aainstitutetext: Theoretical Physics Division, Physical Research Laboratory,
Shree Pannalal Patel Marg, Ahmedabad - 380009, Gujarat, Indiabbinstitutetext:
Institute for Particle Physics Phenomenology, Department of Physics, Durham
University,
Durham DH1 3LE, United Kingdom

#  Interpretable deep learning models for the inference and classification of
LHC data

Vishal S. Ngairangbam b and Michael Spannowsky
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Shower Deconstruction methodology is pivotal in distinguishing signal and
background jets, leveraging the detailed information from perturbative parton
showers. Rooted in the Neyman-Pearson lemma, this method is theoretically
designed to differentiate between signal and background processes optimally in
high-energy physics experiments. A key challenge, however, arises from the
combinatorial growth associated with increasing jet constituents, which
hampers its computational feasibility. We address this by demonstrating that
the likelihood derived from comparing the most probable signal and background
shower histories is equally effective for discrimination as the conventional
approach of summing over all potential histories in top quark versus Quantum
Chromodynamics (QCD) scenarios. We propose a novel approach by conceptualising
the identification of the most probable shower history as a Markov Decision
Process (MDP). Utilising a sophisticated modular point-transformer
architecture, our method efficiently learns the optimal policy for this task.
The developed neural agent excels in constructing the most likely shower
history and demonstrates robust generalisation capabilities on unencountered
test data. Remarkably, our approach mitigates the complexity inherent in the
inference process, achieving a linear scaling relationship with the number of
jet constituents. This offers a computationally viable and theoretically sound
method for signal-background differentiation, paving the way for more
effective data analysis in particle physics.

###### Keywords:

Large Hadron Collider, Jets, Parton Shower

††preprint: IPPP/23/81

##  1 Introduction

Unlike any other discipline, particle physics is blessed with a tremendous
amount of highly complex data that can be theoretically understood and
modelled with high accuracy. Future runs at the Large Hadron Collider (LHC)
will collect even larger amounts of data, allowing for an unprecedented setup
for developing novel data-analysis methods. In recent years, machine learning,
particularly deep learning frameworks applied to raw data, has been the most
popular approach for efficient data processing. Notably, there are already
various deep learning architectures adopted in different stages of the
analysis pipeline of the ATLAS ATL-SOFT-PUB-2018-001 ; ATL-SOFT-PUB-2018-002 ;
ATLAS:2020jip ; Aad:2861008 ; ATL-PHYS-PUB-2023-011 ; ATL-PHYS-PUB-2022-040 ;
app13053282 ; app13010086  and CMS Bols:2020bkb ; CMS:2020poo ; Pata:2750781 ;
Cagnotta:2022hbi ; Lorusso:2022xhq ; May:2022lhr  collaborations. However,
using deep learning techniques for classification tasks, i.e. discrimination
of signal and background events, has been primarily restricted to
phenomenological studies Carrazza:2019efs ; Blance:2019ibf ; Anisha:2023xmh ;
Atkinson:2021jnj ; Konar:2022bgc ; Chiang:2022lsn ; Lv:2022pme ; Badea:2022dzb
; Alhazmi:2022qbf ; GomezAmbrosio:2022mpm ; Freitas:2022cno , with the
community still resolving the more nuanced aspects of these powerful, yet at
times black-box techniques, e.g. uncertainty estimation louppe2016pivot ;
Barnard:2016qma ; Bollweg:2019skg ; Nachman:2019dol ; Golutvin:2023fle ,
reliability DeCastro:2018psv ; Araz:2021wqm ; Ghosh:2021roe ;
Feichtinger:2021uff ; Bright-Thonney:2023gdl ; Metodiev:2023izu  and
interpretability considerations Chang:2017kvc ; Faucett:2020vbu ;
Romero:2021qlf ; Das:2022cjl ; Komiske:2018cqr ; Konar:2021zdg ;
Atkinson:2022uzb ; Athanasakos:2023fhq ; pmlr-v119-bogatskiy20a ; Gong:2022lye
; Bogatskiy:2022czk ; hao2023lorentz .

The interpretation of LHC data rests on how accurately it is reproduced by
simulated pseudo-data. The pseudo-data results from a theoretical calculation
that factorises into the calculation of the so-called hard process, the parton
shower evolution and eventually hadronisation,111For a detailed review, see
for example Buckley:2011ms ; Hoche:2014rga . i.e. the reorganisation of
colour-charged partons into colour-neutral hadrons. To formalise this
interpretation process, matrix element methods have been devised, either
utilising calculations of the hard matrix element interfaced with transfer
functions Kondo:1988yd ; D0:2004rvt ; Artoisenet:2010cn ; Andersen:2012kn ;
Bury:2020ewi ; Maitre:2021uaa ; Dersy:2022bym ; Butter:2022vkj ;
Heimel:2023mvw , or even calculating the full combined hard matrix element and
parton shower process from first principles. The latter method is called
Shower Deconstruction Soper:2011cr ; Soper:2012pb ; FerreiradeLima:2016gcz  or
Event Deconstruction Soper:2014rya ; Prestel:2019neg  and constructs all
possible shower histories that connect the hard process with the measured
final state objects. It calculates the histories’ weights and sums them up to
calculate the likelihood ratio between competing hypotheses, e.g. signal
process versus background process. The Neyman-Pearson Lemma proves formally
that such a likelihood ratio is an ideal classifier. However, while Matrix
Element methods have the advantage over machine learning techniques that do
not require training, applying them to classify measured data can be
computationally costly and time-consuming as the entire likelihood calculation
must be redone for each measured event. The number of shower histories scales
approximately factorially with the number of reconstructed jets.

