aaaffiliationtext: Fellow, Canadian Derivatives Institute and Department of
Finance, HEC Montréal, Canadabbaffiliationtext: GERAD and Department of
Decision Sciences, HEC Montréal, Canadaccaffiliationtext: Concordia
University, Department of Mathematics and Statistics, Canadaddaffiliationtext:
Quantact Laboratory, Centre de Recherches Mathématiques, Canada

# Is the Difference between Deep Hedging and Delta Hedging a Statistical
Arbitrage?††thanks: Gauthier is supported by the Natural Sciences and
Engineering Research Council of Canada (NSERC, RGPIN-2019-04029), a
professorship funded by HEC Montréal, and the HEC Montréal Foundation. Godin
is funded by NSERC (RGPIN-2024-04593).

Pascal François  Geneviève Gauthier  Frédéric Godin  Carlos Octavio Pérez
Mendoza Corresponding author.
Email<EMAIL_ADDRESS>(Pascal François),
<EMAIL_ADDRESS>(Geneviève Gauthier<EMAIL_ADDRESS>(Frédéric Godin<EMAIL_ADDRESS>(Carlos Octavio Pérez
Mendoza).

###### Abstract

The recent work of Horikawa and Nakagawa, (2024) claims that under a complete
market admitting statistical arbitrage, the difference between the hedging
position provided by deep hedging and that of the replicating portfolio is a
statistical arbitrage. This raises concerns as it entails that deep hedging
can include a speculative component aimed simply at exploiting the structure
of the risk measure guiding the hedging optimisation problem. We test whether
such finding remains true in a GARCH-based market model. We observe that the
difference between deep hedging and delta hedging can be a statistical
arbitrage if the risk measure considered does not put sufficient relative
weight on adverse outcomes. Nevertheless, a suitable choice of risk measure
can prevent the deep hedging agent from including a speculative overlay within
its hedging strategy.

JEL classification: C45, C61, G32.

Keywords: Deep reinforcement learning, optimal hedging, arbitrage.

## 1 Introduction

The seminal paper of Buehler et al., (2019), which proposes to use deep
reinforcement learning (RL) methods to obtain optimal hedging procedures for
financial derivatives, initiated a whole strand of literature: Halperin,
(2019), Cao et al., (2020), Du et al., (2020), Carbonneau and Godin, (2021),
Carbonneau, (2021), Horvath et al., (2021), Imaki et al., (2021), Lütkebohmert
et al., (2022), Cao et al., (2023), Carbonneau and Godin, (2023), Marzban et
al., (2023), Mikkilä and Kanniainen, (2023), Raj et al., (2023) and Wu and
Jaimungal, (2023). See also Hambly et al., (2023) and Pickard and Lawryshyn,
(2023) for related surveys. Deep RL methods are particulary well-suited to
solve dynamic hedging problems as such methods can handle the curse of
dimensionality, a problem that more traditional methods (e.g., finite elements
dynamic programming) struggle to overcome. They can also work with very
general dynamics for hedging asset prices, not being limited by mathematical
tractability issues.

While the ability of deep hedging strategies to outperform standard
counterparts is well-documented, the existing literature has not yet
extensively analyzed the structure of optimal policies and explained how such
incremental performance is attained. Neagu et al., (2024) make a step in that
direction by investigating the impact of the various features on the optimal
hedging decision in the presence of illiquidity market impacts.

In their recent work, Horikawa and Nakagawa, (2024) investigate complete
markets that allow for statistical arbitrage with respect to a specific risk
measure $\rho$. They assert that, within this framework, hedging strategies
that minimize the chosen risk metric combine the traditional delta-hedging
approach with a statistical arbitrage overlay. In a more general market
setting, Buehler et al., (2021) find that the optimal hedging strategy
maximizing the entropy utility can also incorporate a statistical arbitrage
component. Such claims raise concern about the suitability of the deep hedging
approach; incorporating speculative or arbitrage-like components that do not
contribute to reducing the risk exposure within hedging portfolios would be
deemed undesirable in practice. Our objective is therefore to assess
empirically whether deep hedging policies minimizing conventional risk metrics
contain a speculative component in a GARCH-based market setting.

The paper is divided as follows. Section 2 provides the hedging problem
formulation. Section 3 describes the deep hedging framework used to solve the
problem, and discusses the common delta hedging benchmark. Numerical
experiments assessing the behavior of the deep versus delta hedging difference
strategy are provided in Section 4.111The Python code which allows replicating
the numerical experiments from this paper can be found at
https://github.com/cpmendoza/DeepHedging_StatisticalArbitrage.git. Section 5
concludes.

## 2 Market model for hedging

This paper considers dynamic hedging strategies for European call options,
which involve the construction of a self-financing hedging portfolio composed
of the underlying asset $S$ and a cash account. The portfolio is rebalanced
daily to optimally offset the net risk exposure at the option maturity,
denoted as $T$ days. The time-$t$ underlying asset price is $S_{t}$. The
hedging strategy is represented by the predictable process
$\delta=\\{\delta_{t}\\}_{t=1}^{T}$, where $\delta_{t}$ is the number of
underlying asset shares held during the interval $(t-1,t]$. The time-$t$
discounted gain made by the hedging portfolio is
$G_{t}^{\delta}=\sum_{k=1}^{t}\delta_{k}(\beta^{k}S_{k}\mbox{e}^{q\Lambda}-\beta^{k-1}S_{k-1})$
with $\beta=\mbox{e}^{-r\Lambda}$, where $r$ is the annualized continuously
compounded risk-free rate, $q$ is the annualized underlying asset dividend
yield, and the period length is $\Lambda=\frac{1}{252}$ years. The time-$t$
self-financing hedging portfolio value is

$V_{t}^{\delta}(V_{0})=\beta^{-t}(V_{0}+G_{t}^{\delta}),$ (1)

where $V_{0}$ is the initial portfolio value that we set to the option price.

