text:
\section{Introduction}
\label{sec:intro}
Deep neural networks have shown great promise in a variety of application domains. The data required for training these models is often generated via devices interaction with edge devices. In a traditional training framework, the data is sent to a central server, where the model gets trained and periodically deployed on the edge devices. The desire to strengthen data privacy and security has driven a surge of interest in developing training methodologies that can operate on local data that is not transmitted to the cloud. A promising framework is Federated Learning (FL), where each device performs local training using its own data and only sends updated model parameters to the server. The server aggregates these updates to build the new model and sends the latest model back to each device.
In order to provide an FL solution that scales to millions of users, in most algorithms (e.g., \texttt{FedAVG} \cite{mcmahan2017fedavg}), the central server first selects a subset of devices uniformly at random, then computes a weighted average of their parameter updates to construct the global model. In real world scenarios with heterogeneous users, the contribution of different devices might significantly vary and uniform sampling might not provide the best strategy. As such, device selection for model aggregation is a critical component of FL algorithms.

Most recent works that study device selection in FL (e.g., \cite{balakrishnan2021diverse,Nagalapatti_Narayanam_2021}) have focused on gradient-based methods, in which the local updates of the devices are compared to decide the selection and aggregation. These methods require the computations to be carried out across all devices in each round. In this paper, we investigate simpler device selection processes that are based on ``device activity'' (i.e. selecting devices based on their number of available training samples). Given the FL server already has access to this information, no additional computation/communication overhead is incurred.
We also study temporal effects of training data in FL, and look at how different device selection and training strategies affect the performance of the model. Most FL papers present results on ``static'' datasets where the time dimension is ignored. \\
\textbf{Our contributions:}
(1) we present the first study of simple device selection based on device activity applied to real-world data, exploring several sampling strategies and show that non-uniform sampling outperforms standard uniform sampling;
(2) we study continual FL to investigate the realistic temporal aspects of FL and address limited data storage on-device. We demonstrate that non-uniform strategies combined with continual learning swiftly catches up with FL methods using all data at once (at the expense of longer delays between updates).

\section{Related Work}
\label{related-workd}
Standard FL algorithms such as \texttt{FedAVG} \cite{mcmahan2017fedavg} and \texttt{FedOpt} \cite{reddi2021fedopt}) have been widely studied.
\texttt{FedAVG} has been proved to converge under a large set of assumptions (e.g., non- I.I.D. data distributions) \cite{yang2021achieving,ruan2021flexible, rizk2020dynamic, acar2021federated}.
The performance of these algorithms has been evaluated in CV (e.g., \cite{mcmahan2017fedavg}), NLP (e.g., \cite{fedNLP}), and graph-structured data (e.g., \cite{fedGraphNN}).
Recently, several works have looked at leveraging the results of previous local device update to compute the sampling probability for the next round.
\cite{chen2020optimal} propose a probability based on the ``importance'' of each device, which is mostly determined by the norm of the latest local update.
In \cite{cho2020client}, the authors show that sampling devices with higher losses leads to faster convergence compared to vanilla \texttt{FedAVG}. Similarly, other methods leverage the latest local gradients to update the device sampling probability \cite{balakrishnan2021diverse,Nagalapatti_Narayanam_2021}. These methods require all the devices to perform local updates which increases the communication overhead. Continual learning (e.g., \cite{Thrun95,Schwarz2018progress,hung2019compacting}) has become increasingly important for learning across a sequence of rounds and tasks.
More recently, continual learning has also been modeled in FL to study how one can leverage learned knowledge from different devices with different tasks~\cite{yoon2021continual}.
In this work, we focus on temporal aspects of continual FL, meaning leveraging the knowledge from past interactions in the future training.

\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.7\textwidth}
  \centering
   \includegraphics[width=\linewidth]{FL_rel_bar_one_shot_FedAVG_all_10k.png}
	\caption{\textit{One-shot FL}. \textbf{Left}: 3 months; \textbf{middle}: 6 months; \textbf{right}: 11 months.}
	\label{fig:one_shot_overall}
\end{subfigure}
\begin{subfigure}[t]{0.29\textwidth}
  \centering
   \includegraphics[width=\linewidth]{FL_rel_barFedOpt_10k_temp.png
	\caption{\textit{Continual FL}. Performance after each of the 3 time periods P1, P2, P3.}
	\label{fig:continual_overall}
\end{subfigure}
\caption{
RCP compared to the the baseline model on the whole test set.
(\ref{fig:one_shot_overall})~Baseline: standard \texttt{FedAVG}  ($f(x)=1$) on 11 months of data; (\ref{fig:continual_overall})~Baseline: \textit{one-shot FL} \texttt{FedAVG} and $f(x)=\log(x+1)$ on 6 months.}
\label{fig:overall}
\end{figure*}

