A New Decision Making Method for Selection of Optimal Data Using the Von Neumann-Morgenstern Theorem

1Introduction

The superiority of artificial neural networks (ANNs) in various tasks (classification, pattern identification, prediction, etc.) has led researchers to focus much of their efforts on the study of the functioning of their components from a theoretical perspective, see Higham and Higham (2019), Smale et al. (2010). It is well known that ANNs have a high capacity for learning, the effectiveness of which depends on many factors. Amongst them, the problem complexity influences to a high degree the ANN performance, which depends not only on the ANN architecture, but also on the accurate and sufficient training data and the efficiency that datasets show throughout the process. Training in recurrent neural networks (RNNs) – ANNs that stand out for their high capacity for learning and recognition of temporal patterns – depends to a large extent on size, type and structure of the selected training sets (Chen, 2006; Zhang and Suganthan, 2016; Zapf and Wallek, 2021): this is such a central point that decisively influences both the speed and the ability to learn. During the training phase, where the unknown parameters are to be determined, the quality and learning capacity of the selected training datasets are of key importance (Mirjalili et al., 2012).

The main objective of this paper is to provide a robust methodology to select optimal training datasets (those with the highest learning capacity) that can be used in any context to maximise the performance of the trained models. This methodology has been designed to run in parallel with RNN learning so that, while the RNN learning evolves progressively only with the optimal training inputs, the data noise gradually disappears. This has a positive impact on the quality of the RNN results while minimising the likelihood of occurrence of overfitting.11 The key idea in the design of the mathematical structure that supports this selection is to define a binary relation that gives preference to those datasets with higher estimator abilities by using Utility theory. A second contribution of our work is to have designed our methodology based on tools that have not been used previously for this. This novelty lies in showing Markov Networks (MNs), widely known as generative models (Gordon and Hernandez-Lobato, 2020), as models with a real discriminative capacity when used in conjunction with the Von Newman Morgenstern theorem (VNM theorem), a cornerstone of Game Theory with an extensive background also in Decision Theory (Machina, 1982; Delbaen et al., 2011). In order to faithfully model the RNN reality, we have used the dynamic version of the MNs (TD-MRFs). It is worth noting the versatility of our proposal, which can be also applied to other data-driven methodologies provided that they are regulated by dynamical systems.

Markov Random Fields (MRFs) are also known as Markov Networks (MNs) in those contexts that require highlighting the undirected graph condition (Dynkin, 1984). MRF-type graphical models have experienced a resurgence in recent years. In its origins, they exclusively performed functions related to image processing such as restoration or reconstructing. Later works such as García Cabello (2021) or Wang et al. (2022) have acknowledged their high predictive capability due to the equivalence between MRFs and Gibbs distributions, which provides an explicit expression of the prior likelihood after appropriate choice of the energy functions. MRF solutions are widely regarded as generative models as opposed to discriminative approaches, more related to tasks which involve classification.

Regarding the literature review, the selection of optimal training sets has not been studied in a general framework so far. To this author’s knowledge, this is the first analysis that aims to provide guidance for a general context. Published papers have studied this issue either only in contexts of ANN classification tasks or in very precise scenarios (electrical, financial or chemical engineering) taking advantage of their specific techniques. Within the first category, the genetic algorithm (GA) is widely used as a tool to create high-quality training sets as a the first step in designing robust ANN classifiers, see Reeves and Taylor (1998), Reeves and Bush (2001) or more recently, the paper (Nalepa et al., 2018). Disadvantages of using GA, apart from slowness, include that it is computationally expensive and too sensitive to the initial conditions. In our proposal, however, the calculation of the probabilities associated with the utility (i.e. efficiency as estimators) of the inputs is very simple and therefore does not add computational cost.

Within the second category, in the paper (Zapf and Wallek, 2021), the authors made a comparison between existing methods in the area of chemical process modelling in order to split a training set from a given data set. In Wong et al. (2016), the authors proposed a data selection for statistical machine translations, based on recursive neural networks which can learn representations of bilingual sentences. The paper (Fernandez Anitzine et al., 2012) analyses through a very context-specific instrument (ray-tracing) the ANN optimal selection of training set in the context of predicting the received power/path loss in both outdoor and indoor links. In Kim (2006), authors propose a GA approach for ANN instance selection for financial data mining.