The hedging problem is a sequential decision problem where the holder of a
short position in a call option seeks for the best sequence of actions
$\delta$ that minimizes the risk associated with the hedging error

$\xi_{T}^{\delta}=\mbox{max}(S_{T}-K,0)-V_{T}^{\delta}(V_{0}),$ (2)

where $K$ is the call option strike price. The hedging problem is formulated
as

$\delta^{*}=\mathop{\arg\min}\limits_{\delta}\rho\left(\xi_{T}^{\delta}\right),$
(3)

where $\rho$ is the risk measure used by the hedging agent to quantify risk.
In this paper we consider the Conditional Value-at-Risk (CVaRα) defined as
$\rho(\xi_{T}^{\delta})=\mathbb{E}[\xi_{T}^{\delta}\mid\xi_{T}^{\delta}\geq\mbox{VaR}_{\alpha}(\xi_{T}^{\delta})]$,
where $\alpha\in(0,1)$ and $\mbox{VaR}_{\alpha}(\xi_{T}^{\delta})$ is the
Value-at-Risk defined as
$\mbox{VaR}_{\alpha}(\xi_{T}^{\delta})=\min_{c}\\{c:\mathbb{P}(\xi_{T}^{\delta}\leq
c)\geq\alpha\\}$.

Each time-$t$ action $\delta_{t+1}$ is a feedback-type decision, being a
function of the information currently available on the market:
$\delta_{t+1}=\tilde{\delta}(X_{t})$ for some function $\tilde{\delta}$ of the
state variable vector $X_{t}$.

## 3 Hedging strategies

### 3.1 Deep hedging

The deep hedging (DH) framework, introduced by Buehler et al., (2019),
provides a solution to the hedging problem (3) by leveraging reinforcement
learning (RL) techniques. The DH policy $\tilde{\delta}$ is approximated with
a neural network $\delta^{DH}_{\theta}$ with parameters $\theta$, which
returns a hedging position $\delta_{t+1}$ when provided with time-$t$ input
features $X_{t}$. The objective function to be minimized is thus

$\mathcal{O}(\theta)=\rho\left(\xi_{T}^{\delta^{DH}_{\theta}}\right).$ (4)

The neural network is optimized with the Mini-batch Stochastic Gradient
Descent method (MSGD). This training procedure relies on updating iteratively
all the trainable parameters of the network based on the recursive equation

$\theta_{j+1}=\theta_{j}-\eta_{j}\nabla_{\theta}\hat{\mathcal{O}}(\theta_{j}),$
(5)

where $\theta_{j}$ is the set of parameters obtained after iteration $j$,
$\eta_{j}$ is the learning rate (step size) which determines the magnitude of
change in parameters on each time step, $\nabla_{\theta}$ is the gradient
operator with respect to $\theta$ and $\hat{\mathcal{O}}$ is the Monte Carlo
estimate of the objective function (4) computed on a mini-batch. Automatic
differentiation packages are used to compute the gradient of
$\hat{\mathcal{O}}$. Additional details are provided in the appendix.

For the neural network, we employ a fully-connected Feedforward Neural Network
(FFNN) architecture with four hidden layers of width 56 using a ReLU
activation function. The output FFNN layer, which maps the output of the
hidden layer $Z$ into the position in the underlying asset position
$\delta_{t+1}^{DH}$, is equipped with a dynamic upper bound on the activation
function to preclude excessive leverage. Indeed, agents have finite borrowing
capacity in practice. We impose that the time-$t$ cash account value
$\phi_{t}$ satisfies $\phi_{t}\geq-B$ for all $t$ and for some threshold
$B>0$. This is achieved by setting the final output layer activation to

$f(Z,t)=\mbox{min}\left(Z,(V_{t}+B)/S_{t}\right),$ (6)

which ensures that the cash amount borrowed in the hedging portfolio remains
below $B>0$ (see François et al., (2024)).

Agents are trained on training sets of 400,000 independent simulated paths
with mini-batch size of 1,000 and a learning rate of 0.0005 that is
progressively adapted with the ADAM (Kingma and Ba,, 2014) optimization
algorithm. The training procedure is implemented in Python, using Tensorflow
and considering the Glorot and Bengio, (2010) random initialization of the
initial parameters of the neural network. Numerical results are obtained from
test sets of 100,000 independent paths.

### 3.2 Delta hedging

Delta hedging aims to reduce the risk associated with price movements of an
underlying asset by adjusting the hedging portfolio positions in the
underlying asset based on the sensitivity ($\Delta$) of the option price to
changes in the price of the underlying asset. Specifically, the time-$t$
position in the underlying asset is defined as the time-$t$ sensitivity
$\Delta_{t}$, which is the partial derivative of the time-$t$ option price
with respect to the underlying asset value.