To address this computation bottleneck while maintaining the high
classification prowess of full-information matrix element methods, we cast the
problem as a game of constructing the best shower history–ShowerMDP. It is a
Markov Decision Process whose optimal policy constructs the most likely shower
history corresponding to a particular process for a measured set of
reconstructed objects. To showcase the method, we apply it to the
discrimination of signal top-quark fat jets versus QCD-induced fat jets,
noting, however, that it can be straightforwardly extended to include the full
event profile, as in Soper:2014rya ; Prestel:2019neg . Furthermore, we expand
this approach by showing the potential of neural networks to learn the optimal
solutions of ShowerMDP with a brute-force supervised learning approach. We
call this method AlphaPS, inspired by references Silver2016 ; Silver2017  on a
neural agent achieving superhuman capabilities for playing the game of Go.
During inference, AlphaPS scales linearly with particle multiplicity while
closely matching the discrimination power of the optimal ShowerMDP policy.
Thus, AlphaPS results in an exponential speedup over matrix element methods,
like Shower Deconstruction, and significantly expand its applicability to
high-multiplicity final states. Being based on the Lagrangian, variations of
the technique could provide ways to encode task-specific physics, like data
generation Finke:2023veq ; Butter:2023fov  and likelihood estimation
Rizvi:2023mws .

The paper is organised as follows: We briefly discuss the Shower
Deconstruction method and show that it maintains its prowess if one only
considers the largest shower history weight in section 2. In section 3, we
give a short account of Markov Decision Processes. We formulate the problem of
finding the highest weighted shower history as a Markov Decision Process in
section 4. The structure of the neural-assisted shower history construction
and numerical results for top classification are presented in section 5.
Finally, we conclude in section 6.

##  2 Event classification using Shower Deconstruction

Shower Deconstruction Soper:2011cr ; Soper:2012pb ; FerreiradeLima:2016gcz ;
Soper:2014rya ; Prestel:2019neg  constructs the likelihood ratio of a measured
set of particles based on the underlying QCD splittings and decay dynamics. To
calculate the likelihood that a signal or a background process induced a
measured final state, we construct all possible shower histories connecting
the hypothesised initial state with the measured final state objects using a
Feynman-diagrammatic approach. Then, we sum overall shower history weights for
signal and background respectively and define the likelihood ratio
$\chi_{D}(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\})$ as

$\chi_{D}(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\})=\frac{P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|S)}{P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|B)}\quad,$
(1)

where $P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|S)$ and
$P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|B)$ are the probabilities that
the reconstructed microjets $\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}$
originated from a signal or background processes. To get the total
probability, we sum over the probabilities of all possible shower histories
for a given set of particles, i.e.

$\begin{split}P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|S)&=\sum_{h}\;p(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|h,S)\quad,\\\
P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|B)&=\sum_{h}\;p(\\{q^{1}_{\mu},p^{2}_{\mu},...q^{N}_{\mu}\\}|h,B)\quad,\end{split}$
(2)

where $h$ denotes a shower history and
$p(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|h,H)$ denotes its associated
probability for a hypothesis $H\in\\{S,B\\}$.

###  2.1 Discrimination for top vs QCD: best history vs sum over histories

Figure 1: The figure shows normalized distributions of the log-likelihood
ratio for the sum over all shower histories $\chi_{D}$, and the one defined
with the ratio of the best shower histories $\chi_{0}$, for top and QCD jets.
The area under the curve (AUC) for the receiver operator characteristics for
top vs QCD discrimination by each variable is also displayed in the green box.

Often, we find a strong hierarchy in the shower history weights, i.e. a large
number of histories are orders of magnitude smaller than a small subset of
histories. Therefore, it is plausible that one can estimate $\chi_{D}$ to good
accuracy without explicitly evaluating the entire ensemble of shower histories
if the ones with large weights can be identified a priori, resulting in a gain
in evaluation time. To take this argument to the extremities, we define a new
likelihood ratio

$\chi_{0}(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\})=\frac{p(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|h^{S}_{0},S)}{p(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|h^{B}_{0},B)}\quad,$
(3)

where $h_{0}^{S}$ and $h_{0}^{B}$ are the histories with the highest
contribution to $P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|S)$ and
$P(\\{q^{1}_{\mu},q^{2}_{\mu},...q^{N}_{\mu}\\}|B)$, respectively in Eq 2.

The left of figure 1 shows the distribution of the log-likelihood ratios222The
dataset is described in Section 5.1. For jets with zero signal weight, we set
the log-likelihood to negative one-thousand which effectively serves as
negative infinity for the numerical evaluations in this work. for top and QCD
jets, showing differences visible after log scaling the $y$-axis with
$\log\chi_{0}$ shifting consistently for both classes to the right side. The
receiver-operator characteristics (ROC) curve and the area under the curve
(AUC) shown on the left exhibit comparable discrimination performance. If an
expert neural agent could construct the best shower history for a set of
microjets for each process without computing all possible shower histories,
there would be a massive reduction in computational time–a single shower
history construction scales as $O(N)$ whereas constructing all shower
histories scales as $O(N!)$, without any loss in the discriminatory power.

The rest of the paper deals with the problem of finding the highest-weighted
shower history for a set of microjets and a given process. As we shall see,
the natural framework for finding the best history is a Markov Decision
Process (MDP); we will first outline the basic machinery of MDPs in the next
section before devising one for our case.

##  3 A brief outline of Markov Decision Processes

A Markov Decision Process (MDP) generalizes Markov processes333With parton
shower generators based on Markov processes, it is self-evident that inverting
such parton showers can be cast as a Markov Decision Process.  to situations
where there is a degree of control of the environment. An agent can (at least
partially) affect its evolution. They are natural frameworks for modelling
real-world multiplayer games involving a discrete, possibly infinite, number
of time steps. The game’s rules define the environment so that an agent can
learn to prioritize actions that lead to favourable outcomes in the game via
interactions with the environment. Although reinforcement learning444See
reference SuttonRichardS2018Rlai  for a pedagogical introduction to
reinforcement learning.  as a means to find such expert agents predate
CAMPBELL200257  the modern deep learning revolution, the availability of
universal function approximators that can learn an environment’s dynamics
efficiently has led to breakthroughs in achieving superhuman proficiency in
games as complex as Go.

