New problems of machine learning theory named randomized machine learning are considered. They are based on the entropy maximization methods, that give the best solutions under maximum uncertainty. In respect to parameterized model we obtain entropy optimized probability density functions of parameters. In machine learning procedures the randomized model is a generator of stochastic ensemble of possible solutions. The problems of classification and dynamic regression are considered.
computed probability, which is computed during the learning process [Hastie et al., 2001]. This result can be achieved by setting of prior probabilistic characteristics of undefined parameterized model followed by their estimation using arrays of real retrospective data. The most common approach of EP-ML is based on Bayes formula. It is known a fact that the method has high sensitivity to prior probabilistic characteristics which are set by experts. This feature can not be treated as positive characteristics of EP-ML, but there exists more important circumstance, which makes EP-ML methodologically flawed under uncertainty: its probabilistic characteristics are “set” and (often the only one) solutions that fit these conditions are generated [Zolotikh, 2013].
In general, the phenomena of the uncertainty in terms of its stochastic representation is much larger. It is a stochastic environment filled with random objects: vectors, trajectories, whose probabilistic characteristics are unknown. So it seems adequate to represent it as a special randomized model. Such a randomization is considered to be optimal under maximum uncertainty. Procedures in which the information entropy is used as the measure of uncertainty we will refer to procedures of Entropy-Randomized Machine Learning (ER-ML) [Popkov & Popkov, 2014]. 2
Statement and Solution of the ER-ML Problems Key blocks of ER-ML-procedure are the model (ERM-ML) whose parameters are randomized, and the algorithm (ERA-ML), which is a composition of mathematical formulation of the problem of estimation of probabilistic characteristics of the model.
Mathematical model is described by a nonrandom vector functional Ωˆ(X˜ϱ(j) | a, P (a)) with random parameters a. For each observation j, the input array (a matrix X˜ϱ(j)) consists of ϱ column vectors x(j − ϱ), x(j − ϱ + 1), . . . , x(j). The model with the above-mentioned properties will be called the randomized parameterized model (RPM). Consequently, the model output at observation j represents an ensemble Yˆ(j | P (a)) of the random vectors yˆ(j | P (a)) relating to the input data and random parameters through the vector functional Ωˆ(X˜ϱ(j) | a, P (a)), i.e., Yˆ(j | P (a)) = Ωˆ(X˜ϱ(j) | a, P (a)), j = 1, s.
The errors in the output data are modeled by an ensemble E (j | Qj(ξ(j))) of the random vectors ξ(j) with the PDF Qj(ξ(j)), which is added to the ensemble of the RPM output: V(j | P (a), Qj(ξ(j))) = Yˆ(j | P (a)) + E (j | Qj(ξ(j))), j = 1, s.
Thus the model is the generator of random vectors with given density.
ERA-ML is formulated as the functional entropy programming problem contained k-balances with real data: under conditions normalized ∫ ∫ j H[P (a), Q(ξ)] = − ∫
A
P (a) ln
P (a)
P 0(a) da − ∑s ∫ j=1 j
Qj(ξ(j))
Qj(ξ(j)) ln Qj0(ξ(j)) dξ(j) ⇒ max, A
P (a)da = 1,
Qj(ξ(j)) dξ(j) = 1, j = 1, s.
(1) (2) (3) (4) and empirical balances
m(k)(j | P (a), Qj(ξ(j))) = y(j), j = 1, s.
Here the vector m(k) contains the components, that are k-roots of k-moments; P 0(a), Qj0(ξ(j)) denote the prior PDFs of the parameters and noises, respectively..
For k = 1 this problem has an analytical solution parameterized by Lagrange multipliers: In these equalities,
P 0(a) exp [− ∑s
j=1⟨θ(j), v(j)(a)⟩ Qj0(ξ(j)) exp [− ∑js=1⟨θ(j), ξ(j)⟩]
[ s ] P 0(a) exp − ∑⟨θ(j), v(j)(a)⟩ da,
[ s ] Qj0(ξ(j)) exp − ∑⟨θ(j), ξ(j)⟩ dξ(j), j = 1, s. (5) (6) (7) (8) (9) 3
where sigm is
P(θ) Qj(θ) j=1
j=1 Here θ = {θ(1), . . . , θ(s)} are Lagrange multipliers. They provide considerably specific system of nonlinear equations consist of so-called integral components: multidimensional definite integrals of the parameters and noises. Nonlinearity of the equations and the availability of integral components lead to the need of exploit numerical methods based on Monte Carlo Method (MMC). We have developed the GFS: generation, ltration, selection algorithm targeted to solving such problems [Popkov et al., 2015].