\section{Problem Statement}
\label{sec:fl}
\textbf{Framework.} \hspace{2.5mm}
In standard machine learning, the central server stores the $n$ user generated entries $X=(x_i)_{i=1\ldots n}$ and the objective is to minimize the average loss $\mathcal{L}$ over these $n$ inputs:
$\min_{w} \mathcal{L}(X, w) = \frac{1}{n} \sum_{i=1}^{n} \ell(x_i, w)$.
In practice, the number of inputs $n$ is large and $\mathcal{L}$ is minimized with stochastic gradient descent (SGD) \cite{bottou2010sgd} where the gradient with respect to a random batch of inputs $B_t$ is used to update the parameters $w$:
$    w_{t+1} = w_{t} - \eta\nabla_w \left(\frac{1}{|B_t|}\sum_{i\in B_t} \ell(x_i, w)\right)$.
In FL, the data is not available to the central server to generate random batches of data $B_t$. If each of the $M$ devices contains $X^m\subset X$ inputs with $\vert X^m\vert=n_m$, $m=1,\ldots,M$, the central objective can be rewritten:
$\mathcal{L}(X, w) = \frac{1}{n} \sum_{m=1}^{M}n_m \mathcal{L}^m(X^m, w)$,
with $\mathcal{L}^m(X^m, w) =\frac{1}{n_m} \sum_{x\in X^m}\ell(x, w)$. In \texttt{FedAVG}, this objective is minimized by iteratively sampling a subset of devices $S_t\subset\{1, \ldots, M\}$, requiring the selected devices to optimize their own objective $\mathcal{L}^m(X^m, w_t)$ locally using the latest version of the model $w_t$ as the starting point. The updated local parameters $w^m_{t+1}$ are sent to the central server for aggregation via:
$w_{t+1} = \sum_{m\in S_t}\frac{n_m}{N_t} w^m_{t+1}$,
where $N_t=\sum_{m\in S_t}n_m$ is the total number of inputs on the selected devices $S_t$.

\noindent \textbf{Device Selection.} \hspace{2.5mm}
In standard FL training, the device selection is made uniformly at random. In a real world scenario with non-I.I.D. data, different device selection strategies could help boost the performance of the resulting model. Previous works used gradient-based selection methods \cite{balakrishnan2021diverse, Nagalapatti_Narayanam_2021} where gradients are computed for all devices before selecting a subset. We look at a simpler/less computationally expensive approach using the number of inputs per device $(n_m)$. We want to understand: 1) change in performance if we put more weight on the active devices, 2) whether we can limit ourselves to only a subset of devices. In our experiments, we explore several selection strategies for $S_t$ that are a function of the number of inputs on each device, namely ${\mathbb{P}[m\in S_t]} \propto f(n_m)$,
with ${f(x)\in\{1, \log(x+1), \sqrt{x}, x, \frac{1}{ \log(x+1)}\}}$, where $f(x)=1$ corresponds to uniform sampling. We experiment with two additional functions referred to as \texttt{heavy} (resp. \texttt{light}) to restrict the optimization to the most (resp. least) active devices: $f(n_m) = 1 $ if $m$ is in top 20\% most (resp. least) used devices, $f(x)=0$ otherwise.

\noindent \textbf{Continual Learning.} \hspace{2.5mm}
In the \textit{one-shot FL} setting, the data is static and used all at once during training. This approach might not be applicable to real-world scenario where users generate data over time due to limited storage on device.
In \textit{continual FL}, we consider a dataset generated over a time period $T$, which is divided into consecutive segments of size $\Delta t$. For each segment, we train a model on the data for that segment, starting from the end model of the previous segment. For the first segment, we use the same starting point as for \textit{one-shot FL}.  Similar to \textit{one-shot}, we compare the different device selection strategies in a \textit{continual} setting.

Our experiments address the following:
\textbf{Device selection:} 1) Is there a non-uniform sampling strategy that outperforms uniform sampling? 2) Can we obtain a similar performance as the standard uniform sampling by restricting training to the most active devices?
\textbf{Temporality:} 1) For \textit{one-shot FL}, does an increase in the amount of historical data have a positive effect on the performance on the next interactions? 2) Does \textit{continual FL} compete with \textit{one-shot FL}, and is there a device selection strategy that boosts \textit{continual FL} performance?

\section{Experimental Setting}
\label{sec:exp}
\textbf{Model:} BERT-\textit{base} model \cite{devlin2019bert} for the masked language modeling (MLM) task that is pre-trained model on the Common Crawl dataset \cite{commoncrawl}.  \\
\textbf{Data:} 12 months of automated transcription of Alexa user request data, that is de-identified to remove information that connects the data to the users. We randomly select 10,000 devices and their utterances. In all experiments, the test set corresponds to the last month of data. We repeat the process 3 times to generate different datasets and evaluate our approaches. In all the figures, the standard deviation across the 3 datasets is displayed.\\
\textbf{One-shot FL:} The FL models are trained on $\vert T\vert =3$ months (months 9 to 11 of the 12 months data) $\vert T\vert =6$ months (months 6 to 11) and $\vert T\vert =11$ months (months 1 to 11).\\
\textbf{Continual FL:} 6 months of data (months 6 to 11 of the whole data) is split into 3 parts of $\Delta t=2$ months. For each time period, the last state of the model from the previous period is used for initialization.\\
\textbf{FL methods:} We compare two of the most studied FL algorithms: \texttt{FedAVG} \cite{mcmahan2017fedavg} and \texttt{FedOpt} \cite{reddi2021fedopt}. For both, we select 800 devices per round and each device performs one local epoch over their data, with standard central aggregation.\\
\textbf{Evaluation:} We use the perplexity metric and study the impact across the different devices by splitting the test set into deciles of devices, based on their number of utterances. For each decile, the perplexity is computed on all the utterances by devices in that decile.
We can't present absolute performance values, instead we present relative change in \% in perplexity (referred to as RCP from this point on) compared to the perplexity on the entire test set for the baseline model.

