# Data Efficient Contrastive Learning in Histopathology using Active Sampling

Tahsin<EMAIL_ADDRESS>and David S. Smith1,2
_1\. Vanderbilt University, Department of Electrical and Computer Engineering;
2. Vanderbilt University Medical Center, Department of Radiology and
Radiological Sciences_

###### Abstract

Deep Learning based diagnostics systems can provide accurate and robust
quantitative analysis in digital pathology. These algorithms require large
amounts of annotated training data which is impractical in pathology due to
the high resolution of histopathological images. Hence, self-supervised
methods have been proposed to learn features using ad-hoc pretext tasks. The
self-supervised training process is time consuming and often leads to subpar
feature representation due to a lack of constrain on the learnt feature space,
particularly prominent under data imbalance. In this work, we propose to
actively sample the training set using a handful of labels and a small proxy
network, decreasing sample requirement by 93% and training time by 62%.

###### Index Terms:

Digital Pathology, Deep Learning, Active Learning, Contrastive Learning, Self
Supervised Learning

##  I Introduction

The recent renaissance of self-supervised learning (SSL) began with
artificially designed pretext tasks, such as relative patch prediction [1],
solving jigsaw puzzles [2], colorization [3] and rotation prediction [4].
Recently, contrastive learning (CL) has emerged as a simple yet promising tool
in SSL [5]. It has been successfully applied to cancer detection, segmentation
in histopathology images [6]. In practice, the CL model model is trained using
a large unlabeled data pool, and the extracted features from this large model
are used to train a smaller linear model, and its prediction is evaluated on a
test set to measure its performance (linear evaluation) [5].

The training phase is slow and learned latent space is unconstrained, which
could be suboptimal for histopathological datasets. Histopathological datasets
often have a lot of confounding image sections consisting of background and
debris alongside normal tissue area. Additionally, the tissue of interest is
normally the cancerous tumor area which is often a minority class. As a result
the model either takes too many iterations to learn the minority class
examples or, even worse, ignores the minority class and uses all its capacity
to learn the non-tumor cells. We propose to solve these inefficiencies by
borrowing techniques from active learning.

Active learning [7, 8] allows humans to interactively improve algorithms. The
algorithm is trained iteratively using an algorithmically sampled data subset
(active sampling) which is labeled by an oracle (e.g., human annotator). In
each iteration, new informative subset of samples is chosen to further train
and improve model predictions Fig. 1. Multiple datasets have been produced by
user refined inputs such as automatic teller machine, self-driving cars,
automatic content tagging in online platforms [9]. Active learning has proven
useful in annotating data when specialist input is required, e.g. medical
images [10].

To decrease data redundancy, we propose to train the network iteratively using
batches of unlabeled data. The first batch is randomly selected and after
training the model on each batch, we select the next batch of data from the
unlabeled pool using a proxy model and a sample selection strategy and append
it to the current batch.

Our work is related to [11] where the latent class centers are approximated
utilizing an external pretrained model CLIP [12] and chooses a subset of
sample which lies near the cluster centers. We do not use such a pretrained
model. Rather we use our CL model features for informative sample selection.
Researchers in [13] used a small proxy network to select samples in an active
learning setup. But they consider active learning in the supervised domain
(they require an oracle for every iteration of the active learning loop). Our
work can be considered as an extension of their framework to the self
supervised domain. Furthermore, this is the first work that considers data
efficiency in CL for the domain of histopathology images (be specific how this
helps to make it novel, does the algorithm change, can we include some domain
related tricks, etc.). The contributions of our work are to

  * •

increase CL efficiency in terms of time and sample complexity,

  * •

provide a method to constrain the model into learning relevant representations
by using active sampling, and

  * •

improve CL efficiency in the histopathology domain.

Figure 1: Active learning loop [14]. An oracle annotates the most informative
samples which is used to refine model predictions.

##  II Method

Figure 2: SimCLR framework introduced in [5]. The model minimizes the distance
(maximizes agreement) between feature representation of two augmented views
$\tilde{x}_{i}$ and $\tilde{x}_{j}$ of the same image. Figure 3: The proposed
framework speeds up the contrastive sampling process by actively selecting
informative samples with the help of a small proxy network.

In this section, we describe the components in our proposed framework and
formalize the methodology.

