minimizes a surrogate loss independent of
the graph optimizer and is not guaranteed to minimize the final tracking
errors that we care about. The other option would be to directly minimize
final tracking errors between optimized and ground truth trajectories [8, 9].
However, many state-of-the-art factor graph optimizers, e.g. iSAM2 [10], are
not natively differentiable due to operations such as dynamic re-
linearizations111See Appendix A for discussion on applying LEO to
differentiable optimizers.. In such cases, we are limited to black-box search
for learning parameters which is very sample inefficient [11, 12, 13].

Figure 1: We show that learning observation models can be viewed as shaping
energy functions that graph optimizers, even non-differentiable ones,
optimize. Inference solves for most likely states $x$ given model and input
measurements $z$. Learning updates observation model parameters $\theta$ from
training data.

Instead of differentiating through the optimization process, we note that what
we ultimately care about is the final solution from the optimizer, which
depends _only on the shape_ of the optimized cost function. We would like a
cost function that has low cost around the observed ground truth trajectories
and high cost elsewhere. This is precisely what energy-based models aim to do
by shaping an “energy” function to be low around observed data and high
elsewhere [14, 15, 16]. We cast our problem of learning observation models for
estimation as energy-based learning.

We propose a novel approach, Learning Energy-based Models in Factor Graph
Optimization (LEO), for learning observation models end-to-end with graph
optimizers that may be non-differentiable. LEO alternates between sampling
trajectories from the graph posterior and updating the model to match the
distribution of samples to ground truth trajectories. We show that we can
generate such samples efficiently using incremental Gauss-Newton graph solvers
that create a Gaussian approximation for the Boltzmann distribution defined
over a continuous space of trajectories. To the best of our knowledge, this is
the first paper to show that even non-differentiable factor graph models can
be learned efficiently. Our main contributions are:

  1. 1.

A novel formulation of learning observation models in estimation as energy-
based learning.

  2. 2.

An algorithm, LEO, to learn observation models that minimize end-to-end
tracking errors in factor graph optimizers that may be non-differentiable.

  3. 3.

Empirical evaluation on two distinct tasks: synthetic navigation and real-
world planar pushing.

##  2 Related Work

Filtering and smoothing based estimation. Observation models represent joint
or conditional distributions between states and measurements. Early estimation
methods utilized these models within filtering contexts such as Kalman filters
and EKFs/UKFs [17, 18]. Recent methods have also looked at making these
filters differentiable for end-to-end learning [19, 20, 21, 22, 23, 24].
However, filtering can be inconsistent for nonlinear estimation problems due
to linearization based on past, marginalized states that cannot be undone
[25]. Instead, problems in simultaneous localization and mapping (SLAM) are
increasingly formulated as smoothing or nonlinear optimization objectives [3,
2, 1]. Unlike filtering, smoothing allows dynamic relinearization of past
states as well as exploits the inherent sparsity resulting in efficient and
more accurate solutions. Factor graphs are an increasingly popular way for
solving such smoothing problems [1, 2, 4, 5, 8, 26, 27, 28, 29], offering a
flexible modeling framework while being efficient to optimize. We use the
factor graph based smoothing framework in this paper.

Learning for smoothing. Typically, observation models used in factor graphs
are analytic models defined a-priori [30, 31, 32]. Recent work has looked at
using learned models [4, 5, 6] or learned measurement representations [27, 33]
either to be used independently or to be plugged into a graph optimizer.
However, these model parameters are learned on surrogate losses independent of
the graph optimizer, and hence do not directly attempt to minimize final
tracking errors.

An alternative approach is to view observation model parameters as hyper-
parameters of the graph and perform a hyper-parameter search. This is usually
done as sampling and derivative-free approaches using black-box solvers [11,
12, 13]. However, the high computational burden associated with sampling and
evaluation can be prohibitive for large parameter spaces.

Differentiable optimization. Instead of a search, one may learn these model
parameters via a continuous gradient-based optimizer matching graph optimizer
output against a desired solution. Minimizing this loss would, however,
require differentiating through the argmin optimization process. One way to do
this is via _unrolling_ where the optimization is represented as a series of
differentiable, gradient based updates [16, 34, 35]. Since smoothing is a
nonlinear least-squares optimization, it too can be unrolled in a similar
fashion via differentiable Levenberg-Marquardt updates [9, 36, 37] or
differentiable Gauss-Newton updates [8, 38, 39]. However, vanilla unrolling
suffers from some drawbacks, notably the learned cost function can be
sensitive to the specific optimization procedure, e.g. number of unrolling
steps [40]. Moreover, this approach cannot be readily applied to commonly used
optimizers relying on non-differentiable heuristics such as those in GTSAM,
SNOPT [41, 42]. An alternate to unrolling is to employ the implicit function
theorem which states that an optimal energy function must satisfy first-order
necessary conditions [43, 44]. While this is promising for convex problems,
for non-convex problems, one may recover an energy function with multiple
minima such that the optimizer at test time may find itself in a different
basin from ground truth.

Energy-based learning. Instead of differentiating through the optimization
process, we note what we ultimately care about is the final solution from the
optimizer, which depends only on the shape of the cost function. This is what
energy-based models aim to do by shaping an “energy” function to be low around
observed data and high elsewhere [14]. Energy-based modeling is a family of
unsupervised learning methods popular in many structured prediction tasks such
as inverse reinforcement learning [45, 46, 47, 48] and computer vision [49,
50]. Recently, a number of approaches have looked at training deep energy-
based models [51, 52, 16]. The challenge with such methods is generating
samples from the energy based model. Methods either rely on MCMC sampling [53,
54, 55] which does not scale well with dimension or resort to learning a
separate generator [56, 57, 15]. We propose a way to generate such samples
efficiently using incremental Gauss-Newton solvers. Recent work [58, 59] has
also looked at learning observation models using variational inference
techniques when ground truth states are not available. However, in our work,
we can assume some ground truth demonstrations from a motion capture system
which makes the problem more tractable.

##  3 Problem Formulation

We formalize our problem of learning observation models under the general
theoretical framework of energy based models. We begin by describing the
observation model, how it is optimized at inference time and finally how its
parameters are learned from training data (Fig. 1).

###  3.1 Model

We define an energy-based model that captures dependencies between variables
in a graph by associating a scalar energy $E(\cdot)$ to each configuration of
the graph. Our goal is to model the likelihood of a sequence of latent states
$x$ given a sequence of measurements $z$. We adopt a factor graph framework
(Fig. 1(a)) for expressing this likelihood as222We use the word likelihood as
in typical learning formulations. Specifically, it refers to the posterior of
states given measurements. Model parameters $\theta$ include covariances
$\Sigma_{k}$ in the norm $||\cdot||_{\Sigma_{k}}$ as well.,

$\begin{split}P_{\theta}(x|z)&=\frac{1}{\mathcal{Z}(\theta;z)}\exp\left\\{-E(\theta,x;z)\right\\}=\frac{1}{\mathcal{Z}(\theta;z)}\exp\left\\{-\frac{1}{2}\sum_{k}||f({\theta,x_{k};z_{k}})||^{2}\right\\}\end{split}$
(1)

where, $x:=\\{x_{1}\dots x_{T}\\}$ is the sequence of latent states, and
$z:=\\{z_{1}\dots z_{T}\\}$ is the sequence of measurements.
$||f({\theta,x_{k};z_{k}})||^{2}$ are the factor costs with
$f({\theta,x_{k}};z_{k})$ being the local observation model mapping a _subset_
of states $x_{k}\subseteq x$ and measurements $z_{k}\subseteq z$ to a cost,
$\theta$ are the learnable model parameters, and
$\mathcal{Z}(\theta;z)=\int_{x}\exp(-E(\theta,x;z))dx$ is the normalization or
partition function.

Note that Eq. 1 is a special case of the Boltzmann distribution where the
energy $E(\cdot)$ is a sum-of-squares. Notably, when $f({\theta,x_{k}};z_{k})$
is linearized, $E(\cdot)$ is quadratic in $x$ and subsequently
$P_{\theta}(x|z)$ becomes a Gaussian distribution. This is the structure that
Gauss-Newton solvers employ, and we will leverage this fact to efficiently
sample trajectories as described later in Section 4.3.

###  3.2 Inference

At inference time, given a sequence of measurements $z$, we wish to solve for
the most likely sequence of latent states $x$. We formulate this as maximizing
the log-likelihood of the model,

$\begin{split}\hat{x}&=\underset{{x}}{\operatorname{argmax}}\ \log
P_{\theta}(x|z)\\\ \end{split}$ (2)

Substituting the model expression from Eq. 1, and dropping constants
$\mathcal{Z}(\theta;z)$ and $\nicefrac{{1}}{{2}}$, results in a nonlinear
least-squares minimization of the form,

$\begin{split}\hat{x}&=\underset{{x}}{\operatorname{argmax}}\
\log\frac{1}{\mathcal{Z}(\theta;z)}\exp\left\\{-\frac{1}{2}\sum_{k}||f({\theta,x_{k}};z_{k})||^{2}\right\\}=\underset{{x}}{\operatorname{argmin}}\sum_{k}||f({\theta,x_{k}};z_{k})||^{2}\\\
\end{split}$ (3)

We use an efficient online Gauss-Newton graph solver for this objective. This
enables both real-time inference as well as efficient sampling for learning
model parameters during training (Section 4.3).

###  3.3 Learning

At train time, our goal is to learn a model that explains training data of
pairs $(x_{gt},z)$. We express this as minimizing a loss $\mathcal{L}(\theta)$
over the training data

$\begin{split}\mathcal{L}(\theta)=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}\mathcal{L}(\theta;x^{i}_{gt},z^{i})\\\
\end{split}$ (4)

where, $\\{x^{i}_{gt},z^{i}\\}\in\mathcal{D}$ is a training dataset of ground
truth trajectories and measurements.