###  3.1 Agent-environment interaction

Figure 2 shows a Markov Decision Process can be encapsulated diagrammatically
within an agent-environment interaction. An agent can access the environment’s
state with a possible set of actions. Although for many MDPs, this choice of
action is not dependent on the state, it is not so for our case, and we will
continue with this added complication in all our discussions. The agent acts
on the environment with a particular action out of these choices, and the
environment changes its state according to a predetermined transition
probability. For the _Markov property_ to hold, this transition probability
should depend only on the current state and not on previously encountered
states. Based on the state transition, the environment also provides a reward
to inform the agent of the suitability of the applied action. This reward is
provided instantly for each step, after fixed intervals, or when the agent
reaches a particular state. In this way, the agent learns to prioritize those
actions that generate more rewards. The agent may have limited information
about the dynamics of the environment, or the dynamics may be too complicated,
making a brute-force search for the best trajectory unfeasible. In the present
case, the infeasibility of the brute-force approach with increasing microjet
multiplicity motivates employing MDPs to solve shower deconstruction.

Figure 2: The figure represents the agent-environment interaction in a Markov
Decision Process.

###  3.2 Policy and State Value function

Based on a model, an environment provides an initial state $\textbf{s}_{0}$
and the possible choices of actions $\mathcal{A}_{0}$. The agent chooses an
action $\mathbf{a}_{1}\in\mathcal{A}_{0}$ according to which the environment
provides a scalar reward $r_{1}$ and the new state $\mathbf{s}_{1}$. Denoting
the discrete time step as $\tau$, the choices of an agent is determined by the
_policy_ $\pi(\mathbf{s}_{\tau},\mathbf{a}_{\tau})$ giving the probability
that an agent chooses action $\mathbf{a}_{\tau}$ given the state
$\mathbf{s}_{\tau}$. The transition probability for a given state-action pair
will be considered deterministic in our case and, therefore, does not warrant
additional discussion. Another critical aspect of MDPs is the finiteness of
the number of steps–episodic, where there is a terminal state, and non-
episodic, where there is no terminal state. A terminal state $\mathbf{s}_{T}$
after some finite timestep $T$ is defined as one where any action produces no
state transitions and zero rewards. As our design will be episodic, the
following discussions will be limited to episodic MDPs.

The main goal of solving an MDP requires finding an optimal policy that
maximizes the long-term reward. Suppose, a given policy $\pi$ produces the
rewards $\\{r_{1},r_{2},.....r_{T}\\}$ for an episode. The total reward is
simply

$R=\sum_{\tau=1}^{\tau=T}\;r_{\tau}$ (4)

The state value $v_{\pi}(\textbf{s}_{\tau})$ of a state $\mathbf{s}_{\tau}$ is
the expected total reward if one follows the policy $\pi$. The optimal policy
$\pi^{*}$ maximizes the function $v_{\pi}$, denoted as $v^{*}$. Reinforcement
learning primarily deals with finding good approximations of the optimal
policy directly or indirectly by different techniques, but most are
essentially based on the agent-environment interaction.

##  4 ShowerMDP: the game of constructing the best shower history

In the previous sections, we outlined Shower Deconstruction and Markov
Decision Processes. One already notices that finding the best shower history
with the maximum contribution to the total weight can be cast as a Markov
Decision Process. In this section, we make this connection concrete and
formulate an MDP whose optimal solution will construct the best history. This
approach will be referred to as ShowerMDP in the following discussions.

###  4.1 Rules of the game: pQCD and inference at LHC

Due to the correspondence of a shower history with a Feynman diagram (although
only at the soft or collinear limits or the narrow width approximation), the
rules of ShowerMDP are also determined by the Standard Model Lagrangian. The
Lagrangian, as we know, is the essential ingredient in Quantum Field Theory,
and this approach, in principle, is not limited to the Standard Model and can
incorporate BSM decays of heavy particles into coloured SM partons.

We will describe the rules by visualizing an episodic construction of the
signal and background shower histories. As shown in figure 3, a shower history
for a process represents a possible realization of a Feynman diagram. At the
start of the game, we are provided with a set containing all the microjets of
the father. The agent should divide this set into two mutually exclusive
subsets whose union gives the parent. The first splitting corresponds to how
the microjets are divided into two mutually exclusive subsets corresponding to
the initial state radiation (ISR) and underlying events (UE) contamination and
those that arise from the hard parton which we call final state radiation
(FSR) set. To simplify the environment, we are only concerned with the
relative importance of the ISR microjets and the exact ISR weights as
described in reference Soper:2011cr  are calculated for the ISR set. Since the
ISR contamination can be absent, we make it possible for this ISR set to be
empty. Depending on the hypothesis, the flavour of the FSR set is assigned–top
quark for the signal (right) and gluon (left) for the background hypothesis.

Figure 3: The figure shows a single shower history of a jet containing six
microjets for the background (left) and signal (right) hypothesis.

Other than the root and the ISR edge, all other shower edges have an assigned
flavour corresponding to a particle of the Standard Model. Therefore, the
game’s rules are the Feynman rules of the Standard Model, where the degrees of
freedom depend on the splittings’ energy scale. At each stage, the player
splits the given microjet constituents into two mutually exclusive subsets
covering the initial set. The game continues until we encounter a zero-
weighted splitting or if all undivided edges contain a single microjet. The
former entails a premature end of the game without reaching the goal if there
are non-zero shower histories, while the latter represents a successful
completion of the goal. The shower histories in figure 3 illustrate completed
episodes for each hypothesis using the constructed Feynman diagrams. Even
though the Feynman rules are self-evident, we will describe them briefly for
completeness.

Two terms in the SM Lagrangian are modelled for the top quark–the $SU(3)_{C}$
gauge interaction for the gluon emissions and the $SU(2)_{L}$ gauge
interaction with the $W$ boson depicting its decay. We group the four light
flavours into a single light quark category for the gluon and treat the bottom
quark interaction separately. Therefore, we have three related terms in the
Lagrangian at $O(\alpha_{s})$–the three gluon self-interaction, $SU(3)_{C}$
gauge interaction of the gluon with light quark-antiquark pair, and with a
pair of bottom quarks. For the top quark and the gluon, the action space is a
product of the corresponding type of interaction (or flavour assignment) and
the partition of the microjet set. The cardinality of the product set will
represent the total number of possible action/splitting choices in the game
tree. In other words, we have separate actions for each flavour assignment for
the same set partition.