###  II-A Contrastive Learning

SimCLR [5] proposed a simple framework for contrastive learning of visual
representations by maximizing agreement between differently augmented views of
the same sample via a contrastive loss in the latent space. Consider an
unlabeled image pool $U$ and $x$ are the image samples $x\in U$. In CL, a
minibatch of $N$ samples are randomly sampled and each sample is applied with
random data augmentation operations, resulting in $N$ pairs of augmented
samples or $2N$ augmented samples in total:

$\displaystyle\tilde{x}_{i}=t(x),\tilde{x}_{j}=t^{\prime}(x),t,t^{\prime}\sim\mathcal{T},$
(1)

where two separate data augmentation operators, $t$ and $t^{\prime}$, are
sampled from the same family of augmentations $\mathcal{T}$. Given one
positive pair, other $2(N-1)$ data points are treated as negative samples. The
representation is produced by a base encoder $f(\cdot)$:

$\displaystyle h_{i}=f(\tilde{x}_{i}),h_{j}=f(\tilde{x}_{j}).$ (2)

The encoded representation is passed through a projection head $g(\cdot)$
which is a series of three fully connected layers. Experiments show that a
projection head improves the quality of the encoded features $h$ [5]:

$\displaystyle z_{i}$ $\displaystyle=g(h),z_{j}=g(h_{j}).$

The CL loss $\mathcal{L}^{i,j}$ is defined using cosine similarity on the
output of the projection head:

$\displaystyle\mathcal{L}^{i,j}$
$\displaystyle=-\log\frac{\exp\left[\operatorname*{sim}(z_{i},z_{j})/\tau\right]}{\sum_{k=1}^{2N}(1-\delta_{ik})\exp\left[\operatorname*{sim}(z_{i},z_{k})/\tau\right]},$

where $\delta_{ik}$ is the Kronecker delta function which evaluates to 1 if
$i=k$ and 0 otherwise. Note that, after training is completed, we throw away
the projection head $g(\cdot)$ and use encoder $f(\cdot)$ and representation
$h$ to train a proxy model.

###  II-B Proxy Model

The proxy model $f_{p}(\cdot)$ is a neural network with a single fully
connected layer. We extract image features using the CL model encoder and
train the proxy model on these features. The output of the proxy model is used
to select informative samples from the unlabeled image pool.

###  II-C Active Sampling

We extract the image features using the CL model and use active sampling
strategies to select most informative samples. For this paper we use
uncertainty sampling and coreset sampling.

####  II-C1 Uncertainty Sampling.

Uncertainty sampling is the most common sampling criteria in which an active
learner selects the instances about which it is least certain how to label.
The uncertain samples normally reside near the class decision boundaries. We
use entropy [15], the most common measure of uncertainty:

$\displaystyle H(x_{i})\equiv\sum_{j}{P(y_{j}\mid x_{i};\theta)\log
P(y_{j}\mid x_{i};\theta)},$ (3)

where $y_{j}$ ranges over all possible labels and $x_{i}$ is data. Entropy is
an information-theoretic measure that quantifies information content.

####  II-C2 Coreset Sampling.

Coreset formulates the active sampling problem as choosing a subset of samples
that spans the dataset [16]. The subset selection problem is formulated as a
k-center problem using a greedy approximation shown in Algorithm 1.

Algorithm 1 k-Center Greedy [16]

Input: data $\mathbf{x}_{i}$, existing unlabeled image pool $\mathbf{s}^{0}$
and a budget $b$

Initialize $\mathbf{s}=\mathbf{s}^{0}$

repeat

$u=\arg\max_{i\in[n]\setminus\mathbf{s}}\min_{j\in\mathbf{s}}\Delta(\mathbf{x}_{i},\mathbf{x}_{j})$

$\mathbf{s}=\mathbf{s}\cup\\{u\\}$

until $|\mathbf{s}|=b+|\mathbf{s}^{0}|$

return $\mathbf{s}\setminus\mathbf{s}^{0}$

Uncertainty sampling focuses on samples in the decision boundary, while
coreset focuses on sample diversity. These are the two most common sampling
methods used in the literature. A complete list of existing strategies can be
found in [14].