The problem of applying MMC to solving of global optimization problems with analytically defined functions is studied in many publications, for instance, [Strongin & Sergeyev, 2000,Zhigliavsky, 2006, Sergeyev & Kvasov, 2008, Polyak & Gryasina, 2008]. GFS -algorithm is oriented to the problems with algorithmically defined functions.
ER-ML procedure is applied to the problem of the soft-binary classi cation. The randomized model (decision rule) bases on a single-layer neural network with random parameters a is used to solve this problem: yˆ(i)(a) = sigm (⟨e(i), a⟩),
i = 1, m, sigm(x) =
1 1 + exp[−α(x − ∆)] with fixed parameters α, ∆. This function has a random argument, as the parameters a of the randomized model are random. The values of sigm(x) from the interval [1/2, 1] correspond to class 1, while the values from the open interval [0, 1/2) to class 2.
i e(1i) 1 0.11 2 0.91 3 0.57 H[P (a)] = −
P (a) ln P (a)da ⇒ max,
P (a)sigm (⟨e(i), a⟩)da = y(i), i = 1, m.
Thus, the “soft-binary” classification problem in terms of ER-ML is stated as (10) (11) (12) (13) (14) (15)
A The solution of this problem has the form where
, W (a) = exp (−⟨θ, yˆ(a)⟩) ,
∫
exp [−⟨θ, yˆ(a)⟩] da.
A Consider classification procedure for an arbitrary document t(j).
Step 1-i. Generate an ensemble Yˆ(i) of the randomized model output (decision rules) (8) with the function P (a) (13). The ensemble contains N random values from the interval [0, 1].
Step 2-i. If a random value from this ensemble exceeds 1/2, then document t(i) is assigned class 1; otherwise, class 2.
Step 3-i Suppose that N1 values are assigned class 1 and N2 values class 2. Since the number of trials N is sufficiently large, the quantities p(i) = N1/N and p(2i) = N2/N yield the empirical 1 probabilities of assigning appropriate classes to document t(i).
By repeating steps 2-i, 3-i for the whole collection T, we obtain the probability distribution of assigning class 1 or 2 to the document.
Example 1. Let us consider a problem of “soft-binary” classification of 3 documents, each of which is characterized by 4 weights.
The dimension of RML-algorithm is 4, the learning collection consists of three documents each described by four weights, see Table 1.
The randomized model (8) has the parameters α = 1.0 and ∆ = 0. The “learner” responses are y = {0.18; 0.81; 0.43} (yi < 0.5 corresponds to class 2, yi ≥ 0.5 to class 1). The parameters belong to the ranges ai ∈ [−10, 10], i = 1, 4. For this learning collection, the entropy-optimal function W (a) (14) takes the form
W (a) = exp (
3 ) − ∑ θiyi(a) ,
i=1 5 a1 0 -5 -10 . v[ih] = Ei(b, m|E0) + ξ[ih],
i ∈ [0, I],
Ei(r, ur | E0) = E0 exp[(r + uri)ih], i ∈ [0, I]. where r means reproduction rate, ur is the velocity of its changing.
The measurement errors are modeled by a random vector ξ = {ξ[0], . . . , ξ[Ih]} with independent interval-type components and a PDF Q(ξ) defined on the set
The ER-ML algorithm yields the following entropy-optimal PDFs:
I Ξ = ∪ Ξj, j=0
Ξj = [ξj , ξj+], p(i) 1 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 • model parameters • noise where and
Class 1 (a) (b) P (r, ur) = 1 I
∏ pi (r, ur|θi),
R(θ|E0) i=0 pi (r, ur|θi) = exp (−θiEi(r, ur|E0)) ;
Q (ξ) = 1 I
∏ qj (ξ[jh]|θj),
Q(θ) j=0 qj (ξ[jh]|θj) = exp (−θjξ[jh]) .
R(θ|E0) = ∫
I ∏ exp (−θiEi(r, ur|E0)) drdur
I i=0 Q(θ) = =
+ ∏I ∫ ξj
exp(−θjξ[jh])dξ[jh] = j=0 ξj
I ∏ 1 (exp(−θjξj ) − exp(−θjξj+)). j=0 θj (19) (20) (21) (22)
Here θ are Lagrange multipliers.
Example 2. Find the entropy-optimal PDFs of the model parameters and noises for the retrospective data corresponding to the period from 1960 to 1995 with step h = 5 years (see Table 2). Using E0 = Ermeall[0] in (19-22) we obtain required PDFs (see Fig. 3–4).
The RPM is used for comparison of UN- and RPM- prognoses for interval 1995–2015 on the base UN-prognosis made at 1985. The relative mean-square deviation between the real trajectory and the ensemble-average one is 0.3%. The relative mean-square deviation for the UN prognosis is 0.8%. This work was supported by the Russian Foundation for Basic Research (project no. 16-0700743).