### 3.3 Statistical arbitrage

Statistical arbitrage strategies, also known as "good deals" according to the
terminology of Cochrane and Saa-Requejo, (2000), are profit-seeking trading
strategies that capitalize on statistical anomalies in the market. Bondarenko,
(2003) defines statistical arbitrage as a trading strategy that makes a profit
on average without requiring any initial capital investment. Assa and Karai,
(2013) extend this definition to offer a more nuanced and comprehensive
evaluation, ensuring that the trading strategy is not only profitable on
average, but also resilient in terms of risk management. As in Assa and Karai,
(2013), we say that $\delta$ is a statistical arbitrage opportunity if

$\rho\left(-V_{T}^{\delta}(0)\right)<0,$ (7)

which entails a trading strategy with zero initial investment that is deemed
strictly less risky than a null investment according to risk measure $\rho$.
Such definition is also in line with that of Buehler et al., (2021), who focus
on the case of the entropy risk measure.

Horikawa and Nakagawa, (2024) claim that in a complete market model that
admits statistical arbitrage, the difference between the deep hedging and the
delta hedging strategies denoted by

$\delta^{-}=\delta^{DH}-\Delta,$ (8)

is a statistical arbitrage strategy according to risk measure $\rho$. We wish
to further extend their study and examine if the trading strategy $\delta^{-}$
behaves like a statistical arbitrage in a GARCH-based market model. In other
words, we investigate whether the deep hedging approach typically incorporates
a speculative arbitrage-like component aimed at exploiting the structure of
the risk measure considered.

## 4 Numerical study

### 4.1 Stochastic market dynamics

We consider market dynamics based on a GARCH process to represent the
underlying asset log-returns. The GJR-GARCH(1,1) model introduced by Glosten
et al., (1993) captures time-varying volatility and accounts for the leverage
effect. For $t=1,\dots,T$, log-returns in the model follow

$R_{t}=\mu+\sigma_{t}\epsilon_{t},\quad\sigma_{t+1}^{2}=\omega+\sigma_{t}^{2}(\alpha+\gamma\mathds{1}_{\\{\epsilon_{t}<0\\}})\epsilon_{t}^{2}+\beta\sigma_{t}^{2},$
(9)

where $\mu,\,\gamma\in\mathbb{R}$, $\omega,\,\alpha,\,\beta$ are positive,
$\mathds{1}_{A}$ is the binary dummy variable indicating if event $A$ occurs
and $\\{\epsilon_{t}\\}_{t=1}^{T}$ are independent standard normal random
variables. Parameter estimates are obtained through maximum likelihood on a
daily time series of the S&P 500 index extending from January 4, 2016, to
December 31, 2020. Estimated parameters are $\mu=0.06\%$, $\omega=0.01\%$,
$\alpha=0.11$, $\gamma=0.20$ and $\beta=0.78$. Furthermore, in all
experiments, the annualized continuously compounded risk-free rate and
dividend yield are assumed to be constant with values set to $r=2.66\%$ and
$q=1.77\%$, respectively.

The initial option price is computed using Monte Carlo simulation based on the
risk-neutral valuation formula

$\mbox{Call}_{0}=\mbox{e}^{-rT\Lambda}\mathbb{E}^{\mathbb{Q}}[\mbox{max}(S_{T}-K,0)],$
(10)

where $\mathbb{Q}$ is a risk-neutral measure.222The $\mathbb{Q}$ risk-neutral
dynamics of the GARCH model are defined by the following equations:
$R_{t}=(r-q)\Lambda-\frac{\sigma^{2}_{t}}{2}+\sigma_{t}\tilde{\epsilon}_{t},\quad\sigma_{t+1}^{2}=\omega+\sigma_{t}^{2}(\alpha+\gamma\mathds{1}_{\\{\tilde{\epsilon_{t}}-\eta_{t}<0\\}})(\tilde{\epsilon_{t}}-\eta_{t})^{2}+\beta\sigma_{t}^{2},$
where $\eta_{t}=(\mu-(r-q)\Lambda+\sigma^{2}_{t}/2)/\sigma_{t}$ and
$\\{\tilde{\epsilon_{t}}\\}_{t=1}^{T}$ are independent standard normal random
variables under $\mathbb{Q}$.

The call option delta is also calculated through Monte-Carlo simulation based
on the relationship

$\Delta_{t}=\mbox{e}^{-r\tau\Lambda}\mathbb{E}^{\mathbb{Q}}\left[\frac{S_{t+\tau}}{S_{t}}\mathds{1}_{\\{S_{t+\tau}>K\\}}\mid\mathcal{F}_{t}\right]$
(11)

where $\tau=T-t$ and $\mathcal{F}_{t}$ represents the available information at
time $t$, i.e., that being generated by the state $X_{t}$.

In this model the state space considered for the RL approach is represented by
the vector $X_{t}=(V_{t}^{\delta},\,\mbox{log}(S_{t}),\,\sigma_{t+1},\,\tau)$.

### 4.2 Comparative analysis of deep hedging and delta hedging strategies

In this section, we study the relationship between delta hedging and deep
hedging. More specifically, we investigate whether the difference between both
strategies represents a speculative overlay reminiscent of a statistical
arbitrage. The comparison is conducted by hedging at-the-money (ATM) European
call options, with $S_{0}=K=100$. The option maturity is $T=63$ days. The
leverage constraint is $B=100$.