\begin{figure*}[t]
\begin{subfigure}[t]{0.74\textwidth}
  \centering
   \includegraphics[width=\linewidth]{utt_test_FL_all_FL_FedAVG.png}
	\caption{\textit{One-shot FL}. \textbf{Left}: 3 months; \textbf{middle}: 6 months; \textbf{right}: 11 months.}
	\label{fig:one_shot_fedavg}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
  \centering
   \includegraphics[width=\linewidth]{utt_test_FL_all_continual_FedAVG.png}
	\caption{\textit{Continual FL}.}
	\label{fig:continual_fedavg}
\end{subfigure}
\caption{\texttt{FedAVG}.
RCP compared to the the baseline model on the whole test set.
(\ref{fig:one_shot_fedavg})  Baseline: standard \texttt{FedAVG} on 11 months of data; (\ref{fig:continual_fedavg}) Baseline: \textit{one-shot FL} \texttt{FedAVG} and $f(x)=\log(x+1)$ on 6 months of data; \texttt{o-s} stands for ``\textit{one-shot}''.}
\label{fig:fedavg}
\end{figure*}

\begin{figure*}[t]
\begin{subfigure}[t]{0.74\textwidth}
  \centering
   \includegraphics[width=\linewidth]{utt_test_FL_all_FL_FedOpt.png}
	\caption{\textit{One-shot FL}. \textbf{Left}: 3 months; \textbf{middle}: 6 months; \textbf{right}: 11 months.}
	\label{fig:one_shot_fedopt}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
  \centering
   \includegraphics[width=\linewidth]{utt_test_FL_all_continual_FedOpt.png}
	\caption{\textit{Continual FL}.}
	\label{fig:continual_fedopt}
\end{subfigure}
\caption{\texttt{FedOpt}.
RCP compared to the the baseline model on the whole test set.
(\ref{fig:one_shot_fedopt}) Baseline: standard \texttt{FedAVG} on 11 months of data; (\ref{fig:continual_fedopt}) Baseline: \textit{one-shot FL} \texttt{FedAVG} and $f(x)=\log(x+1)$ on 6 months of data; \texttt{o-s} stands for ``\textit{one-shot}''.}
\label{fig:fedopt}
\end{figure*}

\begin{figure}[t]
  \centering
  \includegraphics[width=\linewidth]{innovation.png}
  \caption{Average repartition of new utterances over a period.}
  \label{fig:innovation}
\end{figure}

\section{Results}
\noindent \textbf{One-shot FL  -- Figures~\labelcref{fig:one_shot_overall,fig:one_shot_fedavg,fig:one_shot_fedopt}.} \hspace{2mm}
The baseline for the RCP computation is the standard \textit{one-shot} \texttt{FedAVG} trained on 11 months of data.
Increasing the amount of training data is clearly beneficial for \texttt{FedOpt}, but not as much for \texttt{FedAVG} (Figure~\ref{fig:one_shot_overall}). As a result, \texttt{FedOpt} outperforms \texttt{FedAVG} for most of the selection strategies with 11 months of training data. It's the other way around for $|T|\in\{3\text{ months}, 6\text{ months}\}$.
Interestingly, standard \texttt{FedAVG} using the entire 11 months of training data performs worse than almost any other strategy, possibly due to a combination of overfitting and sub-optimal selection strategy.
In Figures~\labelcref{fig:one_shot_fedavg,fig:one_shot_fedopt}, for every method, the perplexity is lower for devices with a higher number of utterances. Every curve sees an uptick in perplexity for the last decile. From the heaviest devices (especially the top 1\%), some also have a high number of utterances in the test set and are harder to predict.\\
\textbf{\texttt{FedAVG}}: the different sampling strategies significantly affect the performance of the resulting model. Overall, for shorter time range (left 2 plots in Figure~\ref{fig:one_shot_fedavg}), using the standard uniform sampling ($f(x)=1$) provides one of the best strategies, on par with the log-based sampling.
However, for longer time horizon (right plot in Figure~\ref{fig:one_shot_fedavg}), non-uniform sampling provides much better results. The model where the probability to select device $m$ is proportional to $f(n_m)= \frac{1}{\log(n_m+1)}$ outperforms all the other models trained using \texttt{FedAVG}. In this strategy we tend to select lighter devices with higher probability.
Conversely, selection strategies that are restricted to a subset of devices (heavy or light devices only and $f(x)=x$) perform poorly. The selection $f(x)=x$ performs worse than when restricting sampling to the top 20\% devices because of the distribution of the devices: their number of utterances varies wildly across the dataset (with a factor of  $>10^4$), and choosing $f(x)=x$ implies the model is likely to select the same few devices (less than 20\% total) having much more utterances than the rest.  In terms of volume, it is not beneficial to extend the size of the data for \texttt{FedAVG}, except for the best strategy $f(x)=\frac{1}{\log(x+1)}$ that performs much better with 6 months over 3 months of data, but only slightly better when using 11 months compared to 6 months. In Figure~\ref{fig:one_shot_fedavg}, we see that more training data mostly benefits light devices. For instance, with 3 months of data and $f(x)=\frac{1}{\log(x+1)}$, the average RCP for the first decile is around 13\% and drops to around 1\% when using 11 months of data. For the last decile, RCP ``only'' drops from 2\% to -5\%.\\
\textbf{\texttt{FedOpt}}: contrary to \texttt{FedAVG}, the performance is similar across the selection strategies, except for selection restricted to \texttt{light} devices that performs poorly in general.  Restricting the training to the top 20\% devices fares as well as the other strategies involving all the devices. As expected, this strategy outperforms on heavier devices and underperforms on lighter devices which have not been seen by the model.
From Figure~\ref{fig:one_shot_fedopt}, uniform sampling provides better performance on light devices, but other strategies provide better performance on the overall test set ($f(x)\in\{\log(x+1), \sqrt{x}\}$).

