Mean-performance of Sharp Restart II: Inequality Roadmap
MMean-performance of Sharp Restart II:Inequality Roadmap
Iddo Eliazar *1 and Shlomi Reuveni †11 School of Chemistry, Center for the Physics and Chemistry of LivingSystems, The Sackler Center for Computational Molecular andMaterials Science, and The Mark Ratner Institute for Single MoleculeChemistry, Tel Aviv University, Tel Aviv 6997801, IsraelMarch 1, 2021
Abstract
Restarting a deterministic process always impedes its completion. However,it is known that restarting a random process can also lead to an opposite outcome– expediting completion. Hence, the effect of restart is contingent on the under-lying statistical heterogeneity of the process’ completion times. To quantify thisheterogeneity we bring a novel approach to restart: the methodology of inequalityindices, which is widely applied in economics and in the social sciences to mea-sure income and wealth disparity. Using this approach we establish an ‘inequalityroadmap’ for the mean-performance of sharp restart: a whole new set of universalinequality criteria that determine when restart with sharp timers (i.e. with fixed de-terministic timers) decreases/increases mean completion. The criteria are based ona host of inequality indices including Bonferroni, Gini, Pietra, and other Lorenz-curve indices; each index captures a different angle of the restart-inequality inter-play. Utilizing the fact that sharp restart can match the mean-performance of anygeneral restart protocol, we prove – with unprecedented precision and resolution –the validity of the following statement: restart impedes/expedites mean completionwhen the underlying statistical heterogeneity is low/high.
Keywords : restart; resetting; first-passage times; inequality indices; Lorenzcurves. * email: [email protected] † email: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Introduction
This paper is the second in a duo of works exploring the mean-performance of sharprestart. In part one we presented a comprehensive statistical analysis of the topic [1].In this part we continue the exploration, doing so from the rather surprising perspec-tive of socioeconomic inequality. Indeed, this paper shall reveal profound connectionsbetween two seemingly unrelated topics: the measurement of socioeconomic inequal-ity on the one hand, and the mean-performance of sharp restart on the other hand.The novel inequality vantage point yields a whole new set of universal inequality cri-teria that pinpoint – with unprecedented precision and resolution – how the mean-performance of general restart protocols is contingent on the underlying statistical het-erogeneity.As explained in part one, sharp restart can be perceived as an algorithm. The al-gorithm’s input is the random duration of a general task to be accomplished – e.g. acompletion time or a first-passage time of a given stochastic process [2]-[6]. As longas the task is not accomplished, the algorithm restarts the stochastic process periodi-cally, using a ‘sharp’ timer – i.e. a fixed deterministic timer. The algorithm’s outputis the random duration it takes to accomplish the task ‘under restart’. The motivationfor using sharp restart, together with a rather rich literature survey (including [7]-[27]),was described in detail in part one [1].Following a main path in restart research, the analysis presented in [1] focused onmean performance: comparing the input’s mean to the output’s mean. Sharp restartwas termed “beneficial” if the output’s mean is smaller than the input’s mean – i.e. ifmean completion is expedited. Conversely, sharp restart was termed “detrimental” ifthe output’s mean is larger than the input’s mean – i.e. if mean completion is impeded.The two principal analytic tools employed in [1] were the residual lifetime of renewaltheory [28]-[30], and the hazard rate of reliability engineering [31]-[33]. Using thesetools, we established a detailed statistical roadmap for the mean-performance of sharp-restart: universal statistical criteria that determine when sharp restart is beneficial, andwhen it is detrimental.A key result in [1] was based on the input’s coefficient of variation (CV), the ra-tio between the standard deviation and the mean of the task’s duration. The CV is aprincipal standardized measure of statistical heterogeneity. Using this measure we es-tablished the following threshold scheme: if the input’s CV is larger than one then thereexist timers with which sharp restart is beneficial; and if the input’s CV is smaller thanone then there exist timers with which sharp restart is detrimental. A similar CV re-sult holds with regard to the introduction of exponential restart – in which the periodsbetween consecutive restart epochs are drawn (independently) from the exponentialdistribution [34]-[41]. Moreover, when adding branching to exponential restart, theinput’s Gini coefficient (to be explained below) enters the picture and joins the input’sCV in determining if the introduction