We will have only the multiplicity-dependent actions for the remaining
flavours as we take only one representative term from the SM Lagrangian. We
have the gluon emission term for the five light quarks. For the $W$ boson, we
decay to a quark-antiquark pair of the same generation, grouping the two
generations into a single representative term. However, there are four edge
classes for the quarks besides the top-quark: quark and antiquark edges for
the first two generations and bottom and antibottom edges. The multiplicity
requirement of the top quark and the $W$ boson edges generate zero weights
when not satisfied and is therefore not implemented as an additional rule.
Consequently, we have a uniform set-partitioning action dependent only on the
microjet multiplicity for all flavours. We can now give precise mathematical
descriptions of the action, the state, and the reward.

###  4.2 Constructing the action space

#### Set partitioning action

Let $S=\\{p_{\mu}^{i}\;|\;i\in I\\}$ be the set of microjets indexed by
$I=\\{1,2,....,n_{e}\\}$ for a given shower edge of $n_{e}$ microjet
multiplicity in the shower history. Independent of the flavour, we need to
divide this mother set into two mutually exclusive subsets which cover the
set. Each subset will be assigned a flavour depending on the context, which
will usually differ. Even if they are the same, for instance, when a gluon
splits into two gluons, the colour flow and hence the left and right colour
connected partners will differ. Therefore, we must distinguish between the two
subsets and designate $L$ and $R$ as the edge that inherits the mother’s left-
and right-color-connected partner, respectively. For the $W$ boson decay, $L$
and $R$ are defined according to which partons have a right and left color-
connected partner, respectively.

Keeping the uniqueness of the two daughter edges in mind, we define the set
partitioning action $\mathbf{a}_{\text{part.}}$, for a single action edge of
$n_{e}$ microjet constituents in the game tree as a boolean vector of $n_{e}$
dimensions. If a particular index of $\mathbf{a}_{\text{part.}}$ is one, then
the element with that label belongs to the $L$ edge. Those with zero will then
naturally belong to the $R$ edge.

#### Flavor assignment action

The flavour assignment action assigns the flavour of the $L$ and $R$ daughter
edges. Other than the top and gluon edges, these are trivial for all other
flavours. Choosing which type of splittings to follow can be cast as a
classification problem where the number of classes is given by the types of
splitting nodes possible for each edge. Therefore, we encode the flavour
assignment for a single splitting as a one-hot encoding. For the top quark, we
define

$\mathbf{a}_{\text{assign.}}^{(t\;\to
W\;b)}=[1,0]\quad,\text{and}\quad\mathbf{a}_{\text{assign.}}^{(t\;\to
t\;g)}=[0,1]\quad.$ (5)

Similarly, for the gluon we define

$\mathbf{a}_{\text{assign.}}^{(g\;\to
g\;g)}=[1,0,0]\;,\;\mathbf{a}_{\text{assign.}}^{(g\;\to
q\;\bar{q})}=[0,1,0]\;,\;\text{and}\quad\mathbf{a}_{\text{assign.}}^{(g\;\to
b\;\bar{b})}=[0,0,1]\quad.$ (6)

###  4.3 State definition

For the Markov property to hold, we need to define the state of ShowerMDP such
that the choices of actions depend entirely on the current state and not on
previously encountered ones. Therefore, the state must contain all the
information that goes into evaluating the splitting kernel and the Sudakov
factor. Here, there is a degree of dependence on the showering model. For the
splitting factors, considering the color-connected partner(s) of the current
edge suffices. In comparison, we need to encode the energy scale at which the
edge was produced for the Sudakov factor, i.e., the mother edge’s information.
These features are enough if we want to generate a parton shower where the
division of the edge’s momentum can be continuous. However, in the present
case, we are confined to discrete divisions given by the edge’s microjet
content. Therefore, this microjet content must also be included in the state
definition. Considering a modular architecture of the neural function
approximation, we call the former state the _tree state_ and the latter the
_microjet state_.

#### Tree State

The number of particles that enter the tree state definition is fixed for a
given flavour. As we will be modularising the agent for each distinct flavour
and hypothesis, we could, in principle, use a single vector representation
containing the feature of all particles. However, we define the tree state as
a point cloud for the efficient extraction and combination of information with
the microjet state, which demands a point cloud representation due to the
varying multiplicity.

For any given flavour, we have at most four shower edges entering the tree
state: the edge which will be split, its mother, and its colour-connected
partners. Except for a gluon with a known mother in the shower history, any
edge will have less than four particles in the tree state definition. Note
that we do not assign the root edge as the mother of the FSR edge for both
hypotheses since we are considering the ISR contamination in the fatjet and
not a proper splitting in the sense of other edges in later stages of the game
tree. We do not consider the tree state for the root edge since there is no
information (as will be defined in Eq. 7) in the root edge’s node feature with
all entries uniformly zero or one for any given microjet set.

Let us denote the shower edge the agent encounters at the particular game
stage by $E$, its mother edge by $M$, the left and right color-connected edges
as $LC$ and $RC$, and the jet by $J$. At this juncture, we must point out that
we are dealing with a second graph when defining the tree state. Each tree
state is a complete graph with a corresponding node for each of the four
shower edges in the game tree. This mapping is shown in figure 4. If the
flavour of the edge $E$ does not require the definition of left or right
colour-connected partners, or if the mother edge $M$ is not defined at the
particular stage in the shower history, they do not enter the tree state
definition. For coloured partons which need colour-connected partners but are
not allocated due to the incomplete structure of the shower histories, we
define soft colour partners with four vectors $q_{\mu}^{LC}$ or $q_{\mu}^{RC}$
with $0.5$ GeV transverse momentum, $0.1$ GeV mass that falls outside the
father. For the left color-connected partner, we set the pseudorapidity
$\eta^{LC}=\eta_{J}-\frac{3}{2\sqrt{2}}R_{jet}$ , and azimuthal angle
$\phi^{LC}=\phi_{J}-\frac{3}{2\sqrt{2}}R_{jet}$. For $q_{\mu}^{RC}$, we set
the angles to $\eta^{RC}=\eta_{J}+\frac{3}{2\sqrt{2}}R_{jet}$, and
$\phi^{RC}=\phi_{J}+\frac{3}{2\sqrt{2}}R_{jet}$, with $\eta_{J}$ and
$\phi_{J}$ denoting the pseudorapidity and the azimuthal angle of the jet.