\noindent \textbf{Continual FL -- Figures~\labelcref{fig:continual_overall,fig:continual_fedavg,fig:continual_fedopt}.}
The baseline is the best \textit{one-shot} model on the same 6 months: \texttt{FedAVG} trained with ${f(x)=\log(x+1)}$.
The advantage of \textit{continual FL} is the possibility to store less data on device but still train a model on the entire time period we consider.
From Figure~\ref{fig:continual_overall}, models trained with most selection strategies improve performance along the time periods, except $f(x)=x$.
Several selection strategies with \textit{continual FL} display less than 5\% relative degradation compared to the best \textit{one-shot FL} model after the third period of training (P3).  \textit{Continual FL} consistently outperforms the \textit{one-shot FL} baseline for \texttt{FedAVG} with $f(x)=\log(x+1)$. However, no other selection strategy beats the baseline with \textit{continual FL}, even if $f(x)=\sqrt{x}$ is on par with the baseline for both \texttt{FedAVG} and \texttt{FedOpt}.  Similar to \textit{one-shot FL}, the difference in performance across the selection strategies is larger for \texttt{FedAVG} than for \texttt{FedOpt}.

\textbf{\texttt{FedAVG}}: The gap between \textit{one-shot FL} and \textit{continual FL} across device deciles for $f(x)=\log(x+1)$ (black and red curves in Figure~\ref{fig:continual_fedavg}) indicates that \textit{continual FL} benefits the heavy devices. This is expected as heavier devices are more likely to have utterances in all the 3 periods used for training.

\textbf{\texttt{FedOpt}}: Even if all the selection strategies used with \texttt{FedOpt} perform worse than the \textit{one-shot} \texttt{FedAVG} baseline, they are consistently on par with the \textit{one-shot} \texttt{FedOpt} model across device deciles. Similar to \texttt{FedAVG}, \textit{continual FL} benefits heavy devices more. For instance,  \textit{continual FL} with \texttt{FedOpt} and $f(x)=\sqrt{x}$ performs worse than the \textit{one-shot} \texttt{FedOpt} baseline on the first device deciles, but outperforms this method on the last deciles.

\noindent \textbf{Insight.} \hspace{2mm} We look at the repartition of utterances for a given period compared to all the previous ones across device deciles. For a given device's set of unique utterances over a time period, we compute 1) the proportion of utterances already seen by this device; 2) the proportion of new utterances for this device but seen by any other device in a previous period of time; 3) the proportion of completely new utterances. In Figure~\ref{fig:innovation}, these metrics are averaged over periods P2 and P3. We observe that heavy devices explore more than light devices, who are mostly ``catching up''. For learning, it seems more beneficial to sample heavier devices who bring additional knowledge.

\section{Conclusion}
\label{sec:conclusion}
We studied popular FL algorithms on non-I.I.D. data borrowed from a real word setup.
Through our experiments, we show that non-uniform device selection outperforms standard uniform sampling of FL and boosts performance for most devices.
We also compare the selection strategies on different amounts of data and observe that \texttt{FedOpt} benefits from more data, but not \texttt{FedAVG}.
Finally, we compare \textit{one-shot FL} with \textit{continual FL} where we run multiple rounds of training on smaller time periods, allowing for lower storage capacity.
We show that \textit{continual FL} outperforms the \textit{one-shot} strategy in some setting, and is overall most beneficial for heavy devices.

\vfill\pagebreak

\bibliographystyle{IEEEbib}


text:
\section{Introduction}
Two Higgs Doublet Models (2HDMs) have been studied for a long time. They represent the next non-trivial step in complication beyond the Standard Model (SM) and yet, the simple addition of one scalar doublet, enriches the phenomenology substantially, leading to a wide spectrum of interesting phenomenological and theoretical possibilities~\cite{Gunion:1989we,Branco:2011iw}.

From the beginning it was realized that discrete symmetries play an important role in the scalar potential. In particular $Z_2$ symmetry has been widely used to restrict the general 2HDM potential leading to interesting and potentially relevant phenomenological consequences. In this letter we explore the general 2HDM with $Z_3$ symmetry and point out some interesting scenarios that could behold a phenomenological interest. In particular we are interested in finding features not present in the $Z_2$ symmetry case that might lead to interesting models. The first and strongest realization of this comes from the fact that the $Z_3$ symmetry may also affect the fermions fields in a non-trivial way, as opposed to the $Z_2$ case, leading to peculiar Yukawa textures~\cite{Ferreira:2010ir}. We find that in fact this happens in the quark sector for a class of models where one of the doublets is restricted to couple to the second and third quark family, while the other doublet couples to everything. Given appropriate $Z_3$ charges for the different quark fields and scalar doublets, acceptable Yukawa textures that lead to the correct Cabibbo-Kobayahi-Maskawa (CKM) matrix can be obtained.

Another interesting and timely venue is that of the so-called inert models, i.e. scenarios where there can be scalar doublets with vacuum expectation values (vevs) equal to zero and a discrete symmetry, usually $Z_2$, that prevents the lightest one from decaying~\cite{Deshpande:1977rw,Barbieri:2006dq}. Since the vevs of these scalar fields are zero, they do not contribute to symmetry breaking. Furthermore, their charges (and that of the fermions in the SM) are chosen so that they do not couple to SM fermions - thus they are {\it inert}. In this case we address the issue of whether or not $Z_3$ symmetry can be used to stabilize a dark matter candidate and if a difference from the $Z_2$ case can be obtained. The answer to both questions is positive and we present the specific case where that happens. It consists on an extension to three scalar doublets: one active and a dark-2HDM with $Z_3$ symmetry.

In section~\ref{sec:analysis} we present the general setup for the scalar potential and the analysis in the Yukawa sector. Section~\ref{sec:yukawatextures} contains different sets of possible models with their generated Yukawa textures. The different inert type scenarios are presented in section~\ref{sec:inert} and we end with some final comments in section~\ref{sec:conclusion}.