There are various choices of losses $\mathcal{L}(\cdot)$ for learning model
parameters $\theta$,

Loss 1: Minimize final tracking loss. A straightforward choice for
$\mathcal{L}(\cdot)$ is to directly minimize final tracking errors between
optimized $\hat{x}^{i}$ and ground truth ${x^{i}_{gt}}$ trajectories, i.e.
$\mathcal{L}(\theta;x_{gt},z)=\nicefrac{{1}}{{|\mathcal{D}|}}\sum_{i}||\hat{x}^{i}\ominus
x^{i}_{gt}||^{2}_{2}$. While this loss directly minimizes final tracking
errors, it may not be differentiable since the graph inference path from
$\theta$ to $\hat{x}$ is not differentiable. In such cases, we are limited to
black-box hyperparameter search which is very sample inefficient.

Loss 2: Minimize surrogate loss. An alternate choice for the loss
$\mathcal{L}(\cdot)$ is to directly map measurements to ground truth states,
for example as a regression or classification problem. Such a surrogate loss
is independent of graph inference, and unlike Loss 1, is differentiable and
easy to optimize. However, this does not minimize the final tracking errors
that we care about.

Loss 3: Minimize energy-based loss. Finally, we consider a loss that is a
function of the energy, i.e. $\mathcal{L}(E(\theta,\cdot);x_{gt}^{i},z^{i})$.
Intuitively, this assigns a low loss to well-behaved energy functions, i.e.
functions that give the lowest energy to training data of ground truth
trajectories (correct answers) and higher energy to unseen data (incorrect
answers).

This energy-based loss, unlike Loss 2, is highly correlated with the final
tracking loss as it shapes the energy (or cost) landscape so as to make the
inference step return trajectories closer to the ground truth. Moreover,
unlike Loss 1, this loss is only a function of energies which is
differentiable _irrespective of inference_. Hence, we use Loss 3 and detail
our approach in the next section.

##  4 Approach

We instantiate the broad framework of energy-based learning within the context
of factor graph estimation in 4.1 and 4.2. In 4.3 we show how incremental
Gauss Newton approaches enable efficient sampling from these energy based
models.

###  4.1 Normalized Negative Log-Likelihood Loss

We define the overall loss as the negative log-likelihood (NLL) of the data
under the energy model333The motivation for NLL loss comes from probabilistic
modeling, notably maximum entropy moment matching. We provide a detailed
derivation along with its Gaussian approximation in Appendix C.:

$\begin{split}\mathcal{L}(\theta)&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}-\log
P_{\theta}(x_{gt}^{i}|z^{i})\\\ \end{split}$ (5)

Substituting $P_{\theta}(x|z)$ as the Boltzman distribution expression from
Eq. 1 we have,

$\begin{split}\mathcal{L}(\theta)=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}E(\theta;x_{gt}^{i},z^{i})+\log\int_{x}\exp(-E(\theta;x,z^{i}))dx\\\
\end{split}$ (6)

The expression above has an intuitive explanation. The first term penalizes
high energy over ground truth training samples. The second term penalizes low
energy everywhere else, thus acting as a “contrastive” term. This has a
desirable effect of globally shaping the energy function so that the minima
lies at the training samples. The contrastive term prevents the energy surface
from becoming flat and thus degenerate [14]. Unfortunately, we cannot directly
plug in Eq. 6 as a standard machine learning loss (e.g. in PyTorch) to
optimize. This is because the second log partition term is over a continuous
space of trajectories $x$, which renders exact computation intractable for
arbitrary nonlinear cost functions [60].

Instead, let us take the gradient of this loss, and substitute
$P_{\theta}(x|z^{i})$ expression from Eq. 1,

$\begin{split}\nabla_{\theta}\mathcal{L}(\theta)=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}\left[\underbrace{\nabla_{\theta}E(\theta;x_{gt}^{i},z^{i}))}_{\text{ground
truth samples}}-\underset{\begin{subarray}{c}x\sim\\\
P_{\theta}(x|z)\end{subarray}}{\operatorname{\mathbb{E}}}\underbrace{\nabla_{\theta}E(\theta;x,z^{i}))}_{\text{learned
distribution samples}}\right]\end{split}$ (7)

We see the log partition term transforms into an expectation over the learned
distribution. The gradient expression above has a similar interpretation as
the loss. The first term pushes down the energy of ground truth training
samples. Interestingly, the second term now pulls up the energy of samples
drawn from the learned Boltzmann distribution. If one is able to generate such
samples, computing the gradient is straightforward.

Generally, sampling from the continuous Boltzmann distribution is intractable,
relying typically on MCMC methods. But as we will detail in Section 4.3, we
can indeed generate such samples efficiently since our graph optimizer
maintains a Gaussian approximation of this distribution. Additionally in
practice, we found that introducing a temperature term $T$ inside the
Boltzmann distribution, i.e.
$P_{\theta}(x|z)\propto\exp(-\nicefrac{{1}}{{T}}E({\theta},x;z))$, was
critical for controlling the variance of the samples. Setting $T=0$
corresponds to only using the mode of the distribution which in this case is
the mean trajectory. This corresponds to minimizing a generalized perceptron
loss [14].

Figure 2: LEO algorithm: For each training example, the zoomed in panel shows
the three main stages: (a) solve graph optimization, (b) sample trajectories
from graph posterior, and (c) compute gradients. Gradients are then pooled
together to update parameters $\theta$. Algorithm 1 LEO algorithm

Input Initial $\theta$, Training data $\\{x^{i}_{gt}$,
$z^{i}\\}^{|\mathcal{D}|}_{i=1}$

while <!converged> do

for each training example i do

$\hat{\mu}^{i},\hat{\Sigma}^{i}=\underset{{x}}{\operatorname{argmin}}\\{E(\theta,x;z^{i})\\}$
$\triangleright$ Solve graph optimizer objective

$\hat{x}^{i}\sim\mathcal{N}(\hat{\mu}^{i},\hat{\Sigma}^{i})$ $\triangleright$
Sample $S$ trajectories from graph posterior

end for

$\theta^{+}\leftarrow\theta-\eta\frac{1}{|\mathcal{D}|}\sum\limits_{i=1}^{|\mathcal{D}|}\left\\{\nabla_{\theta}E(\theta;x^{i}_{gt},z^{i})-\frac{1}{{S}}\sum\limits_{\hat{x}^{i}}\nabla_{\theta}E(\theta;\hat{x}^{i},z^{i})\right\\}$
$\triangleright$ Update model parameters (Eq. 7)

end while

###  4.2 LEO Algorithm

Algorithm 1 summarizes the LEO algorithm, also illustrated in Fig. 2. It takes
as inputs the training data containing ground truth state and measurement
pairs, i.e. $\\{x^{i}_{gt}$, $z^{i}\\}^{|\mathcal{D}|}_{i=1}$, along with
initial values for the learnable observation model parameters $\theta$. The
goal is to converge to a $\theta$ that minimizes the log-likelihood loss in
Eq. 5 over the given training set $i=1\dots|\mathcal{D}|$.

LEO invokes a black-box, non-differentiable graph optimizer iSAM2 in the loop
to generate samples. Within each LEO iteration, for every training example
$i$, we invoke the graph optimizer to obtain a Gaussian approximation
$\mathcal{N}(\hat{\mu}^{i},\hat{\Sigma}^{i})$ to the original Boltzmann
distribution $P_{\theta}(x|z)$ about its mode. We then sample trajectories
$\hat{x}^{i}\sim\mathcal{N}(\hat{\mu}^{i},\hat{\Sigma}^{i})$ from this
distribution.

The next step is to compute gradients of the energy, i.e.
$\nabla_{\theta}E(\theta;{x}_{gt}^{i},z^{i})$ and
$\nabla_{\theta}E(\theta;\hat{x}^{i},z^{i})$, at the ground truth $x^{i}_{gt}$
and the samples $\hat{x}^{i}$, respectively. Note that the gradients here are
only w.r.t. $\theta$, and not $x$, and hence do not require the graph
optimizer to be differentiable. Once we have the two gradients, we use the
difference between them to update the observation model parameters $\theta$.
These updated parameters are then used in the next LEO iteration.

###  4.3 Efficient Inference via Incremental Gauss-Newton

The only thing that remains to be specified in Algorithm 1 is — How do we
generate samples efficiently from the graph inference? In Section 3.2 we noted
that our inference algorithm must satisfy two requirements (a) real-time
inference and (b) sampling from the Boltzmann distribution. We achieve this
using via incremental Gauss-Newton to solve the nonlinear least squares in Eq.
3.

Gauss-Newton proceeds by linearizing observation model functions $f(\cdot)$
about a point $x_{k}^{0}$,

$\begin{split}f({\theta,x_{k}};z_{k})=f(\theta,x_{k}^{0}+\delta
x_{k};z_{k})\approx f(\theta,x_{k}^{0};z_{k})+F_{k}\delta{x_{k}}\end{split}$
(8)

where, ${x_{k}}^{0}$ is the linearization point, $F_{k}=\left.\frac{\partial
f({\theta,x_{k};z_{k}})}{\partial{x_{k}}}\right|_{x_{k}^{0}}$ is the
measurement Jacobian, and $\delta{x_{k}}=x_{k}-x_{k}^{0}$ the state update
vector. Measurements $z$ and learnable parameters $\theta$ are treated as
constant within the graph optimization step. Substituting the Taylor expansion
in Eq. 8 back into Eq. 3,

$\begin{split}\delta
x^{*}&=\underset{\delta{x}}{\operatorname{argmin}}\sum_{k}||F_{k}\delta{x}_{k}-\left(-f(\theta,x_{k}^{0};z_{k})\right)||^{2}=\underset{\delta{x}}{\operatorname{argmin}}\
||A\delta{x}-b||^{2}_{2}\end{split}$ (9)

where, $A$, $b$ are constructed by concatenating all $F_{k}$,
$f(\theta,x_{k}^{0};z_{k})$ together into a single matrix and vector
respectively. We then iterate $x^{0}\leftarrow x^{0}+\delta x^{*}$ until
convergence.