Table 1 presents the hedging performance of the deep hedging agents trained
with the CVaRα risk measure. We consider the following confidence levels
$\alpha$: 1%, 5%, 10%, 20%, 50%, 85%, 90%, and 95%. High values of $\alpha$
only put weight on the most adverse outcomes and entail focusing purely on
risk reduction. Conversely, low values for $\alpha$ both penalize losses and
reward gains, which leads to seeking risk-reward trade-offs. As such, the
CVaRα with a low confidence level does not align with the objective of
limiting the variability of the hedging error. Since the CVaRα is an
increasing function of $\alpha$, there are more statistical arbitrage
strategies becoming available as $\alpha$ decreases.

Columns labeled "Base strategies" display the risk measure applied to the
hedging error for the deep hedging strategy, and the difference between the
risk provided by deep hedging and that of delta hedging. Columns labeled
"Difference strategy" provide statistics (hedging error risk and expectation
of net cash flow) of the trading strategy $\delta^{-}$ representing the
differential position between deep hedging and delta hedging. Since such
strategy reflects a long position on the deep hedge and a short position on
the delta hedge, the option payoffs from the two (long and short) positions
cancel out. Hence, performance is assessed by looking at
$V_{T}^{\delta^{-}}(0)$ rather than $\xi_{T}^{\delta^{-}}$.

Table 1: Performance assessment for deep hedging, delta hedging and their
difference over a short position on an ATM call option with maturity $T=63$
days.

| Base strategies |  | Difference strategy
---|---|---|---
Metric |  $\rho(\xi_{T}^{\delta^{DH}})$ |  $\rho(\xi_{T}^{\delta^{DH}})-\rho(\xi_{T}^{\Delta})$ |  |  $\rho(-V_{T}^{\delta^{-}}(0))$ |  $\mathbb{E}[V_{T}^{\delta^{-}}(0)]$
CVaR1% |  -1.918 |  -1.368 |  |  -1.306 |  -1.372
CVaR5% |  -1.852 |  -1.351 |  |  -1.167 |  -1.372
CVaR10% |  -1.767 |  -1.325 |  |  -1.038 |  -1.371
CVaR20% |  -1.575 |  -1.256 |  |  -0.807 |  -1.371
CVaR50% |  -0.614 |  -0.791 |  |  -0.071 |  -1.370
CVaR85% |  -1.505 |  -0.129 |  |  -1.208 |  -0.039
CVaR90% |  -2.055 |  -0.128 |  |  -1.599 |  -0.225
CVaR95% |  -3.102 |  -0.121 |  |  -2.121 |  -0.369

  * •

Results are computed using 100,000 out-of-sample paths. The initial price of
the option is 3.16. $\xi_{T}^{\delta}$ is the hedging error for trading
strategy $\delta$, with $\delta^{DH}$ being deep hedging and $\Delta$ being
delta hedging. The strategy $\delta^{-}$ uses the underlying asset position
defined by the difference between that of the deep hedging and the delta
hedging strategies.

For all confidence levels $\alpha$ below $50\%$, the strategy $\delta^{-}$
exhibits both positive average profitability
$\mathbb{E}[V_{T}^{\delta^{-}}(0)]$ and a CVaR value
$\rho(-V_{T}^{\delta^{-}}(0))$ that is negative. This corresponds to a formal
statistical arbitrage strategy. Moreover, for the case $\alpha=50\%$, even if
the risk measure is positive, it is nevertheless negligible in comparison to
expected profits. The strategy $\delta^{-}$, though not a statistical
arbitrage from the definition, exhibits a behavior that is quite similar to a
statistical arbitrage. Conversely, difference strategies $\delta^{-}$ using
$\alpha\geq 85\%$ clearly do not qualify as statistical arbitrage; the
associated risk measure is high and the average profitability is negative.

The distribution of the profit and loss (P&L) for the trading strategy
$\delta^{-}$, which is $V_{T}^{\delta^{-}}(0)$, is depicted in Figure 1 for
various confidence levels. This confirms that difference strategies associated
with low confidence levels $\alpha$ (1%, 10% and 50%) are exactly or similar
to statistical arbitrage with very high average profits and a very fat left
tail (high extreme loss potential). The deep hedging agent is incorporating a
strong speculative element in its trading strategy, which is unsuitable in
practice. Conversely, the strategies associated with higher values for
$\alpha$ do not exhibit characteristics of a statistical arbitrage and do not
lead to concerns about the suitability of the deep hedging strategy.

Figure 1: P&L distribution of the strategy $\delta^{-}$.

  * •

Distributions are computed using 100,000 out-of-sample paths. The P&L is
simply defined by the portfolio value $V_{T}^{\delta^{-}}(0)$ at maturity.

We analyze the statistical relationship between deep hedging and delta hedging
strategies through (i) sample Spearman (rank) correlations between underlying
asset positions of both strategies, and (ii) the regression model

$\delta^{DH}=\kappa_{0}+\kappa_{1}\Delta+\epsilon,$ (12)

with $\delta^{DH}$ and $\Delta$ being positions produced by the deep hedging
and delta hedging strategies, respectively. Metrics (regressions and
correlations) are computed across all rebalancing points of all paths in the
test sets.

