Maintaining prediction quality under the condition of a growing knowledge space. A probabilistic dynamic model of knowledge spaces quality evolution

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1 Maintaining redition quality under the ondition of a growing knowledge sae Christoh Jahnz ORCID: jahnz@gmx.de Intelligene an be understood as an agent s ability to redit its environment s dynami by a level of reision whih allows it to effetively foresee oortunities and threats. Under the assumtion that suh intelligene relies on a knowledge sae any effetive reasoning would benefit from a maximum ortion of useful and a minimum ortion of misleading knowledge fragments. It begs the question of how the quality of suh knowledge sae an be ket high as the amount of knowledge kees growing. This artile rooses a mathematial model to desribe general riniles of how quality of a growing knowledge sae evolves deending on error rate, error roagation and ountermeasures. There is also shown to whih extend the quality of a knowledge sae ollases as removal of low quality knowledge fragments ours too slowly for a given knowledge sae s growth rate. A robabilisti dynami model of knowledge saes quality evolution In the age of modern eistemology intelligene is not an indisutable term as the onet of reality is neither. The term intelligene as used in following an aet a Glasersfeld s eistemologial view of onstrutivism [] as far as two onditions are aeted: (I) (II) An agent s intelligene enables it to derive regularities from exeriene in order to guide it to advantageous ations. There is an underlying reality whih is onsistent with itself, i.e. different ersetives on the same subjet-matter do not result ontraditorily [2]. This artile further assumes an agent s intelligene is based on some knowledge sae. A knowledge sae is omosed by knowledge fragments aquired during the agent s ste by ste learning. The knowledge sae is a ruial art of the world model whih the agent is establishing to gain orientation in its dynami environment. Whether the knowledge sae is understood, naively, as a reresentation of reality or whether it just omoses the agent s subjetive reality is not imortant in the ontext of this artile as long as the basi assumtions (I) and (II) mentioned above are aeted. In following, an atomi fragment of this knowledge sae will be named onet. Insired by onstrutivism, eah of suh onets has been established by onets already existing within an agent s knowledge sae enrihed by an agent s sensory information. New onets are either assumed to be reated by dedutive [4][5][6] or by indutive [7][8] inferene. When an agent tries to redit its environment it usually selets a subset from its whole knowledge sae due to some method. Using this knowledge subset, it dedues reditions about the environment. However, the redition quality deends on the quality of onets used. The onets an never be of erfet quality beause maing the world into a limited model means simlifying and losing information. A resulting redition will robably be inorret if relevant information has, for some reason, been lost, i.e. the redition will onflit with the subsequently ereived result of the orresonding world behaviour. Within our robabilisti model, the quality of eah onet an be defined as an abstrat redition suess robability ( 0 ), measuring how often a single onet

2 ontributes to a orret redition result relative to the frequeny of its invoation within any ontext. Within a onet sae, the onets with P lim (P lim 0 P lim ) will, by definition, be alled arasiti and onets with P lim will be alled aurate, whilst P lim an be hosen arbitrarily. Low quality onets may be established for artiular reasons suh as overgeneralization [9] in indutive ognitive roesses or roessing failure or inomlete roessing [0] during dedutive ognitive roesses. Suh ognition failures an lead to the misreation of a new onet. It is assumed that eah new onet is formed as arasiti in onsequene of erroneous logi or bad indution with a robability P err (P err 0 P err ). Referring to the assumtion, that new onets usually are founded in existing ones, there an be identified still another ossibility of a onet beoming arasiti: if at least one of its base onets is arasiti (figure ). A onet is defined as a base onet if it is neessary for the derivation of a hild-onet by ognitive dedution [5]. As suh, using the hild-onet also means inluding the quality of all of its base onets, giving a redition suess robability hild min( base(),..., base(n) ) where base is the redition suess robability of the resetive base onet and n the number of all base onets ontributing to the hild onet. In other words, the quality of a hild-onet annot be better than the worst quality of any of its base onets. This way, quality roblems are roagating throughout the onet sae by inluding base onets of low quality. Hierarhially strutured attern lassifiers whih hel an intelligent agent to determine the semanti meaning of objets it is dealing with [3] are one tyial examle for deely tree strutured onet hierarhies inluding numerous base onets for most of the onets within those lassifiers. Assuming firstly a redition suess quality n of any base onet for a seifi hild onet as statistially indeendent from m of any other base onet and seondly that base onets aear to be randomly hosen over, the resulting robability of forming a arasiti hild-onet by inluding a arasiti base onet is () i ( ) B where ( ) is the total number of onets in, ( ) is the number of arasiti onets in and B (B ) is the number of base onets for eah hild onet. Sine further analysis onsiders a non-disrete growth roess, the first two arameters are not defined as integers. Assuming further that either the orresonding influene of i or P err is suffiient to ause a new onet to beome arasiti, the total robability of a onet beoming arasiti results in (2) Perr i Perr i ) B ( Perr )(. Now the roess of a growing onet sae (3) L t r t leanu (4) L t r t leanu an be desribed as 2