We can solve for the objective in Eq. 9 efficiently by exploiting two
properties of the problem: _sparsity_ and _online_ measurements. For SLAM
problems, both $A$ and $A^{T}A$ are large but sparse matrices, and can hence
be factorized efficiently using sparse QR or Cholesky factorization
respectively [61]. However, this still does not guarantee real-time inference
during test time since $A$ grows with measurements over time, making the
method increasingly slower. Instead, subsequent incremental solvers [10, 62,
63] reuse matrix factorizations from previous time steps. In this work, we use
iSAM2 [10], an incremental Gauss-Newton graph solver whose resulting MAP
solution can be expressed as a Gaussian probability distribution with mean and
covariance,

$\begin{split}\hat{\mu}=\underset{{x}}{\operatorname{argmin}}\sum_{k}||f({\theta,x_{k};z_{k}})||^{2},\hskip
30.0pt\hat{\Sigma}=\left(A^{T}A\right)^{-1}|_{x=\hat{\mu}}\end{split}$ (10)

This is the Gaussian approximation of the original Boltzmann distribution
$P_{\theta}(x|z^{i})$ which we can now sample from efficiently. The graph
posterior covariance $\hat{\Sigma}$ for such a Gauss-Newton solver is obtained
using just Jacobians $A$ without needing to compute a Hessian [2]. Note that
sampling $\hat{x}\sim\mathcal{N}(\hat{\mu},\hat{\Sigma})$ is done at the
tangent space of the linearized graph, i.e.
$\hat{x}\leftarrow\hat{\mu}\oplus\delta x$ where ${\delta
x}\sim\mathcal{N}(0,\hat{\Sigma})$ and $\oplus$ is a retraction [64]. The
Gauss-Newton method for inference is an efficient technique for sampling from
intractable models. A recent overview of the Laplace approximation with Gauss-
Newton method can be found in [65]. Here, we build a specific Laplace
approximation using the incremental solver iSAM2 that scales to sampling
efficiently in real-time from very large graphs.444See Appendix B for run time
comparison to an alternate MCMC sampler.

##  5 Results and Evaluation

We evaluate LEO on synthetic navigation and real-world planar pushing. We
compare against baselines on metrics like final tracking error and sample
efficiency. We implement LEO in PyTorch [66] and interface it with the GTSAM
C++ library [41], which contains several factor graph optimizers. We use the
iSAM2 [10] optimizer for real-time, online optimization.

Figure 3: Real-world planar pushing setup: Learned tactile factors penalize
deviations between predicted relative pose from the tactile factor network and
estimated relative pose using variables in the graph. Tactile factor network
contains an image to keypoint feature encoder pre-trained using a self-
supervised loss (and kept fixed) [4] and a keypoint to relative transform
network that is fine-tuned on the end-to-end tracking errors.

###  5.1 Overview

Experimental setup: For synthetic navigation (Datasets N1–N4), the goal is to
estimate latent robot poses $x_{t}\in SE(2)$ from relative odometry and
absolute GPS measurements. For real-world planar pushing (Datasets P1–P3), the
goal is to estimate latent object poses $o_{t}\in SE(2)$ from tactile image
measurements using a DIGIT tactile sensor [67] as shown in the setup in Fig.
3. Both these problems are modeled as factor graphs with poses as variable
nodes and measurements as factor nodes.

Observation model parameters: Recall factor costs are defined by the local
observation model $f(\cdot)$ and learned model parameters $\theta$. Since we
use Gaussian factors, we will learn their mean and covariance parameters,
$\theta:=\\{\mu,\Sigma\\}$. For the navigation task, we learn (a) fixed
covariances $\theta:=\\{\Sigma_{odom},\Sigma_{gps}\\}$ (N1, N2), and (b)
covariances as a function of measurements
$\theta:=\\{\Sigma_{odom}(z),\Sigma_{gps}(z)\\}$ (N3, N4). For the planar
pushing task, we learn (a) fixed covariances
$\theta:=\\{\Sigma_{tac},\Sigma_{qs}\\}$ (P1, P2), and (b) means as a function
of tactile image measurements $\theta:=\\{\mu_{tac}(z)\\}$ (P3). For all
experiments, we use a diagonal covariance model.

Baselines: We compare against a learned sequence model such as LSTM, and
black-box hyperparameter search methods such as CMAES [11] and Nelder-Mead,
all of which optimize Loss 1. We also compare against a surrogate supervised
learning method [4] which optimizes Loss 2.

Appendix D contains more details on the experimental setup, factor graph
models, and results.

Figure 4: Synthetic Navigation Datasets: Comparison of LEO against baselines.
Each dataset N1-N4 consists of $50$ varying trajectories with a 30/20
train/test split. Translational and rotational trajectory tracking errors
represent average RMSE over the test set. fevals is number of graph optimizer
calls and is computed per data point during training. Runtime $\sim$$10$ms per
optimizer call for a 900-dim state space (3-dof * 300 steps).

###  5.2 End-to-end Tracking Errors

Fig. 4 shows final tracking results for navigation datasets. We initialize LEO
with random $\theta$ such that the graph optimizer returns trajectories far
from the ground truth. At the final iteration, LEO converges to a $\theta$
that matches the optimized trajectories to ground truth. We summarize
quantitative performance against baselines minimizing final tracking loss
(Loss 1). LEO consistently outperforms baselines on final rmse tracking
errors. The difference is more pronounced for datasets N3, N4 where the
varying covariance model has a higher dimensional parameter space.

Fig. 5 shows final tracking results for real-world planar pushing datasets.
Similar to navigation, we show LEO converging to ground truth trajectory for
two different objects, a rectangle and ellipse, with differing local contact
patches. We report quantitative performance on datasets P1, P2. LEO
outperforms different hand-tuned choices of covariances as used in prior work
[4].

Figure 5: Real-world Planar Pushing Datasets: Each dataset P1, P2 consists of
$15$ varying trajectories with a 10/5 train/test split for rect, ellipse
object respectively. Trajectory tracking errors represent average RMSE over
the test set. Runtime $\sim$ $50$ms per optimizer call for a 900-dim state
space (3-dof * 300 steps).

###  5.3 Sample Efficiency and Convergence

Black-box hyper-parameter search is sample inefficient and does not scale well
with increasing dimensionality. LEO, on the other hand, is efficient as it
uses gradients. This can be seen in Fig. 4, where LEO consistently has lower
graph optimizer calls (fevals). Moreover, LEO can also be used to learn models
with larger parameter spaces, e.g. Fig. 7 learns parameters for a tactile
factor neural network mapping keypoint image features to transformations (Fig.
3).

Finally, Fig. 6 shows LEO consistently converging for varying $\theta$
initializations on navigation dataset N1. We further show how the LEO samples
evolve over iterations. Initially, when the cost is incorrect, the samples are
more spread out. Over iterations, as the cost converges, the samples
concentrate about the ground truth trajectory.

Figure 6: (a) LEO convergence for varying initial $\theta$ on dataset N1. (b)
Evolution of graph optimizer trajectory samples over LEO iterations. Figure 7:
(a) Analysis showing that a surrogate loss, independent of the graph
optimizer, may not minimize end-to-end tracking errors. (b) Minimizing only
surrogate loss results in a higher final tracking error than minimizing the
LEO loss.

###  5.4 Surrogate Loss Analysis

We further analyze why a surrogate loss (Loss 2), independent of the graph
optimizer, may not minimize end-to-end tracking errors. We train a network on
surrogate supervised learning loss to directly map a pair of tactile image
features to ground truth poses similar to prior work [4]. Fig. 7(b) shows the
per time step surrogate loss on two trajectories — an initial trajectory that
deviates from ground truth, and a final converged trajectory. While the
tracking errors of final is much less than initial, the surrogate loss does
not strictly improve. This is further reflected in the table in Fig. 7(c)
where the surrogate loss baseline has higher final tracking errors over LEO
for dataset P1.

##  6 Conclusion

We presented an approach, LEO, for learning observation models end-to-end with
graph optimizers that may be non-differentiable. We compared LEO against
baselines for different observation model types on two distinct tasks of
navigation and real-world planar pushing. Overall, LEO is able to learn
observation models with lower errors and fewer samples. While we investigated
LEO with non-differentiable optimizers in this paper, as future work, we would
like to apply LEO to differentiable optimizers and compare against alternate
solutions such as unrolling (see Appendix A for a more detailed discussion).
We would also like to scale to more complex observation models with
applications to 3D object tracking using tactile sensors.

## Acknowledgements

This work was supported through the FRAIM program. We thank Frank Dellaert for
insightful feedback and suggestions on the paper.

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## Appendix

##  Appendix A Applying LEO to Differentiable Optimizers

We discuss here how LEO can be applied to differentiable optimization
libraries. It requires two simple changes, (a) disable gradients from the
inner loop optimizer block, and (b) replace the direct final tracking loss
(Loss 1) with an energy-based loss (Loss 3). To illustrate how LEO performs
compared to alternate solutions such as unrolling, we choose the uni-
dimensional regression task from [40]. This enables us to visualize the
learned 2D energy surfaces.

Setup In the regression task, the goal is to learn a network that maps points
$(x,y)$ to an energy value, given samples from a ground truth function. The
ground truth data is generated from the function $g(x)=x\sin(x)$ for
$x\in[0,2\pi]$. We model $P(y|x)\propto\exp\\{-E(\theta,y;x)\\}$ where
$E(\theta,y;x)=||f(\theta,y;x)||^{2}_{2}$. $E(.)$ is the energy function
expressed as a sum-of-squares of $f(\theta,y;x)$, similar to Section 3.1, to
apply Gauss-Newton solvers. $f(\theta,y;x)$ is modeled as a neural network
with $\theta$ as the learnable network parameters.