Table 2 provides the Spearman correlation coefficient $\varrho$, which
evaluates monotonic relationships between strategies, and the coefficient of
determination $R^{2}$, which measures the strength of the linear association
between the strategies. These metrics are computed for the various CVaR
confidence levels.

Table 2: Statistical relationships between positions of delta hedging and deep
hedging.

| Statistics
---|---
Metric |  $\varrho$ |  $R^{2}$
CVaR1% |  -0.270 |  0.003
CVaR5% |  -0.271 |  0.003
CVaR10% |  -0.272 |  0.003
CVaR20% |  -0.273 |  0.003
CVaR50% |  -0.273 |  0.003
CVaR85% |  -0.939 |  0.773
CVaR90% |  -0.963 |  0.816
CVaR95% |  -0.969 |  0.808

  * •

Results are for a short position on an ATM call option with a maturity of
$N=63$ days. They are computed using 100,000 out-of-sample paths. The metric
$\varrho$ denotes the (unconditional) Spearman correlation between underlying
asset positions of the delta hedging strategy and the deep hedging strategy
across all rebalancing days while $R^{2}$ represents the $R^{2}$ statistic
obtained after regressing deep hedging positions onto delta hedging positions.

Results presented in Table 2 show strong monotonic and linear association
between deep and delta hedging positions for high confidence levels
$\alpha=85\%$, $90\%$ or $95\%$. The deep hedging strategies can therefore be
seen as alterations of the delta hedging procedure that improve hedging
performance. Conversely, for low confidence levels ($50\%$ or below), deep
hedging positions seem completely unrelated to delta hedging positions,
indicating that the agent has mostly abandoned its hedging objective and is
rather attempting to speculate or conduct statistical arbitrage.

## 5 Conclusion

Consider the trading strategy whose underlying asset positions correspond to
the difference between these of deep hedging and delta hedging. What if there
exist market models under which such strategy is a statistical arbitrage? This
would raise concerns about the suitability of deep hedging procedures, as it
raises the possibility that typical deep hedging strategies could consist of
conventional hedging strategies that are enhanced with speculative overlays
which are unrelated to hedging.

Our study shows that these concerns can be mitigated under GARCH-based market
models; if the risk measure considered in the hedging optimization problem
does not sufficiently penalize losses relative to rewards provided for gains,
the deep hedging strategy attaches a statistical arbitrage strategy overlay to
the delta hedging strategy. Nevertheless, when using a proper risk measure
(the CVaR with sufficiently high $\alpha$ in our case) within the optimization
problem, the difference between deep hedging and delta hedging does not
exhibit statistical arbitrage-like behavior and cannot be interpreted as a
speculative strategy reaping profits while exploiting blind spots of the
chosen risk measure.

The two main conclusions from this study are therefore that (i) the objective
function of the deep hedging problem must be carefully selected to prevent the
hedging agent from abandoning its hedging objective and pursuing speculative
behavior, and (ii) deep hedging can soundly achieve its hedging objectives
when provided with a suitable risk measure.

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## Appendix: Details for the MSGD training approach

The MSGD method estimates the penalty function $\mathcal{O}(\theta)$, which is
typically unknown, through small samples of the hedging error called batches.
Let
$\mathbb{B}_{j}=\\{\xi_{T,i}^{\delta^{DH}_{\theta_{j}}}\\}_{i=1}^{N_{\mbox{\scriptsize{batch}}}}$
be the $j$-th batch where $\xi_{T,i}^{\delta^{DH}_{\theta_{j}}}$ denotes the
hedging error of the $i$-th simulated path in the $j$-th batch defined as

$\xi_{T,i}^{\delta^{DH}_{\theta_{j}}}=\mbox{max}(S_{T,ij}-K,0)-V_{T,i}^{\delta^{DH}_{\theta_{j}}}(V_{0})\mbox{,}$

where $S_{T,ij}$ is the price of the underlying asset at time $T$ in the
$i$-th simulated path, and $V_{T,i}^{\delta^{DH}_{\theta_{j}}}$ is the
terminal value of the hedging strategy for that path when $\theta=\theta_{j}$.
The penalty function estimation for the batch $\mathbb{B}$ is

$\hat{C}^{\left(\mbox{\scriptsize{CVaR}}\right)}(\theta_{j},\mathbb{B}_{j})=\widehat{\mbox{VaR}}_{\alpha}(\mathbb{B}_{j})+\frac{1}{(1-\alpha)N_{\mbox{\scriptsize{batch}}}}\sum_{i=1}^{N_{\mbox{\scriptsize{batch}}}}\max\left(\xi_{T,i}^{\delta^{DH}_{\theta_{j}}}-\widehat{\mbox{VaR}}_{\alpha}(\mathbb{B}_{j}),0\right)\,\mbox{,}$

where $\widehat{\mbox{VaR}}_{\alpha}(\mathbb{B}_{j})=\xi_{T,\lceil\alpha\cdot
N_{\mbox{\scriptsize{batch}}}\rceil}^{\delta^{DH}_{\theta_{j}}}$ is the
estimation of the VaR obtained from the ordered sample
$\\{\xi_{T,[i]}^{\delta^{DH}_{\theta_{j}}}\\}_{i=1}^{N_{\mbox{\scriptsize{batch}}}}$
and $\lceil\cdot\rceil$ is the ceiling function. These empirical
approximations are used to estimate the gradient of the penalty function
required in Equation (5).333Details about gradient of the empirical objective
function are provided in Goodfellow et al., (2016).