###  II-D Proposed Framework

The proposed framework is depicted in Fig. 3. Let $U^{t}$ be the unlabeled
image pool at iteration $t$, $S_{L}$ is a small set of images with labels and
$S_{\mathrm{test}}$ is the test set. We sample a initial random pool of images
$S^{t}\subset U^{t}$ of budget $b$ and update the unlabeled image pool,
$U^{t}\leftarrow U^{t}\setminus S^{t}$. The subset $S^{t}$ is used to train
the SimCLR model. We freeze the encoder $f(\cdot)$ and extract features
$h_{i}$, for $x_{i}\in S_{L}$. A proxy model $f_{p}^{t}$ was trained using
features $h_{i}$. This proxy model was used to find the next subset of images
$S^{t+1}$ to be included in the CL training set, $S^{t}\leftarrow S^{t}\cup
S^{t+1}$. If uncertainty is used as sampling criterion, the entropy $H(x_{i})$
is computed using outputs of $f_{p}^{t}(\cdot)$ and we select the top $b$
samples with highest uncertainty. If coreset is used, we select $b$ samples
following the k-center greedy algorithm presented in Algorithm 1. This
iterative process is continued for a given number of iterations $T$. Note that
the weights of the SimCLR model are randomly initialized in the first
iteration and training continues in the subsequent iterations using the
learned weights in the previous iteration. The proxy network, however, is
learned from random weights in each iteration. The set $S^{t}\cup S^{t+1}$ is
used for the next iteration of CL training. This process continues for a set
number of iterations $T$. The pseudocode for the algorithm of our proposed
framework is presented in Algorithm 2.

Algorithm 2 Contrastive Learning with Active Sampling

1: Input: $S_{L}$, $U^{t}$, $g\circ f$ $f_{p}^{t}$, $T$

2: Output: trained encoder $f(.)$

3: Initialize:$S^{0}\leftarrow\text{Random sampling on}~{}U^{0}$

4:

5: for $t=0$ to $T-1$ do

6: Train $g\circ f$ using $S^{t}$

7: Train $f_{p}^{t}$ using $S_{l}$

8: $U^{t}\leftarrow U^{t}\setminus S^{t}$

9: $S^{t+1}\leftarrow\text{Active Sampling on}~{}U^{t}$

10: $S^{t}\leftarrow S^{t}\cup S^{t+1}$

11: end for

##  III Experiment and Results

###  III-A Dataset

We used the Kather-19 dataset presented in [17]. The train set consists of
$100,000$ non-overlapping image patches from hematoxylin and eosin (H&E)
stained histological images of human colorectal cancer (CRC) and normal
tissue. Tissue classes are: adipose (ADI), background (BACK), debris (DEB),
lymphocytes (LYM), mucus (MUC), smooth muscle (MUS), normal colon mucosa
(NORM), cancer-associated stroma (STR), colorectal adenocarcinoma epithelium
(TUM). These images were manually extracted from 86 H&E stained human cancer
tissue slides from formalin-fixed paraffin-embedded (FFPE) samples from the
NCT Biobank (National Center for Tumor Diseases, Heidelberg, Germany) and the
UMM pathology archive (University Medical Center Mannheim, Mannheim, Germany).
Tissue samples contained CRC primary tumor slides and tumor tissue from CRC
liver metastases; normal tissue classes were augmented with non-tumorous
regions from gastrectomy specimen to increase variability. The test set
consists of $7180$ image patches from $50$ patients with colorectal
adenocarcinoma (no overlap with patients in train set) collected from NCT
tissue bank. Both the train and test set patches are $224\times 224$ pixels at
$0.5$ microns per pixel. All images are color-normalized using Macenko’s
method [18]. We created an imbalanced dataset by setting the TUM tissues as
the positive class and combining all the non-tumor (NON-TUM) as negative
class. The train set had $14,317$ TUM patches and $85,683$ NON-TUM patches.
While the the test set had $1233$ TUM patches and $5947$ NON-TUM patches.

###  III-B Experimental Details

The SimCLR model was trained using $224\times 224$ input images. To create two
views of the same image the following augmentations were used: color jitter,
random crop, random grayscale, Gaussian blur, horizontal and vertical flip,
random rotation by $\pm 90$ deg. The image intensities were divided by $255$
and normalized by the ImageNet mean and standard deviation.