LEO minimizes an energy-based loss (Loss 3), i.e.
$\mathcal{L}(E(\theta,\cdot);y_{gt}^{i})$, resulting in a similar update rule
of the form,

$\begin{split}\theta^{+}\leftarrow\theta-\frac{1}{|\mathcal{D}|}\sum^{|\mathcal{D}|}_{i=1}\left\\{{\nabla_{\theta}E(\theta;y^{i}_{gt})-\frac{1}{S}\sum_{\hat{y}}\nabla_{\theta}E(\theta;\hat{y}^{i})}\right\\}\end{split}$
(11)

We solve for this update using the Adam optimizer in Pytorch. We use a batch
Gauss-Newton solver as our underlying inner loop optimizer (LEO GN).

The unrolling baselines minimize a final tracking loss (Loss 1), i.e.
$\mathcal{L}(\theta;y_{gt})=\nicefrac{{1}}{{|\mathcal{D}|}}\sum_{i}||\hat{y}^{i}\ominus
y^{i}_{gt}||^{2}_{2}$, where $\hat{y}:=\underset{y}{\operatorname{argmin}}\
||f(\theta,y;x)||^{2}_{2}$. We use both unrolled gradient descent (Unrolled
GD) and unrolled Gauss-Newton (Unrolled GN) [8].

For all methods, we use two initialization schemes for the underlying
optimizer. The first scheme initializes from $y=0$ for all $x$, as followed in
[40]. The second scheme initializes from the ground truth $g(x)=x\sin(x)$ as
suggested in [8].

Figure 8: Evolution of learned 2D energy surfaces using initialization scheme
1. Contour surfaces show the normalized energy surfaces. Ground truth function
$x\sin(x)$ in green, LEO samples in orange. Lighter colors correspond to lower
energy. Figure 9: Evolution of learned 2D energy surfaces using initialization
scheme 2. Contour surfaces show the normalized energy surfaces. Ground truth
function $x\sin(x)$ in green, LEO samples in orange. Lighter colors correspond
to lower energy.

Results Fig. 8 shows the learned energy surfaces for all 3 methods using
initialization scheme 1, i.e., initializing the optimizer from zero. At
iteration $0$, all 3 approaches are initialized with a random energy function.
While all 3 approaches end up with a low final loss, only LEO converges to an
energy function with a single basin about the ground truth samples. The
unrolled baselines converge to energy functions with multiple minima, not all
of them on the ground truth samples. This is because the energy function need
only be good enough for the optimizer, when initialized from $y=0$ to reach
the ground truth for a fixed number of unrolling steps. It does not guarantee
a similar convergence when executed with different number of unrolling steps
or initialization from different $y$ values.

Fig. 9 shows the learned energy surfaces for initialization scheme 2, i.e.,
initializing the optimizer from ground truth. Similar to the previous figure,
all 3 methods initialize from random energy functions and converge to a low
final loss. But only LEO converges to a similar energy function as before,
i.e., one with a single basin around the ground truth. The unrolling baselines
converge to a different energy function than in scheme 1. In this case, the
energy function need only be good enough for the optimizer, when initialized
from ground-truth, to stay on the ground truth.

For both schemes, LEO alone converges to the same energy function, as it is
less sensitive to the underlying optimization process. In future work, we aim
to apply LEO on differentiable optimizers for robot estimation tasks. This
would require a factor graph optimization library similar to GTSAM but
implemented in a differentiable manner. There are recent efforts on this front
[8, 68, 69], but there isn’t a single, stable open-source library yet. We hope
our analysis on this regression task would encourage others in the community
to try out LEO in their respective differentiable optimizer libraries.

##  Appendix B Comparison to a MCMC sampler

Figure 10: Runtime comparison against an HMC sampler

We compare run times against an HMC sampler implemented in the hamiltorch
library [70]. The sampler simulates a set of Hamiltonian differential
equations which upon simulating generates a sequence of samples.

We run this on the 2D navigation dataset which have a 900-dimensional state
space (3-DOF * 300 steps). The HMC sampler is sampling from the true posterior
distribution while the incremental Gauss-Newton, iSAM2 [10], sampler that we
use samples from a Gaussian approximation that it maintains online. However,
we found similar convergence at train time using both samplers which suggests
that the Gaussian approximation was reasonable for our applications.

In terms of run time, we found the HMC sampler run time to be two orders of
magnitude greater than the iSAM2 sampler as shown in Fig. 10. This is expected
since the HMC sampler evaluates the cost as well the gradients of the cost
every time a sample is generated which requires looping through all the
factors and can be expensive. In contrast, the iSAM2 sampler allows for
directly sampling from the Gaussian that is significantly faster. When used in
an online setting, as new factors get added to the graph, the HMC sampler
would have to recompute the posterior distribution. On the other hand, the
iSAM2 sampler efficiently and incrementally updates the Gaussian approximation
by leveraging sparsity.

##  Appendix C Principle of Maximum Entropy

In Section 4.1, we defined the energy-based loss as the normalized negative
log-likelihood loss. The motivation for such a loss comes from probabilistic
modeling, notably the principle of maximum entropy moment matching.

Figure 11: Ground truth distribution is unknown and only observable via
measurements $z_{1:t}$. The goal is to estimate a distribution $P(x|z)$ that
matches observed moments.

Assume a ground truth distribution $P_{gt}(x,z)$ that generates x and z. We
would like to estimate a distribution $P_{\theta}(x|z)$ that matches this. The
true ground truth distribution is unknown, but we can sample from it. This
allows us to estimate $P_{\theta}(x|z)$ by matching empirical moments computed
from these samples (Fig. 11).

However, there is a continuous space of probability distributions that match
these empirical moments. Which of these distributions do we pick? The maximum
entropy (MaxEnt) principle suggests we pick the distribution which is least
committal, i.e. the one with maximum entropy [71]. We can write out this
objective of maximizing entropy of the distribution $P_{\theta}(x|z)$ subject
to matching moment constraints against the unknown ground truth distribution
as,

$\begin{split}&\operatorname{max}\ -\sum\limits_{x}\ P_{\theta}(x|z)\log
P_{\theta}(x|z)\\\ s.t.\underset{{{\begin{subarray}{c}z\sim
P_{gt}\end{subarray}(z)}}}{\operatorname{E}}&\underset{{{\begin{subarray}{c}x\sim
P_{\theta}(x|z)\end{subarray}}}}{\operatorname{E}}\Phi(x,z)=\underset{{{\begin{subarray}{c}(x,z)\sim
P_{gt}\end{subarray}(x,z)}}}{\operatorname{E}}\Phi(x,z)\end{split}$ (12)

where, $\Phi(\cdot)$ are the moment functions. Applying KKT conditions for Eq.
LABEL:eq:appx:eqA.1 and expanding the dual results in an unconstrained
objective of the form [71, 72] (more details in Eq.
LABEL:eq:end2end_learnopt:appx:moments:eq1),

$\begin{split}&\underset{{\theta}}{\operatorname{max}}\
\underset{{\begin{subarray}{c}x,z\sim\\\
P_{gt}(x,z)\end{subarray}}}{\operatorname{E}}\ \log P_{\theta}(x|z)\\\
&P_{\theta}(x|z)\propto\exp(-\theta^{T}\Phi(x,z))\end{split}$ (13)

where, $\theta$ are the Lagrange multipliers. The term
$\theta^{T}\Phi(x,z)=E(\theta,x;z)$ can be interpreted as a linear “energy
function” such that $P_{\theta}(x)\propto\exp(-E(\theta,x;z))$ puts a high
energy on low probability solutions. This linear class of cost functions can
be lifted to a richer class of parametric nonlinear functions $E(\theta,x;z)$.
Note that this may induce a duality gap in optimizing Eq. LABEL:eq:appx:eqA.1,
but is commonly used in practice when parameterizing energy functions as
neural networks [14, 73, 74]. The maximum entropy solution can then be written
out as,

$\begin{split}&\underset{{\theta}}{\operatorname{max}}\
\underset{{\begin{subarray}{c}x,z\sim\\\
P_{gt}(x,z)\end{subarray}}}{\operatorname{E}}\ \log P_{\theta}(x|z)\\\
&P_{\theta}(x|z)\propto\exp(-E({\theta},x;z))\end{split}$ (14)

Eq. LABEL:eq:appx:eqA.3 asks us to maximize the likelihood of ground truth
trajectories where the likelihood takes the form of a Boltzmann distribution,
where lower energy implies higher probability mass. In other words, we want to
find energy function $E({\theta},x;z)$ that places maximal probability mass on
the ground truth trajectory. Writing out this objective over a training
dataset $(x_{gt},z)\in\mathcal{D}$ results in the NLL loss that we saw earlier
in Section 4.1,

$\begin{split}\mathcal{L}(\theta)&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}-\log
P_{\theta}(x_{gt}^{i}|z^{i})\\\ \end{split}$ (15)

Let’s expand out the steps in Section 4.1 in more detail. Substituting the
expression for Boltzmann distribution $P_{\theta}(x|z)$ results in,

$\begin{split}\mathcal{L}(\theta)&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}-\log(\exp(-E(\theta;x_{gt}^{i},z^{i})))+\log
Z(\theta;z^{i})\\\
\mathcal{L}(\theta)&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}E(\theta;x_{gt}^{i},z^{i})+\log\int_{x}\exp(-E(\theta;x,z^{i}))dx\end{split}$
(16)

Take the gradient of this loss,

$\begin{split}\nabla_{\theta}\mathcal{L}(\theta)&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}\left[\nabla_{\theta}E(\theta;x_{gt}^{i},z^{i}))+\frac{1}{Z(\theta;z^{i})}\int_{x}\nabla_{\theta}E(\theta;x,z^{i})\exp(-E(\theta;x,z^{i}))\right]\\\
\end{split}$ (17)

Substituting in $P_{\theta}(x|z^{i})$ from Section 3.1 we have the resulting
gradient expression that we finally obtained in Section 4.1,

$\begin{split}\nabla_{\theta}\mathcal{L}(\theta)=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}\left[\underbrace{\nabla_{\theta}E(\theta;x_{gt}^{i},z^{i}))}_{\text{ground
truth samples}}-\underset{\begin{subarray}{c}x\sim\\\
P_{\theta}(x|z)\end{subarray}}{\operatorname{\mathbb{E}}}\underbrace{\nabla_{\theta}E(\theta;x,z^{i}))}_{\text{learned
distribution samples}}\right]\end{split}$ (18)