Figure 4: The figure shows a schematic representation of the mapping between
the shower edges of the game tree to the nodes of the tree state. The colors
of the edges are made different to figuratively distinguish the edges in the
tree state. Note that for a gluon, there will be an additional color partner
in the tree graph and we will have a complete graph of four nodes in the tree
state.

We define a five-dimensional node feature for a shower edge $i$ in terms of
the ratios of transverse momentum $p_{T}^{i}/p_{T}^{j}$, and squared mass
$m_{i}^{2}/m^{2}_{j}$; differences in rapidity $\Delta y_{i,j}$, and azimuthal
angle $\Delta\phi_{i,j}$; and an edge embedding scalar $e_{i}$ to distinguish
between the edge’s labels. The node feature of the shower edge $E$ is with
respect to the jet

$\mathbf{h}_{E}=\left[\frac{p_{T}^{E}}{p_{T}^{J}},\frac{m^{2}_{E}}{m^{2}_{J}},\Delta
y_{E,J},\Delta\phi_{E,J},e_{E}\right]\quad,$ (7)

On the other hand, the node features for $a\in\\{M,LC,RC\\}$ is defined with
respect to $E$ as

$\mathbf{h}_{a}=\left[\frac{p_{T}^{a}}{p_{T}^{E}},\frac{m^{2}_{a}}{m^{2}_{E}},\Delta
y_{a,E},\Delta\phi_{a,E},e_{a}\right]\quad,$ (8)

where instead of the jet momentum, we have used $p_{\mu}^{E}$. For the four
shower edges, we set $e_{E}=1$, $e_{M}=-1$, $e_{LC}=-2$ and $e_{RC}=2$.

Since the top and $W$ shower edges are necessary for determining the decay
probabilities, we also include each node’s mass information in the tree
state’s node features since their mass is essential for determining the decay
probabilities. The mass information for all nodes $i$ in the tree state
definition is added as $m_{i}/100$ to keep the range of the node features
uniform.

Another information which is also relevant for the tree state is the flavour
information of the edges. We supply them as the PID Workman:2022ynf  with all
four light quarks assigned the up quark’s PID. Since these are arbitrarily
based on convention and not on some underlying physical principle, we will
utilise a neural network to learn a vector embedding of the flavour
information of each node and supply the concatenated vector with
$\mathbf{h}_{i}$ as inputs to the network processing the tree state.

#### Microjet State

The microjet state is also defined as a point cloud, with each microjet
denoted as a node. Similar to the tree state, the node feature of each
microjet $i$ is defined as

$\mathbf{h}_{i}=\left[\frac{p_{T}^{i}}{p_{T}^{E}},\frac{m^{2}_{i}}{m^{2}_{E}},\Delta
y_{i,E},\Delta\phi_{i,E}\right]\quad,$ (9)

Since we will use point-transformers for the microjet set, we will not put any
additional graph construction on the microjet set. Instead, we will use the
complete graph and learn the importance of the different microjets with the
attention mechanism.

###  4.4 Modeling the reward

For episodic games, the reward of each non-terminal state is determined after
the end of the episode. As we will not be doing exploration-based learning, we
define the reward of each non-terminal state as the shower weight evaluated
after the episode ends (either prematurely via a zero-weight splitting or
after all non-ISR leaves contain exactly one microjet.) Such a reward
structure can extract the optimal shower history $p_{h_{0}}$ for each
hypothesis.

##  5 AlphaPS: solving ShowerMDP with supervised learning

Having laid the groundwork, we present numerical results of the supervised
training procedure used to solve the ShowerMDP, taking top-tagging as an
example. We call this network AlphaPS, directly inspired by references
Silver2016 ; Silver2017  for mastering the game of Go. Our supervised approach
bears similarities to AlphaGo Fan Silver2016 , with the differences arising
because of the variable action space, availability of an exact environment
model, and the single-player nature of ShowerMDP. The reason for reverting to
a supervised training method is as follows. Unlike the application of
reinforcement learning in other domains, we have a concrete benchmark to which
we need to compare–the best shower history given by the brute force search,
which should not be sidestepped in a proof-of-principle analysis. In other
words, checking that a supervised training procedure can learn the optimal set
of splittings when directly provided is necessary before we attempt a
reinforcement learning procedure relying on the agent-environment interaction
where the agent needs to find the best set of actions via explorations.

Exploitation versus exploration–whether to exploit the current best actions or
explore previously unexplored trajectories that may yield higher rewards, is a
subject of close study in any reinforcement learning situation. Situations,
where there is no exact environment model, demand a reinforcement learning
procedure to search for possible optimal policies with explorations of the
environment. In our case, where we have a precise ShowerMDP model, a
reinforcement learning procedure is justifiable only if the exploration stage
uses fewer resources than the brute force strategy. Therefore, we need to
devise an efficient exploration algorithm, leading to lower computational
expenditure with an efficient learning capability. This resource reduction is
all the more important for extending the method to higher multiplicities. In
the present case, we discard exploitation entirely since the supervised
approach can be regarded as the agent acting after a complete exploration of
the environment.

###  5.1 Details of the dataset

We study the ability of the neural function approximation of the optimal
ShowerMDP policy for QCD (background) vs top (signal) jets at 14 TeV LHC.
Parton-level dijet and top pair production processes for the background and
signal were generated using MadGraph5_aMC@NLO (v3.5.0) Alwall:2014hca  in the
five-flavour scheme with a minimum transverse momentum cut of 380 GeV on the
leading jet and top quarks. The top quarks were decayed into the fully
hadronic channel using MadSpin Artoisenet:2012st . These parton-level events
were showered and hadronised with Pythia8 (v8.306) Bierlich:2022pfr  along
with multi-parton effects. To facilitate b-tagging within the jet, which is an
important factor in the discrimination power of shower deconstruction, we keep
all B-mesons stable by setting the user hook mayDecay=off. All stable
particles after hadronisation are clustered into jets of radius R=1.0 with the
anti-$k_{t}$ algorithm implemented in FastJet (v3.4.0) Cacciari:2005hq ;
Cacciari:2011ma . We select the leading jet in an event if it is within
$|\eta|<3$ and $p_{T}\in(400,500)$.