\section{Analysis}
\label{sec:analysis}
\subsection{General scalar sector}
There are two SU(2) doublets $H_i$, $i=1,2$, with the same hypercharge. We impose a $Z_3$ symmetry under which an arbitrary field field ${\cal F}$ transforms as ${\cal F} \rightarrow  {\cal F}^{\prime} = \omega^{n_f} {\cal F}$, where $\omega \equiv \exp(2\pi i/3)$ and $|n_f| \in \{0,1,2\}$. We say the field ${\cal F}$ has {\it charge} $n_f$ under $Z_3$. For simplicity we assume CP conservation in the scalar sector and thus all vevs and couplings are taken to be real.

The general SU(2)$_W\times$U(1)$_Y$ invariant scalar potential can be written as
\begin{eqnarray} \label{potential}
V(H) = \mu_{ij}^2 H_i^{\dagger}H_j + \lambda_{ijkl}\left( H_i^{\dagger}H_jH_k^{\dagger}H_l + h.c.\right) \ ,
\end{eqnarray}
where $i,j = 1,2$. Denoting the $Z_3$ charges of $H_1$ and $H_2$ as $n_{h_1}$ and $n_{h_2}$ respectively we can express the $Z_3$ charges of each term in the potential using the coefficients $\mu$ and $\lambda$. For example the coefficients of the (always) invariant quadratic terms involving $H_1$ or $H_2$ (but not their mixing) are said to have zero charge: $ [\mu_{ii}] = 0$. The other invariant terms are  $\left[ \lambda_{iiii}\right]  = \left[ \lambda_{ijji}\right]   = 0$. The rest are obtained from the following relations $(\ i\neq j )$:
\begin{eqnarray}\label{potential-lambda-terms}
[\mu_{ij}] &=& -[\mu_{ji}]=  [\lambda_{iiij}] = \frac{1}{2} [\lambda_{ijij}] = n_{h_j}-n_{h_i} \pmod{3}  \ ,\\
\left[\lambda_{ijii}\right] &=& \left[\lambda_{iiij}\right]  = -\left[\lambda_{jiii}\right]  = -\left[\lambda_{iiji}\right] \ .
\end{eqnarray}

Note that the particular case where $n_{h_j}-n_{h_i} \neq 0 \pmod{3}$ leads to a $Z_2$ invariant potential: it does not have $H_i^{\dagger}H_j$ quadratic terms and only six quartic terms survive. Also, If both scalars develop nonzero vevs, the gauge symmetry is completely broken and soft breaking terms must be included in order to obtain a $U(1)_{em}$ invariant vacuum~\cite{Gunion:1989we}. Thus we are interested in the situation where $n_{h_j}-n_{h_i} \neq 0 \pmod{3}$ and the possible consequences that this may have in the Yukawa sector. The case $n_{h_1}=n_{h_2}$ is possible but uninteresting.

\subsection{Yukawa sector}

The Yukawa sector of the general scenario is given by

\begin{eqnarray}\label{yukawa}
{\cal L}_Y = {\cal Y}^{ui}_{ab} \overline{Q}_a \widetilde{H}_i u_b + {\cal Y}^{di}_{ab} \overline{Q}_a H_i d_b + h.c. \ ,
\end{eqnarray}
where $i=1,2$, $a,b=1,2,3$, and $\widetilde{H}_i \equiv i\sigma_2H_i^*$.
Denoting the $Z_3$ charges of the different fields by $n_{h_i}$, $n_{q_a}$, $n_{u_a}$, $n_{d_a}$, the "charge" associated to the Yukawas become
\begin{eqnarray}
\label{yukawa-up-charges}
\left[  {\cal Y}^{ui}_{ab}   \right] & = & n_{u_b} - n_{q_a} - n_{h_i}  \pmod{3} \ , \\
\label{yukawa-down-charges}
\left[  {\cal Y}^{di}_{ab}   \right] & = & n_{d_b} - n_{q_a} + n_{h_i}   \pmod{3} \ .
\end{eqnarray}

These expressions can be used to determine the type of Yukawa textures that can be obtained from different charge assignments. Our purpose in this work is to  present a few possibilities that can lead to the construction of  models and so, instead of making a full classification of all possible models (as has been done in~\cite{Ferreira:2010ir}), we explore some specific arrangements that we feel can be of phenomenological interest. We restrict the discussion to the quark sector for the moment (in order not to include the extra fields required to explore the lepton sector~\cite{Aranda:2011rt}) but stress that the possibilities presented here can be successfully extended, as will be shown in a future publication.

\section{Particular cases: Yukawa textures}
\label{sec:yukawatextures}
\subsection{One of up, one for down}
An interesting possibility is whether or not the $Z_3$ symmetry can be used for "separating" the up and down-type sectors. From Eqs.\eqref{yukawa-up-charges} and \eqref{yukawa-down-charges} it can be seen that this happens for the following case: given $n_{q_a}$ for $Q_a$, then $H_1$ couples to the up-type sector and $H_2$ to the down-type one if $n_{h_1} \neq n_{h_2}$, $n_{u_b}=n_{q_a}+n_{h_1}$, and $n_{d_b}=n_{q_a}-n_{h_2}$ ($ \pmod{3}$ is implied in all these relations).