Negative log likelihood approximation

Since directly sampling from the true posterior
$P_{\theta}(x|z)=\frac{1}{\mathcal{Z}(\theta)}e^{-E(\theta,x;z)}$ is
intractable, in LEO, we instead sample from a Gaussian approximation of this
distribution, $\hat{P}_{\theta}(x|z)=\mathcal{N}(\mu,\Sigma)$ where $\mu$,
$\Sigma$ are obtained as shown in Section 4.3. Hence, we effectively minimize
an approximation of the true NLL loss,

$\begin{split}\hat{\mathcal{L}}(\theta)&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}-\log\hat{P}_{\theta}(x_{gt}^{i}|z^{i})\end{split}$
(19)

The difference between the approximated loss and the true loss is,

$\begin{split}\hat{\mathcal{L}}(\theta)-\mathcal{L}(\theta)&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}-\log\hat{P}_{\theta}(x_{gt}^{i}|z^{i})-\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}-\log{P}_{\theta}(x_{gt}^{i}|z^{i})\\\
&=\frac{1}{|\mathcal{D}|}\sum_{(x_{gt}^{i},z^{i})\in\mathcal{D}}\frac{P_{\theta}(x_{gt}^{i}|z^{i})}{\hat{P}_{\theta}(x_{gt}^{i}|z^{i})}\\\
&=\underset{\begin{subarray}{c}(x_{gt},z)\sim\mathcal{D}\end{subarray}}{\operatorname{\mathbb{E}}}\
\ \frac{P_{\theta}(x_{gt}|z)}{\hat{P}_{\theta}(x_{gt}|z)}\
\leq\max_{\begin{subarray}{c}(x_{gt},z)\in\mathcal{D}\end{subarray}}\
\log\frac{P_{\theta}(x_{gt}|z)}{\hat{P}_{\theta}(x_{gt}|z)}\end{split}$ (20)

The difference in losses is bounded by the maximum log density ratio between
the true and approximated distributions. The ratio is bounded as long as the
denominator is not zero, i.e. the approximation has full support over the
dataset.

Derivation from moment matching

When going from Eq. LABEL:eq:appx:eqA.1 to LABEL:eq:appx:eqA.2, we rewrote the
constrained maximum entropy objective as an unconstrained objective. Let’s
derive this for our setting [71]. For simplicity of notations, we’ll drop the
conditional $z$ in the probability expressions.

We would like to maximize (or minimize negative) entropy subject to matching
moments, i.e.,

$\begin{split}&\min_{x}\sum_{x}P_{\theta}(x)\log P_{\theta}(x)\\\ \text{s.t.,
}\ \ \ \ \ &\sum_{x}P_{\theta}(x)\Phi(x)=\sum_{x}P_{gt}(x)\Phi(x)=c\hskip
10.0pt\textit{(moment constraint)}\\\ &\sum_{x}P_{\theta}(x)=1\hskip
90.0pt\textit{(probability sum constraint)}\\\ \end{split}$ (21)

Writing out the Lagrangian for this constrained problem,

$\begin{split}\mathcal{L}(\theta,\lambda,\mu)=\sum_{x}P_{\theta}(x)\log
P_{\theta}(x)+\lambda\left(\sum_{x}P_{\theta}(x)\Phi(x)-c\right)+\mu\left(\sum_{x}P_{\theta}(x)-1\right)\end{split}$
(22)

where, $\lambda$, $\mu$ are the Lagrange multipliers. Taking gradient of
Lagrangian and setting to $0$,

$\begin{split}&\hskip
50.0pt\nabla_{\theta}\mathcal{L}(\theta,\lambda,\mu)=0\\\
\Rightarrow&\sum_{x}\cancel{{P_{\theta}(x)}}\frac{1}{\cancel{P_{\theta}(x)}}\nabla
P_{\theta}(x)+\sum_{x}\log P_{\theta}(x)\nabla_{\theta}P_{\theta}(x)+\\\
&\hskip
50.0pt\lambda\sum_{x}\nabla_{\theta}P_{\theta}(x)\Phi(x)+\mu\sum_{x}\nabla_{\theta}P_{\theta}(x)=0\\\
\Rightarrow&\sum_{x}\nabla P_{\theta}(x)(1+\log
P_{\theta}(x)+\lambda\Phi(x)+\mu)=0\end{split}$ (23)

Eq. LABEL:eq:end2end_learnopt:appx:moments:eq3 goes to $0$ for a probability
distribution $P_{\theta}(x)$ of the form,

$\begin{split}P_{\theta}(x)=\exp^{-(1+\mu+\lambda\Phi(x))}=\frac{1}{{Z}(\theta,\lambda)}\exp(-\lambda\Phi(x))\end{split}$
(24)

where, ${Z}(\theta,\lambda)=\sum_{x}\exp(-\lambda\Phi(x))$ such that
$\sum_{x}P_{\theta}(x)=1$. We can subsume $1+\mu$ since that’s a constant.

Writing out the dual form $\max_{\lambda}\mathcal{L}(\theta,\lambda,\mu)$ of
the Lagrangian with $P_{\theta}(x)$ expression substituted from Eq. 24,

$\begin{split}\hskip 30.0pt\max_{\lambda}\ \
&\sum_{x}\frac{1}{Z(\theta,\lambda)}\exp\left(-\lambda\Phi(x)\right)\log\left[\frac{1}{Z(\theta,\lambda)}\exp(-\lambda\Phi(x))\right]+\\\
&\lambda\left[\sum_{x}\frac{1}{Z(\theta,\lambda)}\exp(-\lambda\Phi(x))\Phi(x)-c\right]+\mu\cancel{\left[\sum_{x}\frac{1}{Z(\theta,\lambda)}\exp(-\lambda\Phi(x))\Phi(x)-1\right]}\\\
\Rightarrow\hskip 30.0pt\max_{\lambda}\ \
&\cancel{\sum_{x}\frac{1}{Z(\theta,\lambda)}\exp\left(-\lambda\Phi(x)\right)\left[-\lambda\Phi(x)\right]}+\sum_{x}\frac{1}{Z(\theta,\lambda)}\exp\left(-\lambda\Phi(x)\right)\left[-\log
Z(\theta,\lambda)\right]+\\\
&\cancel{\lambda\sum_{x}\frac{1}{Z(\theta,\lambda)}\exp\left(-\lambda\Phi(x)\right)\Phi(x)}-\lambda
c\\\ \Rightarrow\hskip 30.0pt\max_{\lambda}\ \ &-\log
Z(\theta,\lambda)\left(\sum_{x}\frac{1}{Z(\theta,\lambda)}\exp\left(-\lambda\Phi(x)\right)\right)-\lambda
c\\\ \Rightarrow\hskip 30.0pt\max_{\lambda}\ \ &-\log
Z(\theta,\lambda)-\lambda c\end{split}$ (25)

Introducing true distribution $P_{gt}(x)$ and rearranging terms results in the
unconstrained objective that we saw in Eq. LABEL:eq:appx:eqA.2,

$\begin{split}&\max_{\lambda}\ \ -\log Z(\theta,\lambda)-\lambda c\\\
\Rightarrow\hskip 30.0pt&\max_{\lambda}\ \ \sum_{x}P_{gt}(x)\left[-\log
Z(\theta,\lambda)\right]-\lambda\sum_{x}P_{gt}(x)\Phi(x)\\\ \Rightarrow\hskip
30.0pt&\max_{\lambda}\ \ \sum_{x}P_{gt}(x)\left[-\log
Z(\theta,\lambda)-\lambda\Phi(x)\right]\\\ \Rightarrow\hskip
30.0pt&\max_{\lambda}\ \
\sum_{x}P_{gt}(x)\log\left[\exp\left(\frac{-\lambda\Phi(x)}{Z(\theta,\lambda)}\right)\right]\\\
\Rightarrow\hskip 30.0pt&\max_{\lambda}\ \ \sum_{x}P_{gt}(x)\log
P_{gt}(x)\end{split}$ (26)

##  Appendix D Experimental Evaluation: Additional Details

###  D.1 Setup and Factor Graph Models

Fig. 12 shows the two factor graphs modeling the (a) synthetic navigation and
(b) real-world planar pushing tasks. For both graphs, poses are modeled as
variable nodes and measurements as factor nodes. The graph inference objective
is to solve for the latent poses (variables) given measurements (factors).

Figure 12: Factor graphs for (a) synthetic navigation (b) real-world planar
pushing.

Synthetic Navigation To replicate a typical robot navigation setup, we
generate odometry measurements by adding Gaussian noise to relative ground
truth state estimates and GPS measurements by adding Gaussian noise to
absolute ground truth estimates. We generate 4 datasets N1–N4, each with a
different covariance setting. N1, N2 has measurements generated from fixed
covariances $\Sigma:=\\{\Sigma_{odom},\Sigma_{gps}\\}$. N3, N4 has
measurements generated from covariances varying between two sets of values,
i.e. $\Sigma:=\\{\Sigma_{odom}(z),\Sigma_{gps}(z)\\}$, where $z$ are binary
measurements simulating an ambient light detector that determines whether the
robot is in an indoor or outdoor environment.

To solve for the sequence of states $x:=\\{x_{1}\dots x_{T}\\}$ from
measurements $z$, we provide the following objective for the graph optimizer
to minimize,

$\begin{split}\hat{x}=&\underset{x}{\operatorname{argmin}}\sum_{k}\left\\{\right.||f_{odom}(x_{k-1},x_{k})-z^{odom}_{k-1,k}||^{2}_{\Sigma_{odom}(z)}+||f_{gps}(x_{k})-z^{gps}_{k}||^{2}_{\Sigma_{gps}(z)}\left.\right\\}\end{split}$
(27)

The observation model parameters that we learn for this task are the fixed and
varying covariances, i.e.
$\theta:=\\{\Sigma_{odom},\Sigma_{gps}\\},\\{\Sigma_{odom}(z),\Sigma_{gps}(z)\\}$.