3 where and (, 0, ) are the hange rates of the resetive onet ounters for eah time ste t. L desribes the total number of new onets generated eah time ste, and for simliity, is assumed to be. This means, that eah time ste one new onet is generated. The fator r leanu (r leanu ) desribes a new aset: Eah time ste there is not only reated a otion of arasiti onets determined by, but it is also assumed that by some roedure eah time ste a ortion of arasiti onets is removed from in order to leanse the onet sae of onets with low redition quality. For now, with L= there an be derived the differential equation from (3), (4) and (2) as (5) r leanu r leanu r leanu whih here will be named equation of arasiti ontamination. It desribes the evolution of the amount of arasiti onets relative to the evolution of the total amount of onets within an intelligent agent. Its solution rovides us with an understanding of how ontamination evolves when inreases. However, before solving the equation, r leanu must be examined more losely. Two generi methods of identifying arasiti onets will be introdued. Generally, it is assumed, that finding arasiti onets is a non-trivial roess within the intelligent agent. Otherwise the agent ould have avoided establishing suh onets at reation time. Both lean-u aroahes are based on statistial methods. The first method rooses a lean-u by exeriene. This means that onets are somehow samled over time for their quality and if they are found to have insuffiient quality, they will be removed from. One way would be to rovide eah onet with a umulative quality rate, whih is inreased eah time the onet artiiates in the satisfatory solution of some ognitive roblem. Conversely, the umulative quality rate is dereased if the solution is found to be unsatisfatory. If the aggregate of quality rate stabilises below a ertain limit, the onet will be removed. The general aroah here is named ragmati redution and is modelled by (6) r rag Prag _ ident Rrag _ freq Rrag where Prag _ ident (P rag_ident 0 P rag_ident ) reresents the robability of identifying one arasiti onet by samling during one time ste and R _ (R rag_freq ) reresents the frequeny by whih this identifiation roess ours during one time ste t. Both as onstant assumed system arameters are ombined to an abstrat redution arameter R whih determines the erformane of ragmati redution. Hereafter, as indeendent of. rag freq rag Rrag will be onsidered The seond method rooses a lean-u by ometition. If there are resent arasiti onets, it is likely that ontains logial inonsistenies, sine it is ostly to hek eah onet at reation time for logial onsisteny against all onets established in []. Suh inonsistenies will result in ontraditory inferenes about one and the same ognitive subjet-matter [2]. Conets whih model reality well ause different inferenes resulting onsistently beause as a basi assumtion in sientifi theory reality is assumed to be 3