As for the use of MNs/MRFs (prior probability) for problems which involve probability a posteriori, in the literature the terms “MNs/MRFs” and “discriminative” appear together only and exclusively to refer to discriminative random fields (DRFs) or equivalently conditional random fields (CRFs), both type of random fields which provide by definition a posterior probability.

The rest of the paper is structured as follows: preliminaries of Section 2 include basic knowledge on preference relations and VNM theorem, MNs and RNN functioning. Section 3 structures the steps to be followed to reach a solution to the proposed problem. The design of an abstract TD-MRF-based framework is performed in Section 4 which will subsequently allow the computation of prior probabilities associated with the VNM theorem. A TD-MRF structure for the input sets is also provided here. In Section 5, the expected utility theorem is applied after proving that the conditions for doing so are met. Section 6 highlights (and proves) the main results of our work. In Section 7, an example of the method application is developed. Section 8 finally concludes the paper.

2Preliminaries

2.3Functioning of Recurrent Neural Networks RNNs

Neural networks (NNs) have a general functional definition as composition of parametric functions which disaggregates the linear component of the non-linear activation function (see García Cabello, 2023). Recurrent Neural networks, RNNs (Chou et al., 2022; Zhang et al., 2014), are a particular case of NNs which operate on time sequences and exhibit a special ability for learning lengthy-time period dependencies. Their functioning lies in an intermediary layer h=(h0,…,hT) of hidden states ht in such a way that the data cycle through a loop to this layer. The output is reached by recursive applying the following functions:

(2)

h1=σ(Wxhx1+Whhh0+bh)⋮⋮ht=σ(Wxhxt+Whhht−1+bh)zt=σ˜(Whzht+bz)

for weighted matrices, Wxh,Whh,Whz, and bias vectors, bh,bz, and where zt is the final output and σ,σ˜ are point-wise nonlinearities (activation functions which are applied component-wise). In RNNs, activation σ˜ is the sigmoid, monotonically increasing with range (0,1). For any RNN input xt, its corresponding output is z[xt]∈(0,1), denoted as zt for simplicity.

The loss function loss is chosen depending on the RNN task: usually it is Mean Square Error MSE = 1n∑i=1n(zi−yi)2, often used as SE=12n∑i=1n(zi−yi)2 for cancelling the constant when computing the gradient, where zi and yi represent the RNN output and the target value respectively. RNN functioning is shown in Fig. 1.

Fig. 1

RNN learning process.

3Problem Formulation

In this section we will develop the theoretical framework that will capacitate Markov Network-methodology to perform discriminative tasks based on prior probability and the VNM Theorem 2.1. Specifically, our aim is to design a mathematical model which enables MNs to identify the optimal RNN training sets, i.e. those that produce better estimates in forecasting taks. For a better understanding, we will make reference to a real example of a RNN forecasting process:

Recall that, in the networks as a whole, Recurrent Neural Networks, RNNs, stand out for their predictive abilities based on their potential in processing temporal data. Thus, we are facing some time-dependent learning process by using RNNs where the superscript t denotes time in all cases (no distinction will be made between matrices and their transpose in order to avoid confusion between t transpose and t time).

As is well known, in RNN prediction tasks, training sets are fed with temporal sequences composed with pairs of previous data of the form (input, labels) with the objective of predicting the future, referred to as future target value, y. Thus, Xt is the time-varying variable which needs to be estimated, Zt is the set of RNN outputs and Yt the set of labels (i.e. past target values for training). In terms of the illustrative example: to predict the electrity price one month ahead (target value y, t=+1), the training sets are made up of temporal sequences (xit,yit) whose first component contains different factors that influence electricity prices (fuel prices, transmission and distribution costs, weather conditions, etc.) and whose second component is the electrity price, both xit, yit in site i and prior to present time (t=−9,…,−2,−1,0).

Let In be the (time-dependent) input RNN dataset formed by n vectors:

The objective is to select from the n inputs in In, D<n vectors that will make up the training set: the selection criterion is to choose those which produce the best estimates in the RNN learning process. The choice will be conducted by only using prior probability –instead of posterior– and the VNM Theorem 2.1.