# Fast GraspNeXt: A Fast Self-Attention Neural Network Architecture for Multi-
task Learning in Computer Vision Tasks for Robotic Grasping on the Edge

Alexander Wong1,2,3 Yifan Wu1 Saad Abbasi1,3 Saeejith Nair1
Yuhao Chen1 Mohammad Javad Shafiee1,2,3
1 Vision and Image Processing Research Group, University of Waterloo
2 Waterloo Artificial Intelligence Institute, Waterloo, ON
3 DarwinAI Corp., Waterloo, ON
{a28wong, yifan.wu1, srabbasi, smnair, yuhao.chen1<EMAIL_ADDRESS>
###### Abstract

Multi-task learning has shown considerable promise for improving the
performance of deep learning-driven vision systems for the purpose of robotic
grasping. However, high architectural and computational complexity can result
in poor suitability for deployment on embedded devices that are typically
leveraged in robotic arms for real-world manufacturing and warehouse
environments. As such, the design of highly efficient multi-task deep neural
network architectures tailored for computer vision tasks for robotic grasping
on the edge is highly desired for widespread adoption in manufacturing
environments. Motivated by this, we propose Fast GraspNeXt, a fast self-
attention neural network architecture tailored for embedded multi-task
learning in computer vision tasks for robotic grasping. To build Fast
GraspNeXt, we leverage a generative network architecture search strategy with
a set of architectural constraints customized to achieve a strong balance
between multi-task learning performance and embedded inference efficiency.
Experimental results on the MetaGraspNet benchmark dataset show that the Fast
GraspNeXt network design achieves the highest performance (average precision
(AP), accuracy, and mean squared error (MSE)) across multiple computer vision
tasks when compared to other efficient multi-task network architecture
designs, while having only 17.8M parameters (about $>$5$\times$ smaller), 259
GFLOPs (as much as $>$5$\times$ lower) and as much as $>$3.15$\times$ faster
on a NVIDIA Jetson TX2 embedded processor.

(a) Visible mask.

(b) Amodal mask.

(c) Center of mass.

(d) Suction grasp heatmap.

Figure 1: Example multi-task outputs from Fast GraspNeXt. (a) and (b) Detected
occluded objects are shown in blue and non-occluded objects are shown in red.
(c) The detected center of mass of each object is shown in blue. (d)
Applicability of suction grasp is labelled from high to low in red, green, and
blue as a heatmap.

##  1 Introduction

Figure 2: Overall network architecture design for Fast GraspNeXt, which
possess a self-attention neural network architecture with highly optimized
macroarchitecture and microarchitecture designs for all components. Fast
GraspNeXt consists of a generated self-attention backbone architecture feeding
into a generated feature pyramid network architecture followed by generated
head network architecture designs for multi-task learning. The numbers in
brackets are channel sizes of the feature maps in the heads.

Significant advances have been made in recent years to take advantage of deep
neural networks for robotic grasping. In particular, multi-task learning has
shown considerable promise for improving the performance of deep learning-
driven vision systems for robotic grasping [7, 8, 3], where the underlying
goal is to learn to perform additional tasks during the model training
process. Multi-task learning has enabled not only greater precision and
versatility in deep learning-driven vision systems for robotic grasping, but
also enabled such systems to perform a wide range of computer vision tasks
that are important for robotic grasping (see Fig. 1 for example tasks that
need to be performed by such deep learning-driven vision systems for robotic
grasping such as visible object mask detection, amodal object detection [1],
center of mass prediction, and suction grasp heatmap generation [2]). However,
while multi-task learning can greatly improve the performance of computer
vision tasks for robotic grasping, high architectural and computational
complexity can limit operational use in real-world manufacturing and warehouse
environments on embedded devices.

Motivated to address these challenges with embedded deployment for robotic
grasping in real-world manufacturing and supply chain environments, we
leverage a generative network architecture search strategy with a set of
architectural design constraints defined to achieve a strong balance between
multi-task learning performance and embedded operational efficiency. The
result of this generative network architecture search approach is Fast
GraspNeXt, a fast self-attention neural network architecture tailored
specifically for multi-task learning in robotic grasping under embedded
scenarios.

The paper is organized as follows. Section 2 describes the methodology behind
the creation of the proposed Fast GraspNeXt via generative network
architecture search, as well as a description of the resulting deep neural
network architecture. Section 3 describes the dataset used in this study, the
training and testing setup, as well as the experimental results and complexity
comparisons.