Real-world planar pushing For the planar pushing setup, states
$x:=\\{x_{1}\dots x_{T}\\}$ in the graph are the planar object and end-
effector poses at every time step, with $x_{t}=[o_{t}\ e_{t}]^{T}$, where
$o_{t},e_{t}\in SE(2)$. Factors in the graph incorporate tactile observations
$f_{tac}(\cdot)$, quasi-static pushing dynamics $f_{qs}(\cdot)$, geometric
constraints $f_{geo}(\cdot)$, and priors on end-effector poses
$f_{eff}(\cdot)$.

To solve for the sequence of state $x:=\\{x_{1}\dots x_{T}\\}$ from
measurements $z$, we pass in the following objective for the graph optimizer
to minimize,

$\begin{split}\hat{x}=\underset{x}{\operatorname{argmin}}\sum_{k}&\left\\{\right.||f^{\phi}_{tac}(o_{k-w},o_{k},e_{k-w},e_{k})||^{2}_{\Sigma_{tac}}+||f_{qs}(o_{k-1},o_{k},e_{k-1},e_{k})||^{2}_{\Sigma_{qs}}+\\\
&||f_{geo}(o_{k},e_{k})||^{2}_{\Sigma_{geo}}+||f_{eff}(e_{k})||^{2}_{\Sigma_{eff}}\left.\right\\}\end{split}$
(28)

The observation model parameters that we learn for this task are the tactile
factor network weights and the tactile and quasi-static factor covariances,
i.e. $\theta:=\\{\phi,\Sigma_{tac},\Sigma_{qs}\\}$. Tactile factor network
contains an image to keypoint feature encoder pre-trained using a self-
supervised image reconstruction loss and a fully connected network mapping
keypoint features to relative transforms. We fine tune weights for the
keypoint to relative transform portion of the network and use fixed, pre-
trained weights for feature encoder. Tactile factors are added between non-
consecutive poses $\\{k-w,k\\}$ over a window with the window length chosen
heuristically, and quasi-static motion factors are added between consecutive
poses $\\{k-1,k\\}$ (Fig. 12). More details on the factor graph models along
with the network architecture can be found in prior work [4].

Figure 13: Qualitative LSTM results on (a) a synthetic navigation robot
trajectory (b) planar pushing object trajectories.

###  D.2 Baselines

For the hyper-parameter search baselines we use off-the-shelf solvers like
CMA-ES [11], and scipy optimizers such as Nelder-Mead. For the learned
sequence model baseline, we use an LSTM.

LSTM architecture For the synthetic navigation task, we directly regresses to
absolute poses. At each time step, the absolute pose and odometry measurements
are concatenated as inputs, and the network predicts the 2D position and the
sin and cos of the rotation angle. We use a 2-layer LSTM with 64 hidden units,
followed by a 2 dense layers of 32 hidden units before decoding into the 4D
output. The loss is a weighted sum of translation and rotation errors. In the
real-world planar pushing datasets, we do not have direct measurements of the
object pose. Therefore, we transform all end-effector trajectories to start at
the origin, which improves the generalization. Each input consists of the end-
effector pose observation and the 2d object contact observation, and the same
architecture as the navigation dataset is used. In addition, we found that
increasing the sequence length with each epoch improved performance.

Fig. 13 shows qualitative results for the LSTM on the (a) synthetic navigation
and (b) real-world planar pushing datasets. The performance on the planar
pushing object trajectories is not as good since, unlike navigation where we
had noisy GPS measurements, tactile measurements only give partially
observable information about the object state.


EPJ Web of Conferences CONF12

11institutetext: Department of Physics, University of Cyprus, CY-1678 Nicosia,
Cyprus

# One-loop calculations in Supersymmetric Lattice QCD

M. Costa 11<EMAIL_ADDRESS>H. Panagopoulos 11

###### Abstract

We study the self energies of all particles which appear in a lattice
regularization of supersymmetric QCD (${\cal N}=1$). We compute,
perturbatively to one-loop, the relevant two-point Green’s functions using
both the dimensional and the lattice regularizations. Our lattice formulation
employs the Wilson fermion acrion for the gluino and quark fields. The gauge
group that we consider is $SU(N_{c})$ while the number of colors, $N_{c}$ and
the number of flavors, $N_{f}$, are kept as generic parameters. We have also
searched for relations among the propagators which are computed from our one-
loop results. We have obtained analytic expressions for the renormalization
functions of the quark field ($Z_{\psi}$), gluon field ($Z_{u}$), gluino field
($Z_{\lambda}$) and squark field ($Z_{A_{\pm}}$).

We present here results from dimensional regularization, relegating to a
forthcoming publication HP:2016  our results along with a more complete list
of references. Part of the lattice study regards also the renormalization of
quark bilinear operators which, unlike the non-supersymmetric case, exhibit a
rich pattern of operator mixing at the quantum level.

##  1 Introduction

In recent years, methods of extracting nonperturbative information for
Supersymmetric Theories through lattice simulations are being studied
extensively Feo ; Catterall:2011 ; Kaplan:2009 , from a number of viewpoints.
There are a number of important physical questions regarding SUSY to be
ultimately addressed on the lattice, such as the nature of SUSY breaking, and
the phase diagram of SUSY models. Such questions have become increasingly
relevant in recent years, in the context of investigations of BSM Physics.
Many notorious problems arise when formulating SUSY models on the lattice
Catterall:2014 , such as the emergence of a plethora of counterterms in the
action and the need for fine-tuning of masses and coupling constants
Jack&Jones ; Taniguchi:2000 ; Giet&Poppitz . The present work investigates
these problems using, as a representative nontrivial model, supersymmetric
${\cal N}=1$ Quantum Chromodynamics (SQCD).

The fundamental fields involved in the construction of the SQCD Lagrangian are
well-known and we present them below for completeness. There are two types of
fields, chiral superfields and vector superfields Weinberg:1980bf ;
Wess&Bagger:1992 . The physical components of a chiral superfield $\Phi$, are
the matter fields: $A$ which represents a scalar boson (squark), $\psi$ which
is a two-component spinor (quark - spin $\frac{1}{2}$) and $F$ which is an
auxiliary complex scalar field. In superspace notation ($x$: spacetime
coordinates, $\theta$/$\bar{\theta}$: anticommuting coordinates) the chiral
superfield $\Phi$ in terms of the above component fields is:

$\displaystyle\Phi(x,\theta,\bar{\theta})$ $\displaystyle=$ $\displaystyle
A(x)+\sqrt{2}\,\theta\psi(x)+\theta\theta\,F(x)+i\theta\,\sigma^{\mu}\,\bar{\theta}\,\partial_{\mu}A(x)$
$\displaystyle+$
$\displaystyle\frac{i}{\sqrt{2}}\theta\theta\,\bar{\theta}\,\bar{\sigma}^{\mu}\,\partial_{\mu}\psi(x)+\frac{1}{4}\;\theta\theta\,\bar{\theta}\bar{\theta}\,\partial_{\mu}\partial^{\mu}A(x)$

The general form of a vector superfield $V(x,\theta,\bar{\theta})$ is:

$\displaystyle V(x,\theta,\bar{\theta})$ $\displaystyle=$ $\displaystyle
C(x)+i\theta\chi(x)-i\bar{\theta}\bar{\chi}(x)+\frac{i}{2}\theta\theta\,\big{[}M(x)\,+\,iN(x)\big{]}-\frac{i}{2}\bar{\theta}\bar{\theta}\big{[}M(x)\,-\,iN(x)\big{]}$
$\displaystyle-$
$\displaystyle\theta\,\sigma^{\mu}\,\bar{\theta}\,u_{\mu}(x)+i\theta\theta\,\bar{\theta}\left[\bar{\lambda}(x)\,+\,\frac{i}{2}\bar{\sigma}^{\mu}\partial_{\mu}\chi(x)\right]-i\bar{\theta}\bar{\theta}\,\theta\left[\lambda(x)\,+\,\frac{i}{2}\sigma^{\mu}\partial_{\mu}\bar{\chi}(x)\right]$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\theta\theta\,\bar{\theta}\bar{\theta}\,\left[D(x)\,+\,\frac{1}{2}\partial_{\mu}\partial^{\mu}C(x)\right]\
.$

We can choose a special gauge where the components $C,\chi,M,N$ are zero. This
defines the Wess$-$Zumino (WZ) gauge. A vector superfield in Wess$-$Zumino
gauge reduces to the form:

$V(x,\theta,\bar{\theta})=-\theta\,\sigma^{\mu}\,\bar{\theta}\,u_{\mu}(x)+i\theta\theta\,\bar{\theta}\bar{\lambda}(x)-i\bar{\theta}\bar{\theta}\,\theta\lambda(x)+\
\frac{1}{2}\,\theta\theta\,\bar{\theta}\bar{\theta}\,D(x)\ .$ (3)

where $u_{\mu}=u_{\mu}^{(\alpha)}T^{(\alpha)}$ is the gluon field,
$\lambda=\lambda^{(\alpha)}T^{(\alpha)}$ is the gluino field and
$D=D^{(\alpha)}T^{(\alpha)}$ is an auxiliary real scalar field; $(\alpha)$ is
a color index in the adjoint representation of the gauge group, and the
generators $T^{(\alpha)}$ satisfy ${\rm
Tr}(T^{\alpha}T^{\beta})=1/2\,\delta^{\alpha\,\beta}$.