We will list the steps to be taken:

4The Abstract TD-MRF-Based Framework

Here, we first design an abstract graph-based framework that shall provide a model for a dynamic context of k sites and r filters for data discrimination (corresponding to r cliques) for which it will be shown that it is an TD-MRF under certain mild conditions. Such TD-MRF will be the core in the computation of prior probabilities po[Xt=xit] of the VMN Theorem 2.1.

Let S be a set of k sites si, S={si|i=1,2,…,k} such that the variable may take different values depending on the site, Xsit. For each site si, the variable Xsit can be disaggregated into a set of d characteristics which fully describes Xsit in the instant of time t: Xsit=(Xsi1,t,Xsi2,t,…,Xsid,t), with lower case xsit=(xsi1,t,xsi2,t,…,xsid,t) for the feature vector (i.e. a numerical vector corresponding to a realisation of the variable). We shall use indistinctly xsit=(xsi1,t,xsi2,t,…,xsid,t) or xit=(xi1,t,xi2,t,…,xid,t) either for upper and lower case. In a compact form, xij,t∈R,i=1,…,k;j=1,…,d.

Remember that two random variables are equivalent if they have identical distribution. Then, we will define the edges by equivalently defining the neighbourhood N of a site: N(Xsit)={Xsjt|Xsjt,Xsitare equivalent}={Xsjt|P[Xsjt⩽x]=P[Xsit⩽x]∀x}.

Recall that marginal distribution is also known as prior probability in contrast with the posterior distribution (the conditional one). From the former definition, sites in the same neighbourhood have the same prior probability. Moreover,

The following theorem proves then that the DGM defined in Definition 4.1 is a TD-MRF by equivalently showing that the Markov condition:

Insofar as the distribution function is the tool used to make estimates, previous Theorem 4.4 provides a joint measure of how close the variable is to taking a particular value.

Each clique has its own common estimation function:

In discrimination/classification works, the commonly used probability is the conditional or posterior probability P[Xt=x|y]=1Z∏cliquesCjj=1re−ψC(Xt=x|y) where x stands for RNN input, y represents the corresponding target value and ψC(Xt=x|y) means that the energy function ψC only assigns to those inputs x that correspond to y.

4.1Visual Flowchart of the TD-MRF Operational Process

Inputs for a TD-MFRs are specific values of the stochastic variable Xt in a particular time instant t and a site si, xsit. Depending on the context, Xt can be considered as disaggregated into a set of d features for each site si. Thus, each input is xit=(xi1,t,xi2,t,…,xid,t), xij,t∈R, i=1,…,k, j=1,…,d such that each univariate component xij,t∈R can be regarded as input of a single-variable for d processes. The TD-GDM operational process is as follows: by assuming there are r cliques, C1,…,Cr, each input is processed in a clique Cj,j=1,…,r, through the corresponding clique potential e−ψCj([.]),j=1,…,r, such that the output is the estimate made by the corresponding clique (according to Proposition 4.7). Each of these is thus aggregated in a final output by means of the expression given in Corollary 4.5. If, on the other hand, the variable is not considered disaggregated, multivariate energy functions will process the information in a single multivariate execution.

Fig. 2

TD-MRF operational process.

Figure 2 provides a visual representation of a TD-MRF operation in the univariate scenario, where the energy functions ψ are of Gaussian type, i.e. ψCj=([.]−μj)2σj2, j=1,…,r. Note that for xit=(xi1,t,xi2,t,…,xid,t), the corresponding probability po[xit] is po[xit]=(po[xi1,t],po[xi2,t],…,po[xid,t]).

Moreover, the operation of an MRF is equally applied in the calculation of following probabilities:

PCj[Xt⩽xt]=∑x⩽xtP[Xt=x],PCj[xpr<Xt⩽xim]=∑x⩽ximP[Xt=x]−∑i⩽xprP[Xt=x],PCj[Xt>xt]=1−PCj[Xt⩽xt]=1−∑x⩽xtP[Xt=x].