Analyzing the different charge combinations leads to Yukawa matrices that are either diagonal or block-diagonal, i.e., the possible textures are
\begin{eqnarray}
\label{yukawas-diagonal}
{\cal Y} \sim
\left( \begin{array}{ccc}
\star & 0 & 0 \\
0& \star & 0 \\
0&0&\star
\end{array}\right), \ \
{\cal Y} \sim
\left( \begin{array}{ccc}
\star & \star & 0 \\
\star & \star & 0 \\
0&0&\star
\end{array}\right), \ \
{\cal Y} \sim
\left( \begin{array}{ccc}
\star & 0 & 0 \\
0& \star & \star \\
0&\star&\star
\end{array}\right).
\end{eqnarray}
These textures are not useful as they do not lead to the observed CKM matrix~\cite{Ramond:1993kv,Roberts:2001zy}.

\subsection{Avoiding one and one for all} \label{oneforthirdoneforall}
Another possibility consists on having a doublet, say $H_1$, coupling to all three families while the other couples only to the $2-3$ sector (thus avoids one!). This happens when the following conditions are met:
\begin{eqnarray}\label{conditions-23} \nonumber
\left[n_{q_2} \right] &=& \left[n_{q_3} \right] , \ \left[n_{u_2} \right]  = \left[n_{u_3} \right], \
\left[n_{q_1} \right]  \neq \left[n_{u_3} \right] -\left[n_{h_2} \right], \\ \left[n_{u_1} \right]  &\neq& \left[n_{q_3} \right] +\left[n_{h_2} \right], \left[n_{u_1} \right]  \neq \left[n_{q_1} \right] +\left[n_{h_2} \right] \  \pmod{3}.
\end{eqnarray}

Given a value for $n_{h_2}$ there are six possible combinations that satisfy the conditions above. The 18 combinations are given in table~\ref{table:oneforallonefor3}.

\begin{center}
\begin{table}[ht]
\begin{tabular}{|ccccc||ccccc||ccccc|}
\hline
$n_{h_2}$ & $n_{u_3}$ & $n_{q_3}$ & $n_{u_1}$ & $n_{q_1}$  &
$n_{h_2}$ & $n_{u_3}$ & $n_{q_3}$ & $n_{u_1}$ & $n_{q_1}$  &
$n_{h_2}$ & $n_{u_3}$ & $n_{q_3}$ & $n_{u_1}$ & $n_{q_1}$ \\
\hline
\hline
0 & 0 & 0 & 1 & 2 & 1 & 0 & 2 & 1 & 1 & 2 & 0 & 1 & 1 & 0\\
0 & 0 & 0 & 2 & 1 & 1 & 0 & 2 & 2 & 0 & 2 & 0 & 1 & 2 & 2\\
{\bf 0} & {\bf 1} & {\bf 1} & {\bf 0} & {\bf 2} & 1 & 2 & 1 & 0 & 0 & 2 & 1 & 2 & 0 & 0\\
0 & 1 & 1 & 2 & 0 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 1\\
0 & 2 & 2 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 2 & 2 & 0 & 0 & 1\\
0 & 2 & 2 & 1 & 0 & 1 & 1 & 0 & 2 & 2 & 2 & 2 & 0 & 1 & 2\\
\hline
\end{tabular}
\caption{Given a charge for $n_{h_2}$, this table lists the 18 possible charge combinations consistent with Eqs.\eqref{conditions-23} for all remaining fields. The case in bold case is used later in the text, this is why it has been singled out.}
\label{table:oneforallonefor3}
\end{table}
\end{center}

As an example consider the (bold case) case in the third row of the first column of Table~\ref{table:oneforallonefor3}. It corresponds to the following charge assignments: $n_{u_1}=0$, $n_{u_2}=n_{u_3}=1$, $n_{q_1}=2$, $n_{q_2}=n_{q_3}=1$, and $n_{h_2}=0$. Then, writing the up-type quark mass matrix as ${\cal Y}^u \equiv {\cal Y}^{u1} + {\cal Y}^{u2}$, we get
\begin{eqnarray}
{\cal Y}^{u2} =
\left( \begin{array}{ccc}
0 & 0 & 0 \\
0& \star & \star \\
0&\star&\star
\end{array}\right), \ \
{\cal Y}^{u1}[n_{h_1}=1] =
\left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right), \ \
{\cal Y}^{u1}[n_{h_1}=2]  =
\left( \begin{array}{ccc}
0 & * & * \\
* & 0 & 0 \\
* & 0 & 0
\end{array}\right), \ \
\end{eqnarray}
and so the two possibilities
\begin{eqnarray}\label{finalUyukawas-case1}
{\cal Y}^u[n_{h_1}=1] =
\left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & \star & \star \\
0 & \star & \star
\end{array}\right), \ \
{\cal Y}^u[n_{h_1}=2] =
\left( \begin{array}{ccc}
0 & * & * \\
* & \star & \star \\
* & \star & \star
\end{array}\right).
\end{eqnarray}

In these expressions the symbols $\star$ and $*$ represent products of unknown dimensionless coefficients with the vevs of $H_2$ and $H_1$ respectively. The second possibility in Eq.~\eqref{finalUyukawas-case1}, corresponding to $n_{h_1}=2$, is clearly viable (of course one now has to determine the down-type sector as well as leptons) while the first one does not work.