The training set is randomly split into two sets: an unlabeled pool $U$ of
$99,000$ images and $1000$ to train a proxy model (randomly sampled from the
full train set). The original test set of size 7180 is used to evaluate the
proxy model. The size of the subset sampled at each iteration for SSL is 1000,
and the number of iterations, $T$, is 20. The CL model $f_{\theta}$ was a
Resnet18 encoder [19] connected to a projection head that consisted of 3 fully
connected layers with 128 units, the proxy model $f_{p}$ was a linear layer
with 512 units. Note that at each iteration, active sampling is done on a
randomly selected subset of 10,000 to reduce computation and data redundancy.
As the benchmark, we trained the SimCLR model for 100 epochs using the full
unlabeled pool $U$ and evaluated its feature quality by training the proxy
model on the small labeled dataset. The proxy model was trained for 200
epochs, and its performance was evaluated at 20 epoch intervals on the test
set. Then we trained the SimCLR model following our proposed framework using a
subset of images. We considered uncertainty, coreset sampling as our active
sampling methods. Also, random sampling of subsets was included as an ablation
study to observe the absence of active sampling. At each iteration we trained
the SimCLR model for 100 epochs and the proxy model for 40 epochs. The
performance of the proxy model was evaluated at the end of each iteration. For
both the experiments, the SimCLR model was trained with a learning rate of
0.0001 and the proxy model with a learning rate of 0.001. The loss used for
SimCLR was normalized cross-entropy with temperature 0.1, and for the
classifier we used binary cross-entropy loss. For both models we used the Adam
optimizer [20] with $\beta_{1}=0.9$, $\beta_{2}=0.999$. The training batch
size is 128. Each experiment was run three times, and the average performance
is reported.

###  III-C Metrics

The evaluation metric used is the F1 score which accounts for the imbalance in
the dataset. The F1 score is the harmonic average of precision (P) and recall
(R).

$\displaystyle
F1=\frac{2\mathrm{RP}}{\mathrm{R}+\mathrm{P}};R=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}},P=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}$
(4)

Here, TP is true positive, FN is false negative, FP is false positive. F1
score is chosen due to the imbalance in the positive and negative classes in
the images. We also computed the average run time for the benchmark and
proposed method to reach a certain F1 score.

###  III-D Results

For the benchmark experiment we observed a highest F1 of 0.84. To reach
similar score the uncertainty and coreset based sampling strategy takes 7000
and 11,000 samples respectively (Fig. 4). Thus sample requirement is decreased
by 93% and 89% respectively. Also, we observed that random sampling performs
worse compared to the active sampling methods. The benchmark achieved its
highest score at 100th epoch of SimCLR training and at the 150th epoch of the
proxy model training. The average runtime for 100 epochs of SimCLR training is
538.70 min. To reach similar scores, the uncertainty based active sampling
experiment required 7 iterations (7000 samples). The average time to train the
SimCLR model for 7 iterations was 196.88 min. Average time to train proxy
model and sample selection was 7.16 min. This resulted in a total average
runtime of 204.4 min.

Similarly, k-center based sampling experiment required 11 iterations (11,000
samples). The average time to train the SimCLR model was 514.95 min. Average
time to train proxy model and sample selection was 23.70 min. This resulted in
a total average runtime of 538.65 min. Due to uncertainty sampling there is a
time reduction of 62% while coreset sampling requires similar amount of time.
Further hyperparameter (such as SimCLR learning epochs, proxy model learning
epochs, etc.) tuning might decrease these time requirements which is left for
future research. The runtime comparisons are summarized in Table I.

Figure 4: Comparison of sampling strategies. The active sampling methods
(uncertainty and coreset) requires less samples to reach the CL model trained
on full set of images (benchmark). TABLE I: Runtime comparison

Method |  | Avg Runtime
---
(min)
| Time Reduction
---
%
Benchmark | 538.70 | NA
Uncertainty | 204.4 | 62
Coreset | 538.65 | 0

##  IV Conclusion

In this paper, we have demonstrated that active learning strategies can be
used in CL to speed up training using less data. Also, using the uncertainty
sampling we have proposed a feedback system for the contrastive learning
framework to select the subset with the most amount of information related to
the downstream task. We showed for histopathological cancer classification, we
can achieve a speedup of 62% while using 93% less data. This method requires
only a small labeled set. Future work could explore whether this number can be
reduced and how that would affect contrastive training. This research focused
on classification as the downstream task. Future research can extend this to
other tasks such as regression or segmentation.

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# Prospects for kaon physics at LHCb

Francesco Dettori 111On behalf of the LHCb collaboration  Università degli
Studi di Cagliari and INFN, Cagliari, Italy<EMAIL_ADDRESS>
###### Abstract

Despite not being designed for it, the LHCb experiment has given world-leading
contributions in kaon and hyperon physics. In this contribution I review the
prospects for kaon physics at LHCb exploiting the already acquired data and
the current and future Upgrade scenarios.