4 onsistent with itself [2]. Thus, only any remaining onets of low quality an be resonsible for ontraditions. The lean-u method is suosed to loate ontraditive onets somehow and to identify the lower quality onets by a suess rate of statisti signifiane. This is similar to Darwinian seletion as some survival-of-the-fittest method is used to deide about whih of the ontraditive onets has to be removed from. The general aroah here is named ometing redution and is defined by 2 (7) rom Pom _ math Pom _ ident Rom _ freq ( ( ) ) Rom where P om _ math (P om_math 0 P om_math ) reresents the robability of one arasiti onet mathing one ontraditive aurate onet er time ste t. P om _ ident (P om_ident 0 P om_ident ) reresents the robability of the suessful identifiation of suh a arasiti onet during t and R om _ freq (R om_freq ) reresents the frequeny by whih suh identifiation roess ours during t. All different robabilities are assumed to be indeendent from eah other. The three as onstant assumed system arameters are ombined to one abstrat redution arameter R whih determines the erformane of ometing redution. Hereafter, om Rom will be onsidered as indeendent of. Finally, the indeendent ontributions to arasiti redution for eah time ste by (6) and (7) lead to 2 (8) rleanu rrag rom Rrag Rom ( ( ) ) aomlishing the definition of all arameters ontained within equation (5). Different solutions for the evolution of arasiti ontamination resulting from (5) for different system arameters are resented in figure 2 together with the results of a Monte Carlo simulation modelling the same senario [3], but without using equation (5). The simulation was ontributed in order to hek if the robabilisti theory is orret. The results roved to be onsistent as shown in figure 2. The Monte Carlo simulation uses a B whih is not onstant for eah onet but is, more realistially, hosen randomly with binominal distribution. As shown in figure 2, assuming the mean value of the binominal distribution B for solving (5) aroximates fairly well the results of the Monte Carlo simulation within a ertain range of. Now, we are able to answer the question, whether and if yes at what ontamination the intelligent agent will stabilise deending on different sets of the onstant arameters {P err, B, R rag, R om } for an ever-inreasing. Or, in other words, now we are able to alulate, what efforts of arasiti redution are exeted by our robabilisti model to reserve the onet sae from beoming entirely ontaminated by arasiti onets. Defining the ontamination arameter k and it an be roven [4] that 4

5 2 3 B (9) ( R R ) k ( R 2R ) k R k ( P )( k) 0 rag om rag om exists as ondition for a growing ontamination and 2 3 B (0) ( R R ) k ( R 2R ) k R k ( P )( k) 0 rag om rag for a dereasing or onstant ontamination assuming an ever-inreasing. om It an be easily verified that (9) is given for k=0 and P err >0. This means that even ontamination free onet saes will start to beome ontaminated with a growing. As soon as the growing ontamination k auses (0) to be fulfilled, the ontamination stabilises at this k. Thus, the boundary ontamination whih is aroahed by the intelligent agent for an ever-growing is determined by searhing for the zero in (0) by starting at 0- ontamination. Conversely, setting a maximum ontamination k at slightly below and traing down (0) until the first zero is enountered, will identify the k at whih the intelligent agent will stabilise for an ever growing after desending from a maximum ontamination. If the initial ontamination is total, i.e. equal to, the intelligent agent will remain stable at this ontamination for an ever-growing. Figure 3 shows the final arasiti ontamination for an infinitely growing as funtion of ragmati and ometing redution for different sets of {P err, B}. For eah onfiguration one senario starts at an initial ontamination of 0 and the other at nearly. Interretation and Conlusion Figure 3 shows a funtion tye whih for its harateristi shae here is named lateau funtion. Reduing the efforts for ragmati and ometing redution suffiiently auses final arasiti ontamination to aroah towards a maximum of whih is on to of the lateau. Intelligent agents residing in suh state will suffer from an ever dereasing quality within their knowledge sae and therefore will be inreasingly less able to make satisfying reditions about their environment. The effet of ragmati and ometing redution effort is asymmetri: If the effort for ragmati redution is ritially low, not any level of ometing redution is able to imrove knowledge sae s quality evolution. In ontrary, a suffiiently high ragmati redution effort is able to imrove knowledge sae s quality evolution even without any ometing redution effort. As the threshold-like liff edge of the lateau shows, there exists an area where inreasing ragmati or ometing redution effort only slightly will lead to a tremendous imrovement of the knowledge sae s quality evolution. Contrariwise, a slight derease of either redution will ause knowledge quality ollasing. Interestingly, neither the amount of base onets B nor the error robability P err signifiantly determine the quality evolution of the knowledge sae. Instead, the quality evolution is mainly determined by the ragmati and ometing redution efforts. Outlook The mehanisms of onet sae ontamination mitigated by ragmati and ometing redution seem to not be limited to the disiline of mahine learning. Instead, they might be suseted to also lay a role for learning in biologial organisms and in human organisations. Further researh might show whether there is any universality about this theory. om om err err 5