The minimal content of the SQCD Lagrangian, in order to contain gluons and
4-component massive quarks is two chiral super$-$multiplets
$\Phi_{+}=\\{A_{+},\psi_{+},F_{+}\\}$ and
$\Phi_{-}=\\{A_{-},\psi_{-},F_{-}\\}$ and a vector super$-$multiplet
$V=\\{u_{\mu},\lambda,D\\}$. We combine the components $\psi_{+}$, $\psi_{-}$
to create Dirac spinors. In order to obtain a renormalizable theory, we need
to construct a Lagrangian with products of superfields having dimensionality
$\leq 4$; in addition, we require Lorentz invariance as well as invariance
under supersymmetric gauge transformations:

$\Phi^{\prime}_{+}=e^{-i\Lambda}\Phi_{+}\,,\,\,\,\,\Phi^{\prime}_{-}=\Phi_{-}e^{i\Lambda},\,\,\,\,e^{V^{\prime}}=e^{-i\Lambda^{\dagger}}e^{V}e^{i\Lambda},$
(4)

where $\Lambda$ is an arbitrary chiral superfield. The Lagrangian takes the
form:

${\cal L}=\frac{1}{8\,g}{\rm
Tr}\big{(}W^{\alpha}\,W_{\alpha}|_{\theta\theta}+\bar{W}_{\dot{\alpha}}\,\bar{W}^{\dot{\alpha}}|_{\bar{\theta}\bar{\theta}}\big{)}+\big{(}\Phi^{\dagger}_{+}\,e^{V}\,\Phi_{+}+\Phi_{-}\,e^{-V}\,\Phi^{\dagger}_{-}\big{)}|_{\theta\theta\bar{\theta}\bar{\theta}}+m\big{(}\Phi_{-}\Phi_{+}|_{\theta\theta}+\Phi^{\dagger}_{+}\Phi^{\dagger}_{-}|_{\bar{\theta}\bar{\theta}}\big{)}$
(5)

where $W_{\alpha}=-\frac{1}{4}\bar{{\cal D}}\bar{{\cal D}}\,e^{-V}\,{\cal
D}_{\alpha}\,e^{V}$ is the supersymmetric field strength and the
supersymmetric covariant derivative is defined as: ${\cal
D}_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}\ +\
i\sigma^{\mu}_{\alpha\dot{\alpha}}\,\bar{\theta}^{\dot{\alpha}}\,\partial_{\mu}\,\,,\bar{{\cal
D}}_{\dot{\alpha}}=-\frac{\partial}{\partial\bar{\theta}^{\dot{\alpha}}}-\
i\theta^{\alpha}\,(\sigma^{\mu})_{\alpha\dot{\alpha}}\,\partial_{\mu}$.
Combining the components of $\Phi_{+}$ with $\Phi_{-}$ we can construct a 4
component Dirac Spinor ($\psi_{D}$). Upon functionally integrating over the
auxiliary fields, rescaling $V\rightarrow 2\,g\,V$ and after a Wick rotation,
the form of the Euclidean action in 4 dimensions in Dirac notation ${\cal
S}^{E}_{\rm SQCD}$ is:

$\displaystyle{\cal S}^{E}_{\rm SQCD}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\Big{[}\frac{1}{4}u_{\mu\nu}^{(\alpha)}u_{\mu\nu}^{(\alpha)}+\frac{1}{2}\bar{\lambda}^{(\alpha)}_{M}\gamma^{E}_{\mu}{\cal{D}}_{\mu}\lambda^{(\alpha)}_{M}$
(6) $\displaystyle+$
$\displaystyle{\cal{D}}_{\mu}A_{+}^{\dagger}{\cal{D}}_{\mu}A_{+}+{\cal{D}}_{\mu}A_{-}{\cal{D}}_{\mu}A_{-}^{\dagger}+\bar{\psi}_{D}\gamma^{E}_{\mu}{\cal{D}}_{\mu}\psi_{D}$
$\displaystyle+$ $\displaystyle
i\sqrt{2}g\big{(}A^{\dagger}_{+}\bar{\lambda}^{(\alpha)}_{M}T^{(\alpha)}P_{+}^{E}\psi_{D}-\bar{\psi}_{D}P_{-}^{E}\lambda^{(\alpha)}_{M}T^{(\alpha)}A_{+}+A_{-}\bar{\lambda}^{(\alpha)}_{M}T^{(\alpha)}P_{-}^{E}\psi_{D}-\bar{\psi}_{D}P_{+}^{E}\lambda^{(\alpha)}_{M}T^{(\alpha)}A_{-}^{\dagger}\big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2}g^{2}(A^{\dagger}_{+}T^{(\alpha)}A_{+}-A_{-}T^{(\alpha)}A^{\dagger}_{-})^{2}-m(\bar{\psi}_{D}\psi_{D}-mA^{\dagger}_{+}A_{+}-mA_{-}A^{\dagger}_{-})\Big{]}\,.$

where $\lambda_{M}=\left({\begin{array}[]{c}\lambda_{a}\\\
\bar{\lambda}^{\dot{a}}\end{array}}\right)$ and
$\psi_{D}^{T}=\left({\begin{array}[]{c}\psi_{+a}\\\
\bar{\psi}_{-}^{\dot{a}}\end{array}}\right)$,
$P_{\pm}^{E}=\frac{1\pm\,\gamma_{5}^{E}}{2}$ ,
$\gamma_{5}^{E}=\gamma_{0}^{E}\gamma_{1}^{E}\gamma_{2}^{E}\gamma_{3}^{E}$.
${\cal S}^{E}_{\rm SQCD}$ is invariant under supersymmetric transformations
($\xi_{M}$: Majorana spinor parameter):

$\displaystyle\delta_{\xi}A_{+}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\bar{\xi}_{M}P_{+}^{E}\psi_{D}\,,$
$\displaystyle\delta_{\xi}A_{-}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\bar{\psi}_{D}P_{+}^{E}\xi_{M}\,,$
$\displaystyle\delta_{\xi}(P_{+}^{E}\psi_{D})$ $\displaystyle=$
$\displaystyle\sqrt{2}({\cal{D}}_{\mu}A_{+})P_{+}^{E}\gamma_{\mu}^{E}\xi_{M}-\sqrt{2}mP_{+}^{E}\xi_{M}A_{-}^{\dagger}\,,$
$\displaystyle\delta_{\xi}(P_{-}^{E}\psi_{D})$ $\displaystyle=$
$\displaystyle\sqrt{2}({\cal{D}}_{\mu}A_{-})^{\dagger}P_{-}^{E}\gamma_{\mu}^{E}\xi_{M}-\sqrt{2}mA_{+}P_{-}^{E}\xi_{M}\,,$
$\displaystyle\delta_{\xi}u_{\mu}^{(\alpha)}$ $\displaystyle=$
$\displaystyle-\bar{\xi}_{M}\gamma_{\mu}^{E}\lambda^{(\alpha)}_{M},$
$\displaystyle\delta_{\xi}\lambda^{(\alpha)}_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{4}u_{\mu\nu}^{(\alpha)}[\gamma_{\mu}^{E},\gamma_{\nu}^{E}]\xi_{M}-2ig\gamma_{5}^{E}\xi_{M}(A^{\dagger}_{+}T^{(\alpha)}A_{+}-A_{-}T^{(\alpha)}A^{\dagger}_{-})\,.$
(7)

##  2 The Calculation

We calculate perturbatively a number of 2-pt Green’s functions up to one-loop,
both in the continuum and on the lattice. The quantities that we study are the
self energies of the quark ($\psi$), gluon ($u_{\mu}$), squark ($A$) and
gluino ($\lambda$) fields, as well as the 2-pt Green’s functions of the
quantities ${\cal O}_{i}(x)=\bar{\psi}(x)\Gamma_{i}\psi(x)$, using both
dimensional regularization (DR) and lattice regularization (L). The index
"$i$” refers to the possibilities of the gamma matrices: (scalar)
$\Gamma_{S}=\leavevmode\hbox{\small 1\kern-4.0pt\normalsize 1}$,
(pseudoscalar) $\Gamma_{P}=\gamma_{5}$, (vector) $\Gamma_{V}=\gamma_{\mu}$,
(axial) $\Gamma_{A}=\gamma_{5}\gamma_{\mu}$, (tensor)
$\Gamma_{T}=\gamma_{\mu}\gamma_{\nu}$.

The first step in our perturbative procedure is to calculate the 2-pt Green’s
functions in the continuum where we regularize the theory in D-dimensions
($D=4-2\,\epsilon$). The continuum calculations are necessary in order to
compute the ${\overline{\rm MS}}$-renormalized Green’s functions; these enter
the subsequent calculation of the corresponding Green’s functions using
lattice regularization and ${\overline{\rm MS}}$ renormalization. The
continuum results also provide the renormalization functions of the quark
field ($Z_{\psi}$), squark field ($Z_{A_{\pm}}$), gluon field ($Z_{u}$) and
gluino field ($Z_{\lambda}$) and of the complete set of quark bilinear
operators ${\cal{O}}_{i}$ ($Z_{i}$). For the extraction of the renormalization
functions, we applied the ${\overline{\rm MS}}$ scheme at a scale $\bar{\mu}$.
Once we have computed the renormalization functions in the ${\overline{\rm
MS}}$ scheme we can construct their RI′ counterparts using conversion factors
which are immediately extracted from our computation to the required
perturbative order. Being regularization independent, these same conversion
factors can then be also used for lattice renormalization functions.

The aforementioned renormalization functions are defined as follows:

$\displaystyle\psi^{R}=\sqrt{Z_{\psi}}\,\psi^{B},\,\,\,A^{R}_{\pm}=\sqrt{Z_{A_{\pm}}}\,A^{B}_{\pm},\,\,\,u_{\mu}^{R}=\sqrt{Z_{u}}\,u^{B}_{\mu},\,\,\,\lambda^{R}=\sqrt{Z_{\lambda}}\,\lambda^{B},\,\,\,{\cal{O}}_{i}^{R}=Z_{i}\,{\cal{O}}_{i}^{B}.$
(8)

The one-loop Feynman diagrams (one-particle irreducible (1PI)) contributing to
$\langle\psi(x)\bar{\psi}(y)\rangle$ are shown in Fig. 1, those contributing
to $\langle A(x)A^{\dagger}(y)\rangle$ in Fig. 2. One-loop Feynman diagrams
contributing to the Green’s function $\langle
u_{\mu}^{(\alpha)}(x)u_{\nu}^{(\beta)}(y)\rangle$ and
$\langle\lambda^{(\alpha)}(x)\bar{\lambda}^{(\beta)}(y)\rangle$ are shown in
Fig. 3 and Fig. 4, respectively. The Feynman diagram that enters the
calculation of the Green’s function $\langle\psi(x){\cal
O}_{i}(z)\bar{\psi}(y)\rangle$ up to one-loop is shown in Fig. 5.