The LHCb experiment [1] at the LHC is designed and optimised to study _beauty_
and _charm_ decays but has proven to be suitable also for strange physics.
This, despite the average decay length of $K_{S}^{0}$ and hyperons at LHC
energies is of the order of one metre, and it is tens of metres for
$K_{L}^{0}$ and $K^{+}$, which are essentially considered stable in LHCb (See
Figure 1). Nevertheless the huge production of these hadrons at LHC, two to
three orders of magnitude larger than that of heavy flavours, makes strange-
hadron physics an increasingly exiting field at LHCb [2].

The LHCb Collaboration has already performed two world best results in this
realm: the search and best upper limit on the branching fraction of
$K^{0}_{S}\rightarrow\mu^{+}\mu^{-}$ decays [3, 4, 5, 6] and the evidence for
the $\Sigma^{+}\rightarrow\mathrm{p}\mu^{+}\mu^{-}$ decays [7]. These two
being decays containing muons, they are probably the low-hanging fruits,
nevertheless demonstrating that $K^{0}_{S}$ and hyperons can be used at LHCb
for rare decays searches.222Incidentally, note that $K^{0}_{S}$ and $\Lambda$
hyperons are also used in LHCb as final states in many $B$ and $D$ physics
analyses, in searches for CP violation, exploiting their dominant decay modes.
In this contribution I present some prospects for different strange-hadron
searches using the LHCb data, already collected in Run 1 and 2 or to be
collected in the future.

The reconstruction of strange-hadrons in LHCb is limited by the tracking of
their decay products and by the trigger. Tracking a charged particle in LHCb
usually requires to have hits in all the tracking stations (_long track_) or
in all but the VELO (See Fig. 1) tracking stations (_downstream track_). The
former limits the decay length to 1 metre while the second allows up to 2
metres. Decays further downstream, albeit reconstructible, would not allow a
momentum measurement of the tracks. In addition to this, the trigger imposes
constraints to the transverse momenta of the particles in an event. In
particular during Run 1 and Run 2, typical thresholds at the level of
$\mbox{$p_{\mathrm{T}}$}>1-5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ were
required for at least a muon or a hadron in the event in the first hardware
level of the LHCb trigger (L0). Strange-hadrons are often too soft for these
L0 thresholds, nevertheless given the high production rates it is sufficient
that a small fraction of the events are triggered by the signal in question
(Triggered On Signal, TOS), and in addition one can exploit event triggered by
other particles in the same event (Triggered Independently of Signal, TIS). On
top of L0, LHCb has two software trigger layers, HLT1 and HLT2, which perform
a coarse and a more refined event reconstruction, and select interesting
events for subsequent offline analysis. These levels are more customisable
than the L0, albeit having to respect tight rate requirements. During Run 1
(2010-2012) no dedicated trigger was present for strange physics at HLT1 and
only for part of it a dedicated line at HLT2 was added for
$K^{0}_{S}\rightarrow\mu^{+}\mu^{-}$ decays. In preparation for Run 2
(2015-2018), different inclusive and exclusive lines were studied and included
in both HLT1 and HLT2 [8] allowing about an order of magnitude increase in
trigger efficiency for strange decays.

\begin{overpic}[width=294.85841pt]{lhcblifetimesmall.pdf}
\put(20.0,95.0){\small(b)} \put(20.0,30.0){\small(a)}
\end{overpic}\begin{overpic}[width=138.76157pt]{production_minbias.pdf}
\put(20.0,85.0){\small(c)} \end{overpic}

Figure 1: In (a) decay length of strange-hadrons produced in $pp$ collisions
at $\sqrt{s}=13$ TeV, compared with (b) the LHCb detector scheme. In (c) the
average production multiplicity of different hadrons in the same kind of
events. Both plots are obtained with PYTHIA Monte Carlo simulations, and are
from Ref [2].