To keep the multiplicity under control for inputs to the brute-force
extraction of shower weights, we recluster the jet constituents with the
Cambridge-Aachen Dokshitzer:1997in  algorithm and a seed radius of
$R_{0}=0.1$, which is increased in steps of 0.01 until the clustering yields
at most eight inclusive microjets with $p_{T}>5$ GeV. Jets with less than
three microjets are not selected as they trivially yield zero signal weights
under shower deconstruction. After neglecting these jets, we have 500k, 200k
and 100k jets for each class in the training, validation, and test datasets,
respectively. We implement bottom-tagging in these microjets by assigning a
b-tag to the nearest microjet within $\Delta R<0.2$ of a B-meson. These
microjets go into the ShowerMDP. Many jets will produce zero signal weights as
combinatorial realisation may not satisfy the top and $W$ mass windows. Due to
the nature of the physics-transparent weight construction, these
configurations are statistically more likely for QCD jets and almost
negligible for top jets. We do not extract the shower history states for both
hypotheses and will employ a value network to filter these jets with zero
signal weights.

For jets passing the zero-weight check, we evaluate the $\log\chi_{D}$ and
$\log\chi_{0}$ and extract the shower history corresponding to $\chi_{0}$ for
both hypotheses used to train the policy network modules.

###  5.2 Neural assisted construction of the best shower history

To construct the best shower history, we learn two parametrised functions with
neural networks–the value network and a composite policy network. Here, we
concentrate on the essential features and expand on the architecture’s detail
and training methodology in appendix A.

The value network takes the initial state $s_{0}$ of the root edge, which
consists of the microjet state alone and is trained on all jet samples. We
teach the network to predict -1 (+1) for those jets with zero (non-zero) total
weight. During inference, if the network output is negative for a given jet,
we set $\chi_{0}$ to zero, and the microjet set does not undergo any history
construction. Given the complexity of the brute-force construction of all
background histories and the combinatorial nature of the assigned divisions
for the signal histories, which may go till the $W$ boson decay (on the third
level of the tree), this step already saves a considerable computational cost,
dependent, of course, on the complexity of the value network.

The policy network consists of different submodules for specific hypothesis-
flavor pairs. For the background hypothesis, we combine the bottom,
antibottom, quark, and antiquark flavours into a single module due to their
similarity. Since the bottom quark originates from the top quark for the
signal, we separate the bottom module for the signal while still using a
combined module for the other three flavours.

To better handle the variable multiplicities, we use message-passing neural
networks, and therefore, the network supplies a binomial probability of the
microjet assignment action for each node in the microjet state. The gluon and
top quark modules also have an additional softmax output for the flavour
assignment actions, which has a constant dimension regardless of the microjet
multiplicity. The extracted state-action pair for all non-zero histories of
both classes are divided into their specific hypothesis-flavour pair, and the
modules are trained independently.

We have two episodes for each hypothesis on those jets with positive
predictions by the trained value network during inference. Starting from the
root node, we consider the assignment of each microjet as a Bernoulli
distribution given by the policy network’s microjet assignment action. An
action is sampled from this distribution and applied to the initial microjet
set. For the top quark and gluon sets, we independently sample the flavour
assignment action via the split action output of the policy. The process
continues until we get a complete shower history or a zero-weighted action.
The latter ends the episode prematurely. Due to the stochastic nature of the
NN-assisted construction, we extract the shower weights for one thousand
episodes and study the dependence of the maximum shower weights on the number
of repetitions. Due to the independence of the Bernoulli probabilities, we can
have instances (although relatively rare) where all of them are either very
close to zero or one. A random sampling using these probabilities would not
produce a correct set partition, and we set the upper limit to produce viable
set partitions to one hundred trials. Suppose we cannot get a proper set
partitioning action after a hundred trials. In that case, a possible action is
implemented by either setting the maximum (minimum) valued probability to one
(zero) when all sampled actions are zero (one).

Figure 5:  We show the percentage of non-zero signal weighted jets with
$\delta<10^{-8}$ varying with the microjet multiplicity $N_{\text{micro.}}$,
for the signal (right) and background (left) hypothesis.

####  5.2.1 Comparisons to the optimal policy

As AlphaPS aims to approximate the optimal ShowerMDP policy rather than
finding the best possible discriminant, we first compare the accuracy of the
approximated best shower history with the optimal one. Denoting the
approximated history as $h_{*}$, the relative error of the
approximation555Although $p_{h_{0}}\geq p_{h_{*}}$, we still define it with
the absolute value as precision errors may numerically violate this condition.
is

$\delta=\frac{|p_{h_{0}}-p_{h_{*}}|}{p_{h_{0}}}$

The percentage of events666These percentages practically stays constant from
$\delta<10^{-2}$ to $\delta<10^{-15}$ decreasing thereafter due to precision
errors. with $\delta<10^{-8}$ as a function of the microjet multiplicity of
the fatjet $N_{\text{micro.}}$, for different number of episodes
$N_{\text{episodes}}$, is shown in figure 5. The relatively lower starting
position for the QCD samples may be attributed to the relatively higher number
of non-zero shower histories for the top dataset biasing the learning process.
Due to this inherent bias, we will consider the features of the top datasets
while discussing the overall features, as unbiased training should ideally
result in the QCD sample’s behaviour resembling that of the top samples.
Additionally, the neural agent does not explore all possible histories over
different episodes as they are independent, dictated only by the neural
agent’s policy for each encountered state. A fallback to the brute-force
approach should be the preferred option for the low multiplicity cases where
the total shower histories are less than the $N_{\text{episodes}}$.