6 Referenes [] E. v. Glasersfeld; Le Moigne s Defense of Construtivism Entre systémique et omlexité, hemin faisant (Between systemi and omlexity, making the way); Presses Universitaires de Frane 999; [2] N. J. Laey, M. H. Lee; The Eistemologial Foundations of Artifiial Agents; Minds and Mahines; August 2003; Volume 3; Issue 3; [3] E. Rih, K. Knight; Artifiial Intelligene; MGraw-Hill 99; 2 nd ; Chater 9 [4] St. Russel; P. Norvig; Artifiial Intelligene A modern aroah; Prentie Hall 2003; 2 nd ; Chater 9 [5] R. Soher-Ambrosius, P. Johann; Dedution Systems; Sringer 999;.4 [6] H. Seiffert; Einführung in die Wissenshafts-Theorie; C.H. Bek 975; Setion 2 [7] St. Russel, P. Norvig; Artifiial Intelligene A modern aroah; Prentie Hall 2003; 2 nd ; Chater 8.2 [8] H. Seiffert; Einführung in die Wissenshafts-Theorie; C.H. Bek 975; Setion 3 [9] St. Russel; P. Norvig; Artifiial Intelligene A modern aroah; Prentie Hall 2003; 2 nd ; Chater 8.3; Noise and overfitting [0] St. Russel, P. Norvig; Artifiial Intelligene A modern aroah; Prentie Hall 2003; 2 nd ; Chater 6.4 [] St. Russel, P. Norvig; Artifiial Intelligene A modern aroah; Prentie Hall 2003; 2 nd ; Chater 0.8 [2] E. Rih, K. Knight; Artifiial Intelligene; MGraw-Hill 99; 2 nd ; Chater 7 [3] Suorting Material: random generator driven simulation of ontamination [4] Suorting Material: Boundary value solution for arasiti ontamination 6

7 onentration of arasitary onets onentration of arasitary onets onentration of arasitary onets onentration of arasitary onets onentration of arasitary onets Figures Fig. : arasiti roagation within a onet tree. Fig. 2: Solution of equation (5) (blak). The y- axis is normalized to total number of onets. The x-axis is normalized to initial number of onets. The asymtote is determined by equation (0) (blue). The Monte Carlo simulation results max. (red) and min. (green) from 20 simulation eohs. The initial number of onets is 0=000 and of arasiti onets 0: (A) Perr=0.05, Bmin=2, Bmax=20, Rrag=0, Rom=0, 0=0 (B) Perr=0., Bmin=7, Bmax=7, Rrag=2, Rom=2, 0=200 (C) Perr=0., Bmin=2, Bmax=2, Rrag=2, Rom=2, 0=200 (D) Perr=0., Bmin=5, Bmax=5, Rrag=2, Rom=2, 0=200 (E) Perr=0., Bmin=2, Bmax=8, Rrag=2, Rom=2, 0=200 B evolution of arasitary onet ontamination C evolution of arasitary onet ontamination total no. of onets as multiles of initial no. of onets total no. of onets as multiles of initial no. of onets A evolution of arasitary onet ontamination D 2 evolution of arasitary onet ontamination E 2 evolution of arasitary onet ontamination total no. of onets as multiles of initial no. of onets total no. of onets as multiles of initial no. of onets total no. of onets as multiles of initial no. of onets 7

8 P err=0., B=5, k 0=0 P err=0., B=50, k 0=0 P err=0.9, B=50, k 0=0 P err=0., B=5, 0.99<k 0< P err=0., B=50, 0.99<k 0< P err=0.9, B=50, 0.99<k 0< Fig. 3: Final arasiti ontamination resulting from equation (0) with k 0 as initial arasiti ontamination. 8