The data collected so far by LHCb correspond to about 10 $\mbox{\,fb}^{-1}$.
About 50 $\mbox{\,fb}^{-1}$ are expected to be collected in the LHCb Upgrade
phase from 2021 on, and there is interest to continue the experiment at high
luminosity with a future Upgrade, possibly reaching 300 $\mbox{\,fb}^{-1}$
[9]. This will allow unprecedented statistics to be collected for heavy-
flavours; while it will be represent a challenge to keep triggering also
strange physics, the highly customisable software trigger from the Upgrade
phase on will allow to target with exclusive lines all the interesting decays,
reaching efficiencies which could be up to 100% evaluated on top of offline
selected signals. Therefore prospects for different analyses have been
prepared taking into account these possible luminosities and efficiencies, and
some of these will be presented in the following.

One of the most interesting decays in the short term, will be the
$K^{0}_{S}\rightarrow\pi^{0}\mu^{+}\mu^{-}$ decay. Its $K_{L}$ analogous,
$K^{0}_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-}$ is very sensitive to physics
beyond the SM, for example it has been studied in models with extra-dimensions
[10]. However the SM prediction still suffers of a large uncertainty yielding
$\mathcal{B}_{SM}(K^{0}_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-})=\\{1.4\pm
0.3,0.9\pm 0.2\\}\times 10^{-11}$. This uncertainty stems from a limited
knowledge of Chiral Perturbation Theory parameter $|a_{S}|$. Luckily, this
parameter can be extracted from a measurement of the
$K^{0}_{S}\rightarrow\pi^{0}\mu^{+}\mu^{-}$ branching fraction. The latter is
currently known with about 50% uncertainty from a measurement by the NA48
Collaboration,
$\mathcal{B}(K^{0}_{S}\rightarrow\pi^{0}\mu^{+}\mu^{-})=(2.9^{+1.5}_{-1.2}\pm
0.2)\times 10^{-9}$ [11]. Any new measurement improving this branching
fraction would reflect in the $K^{0}_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-}$
sensitivity. The sensitivity to $K^{0}_{S}\rightarrow\pi^{0}\mu^{+}\mu^{-}$
decays at LHCb has been studied in Ref. [12], with two different
reconstruction methods: one with a full reconstruction and one without
reconstructing the soft $\pi^{0}$, which poses difficulties in the LHCb
detector. This partial reconstruction still allows a good invariant mass
reconstruction assuming a dummy $\pi^{0}$, given the very low $q$ value of the
decay. Signal efficiency is studied with Monte Carlo simulations, while
backgrounds are studied in real data. It is found that, depending on the
trigger efficiency, LHCb can be competitive and improve on the NA48
measurement with about 10 $\mbox{\,fb}^{-1}$ of Upgrade data.

A second group of decays which will become gradually more promising are 4-body
leptonic decays. Again, it is the $K^{0}_{L}$ mode to be the most interesting:
the short distance contribution to
$K^{0}\rightarrow\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ decays is sensitive to new
physics, but, as often happens in strange physics, it is dominated by the long
distance contribution uncertainty. However a measurement of the interference
of $\mathcal{A}(K^{0}_{S}\rightarrow\ell^{+}\ell^{-}\ell^{+}\ell^{-})$ and
$\mathcal{A}(K^{0}_{L}\rightarrow\ell^{+}\ell^{-}\ell^{+}\ell^{-})$ would give
a measurement of the sign of $\mathcal{A}(K^{0}_{L}\rightarrow\gamma\gamma)$
which is a stringent test of CKM [13, 14]. While the
$K^{0}_{L}\rightarrow\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ have been studied by
different experiments, no experimental constraints are present on the
$K^{0}_{S}$ modes. Therefore despite the rates are expected to be very low
($\mathcal{B}(K^{0}_{S}\rightarrow e^{+}e^{-}e^{+}e^{-})\sim 10^{-10}$
$\mathcal{B}(K^{0}_{S}\rightarrow\mu^{+}\mu^{-}e^{+}e^{-})\sim 10^{-11}$,
$\mathcal{B}(K^{0}_{S}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-})\sim 10^{-14}$)
any limit approaching the SM rates would be a test of new physics.