For the signal hypothesis on the right, one sees a decreasing trend, which can
be understood via the increasing search space complexity with increasing
microjet multiplicity. The situation persists to a weaker extent for the
background hypothesis on the left for $N_{\text{episode}}\in\\{10,100\\}$,
with the curve almost flat for $N_{\text{episodes}}=1000$. This trend can be
understood from the relative differences in the distribution of non-zero
weighted history for the background hypothesis compared to that of the signal.
Imposing the Breit-Wigner shape on the top and W decays makes the shower
weights fall relatively faster than the imposition of QCD splittings alone for
the background hypothesis. Therefore, for the same amount of episodes, an
agent can find shower histories which closely approximate the optimal shower
history’s weights and with $N_{\text{episodes}}=1000$, the neural agent has
enough trials to have a very weak dependence on the scaling of the microjet
multiplicity till $N_{\text{micro.}}=8$. It is important to reiterate that
such a situation arises because the policy network has inductively learned the
characteristics of the optimal policy, which reduces the sampling process to
the vicinity of the highest weighted history.

####  5.2.2 Discrimination with neural agent

Figure 6: The figure shows the distribution of $\log\chi_{0}$ of the optimal
history (after a brute force search), the neural network assisted history
construction, and randomly constructed history of QCD and top jet samples.

Figure 6 shows the distribution of $\log\chi_{0}$ evaluated with the best
shower histories and the neural-assisted agent for different number of
episodes $N_{\text{episodes}}$. The overlap between the optimally extracted
$\log\chi_{0}$ increases with increasing $N_{\text{episodes}}$, pointing
towards the agent learning to construct the individual shower histories
correctly. The corresponding ROC curves and the AUC are shown on the right. As
expected, the power of the constructed history increases with increasing
$N_{\text{episodes}}$. The scaling of the AUC with increasing
$N_{\text{episodes}}$ is relatively fast with $N_{\text{episodes}}=30$ giving
an AUC of 0.842, which is around 1.5% lower than the optimal AUC of 0.855.
However, there is a limit to the possible discrimination due to the neural
approximation as the AUC has already reached a plateau for
$N_{\text{episodes}}=500$ with a slight increase to for
$N_{\text{episodes}}=1000$.

##  6 Conclusions

Our study presents a novel approach to the classification and inference of LHC
data, addressing the computational challenges posed by the combinatorial
explosion in the Shower Deconstruction method. By casting the problem of
identifying the highest weighted shower history as a Markov Decision Process
(MDP), we have demonstrated the practical application of a modular point-
transformer architecture for learning the optimal policy. This neural agent,
AlphaPS, has shown remarkable capability in constructing the highest weighted
shower history, generalising well to unseen test data and significantly
reducing complexity during inference.

The results from AlphaPS indicate a substantial improvement in computational
efficiency, scaling linearly with constituent multiplicity. This significantly
advances traditional matrix element methods, enabling practical applicability
to high-multiplicity final states. Furthermore, our method maintains
discrimination power comparable to the optimal ShowerMDP policy, showcasing
the potential of deep learning techniques in particle physics data analysis in
a physics-transparent approach.

This study opens several avenues for future research. Extending this framework
to include a broader range of particle physics phenomena could provide deeper
insights into LHC data interpretation. Secondly, further refinement of the
neural network architecture and training process with exploration-based
reinforcement learning could yield more efficient, accurate models and
generalisations to higher multiplicities. Lastly, exploring the integration of
this approach with other machine-learning techniques could lead to a more
holistic and powerful tool for particle physics data analysis.

We demonstrate the promising synergy between deep learning and particle
physics, offering a novel perspective in interpreting complex LHC data and
setting a precedent for future explorations in this interdisciplinary field.

## Acknowledgements

M.S. and V. S. N. are supported by the STFC under grant ST/P001246/1.
Computational work were performed on the Param Vikram-1000 High Performance
Computing Cluster and TDP resources at the Physical Research Laboratory (PRL).

##  Appendix A Details of Network architecture and training

In the appendix, we detail the network architecture and the training procedure
for the neural function approximation of the optimal policy. Independent of
the input and output dimensions, all multilayer-perceptrons (MLPs) in our
architecture have two hidden layers of 128 nodes and ReLU activation
functions. All networks are implemented and trained using PyTorch-Geometric
Fey/Lenssen/2019

###  A.1 Value Network

The value network determines whether a given fatjet, with its microjet set,
should undergo the construction of shower histories based on the finiteness of
its signal weight. Since the task of identifying microjet sets with no
combinatorial divisions in the top or $W$ mass window is relatively easier
than understanding the evolution of the microjets based on perturbative QCD
splittings, we employ a simple deep sets architecture for the value network.
It consists of two MLPs: the first maps the microjet features to a
128-dimensional latent node representation. We perform a mean readout of these
node features, which feeds to the second MLP. The output of this second MLP is
made to fall within $(-1,1)$ via a hyperbolic tangent activation function.

###  A.2 Policy Network

The flavour policy modules have two distinct parts: a message-passing block
and downstream processing of the message-passing block’s output specific to
the nature of the action space. In the following passage, we discuss these two
stages separately.

##### Message Passing Blocks:

Two message-passing-neural-network(MPNN)-blocks are common for the policy for
all flavours except the root edge, one each for the microjet state (micro-
MPNN) and the tree state (tree-MPNN). A flavor-embedding MultiLayer Perceptron
(FE-MLP), which has two hidden layers of thirty-two nodes and ReLU activation,
takes the PIDs of the tree state’s nodes and embeds them into a four-
dimensional vector. The concatenated vector formed by the tree state’s node
features (Eq 9) and the four-dimensional flavour embedding feeds as the node
features to the tree-MPNN. The policy for the root edge has a single MPNN for
the microjet state.