Lets check the down-type sector. Concentrating on the second case, i.e. the one corresponding to $n_{h_1}=2$, and using Eq.~\eqref{yukawa-down-charges}, we obtain (expressing the charges of each entry):
\begin{eqnarray}\label{Dyukawa-case1} \nonumber
\left[{\cal Y}^{d1}[n_{h_1}=2]\right] = \left[
\left( \begin{array}{ccc}
n_{d_1} & n_{d_2} &n_{d_3} \\
n_{d_1}+1 & n_{d_2} +1&  n_{d_3} +1 \\
n_{d_1}+1 & n_{d_2}+1 &  n_{d_3} +1
\end{array}\right) \right], \ \
\left[{\cal Y}^{d2}[n_{h_1}=2]\right] = \left[
\left( \begin{array}{ccc}
n_{d_1} -2 & n_{d_2} -2 &n_{d_3} -2 \\
n_{d_1}-1 & n_{d_2} -1&  n_{d_3} -1 \\
n_{d_1}-1 & n_{d_2}-1 &  n_{d_3} -1
\end{array}\right)\right]. \\
\end{eqnarray}

Now lets choose as an example the following charge assignments: $n_{d_1}=1$ and $n_{d_2}=n_{d_3}=2$. Thus, we obtain
\begin{eqnarray}
{\cal Y}^{d2} =
\left( \begin{array}{ccc}
0 & \star & \star \\
\star & 0 & 0 \\
\star & 0 & 0
\end{array}\right), \ \
{\cal Y}^{d1}[n_{h_1}=2] =
\left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & * & * \\
0 & * & *
\end{array}\right),
\end{eqnarray}
leading to
\begin{eqnarray}\label{finalDyukawa-case1}
{\cal Y}^d[n_{h_1}=2] =
\left( \begin{array}{ccc}
0 & \star & \star \\
\star & * & * \\
\star & * & *
\end{array}\right).
\end{eqnarray}

Note that the role of $H_1$ and $H_2$ is inversely correlated between ${\cal Y}^u$ and ${\cal Y}^d$ in Eqs.~\eqref{finalUyukawas-case1} and~\eqref{finalDyukawa-case1}.

This example is one of several that can lead to interesting Yukawa textures in agreement with the CKM matrix. A complete phenomenological analysis of these models, including the lepton sector, is beyond the scope of this letter and will be presented in a future publication.

\subsection{One up and one for all}
One can also look for a case where one of the doublets (say $H_2$) couples to the up-type sector only while the other couples to everything. From Eq.~\eqref{yukawa-down-charges} we obtain the restriction $n_{d_b} \neq n_{h_2} + n_{q_a} \pmod{3}$. There are three possibilities: i) all $n_{q_a}$ are different and thus we cannot satisfy the condition, ii) two equal, i.e. WLOG $n_{q_1} = n_{q_2} \neq n_{q_3}$, then all three $n_{d_a}$ must be equal leading to unacceptable Yukawa textures. Finally iii) where the three $n_{q_a}$ are equal. This last case leads to Yukawa matrices with either no zeros or complete columns equal to zero. These textures are not useful.

\section{Inert models}
\label{sec:inert}

The inert model corresponds to the situation where all fields have zero charge under $Z_3$ except for one of the scalar doublets, say $H_2$, and only $H_1$ gets a nonzero vev. In this case the $Z_3$ symmetry is not broken and there is a stable neutral particle in the spectrum. Note that this case would correspond to the situation described right after Eq.~\eqref{potential-lambda-terms} where $n_{h_2}-n_{h_1}\neq 0$, except that the correct breaking of the gauge symmetry is accomplished without any soft breaking terms.

The potential in this case is given by
\begin{eqnarray} \label{inertpotential} \nonumber
V(H_1 ,H_2) & = & \mu^2_1 H_1^{\dagger}H_1+\mu^2_2 H_2^{\dagger}H_2 + \lambda_1 \left( H_1^{\dagger}H_1 \right)^2  +\lambda_2 \left( H_2^{\dagger}H_2 \right)^2 + \\
& + & \lambda_3 \left( H_1^{\dagger}H_1 \right)\left( H_2^{\dagger}H_2 \right) + \lambda_4 \left( H_1^{\dagger}H_2 \right)\left( H_2^{\dagger}H_1 \right) \ ,
\end{eqnarray}
and it corresponds to the general (inert) case with arbitrary $Z_n$ (except for $n=2$ where the term $\mathfrak{Re} \left( (H_1^{\dagger}H_2)^{2}\right)$ is also present). The $Z_3$ case does not contribute anything new.

A potentially interesting extension is to consider a third Higgs doublet (for a similar case with $Z_2$ see~\cite{Keus:2014jha} and for recent work where multi-inert doublets are explored see~\cite{Keus:2014isa}). There are two possibilities: either two of them are neutral under $Z_3$ and the third one does not acquire a vev (hence this third one would contain the dark matter candidate) or just one of the doublets is neutral and gets a vev with an additional "Dark Two Higgs Doublet model" D2HDM (note that this is different from the model presented in~\cite{Lee:2013fda}, where the authors used the same name for a model with a normal 2HDM plus an inert singlet scalar.). Of course, in either case, the $Z_3$ symmetry does not play any role in the Yukawa sector and would be useful only in order to stabilize the dark matter candidate. The idea would be to determine if using $Z_3$ would give something different from the $Z_2$ case.

Lets consider the first case and call it the Inert 3HDM. We introduce a third scalar doublet $H_3$ charged under $Z_3$ with charge $n_{h_3}$ and let $H_1$ and $H_2$ be neutral. The potential is given by
\begin{eqnarray} \label{I3HDMpotential} \nonumber
V(H_i) & = &  \mu^2_i H_i^{\dagger}H_i + \lambda_i \left( H_i^{\dagger}H_i \right)^2  + \lambda_{\substack{ij \\
        i < j}} \left( H_i^{\dagger}H_i \right)\left( H_j^{\dagger}H_j \right)  +  \lambda^{\prime}_{\substack{ij \\
        i < j}} \left( H_i^{\dagger}H_j \right)\left( H_j^{\dagger}H_i\right)  + \\ \nonumber
& + & \left\{ \lambda_4 \left( H_3^{\dagger}H_3 \right)\left( H_1^{\dagger}H_2 \right) + \lambda_{123} \left( H_1^{\dagger}H_3 \right)\left( H_3^{\dagger}H_2 \right)
+ \lambda^{\prime\prime}_{\substack{ab \\
        a < b}}  \left( H_a^{\dagger}H_b \right)\left( H_b^{\dagger}H_b \right)  + h.c. \right\}, \\
\end{eqnarray}
where $i,j=1,2,3$ and $a,b=1,2$. Again, this potential is quite general and not specific to $Z_3$. It  is however interesting and can, in principle, contain regions of parameter space where a viable dark matter candidate might exist. In order to determine that one must check all appropriate conditions. At this moment we leave it open as a possibility and jump to the other, namely, the D2HDM where there are two scalar SU(2) doublets charged under $Z_3$ and one neutral.