As a proxy for these decays, the
$K^{0}_{S}\rightarrow\pi^{+}\pi^{-}e^{+}e^{-}$ has been studied at LHCb [15].
If one considers Run 1 conditions, as expected, the efficiency is limited
tightly by the L0 trigger. Nevertheless, given the high branching fraction,
$\mathcal{B}(K^{0}_{S}\rightarrow\pi^{+}\pi^{-}e^{+}e^{-})=(4.79\pm
0.15)\times 10^{-5}$, a yield of a hundred signal events per
$\mbox{\,fb}^{-1}$ could be expected, with a very simple selection. Dedicated
HLT2 trigger lines have been deployed in Run 2 and will allow for a first
proof of concept. Upgrade trigger will instead improve the efficiency on this
and related decays sensibly. In the ideal scenario of $\sim 100\%$ trigger
efficiency, with respect to offline selected events, of the order of $5\cdot
10^{4}$ per $\mbox{\,fb}^{-1}$ $K^{0}_{S}\rightarrow\pi^{+}\pi^{-}e^{+}e^{-}$
decays could be observed. Since one expects similar or higher efficiencies for
the $K^{0}_{S}\rightarrow\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ rare decays single
event sensitivities of order $9.6\cdot 10^{-10}$ over each $\mbox{\,fb}^{-1}$
in Upgrade conditions are expected.

More comprehensive studies, albeit less detailed than the ones presented
above, have been done in Ref. [2], using approximate simulations of the LHCb
detector. Dozens of decays have been studied between $K^{0}_{S}$, and hyperons
showing that LHCb will be in the position to give many different contributions
to strange-hadron physics in the near future. Interestingly, one can also use
charged kaons at LHCb. While their decay length is very long compared to LHCb
acceptance, it has already been shown [16] that high yield decays such as
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ decays, can be reconstructed and used
for physics calibration. This same decay mode has been proposed for the
measurement of the $K^{+}$ mass, that suffers a long-standing discrepancy
among the most precise measurements [17]. In addition, with the full software
trigger of the Upgrade it will be possible to study $K^{+}$ rare decays, such
as $K^{+}\rightarrow\pi^{+}\mu^{-}\mu^{+}$, albeit not being competitive with
the NA62 experiment. Finally, a field in which LHCb will also be able to
contribute, are the lepton-flavour-violating decays. Clearly these are
extremely interesting being a null test of the SM and not suffering of its
uncertainties. As such, they have been studied by several experiments and the
current upper limits read: $\mathcal{B}(K_{L}\rightarrow
e^{\pm}\mu^{\mp})<4.7\times 10^{-12}$ [18],
$\mathcal{B}(K_{L}\rightarrow\pi^{0}e^{\pm}\mu^{\mp})<7.6\times 10^{-11}$
[19], $\mathcal{B}(K^{+}\rightarrow\pi^{+}e^{-}\mu^{+})<1.3\times 10^{-11}$
[20], $\mathcal{B}(K^{+}\rightarrow\pi^{+}e^{+}\mu^{-})<5.2\times 10^{-10}$
[21]. However, using $B$-physics lepton (non) universality constraints one can
predict branching fractions of order $10^{-13}$ [22]. LHCb will be possibly
able to reach this level exploiting the full Upgrade dataset [22, 2].

En passant I would like to mention also the possibilities for hyperons at
LHCb. Concerning $\Sigma^{+}$ baryons, besides the
$\Sigma^{+}\rightarrow\mathrm{p}\mu^{+}\mu^{-}$ decay, LHCb could improve the
$\Sigma^{+}\rightarrow p\gamma$ decay and try to access the
$\Sigma^{+}\rightarrow pe^{+}e^{-}$ decay. On $\Lambda$ hyperons, the
branching fraction of the radiative $\Lambda\rightarrow p\pi\gamma$ could also
be improved and possibly access the $\Lambda\rightarrow p\pi e^{+}e^{-}$
decay. A large number of baryon number or lepton flavour violating decays have
been recently constrained by the CLAS collaboration [23]: these could also be
tested at LHCb. Higher strange-number baryons could also be probed at LHCb in
order to test $\Delta S=2$ processes, such as $\Xi^{0}\rightarrow p\pi$ and
$\Omega\rightarrow\Lambda\pi$, possibly improving limits by orders of
magnitude.

In summary, the LHCb experiment has demonstrated the capability to study rare
decays of $K^{0}_{S}$ and hyperons, and has the possibility to improve many
new physics sensitive decays as well as help reduce uncertainties on decays
needed for $K^{0}_{L}$ physics. With the Upgrade, the flexibility of the new
software trigger will allow unprecedented rates for different decays, possibly
establishing LHCb as a strange factory. I would like to underline here, that
the decay modes mentioned in this contribution are just examples of the
possibilities and benchmarks; however new ideas area always welcome to exploit
the full potential of these data.

## References

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