The MPNN blocks are based on the PointTransformerConv Zhao_2021_ICCV , which
automatically learns vector attention for each node with every other node in
the graph. This attention mechanism is advantageous for the microjet set,
where, without any input in the graph construction, the network automatically
learns the interrelations needed for the division of the microjets. The micro-
MPNN and the tree-MPNN have two message-passing operations for all relevant
flavours and hypotheses except the top quark’s micro-MPNN and the micro-MPNN
of the gluon flavour module for the background hypothesis, where we found an
improved accuracy when utilizing three message-passing operations. Due to the
computational demands of increasing message-passing operations, we did not
consider more operations for other flavours.

The point transformer operation updates the node features
$\mathbf{h}_{i}^{(l)}$ at a message-passing stage $l$ to

$\mathbf{h}_{i}^{(l+1)}=\sum_{j\in\mathcal{N}(i)}\,\mathbf{w}_{ij}\,.\,\left(\alpha(\mathbf{h}^{(l)}_{j})+\delta(\mathbf{p}_{i}-\mathbf{p}_{j})\right)\quad,$
(10)

where $\mathcal{N}(i)$ is the neighbourhood of the node $i$, $\mathbf{w}_{ij}$
are the attentions weights, $\alpha$ is a linear layer, and $\mathbf{p}_{i}$
and $\mathbf{p}_{j}$ are the coordinate inputs for the position MLP $\delta$.
The weights $\mathbf{w}_{ij}$ are calculated as

$\mathbf{w}_{ij}=\rho\left(\gamma(\,\phi(\mathbf{h}^{(l)}_{i})-\psi(\mathbf{h}^{(l)}_{j})+\delta(\mathbf{p}_{i}-\mathbf{p}_{j})\,)\right)\quad,$
(11)

with $\phi$ and $\psi$ denoting linear layers, and $\gamma$ denoting our
choice of mapping function. $\rho$ is a SoftMax normalization over the $j$
axis. All point-transformer convolutions in the policy and the value network
outputs a 128-dimensional node feature. The $\delta$ MLPs take the input node-
features’ two-dimensional rapidity-azimuth information and map it to the
128-dimensional output feature space. For $\gamma$, we use MLPs that map the
128-dimensional vector in the argument to a 128-dimensional output node
feature space through which the attention weights are calculated for each
feature dimension separately. To enable the downstream network to segregate
information at different scales, we take the output of each message-passing
operation and form a concatenated $L\times 128$ dimensional node feature
vector as the output of an MPNN block, with $L$ being the number of message-
passing operations.

If present, the output node features of the tree-MPNN undergo a mean graph
readout. This graph representation gets concatenated with each node feature
output of the micro-MPNN, which acts as the input to all the downstream
networks.

##### Downstream Architecture:

For all flavours, the policy network has a microjet-assignment-MLP (MA-MLP),
which takes the combined node representation and outputs a single value in
$(0,1)$ as the node’s set partitioning probability. We implement a sigmoidal
activation for the output of this MLP to have a probabilistic interpretation.
We have two additional MLPs for the flavour assignment of the gluon and top
quark. The first MLP takes the combined representation and outputs a node-
level feature embedding. We perform a mean graph readout of the node
representations and feed it into the second MLP, which outputs a fixed
dimensional array of $2$ or $3$-dimensional matrix depending on the flavour.
These matrices undergo a softmax normalization for the probabilistic
interpretation.

###  A.3 Training

Flavor |  | Best Accuracy
---|---|---
Value |  |
NA |  | 0.9708
Signal Hypothesis |  |
Root |  | 0.8274
Top |  | 0.6638
$W$ |  | 0.7575
Gluon |  | 0.9077
Bottom |  | 0.8863
Quarks |  | 0.9357
Background Hypothesis |  |
Root |  | 0.5533
Gluon |  | 0.7513
Quarks |  | 0.9057
Table 1: The table shows the accuracy of the best epoch on the validation
dataset for the value network and the action prediction of the flavour policy
modules.

All networks are trained on the extracted data independently for each flavor-
hypothesis pair for a hundred epochs using the Adam optimizer
DBLP:journals/corr/KingmaB14  initialized with a learning rate of 0.001. The
learning rate decreases by half if the validation loss has not decreased for
three epochs. We use the binary and categorical cross-entropy as the loss
function for the microjet and flavour assignment actions, respectively, while
for the value network, we use mean squared error. We use the epoch with the
minimum validation loss for inference.

We can estimate the learned understanding for the different modules by
extracting the accuracy of the predicted value, the predicted set
partitioning, and flavour assignment actions on the validation dataset. For
the value, this is done by assigning -1 to negative outputs and +1 to positive
outputs. We convert the fuzzy logical output of a sigmoid set partitioning
probability $p$ via the boolean expression $p<0.5$. At the same time, for the
split assignment action, we transform the softmax output vectors to one-hot
encoded vectors by assigning one to the index with the largest entry.

The accuracy of the best epoch on the validation dataset for the various
modules is shown in Table 1. Due to the relative ease of identifying jets
falling outside the top and $W$ mass window, the value network has a high
accuracy of 0.9604. The relative accuracy of the various policy modules helps
estimate the difficulty level in learning the optimal actions for the
respective flavour-hypothesis pair. For each hypothesis, the flavour of the
first FSR set, the top quark for the signal, and the gluon for the background
have the least accuracy. To understand this, we note that the action choice
for these FSR flavours depends on the tree’s full depth, which would be more
complex than other flavours. Consequently, the signal hypothesis’s gluon
accuracy is relatively much better due to its occurrence at later stages of
the shower tree than the top quark. For all other flavours, the best accuracy
is somewhat similar, pointing towards the easier choice of optimal actions due
in part to the lower diversity of the encountered edges further down the
shower history.

Note that the accuracy of the root edge is much better than the FSR edge’s
flavour for the signal hypothesis, which can be understood if we recall that
the larger multiplicity of the top decay demands more microjets in the FSR
sets, with the highest factor mainly arising from the initial set’s
constituents going to the FSR set, which is easily learned by the policy
module. On the other hand, the division of the background root edge into ISR
and FSR subsets is highly non-trivial, and the root edge’s accuracy is not as
good as that of the signal. This accuracy, being close to 50%, is a
significant improvement from a randomly acting agent whose accuracy scales as
the inverse of the number of possible combinatorial divisions of the microjet
set into two distinct subsets.

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