5Application of the VNM Utility Theorem

In this section, we will further develop equation (3),

Recall that in the preceding sections we have considered a second lottery which arises naturally when we test how good the ouput zit∈Zt[In] is as estimate. If M is the threshold below which the loss function is considered acceptable, the secondary lottery is whether the loss function loss(zit)<M or not. With the intuitive notation of the paper (Machina, 1982), our secondary lottery is a probability distribution function over the interval [0,M] (the set of all lotteries over [0,M] is denoted in Machina (1982) by D[0,M]) and the preference functional V(·) on D[0,M] in our case is loss(·). Note that Definition 3.1 of preference over the set Zt[In] can be additionally considered over In for the deterministic assignment xit→zit: for xit,xjt∈In

First of all, it has to be shown that it is a preference relation:

Moreover, former preference relation satisfies the following conditions:

Therefore, the preference relation stated in Definition 3.1 verifies the conditions required for the application of the VNM Theorem 2.1. Thus, there exists an utility function u:In→[0,1] which quantifies the preferences: x1t⪰x2t iff u(x1t)⩾u(x2t). Moreover, the Von Neumann-Morgenstern Expected Utility theorem also provides guidance for computing the utility of the set of inputs through the explicit formula given in equation (3):

6Main Results

The objective of this section is to highlight (after proving) the main results of our proposal. First of all, we define the level of efficiency of an input set In, Eff[In]. Thus, next Theorem 6.2 proves the existence of a formula for determining the efficiency of a training set while Theorem 6.3 gives an explicit expression for computing Eff[In].

7A Case Application

This section is aimed at developing an example of the method application. In order to make a choice between two real data sets, their level of efficiency will be computed. As stated in Definition 6.1, the level of efficiency of an input set Eff[In] measures the learning capacity of its inputs on the basis of the preference relation of Definition 3.1 that prioritises those inputs whose outputs have a lower associated loss (better estimates). To compute Eff[In] we shall apply the formula proved in Theorem 6.3 which follows from Theorem 6.2, in which the existence of an utility function u which quantifies the preference relation was proved. This is

The data sets we shall use here contain prices (€/1 kg) over time for the most common olive oil varieties. These data are available on the websites of the Government of Spain:

https://www.mapa.gob.es/es/agricultura/temas/producciones-agricolas/aceite-oliva-y-aceituna-mesa/Evolucion_precios_AO_vegetales.aspx,

where prices are published on a weekly basis. Specifically, we will consider data sets corresponding to weeks 28/2023 and 39/2022 (the reasons for this choice will be explained later). We shall thus apply the above formula taking into account the following considerations. On one hand, note that the choice of the threshold M will depend on the context. In the olive oil market scenario, assuming a deviation from the olive oil price of 10% is acceptable. Hence, M=0.5. On the other hand, the value of the parameter p will depend on several market features which vary over time depending on physical (rainfall, pollen levels…) and socio-economic (Government regulations) circumstances. When real prices are subject to unusual fluctuations (due to changes in the aforementioned circumstances), such prices used in training tasks will deviate more from the real ones. In consequence, the probability p, such that |zit−y|<M,∀zit, will decrease. Either way, it should be noticed that p is known as soon as the value of M is fixed, but it is not the same for all input sets.

From the above formula, since M is known (and therefore so is p), we must focus on computing po[Xt=xit] for each input xit. To that end, we will follow Fig. 2 of the TD-MRF operational process given in Section 4.1. Since each input xit may be disaggregated into d features xik,t (k=1,…,d), xit=(xi1,t,xi2,t,…,xid,t), the same is true for the corresponding probability po[Xt=xit]=po[xit], which is po[xit]=(po[xi1,t],po[xi2,t],…,po[xid,t]).

We assume that the energy functions ψ corresponding to the r cliques are of Gaussian type, i.e. ψCj=([.]−μj)2σj2, j=1,…,r, since Gaussian distributions are suitable in the olive oil scenario. Actually, given that Gaussian distributions portray those data sets whose majority of elements revolves around the centre, energy functions of Gaussian type are particularly suited for goods whose price takes values in an interval of small length and do not suffer very sharp price variations.

In order to achieve our goal, each input xit must be first processed in each clique Cj,j=1,…,r, through the corresponding clique potential, whose result is