##  2 Methods

###  2.1 Generative Network Architecture Search

In this paper, we take a generative network architecture search approach to
creating the optimal multi-task deep neural network architecture for Fast
GraspNeXt. More specifically, we leveraged the concept of generative synthesis
[13], an iterative method that generates highly tailored architectural designs
that satisfy given requirements and constraints (e.g., model performance
targets). Generative synthesis can be formulated as a constrained optimization
problem:

$\mathcal{G}=\max_{\mathcal{G}}\mathcal{U}(\mathcal{G}(s))\;\;\;\textrm{
subject to}\;\;\;1_{r}(G(s))=1,\;\;\forall\in\mathcal{S}.$ (1)

where the underlying objective is to learn an expression $\mathcal{G}(\cdot)$
that, given seeds $\\{s|s\in S\\}$, can generate network architectures
$\\{N_{s}|s\in S\\}$ that maximizes a universal performance metric $U$ (e.g.,
[11]) while adhering to operational constraints set by the indicator function
$1_{r}(\cdot)$. This constrained optimization is solved iteratively through a
collaboration between a generator $G$ and an inquisitor $I$ which inspects the
generated network architectures and guides the generator to improve its
generation performance towards operational requirements (see [13] for
details).

To build Fast GraspNeXt, we enforce essential design constraints through
$1_{r}$(·) in Eq. 1 to achieve the desired balance between i) accuracy, ii)
architectural complexity, and iii) computational complexity to yield high-
performance, compact, and low-footprint neural network architectures such as:

  1. 1.

Encouraging the implementation of anti-aliased downsampling (AADS) [14] to
enhance network stability and robustness.

  2. 2.

Encouraging the use of attention condensers [12], which are highly efficient
self-attention mechanisms designed to learn condensed embeddings
characterizing joint local and cross-channel activation relationships for
selective attention. They have been shown to improve representational
performance while improving efficiency at the same time.

  3. 3.

Enforce a FLOPs requirement of less than 300B FLOPs and an accuracy
requirement of no lower AP across all assessable tasks than a ResNet-50
variant of the multi-task network for robotic grasping (which we call ResNet-
GraspNeXt) by 0.5%.

###  2.2 Network Architecture

The resulting Fast GraspNeXt network architecture design is shown in Fig. 2.
It possesses a self-attention neural network architecture with highly
optimized macroarchitecture and microarchitecture designs for all its
components. The network architecture adheres to the constraints we imposed,
with the generated backbone architecture feeding into a generated feature
pyramid network architecture design followed by generated head network
architecture designs for predicting the multi-task outputs: i) amodal object
bounding boxes, ii) visible object masks, iii) amodal object masks, iv)
occlusion predictions, v) object center of mass, vi) and suction grasp
heatmap.

More specifically, the multi-scale features from the generated backbone
architecture are provided as input directly to each level of the generated
feature pyramid network architecture, followed by the generated bounding box
head, visible mask head, amodal mask head and occlusion prediction head. Each
level of the feature pyramid network are also upsampled to reach the same
scale and summed as input for the center of mass head and suction grasp
heatmap head.

The multi-task training loss, denoted as $L_{mt}$, used to train Fast
GraspNeXt is a weighted combination of task-specific losses and can be
expressed by

$\begin{split}L_{mt}&={l_{rpn}+\lambda_{1}l_{abox}+\lambda_{2}l_{segm\\_v}+\lambda_{3}l_{segm\\_a}}\\\
&+\lambda_{4}l_{occ}+\lambda_{5}l_{com}+\lambda_{6}l_{suc}\end{split}$ (2)

where $\lambda_{1},\lambda_{2},\ldots,\lambda_{6}$ denote task-specific weight
coefficients used to balance the contribution of individual task-specific
losses. The individual task-specific losses are defined as follows:

  * •

$l_{rpn}$: Region Proposal Network loss [9]

  * •

$l_{abox}$: Amodal bounding box prediction loss [1]

  * •

$l_{segm\\_v}$: Visible mask segmentation loss [1]

  * •

$l_{segm\\_a}$: Amodal mask segmentation loss [1]

  * •

$l_{occ}$: Occlusion classification loss [1]

  * •

$l_{com}$: Center of mass heatmap prediction loss implemented with the
modified focal loss proposed by CenterNet [15]

  * •

$l_{suc}$: Suction grasp heatmap prediction loss implemented with pixel-wise
averaged mean squared error (MSE) loss

It can be observed that the architecture design is highly heterogeneous and
columnar for high architectural and computational efficiency. It can also be
observed that the architecture design possesses attention condensers at
different stages of the architecture for improved attentional efficacy and
efficiency. Furthermore, the architecture design possesses AADS at strategic
locations for greater robustness. Finally, it can be observed that the
macroarchitecture for each task-specific head is unique, thus tailored around
the specific balance between accuracy and efficiency for each individual task.
As such, these characteristics make the Fast GraspNeXt architecture design
well-suited for high-performance yet highly efficient multi-task robotic grasp
applications on the edge.

##  3 Experiments

Table 1: Summary of quantitative performance results on MetaGraspNet dataset
and network complexity.

Model |  | Inf. Time
---
(ms)
| Amodal
---
Bbox AP
| Visible
---
Mask AP
| Amodal
---
Mask AP
| Occlusion
---
Accuracy
| Center of
---
Mass AP
| Heatmap
---
MSE
| Parameters
---
(M)
| FLOPs
---
(B)
ResNet-GraspNeXt | 3501 | 85.0% | 84.9% | 84.1% | 77.2% | 75.3% | 0.0113 | 92.1 | 1314
EfficientNet-GraspNeXt | 2972 | 84.6% | 85.0% | 83.8% | 81.7% | 82.6% | 0.0189 | 72.0 | 1183
MobileNet-GraspNeXt | 2712 | 84.3% | 84.6% | 83.7% | 80.7% | 81.2% | 0.0104 | 70.9 | 1189
Fast GraspNeXt | 1106 | 87.9% | 85.4% | 85.0% | 85.1% | 84.6% | 0.0095 | 17.8 | 259

Figure 3: (top) Predicted suction grasp heatmaps produced by the proposed Fast
GraspNeXt. (bottom) Example ground truth suction grasp heatmaps.