Let $H_1$ denote the $Z_3$ neutral scalar. Then its vev breaks electroweak symmetry but leaves $Z_3$ unbroken. The potential in this case is more constrained and the number of terms depends on whether $H_2$ and $H_3$ have the same non-zero $Z_3$ charge or not:
\begin{eqnarray} \label{D2HDMpotential} \nonumber
V(H_i) & = &  \mu^2_i H_i^{\dagger}H_i + \lambda_i \left( H_i^{\dagger}H_i \right)^2  + \lambda_{\substack{ij \\
        i < j}} \left( H_i^{\dagger}H_i \right)\left( H_j^{\dagger}H_j \right)  +  \lambda^{\prime}_{\substack{ij \\
        i < j}} \left( H_i^{\dagger}H_j \right)\left( H_j^{\dagger}H_i\right)  + \\ \nonumber
& + & \left[ \lambda^{\prime \prime}_{ab} \left( H_1^{\dagger}H_a \right)\left( H_b^{\dagger}H_a \right)  + h.c. \right]+
 \left\{ \mu^2_{23} H_2^{\dagger}H_3  + \lambda_{123} \left( H_1^{\dagger}H_1 \right)\left( H_2^{\dagger}H_3 \right)
+ h.c. \right\}, \\
\end{eqnarray}
with $a = 2, 3$ and where the terms inside the curly brackets survive in the case where $H_2$ and $H_3$ have the same $Z_3$ charge, and so, if they do not, the potential without those terms is different from the one that we would obtained using $Z_2$ instead. .

The interesting (different from $Z_2$) case then corresponds to $n_{h_3}\neq n_{h_2}$ leading to the $Z_3$ invariant potential
\begin{eqnarray} \label{D2HDMZ·potential} \nonumber
V(H_i) & = &  \mu^2_i H_i^{\dagger}H_i + \lambda_i \left( H_i^{\dagger}H_i \right)^2  + \lambda_{\substack{ij \\
        i < j}} \left( H_i^{\dagger}H_i \right)\left( H_j^{\dagger}H_j \right)  \\
& + &   \lambda^{\prime}_{\substack{ij \\
        i < j}} \left( H_i^{\dagger}H_j \right)\left( H_j^{\dagger}H_i\right)  +
 \left[ \lambda^{\prime \prime}_{ab} \left( H_1^{\dagger}H_a \right)\left( H_b^{\dagger}H_a \right) + h.c. \right] \ .
\end{eqnarray}

The "dark" phenomenology of this model can be substantially different from the case with one inert doublet. This D2HDM has two charged dark scalars, two dark pseudoscalars and two CP-even dark scalars. Depending on the mass spectrum for the dark sector there will be interesting cascade and radiative processes not present in the case where the is only on inert doublet~\cite{Keus:2014isa}. A complete study of the properties and signals for this model along the lines of~\cite{Keus:2014isa} will be presented in a forthcoming publication.

\section{Conclusion}
\label{sec:conclusion}
We have explored some consequences of imposing $Z_3$ symmetry to the scalar potential of the Two Higgs Doublet Model. The exploration went into two main venues: on one hand, by charging the Standard Model fermions under the $Z_3$, it is possible to generate interesting Yukawa textures, something that the usual case with $Z_2$ symmetry cannot do. We have presented some cases that can lead to phenomenological viable models, namely cases that can reproduce the CKM matrix and, if extended as in~\cite{Aranda:2011rt}, the PMNS matrix in the lepton sector. The successful case corresponds to a situation where one of the doublets is free to couple to all generations whereas the other is restricted to couple to the $2-3$ sector only. Depending on the specific charges, several models can emerge with suitable Yukawa textures and a full phenomenological analysis of those models will be presented elsewhere. We discussed other two scenarios that however did not succeed in providing useful Yukawa textures: i) requiring one doublet to couple to the up-type sector and the other to the down-type sector and ii) one doublet free to couple to everything while the other restricted to the up-type sector.

The second venue of exploration consisted in the implementation of the so-called inert models within the framework of $Z_3$. In this case the doublet(s) that can get a vev must be neutral under $Z_3$ (or any other symmetry that might be used in the scalar sector) and so after electroweak symmetry breaking the $Z_3$ si preserved. If there is a doublet (or several) that is charged nontrivially under the $Z_3$ (and thus is not allowed to have a vev), then there is a stable particle in the spectrum that can in principle become a dark matter candidate. We showed that a particularly promising scenario is the one in which there is the usual Standard Model Higgs doublet plus two inert doublets with $Z_3$ symmetry, a Dark Two Higgs Doublet Model. This scenario can be different from the one with $Z_2$ symmetry and can lead to interesting phenomenology. A full analysis of this model is under investigation.

\acknowledgments
This work was supported in part by CONACYT (M\'exico) and by PROMEP (M\'exico) under the grant: Red Tem\'atica: "F\'isica del Higgs y del sabor."

\bibliographystyle{ieeetr}

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