of exponential restart is detrimental or beneficial[42].In economics, the Gini coefficient is a principal inequality index [43]-[49]. Widely The CV threshold scheme holds when the input’s mean is finite. If the input’s mean is infinite then sharprestart is always beneficial. With regard to these indices we establish existence criteria similar to thatof the aforementioned CV result [1]. Namely, each index has a corresponding thresh-old level, and: if the index is larger/smaller than its threshold level then there existtimers with which sharp restart is beneficial/detrimental. These three existence criteriaare different, as their indices capture different ‘angles’ of inequality. Consequently,each of the existence criteria provides a different type of information regarding therestart-inequality interplay.While existence criteria are important, they cannot assert if sharp restart with aspecific timer is beneficial or detrimental. For practical uses, it is evident that timer-specific criteria are essential. This practical need leads to a profound theoretical ques-tion: can inequality provide timer-specific information? To that end, we first focuson two sharp-restart timers of special interest – one that equals the input’s mean, andone that equals the input’s median. We show that two inequality indices, the Pietraindex and vertical-diameter (Vdiam) index, relate to these timers respectively. Withregard to these indices we establish timer-specific criteria for the mean and mediantimers. Namely, each index has a corresponding threshold level, and: if the index islarger/smaller than its threshold level then sharp restart with the compatible timer isbeneficial/detrimental.From the mean and median timers we carry on to general timers. To that end, weconsider two continuums of vertical and horizontal Lorentz-curve inequality indices.Given a general timer, we show that this timer is related to a unique index of the verticalcontinuum, as well as to a unique index of the horizontal continuum. With regard tothese unique indices we establish timer-specific criteria. Namely, each unique index The terms “Gini coefficient” and “Gini index” are used interchangeably in the literature. E [ ξ ] denotes the expectation of a (non-negative) random variable ξ ; and IID is acronym for independent and identically dis-tributed (random variables). This section sets the stage for a comprehensive mean-performance analysis of sharprestart – to be based on socioeconomic inequality indices – that will be carried out inthe next sections. Subsections 2.1 and 2.2 recap material that was presented in [1].Subsection 2.3 and 2.4 review, respectively, the socioeconomic notions of inequalityindices and the Lorenz curve.
Sharp restart admits the following algorithmic description. There is a general task withcompletion time T , a positive-valued random variable. To this task a three-steps algo-rithm, with a positive deterministic timer τ , is applied. Step I: initiate simultaneouslythe task and the timer. Step II: if the task is accomplished up to the timer’s expiration –i.e. if T ≤ τ – then stop upon task completion. Step III: if the task is not accomplishedup to the timer’s expiration – i.e. if T > τ – then, as the timer expires, go back to StepI and start afresh (independently of the past).The sharp-restart algorithm generates an iterative process of independent and sta-tistically identical task-completion trials. This process halts during its first successfultrial, and we denote by T R its halting time. Namely, T R is the overall time it takes –when the sharp-restart algorithm is applied – to complete the task. The sharp-restartalgorithm is a non-linear mapping whose input is the random variable T , whose outputis the random variable T R , and whose parameter is the deterministic timer τ .This paper shall use the following notation regarding the input’s statistics: distribu-tion function, F ( t ) = Pr ( T ≤ t ) ( t ≥ F ( t ) = Pr ( T > t ) ( t ≥ f ( t ) = F (cid:48) ( t ) = − ¯ F (cid:48) ( t ) ( t > µ = E [ T ] = (cid:82) ∞ t f ( t ) dt .The input’s density function is henceforth considered to be positive-valued over thepositive half-line: f ( t ) > t > This is merely a technical assumption, which is introduced in order to assure that all positive timers τ M ( τ ) = E [ T R ] shall denote the output’s mean; this notationunderscores the fact that the output’s mean is a function of the timer τ , the parameter ofthe sharp-restart algorithm. In terms of the input’s distribution and survival functions,the output’s mean is given by [1, 31, 34]: M ( τ ) = F ( τ ) (cid:90) τ ¯ F ( t ) dt . (1)As the input’s mean is the integral of the input’s survival function, µ = (cid:82) ∞ ¯ F ( t ) dt , Eq.