Figure 1: One-loop Feynman diagrams contributing to the 2-pt Green’s function
of the quark propagator, $\langle\psi(x)\bar{\psi}(y)\rangle$. Figure 2: One-
loop Feynman diagrams contributing to the 2-pt Green’s function of the squark
propagator, $\langle A_{+}(x)A_{+}^{\dagger}(y)\rangle$. Similar diagrams
apply to the propagator of the $A_{-}$ field.  Figure 3: One-loop Feynman
diagrams contributing to the 2-pt Green’s function of the gluon propagator,
$\langle u_{\mu}^{(\alpha)}(x)u_{\nu}^{(\beta)}(y)\rangle$. The last Feynman
diagram is the one with a closed ghost loop coming from the ghost part of the
action.  Figure 4: One-loop Feynman diagrams contributing to the 2-pt Green’s
function of the gluino propagator,
$\langle\lambda^{\alpha}(x)\bar{\lambda}^{\beta}(y)\rangle$.  Figure 5: One-
loop Feynman diagram contributing to the 2-pt Green’s function of
$\langle\psi(x){\cal O}_{i}(z)\bar{\psi}(y)\rangle$, where ${\cal O}_{i}$ are
all local quark bilinear operators. The circled cross denotes the quark
bilinear operator insertion.

##  3 Continuum Results

Here we can collect all our results for the 2-pt Green’s functions 111A
Fourier transformation, with momentum $q$, is intended for all Green’s
functions appearing on the lhs of Eqs. (9-17) and (21).:

$\langle\psi(x)\bar{\psi}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}=i{\not{q}}\left[1-\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\left(\frac{2+\alpha}{\epsilon}+4+\alpha+(2+\alpha)\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\right]$
(9)

where $C_{F}=(N_{c}^{2}-1)/(2\,N_{c})$ is the quadratic Casimir operator in
the fundamental representation, $\alpha$ is the gauge parameter ($\alpha=1(0)$
corresponds to the Feynman (Landau) gauge). A Kronecker delta for color
indices is understood in Eqs. (9) and (10).

$\langle
A_{\pm}(x)A_{\pm}^{\dagger}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}=q^{2}\left[1-\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\left(\frac{1+\alpha}{\epsilon}+\frac{16}{3}+(1+\alpha)\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\right]$
(10) $\displaystyle\langle
u_{\mu}^{(\alpha)}(x)u_{\nu}^{(\beta)}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}$
$\displaystyle=$
$\displaystyle\frac{1}{\alpha}\delta^{(\alpha)\,(\beta)}q_{\mu}q_{\nu}+\delta^{(\alpha)\,(\beta)}\left(q^{2}\delta_{\mu\nu}-q_{\mu}q_{\nu}\right)\times$
$\displaystyle\Big{[}1-\frac{g^{2}\,N_{f}}{16\,\pi^{2}}\left(\frac{1}{\epsilon}+2+\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)$
$\displaystyle-\frac{g^{2}\,N_{c}}{16\,\pi^{2}}\frac{1}{2}\left(\frac{1+\alpha}{\epsilon}+\frac{7}{2}-\alpha-\frac{\alpha^{2}}{2}+(1+\alpha)\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\Big{]}$
$\displaystyle\langle\lambda^{(\alpha)}(x)\bar{\lambda}^{(\beta)}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}$
$\displaystyle=$
$\displaystyle\frac{i}{2}\delta^{(\alpha)\,(\beta)}\,{\not{q}}\Big{[}1-\frac{g^{2}\,N_{f}}{16\,\pi^{2}}\left(1+\frac{1}{\epsilon}+\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)$
$\displaystyle-\frac{g^{2}\,N_{c}}{16\,\pi^{2}}\left(\alpha+\frac{\alpha}{\epsilon}+4\,\alpha\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\Big{]}$
$\displaystyle\langle\psi(x){\cal
O}_{S}(z)\bar{\psi}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}$ $\displaystyle=$
$\displaystyle\leavevmode\hbox{\small 1\kern-4.0pt\normalsize
1}\left[1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\left(\frac{3+\alpha}{\epsilon}+4+2\alpha+(3+\alpha)\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\right]$
(13) $\displaystyle\langle\psi(x){\cal
O}_{P}(z)\bar{\psi}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}$ $\displaystyle=$
$\displaystyle\gamma_{5}\left[1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\left(\frac{3+\alpha}{\epsilon}+12+2\alpha+(3+\alpha)\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\right]$
(14) $\displaystyle\langle\psi(x){\cal
O}_{V}(z)\bar{\psi}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}$ $\displaystyle=$
$\displaystyle\gamma_{\mu}\left[1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\alpha\left(\frac{1}{\epsilon}+1+\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\right]-2\alpha\frac{q_{\mu}{\not{q}}}{q^{2}}\frac{g^{2}\,C_{F}}{16\,\pi^{2}}$
(15) $\langle\psi(x){\cal
O}_{A}(z)\bar{\psi}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}=\gamma_{5}\gamma_{\mu}\left[1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\left(\frac{\alpha}{\epsilon}+4+\alpha+\alpha\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\right]-2\alpha\gamma_{5}\frac{q_{\mu}{\not{q}}}{q^{2}}\frac{g^{2}\,C_{F}}{16\,\pi^{2}}$
(16) $\langle\psi(x){\cal
O}_{T}(z)\bar{\psi}(y)\rangle^{DR}_{\rm{amp}}\Big{|}_{m=0}=\gamma_{\mu}\gamma_{\nu}\left[1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\left(\alpha-1\right)\left(\frac{1}{\epsilon}+\log\left(\frac{\bar{\mu}^{2}}{q^{2}}\right)\right)\right].$
(17)

One can observe that there is no one-loop longitudinal part for the gluon
self-energy. Thus the renormalization function for the gauge parameter
receives no one-loop contribution. From the above results we can extract the
renormalization functions:

$\displaystyle Z_{\psi}^{DR,{\overline{\rm MS}}}$ $\displaystyle=$
$\displaystyle
1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\frac{1}{\epsilon}\left(2+\alpha\right),\,\,Z_{A_{\pm}}^{DR,{\overline{\rm
MS}}}=1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\frac{1}{\epsilon}\left(1+\alpha\right)$
(18) $\displaystyle Z_{u}^{DR,{\overline{\rm MS}}}$ $\displaystyle=$
$\displaystyle
1+\frac{g^{2}\,}{16\,\pi^{2}}\frac{1}{\epsilon}\left(\frac{1+\alpha}{2}N_{c}+N_{f}\right),\,\,Z_{\lambda}^{DR,{\overline{\rm
MS}}}=1+\frac{g^{2}\,}{16\,\pi^{2}}\frac{1}{\epsilon}\left(4\,\alpha\,N_{c}+N_{f}\right)$
(19) $\displaystyle Z_{S,P}^{DR,{\overline{\rm MS}}}$ $\displaystyle=$
$\displaystyle
1-\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\frac{1}{\epsilon},\,\,Z_{V,A}^{DR,{\overline{\rm
MS}}}=1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\frac{2}{\epsilon},\,\,Z_{T}^{DR,{\overline{\rm
MS}}}=1+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\frac{3}{\epsilon}.$ (20)

Calculating the same Green’s functions as before on the lattice, and combining
them with our results from the continuum, we will be able to extract
$Z_{\psi}^{L}$, $Z_{u}^{L}$, $Z_{\lambda}^{L}$, $Z_{A_{\pm}}^{L}$ and
$Z_{\Gamma}^{L}$ in the ${\overline{\rm MS}}$ scheme and on the lattice. On
the lattice we have to calculate all the diagrams which were presented here as
well as further tadpole diagrams containing closed gluon, fermion, ghost loops
and for gluon propagator we have also take into account the contribution which
comes from the measure part of the action. For the algebraic operations
involved in evaluating Feynman diagrams, we make use of our symbolic package
in Mathematica.

##  4 Future Plans – Conclusion

This will be the first calculation of the renormalization functions for SQCD
on the lattice, providing a thorough set of results for all counterterms,
mixing coefficients and parameter fine-tuning. The determination of the
Renormalization Functions of all fields and parameters which appear in ${\cal
S}^{E}_{\rm SQCD}$ and of a complete set of quark bilinear operators on the
lattice are necessary ingredients in the prediction of physical probability
amplitudes from lattice matrix elements. As a first example of our lattice
results, we find for the inverse quark propagator:

$\langle\psi^{B}(x)\bar{\psi}^{B}(y)\rangle^{L}_{\rm{amp}}\Big{|}_{m=0}=i{\not{q}}+\frac{g^{2}\,C_{F}}{16\,\pi^{2}}\left[i{\not{q}}\left(12.8025-4.79201\alpha+(2+\alpha)\log\left(a^{2}\,q^{2}\right)\right)-\frac{1}{a}51.4347\,r\right]$
(21)

where $a$ is the lattice spacing. From Eqs. (21) and (9) we extract the
renormalization for the quark field on the lattice:
$Z_{\psi}^{L,{\overline{\rm
MS}}}=1+g^{2}\,C_{F}/(16\,\pi^{2})\left(-16.8025+3.79201\alpha-(2+\alpha)\log\left(a^{2}\,\bar{\mu}^{2}\right)\right)$.
In additional the quark critical mass can read from Eq. (21):
$m^{quark}_{crit.}=-g^{2}\,C_{F}/(16\,\pi^{2})\,51.4347\,(1/a)\,r$. A further
extension of the present work would be to investigate of relationships between
different Green’s functions involved in SUSY Ward identities and to study the
mixing among operators.

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