###  3.1 Dataset

We evaluate the performance of the proposed Fast GraspNeXt on the MetaGraspNet
[4] benchmark dataset to explore the efficacy. This large-scale robotic
grasping benchmark dataset contains 217k images across 5884 scenes featuring
82 different objects. We use 60%, 20%, and 20% of the scenes for training,
validation, and testing respectively. Average precision (AP) evaluation was
conducted for amodal object bounding box, visible object mask, amodal object
mask, and object center of mass. Occlusion accuracy evaluation was conducted
to evaluate occlusion predictions, while mean squared error (MSE) evaluation
was conducted to evaluate suction grasp heatmap predictions. Our experiments
use the class agnostic labels which put all objects into one class category,
so that it can be readily deployed in industrial scenarios with novel, unseen
items.

###  3.2 Training and Testing Setup

In addition to the proposed Fast GraspNeXt, we evaluated the performance of
efficient multi-task network designs leveraging ResNet-50 [5], EfficientNet-B0
[10], and MobileNetV3-Large [6] as backbones paired with our multi-task
network architecture design but without utilizing the generative network
architecture search strategy. Both EfficientNet and MobileNetV3 are widely-
used, state-of-the-art efficient backbones, making them well-suited for this
comparison. Those network architectures are designated as ResNet-GraspNeXt,
EfficientNet-GraspNeXt, and MobileNet-GraspNeXt, respectively.

For training, we use a base learning rate of 0.03, SGD optimizer with momentum
of 0.9, and weight decay of 0.0001 for all experiments. Learning rate step
decay are performed at 67% and 92% of the total epochs with gamma of 0.1. All
network architectures are trained with the full image size of 1200$\times$1200
pixels with batch size of 2. Empirical results found that the above training
strategy yielded the best performance for all tested architectures.

Inference time evaluations are executed with batch size of 1 to reflect the
robotic grasping environment which prioritise lowest possible inference
latency instead of potential speed benefit of batched inference. We evaluate
the inference time on the NVIDIA Jetson TX2 embedded processor with 8 GB of
memory, which is widely used for embedded robotics applications in
manufacturing and warehouse scenarios.

###  3.3 Results and Analysis

Tab. 1 shows the quantitative performance results and model complexity of the
proposed Fast GraspNeXt compared to ResNet-GraspNeXt, EfficientNet-GraspNeXt,
and MobileNet-GraspNeXt. We can observe that leveraging state-of-the-art
efficient backbone architectures EfficientNet-B0 and MobileNetV3-Large enables
noticeably faster inference time and lower architectural complexity when
compared to leveraging ResNet-50 but results in a noticeable drop in amodal
bbox AP and amodal mask AP performance. In contrast, the proposed Fast
GraspNeXt is $>$3.15$\times$, $>$2.68$\times$, and $>$2.45$\times$ faster on
the Jetson TX2 embedded processor compared to ResNet-GraspNeXt, EfficientNet-
GraspNeXt, and MobileNet-GraspNeXt, respectively, while improves the
performance across all test tasks. Specifically, Fast GraspNeXt improves the
amodal bbox AP, visible mask AP, amodal mask AP, occlusion accuracy, center of
mass AP, and averaged heatmap MSE by 2.9%, 0.4%, 0.6%, 3.4%, 2.0%, and 8.7%
respectively compared to the second best results.

In terms of architectural complexity, Fast GraspNeXt is 5.2$\times$ smaller
then ResNet-GraspNeXt which has the second best amodal bbox AP and amodal mask
AP, 4$\times$ smaller then EfficientNet-GraspNeXt which has the second best
visible mask AP and center of mass AP, and 4$\times$ smaller then MobileNet-
GraspNeXt. In terms of computational complexity, Fast GraspNeXt is
5.1$\times$, 4.6$\times$, and 4.6$\times$ lower FLOPs than ResNet-GraspNeXt,
EfficientNet-GraspNeXt, and MobileNet-GraspNeXt respectively. Example ground
truth suction grasp heatmaps along with the predicted suction grasp heatmaps
produced by proposed Fast GraspNeXt are shown in Fig. 3.

As such, the above experimental results demonstrated that the proposed Fast
GraspNeXt achieves significantly lower architectural complexity and
computational complexity while possessing improved AP across test tasks
compared to designs based on state-of-the-art efficient architectures.
Furthermore, these experiments demonstrated that Fast GraspNeXt achieves
significantly faster inference time on the NVIDIA Jetson TX2 embedded
processor, making it well-suited for robotic grasping on embedded devices in
real-world manufacturing environments. Future work involves exploring this
generative approach to network architecture search for other embedded robotics
applications in manufacturing and warehouse scenarios.

#### Acknowledgements

This work was supported by the National Research Council Canada (NRC) and
German Federal Ministry for Economic Affairs and Climate Action (BMWK) under
grant 01MJ21007B.

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