(1) implies that: in the limit τ → ∞ , the output’s mean coincides with the input’s mean,lim τ → ∞ M ( τ ) = µ .Examining the sharp-restart algorithm from a mean-performance perspective, it iskey to determine if the application of the algorithm will expedite task-completion, orif it will impede task-completion. To that end the following terminology shall be used[1]: • Sharp restart with timer τ is beneficial if it improves mean-performance, M ( τ ) < µ . • Sharp restart with timer τ is detrimental if it worsens mean-performance, M ( τ ) > µ .Eq. (1) implies that the output’s mean is always finite: M ( τ ) < ∞ for all timers τ . Thus,if the input’s mean is infinite, µ = ∞ , then the application of the sharp-restart algorithmis highly beneficial – as it reduces the input’s infinite mean to the output’s finite mean: M ( τ ) < µ = ∞ . Having resolved the case of infinite-mean inputs, we henceforth setthe focus on the case of positive-mean inputs, 0 < µ < ∞ . The variance of the input T is the input’s mean square deviation from its mean, σ = E [ | T − µ | ] . The input’s standard deviation σ is the square root of its variance. And,the input’s coefficient of variation (CV) is the ratio of its standard deviation to its mean, σ / µ . The input’s CV is a ‘normalized’ version of its standard deviation σ .In the first part of this duo, based on the input’s CV, the following pair of CV criteria was established [1]: • If the CV is smaller than one – which is equivalent to σ < µ – then there existtimers τ for which sharp restart is detrimental. • If the CV is larger than one – which is equivalent to σ > µ – then there existtimers τ for which sharp restart is beneficial.The CV criteria stem from the following formula: (cid:90) ∞ [ M ( τ ) − µ ] F ( τ ) d τ = (cid:0) µ − σ (cid:1) . (2) are admissible. In general, admissible timers are in the range t low < τ < ∞ , where t low is the lower bound ofthe support of the input’s density function: t low = inf { t > | f ( t ) > } . σ versus µ – affects the output’s mean M ( τ ) . Equivalently, Eq. (2) manifeststhe effect of the input’s CV on the output’s mean M ( τ ) . The derivation of Eq. (2) isdetailed in the Methods.The CV criteria have the following threshold form. The input’s CV is compared tothe threshold level 1. To determine the existence of timers for which the application ofthe sharp-restart algorithm is detrimental or beneficial, one needs to check if the input’sCV is below or above the threshold level, respectively. Evidently, in the CV criteria,one can replace the CV by its square, σ / µ . Consider a human society comprising of members with non-negative wealth values.Such a society has two socioeconomic extremes: perfect equality and perfect inequal-ity. In a perfectly equal society all members share a common positive wealth value. Ina perfectly unequal society 0% of the members possess 100% of the overall wealth; inthis socioeconomic extreme the society’s population is assumed infinitely large [55].An inequality index I is a measure of the society’s socioeconomic inequality: thelarger the index – the more socioeconomically unequal the society. More specifically,an inequality index I takes values in the unit interval, 0 ≤ I ≤
1, and it has three basicproperties [50]-[53]. (I) The index meets its zero lower bound I = I =
1. (III) The index is invariant with respect to the currencyvia which wealth is measured.While widely used in economics and in the social sciences [50]-[53], the applica-tion of inequality indices is not confined to these fields alone. Indeed, an inequalityindex I can be used to measure the inherent ‘socioeconomic inequality’ of any givennon-negative random variable with a finite mean [55, 56]. To that end, deem the randomvariable under consideration to represent the wealth of a randomly-sampled member ofa virtual society. Then, the socioeconomic inequality of the random variable is that ofits corresponding virtual society.Henceforth, the random variable under consideration is the input T of the sharp-restart algorithm. An illustrative example of an inequality index of the input T is whatshall be referred to here as CV index . This inequality index was introduced in [58], andit is a special case of the input’s R´enyi spectra [56]-[58]. In terms of the input’s CV, σ / µ , the input’s CV index admits the following representation [58]: I CV = − + ( σ / µ ) . (3)The input’s CV index I CV is a monotone increasing function of the input’s CV.This index meets its zero lower bound if and only if the input’s CV vanishes, i.e. –for a given positive-mean input – if and only if the input’s standard deviation vanishes, σ =
0. And, this index meets its unit upper bound if and only if the input’s CV diverges,i.e. – for a given positive-mean input – if and only if the input’s standard deviationdiverges, σ = ∞ . For a more detailed account of the CV index and its properties, see[56]. 6n terms of the input’s CV index I CV , the pair of the CV criteria of subsection 2.2can be re-formulated as the following pair of CV-index criteria : • If the input’s CV index is smaller than half, I CV < , then there exist timers τ for which sharp restart is detrimental. • If the input’s CV index is larger than half, I CV > , then there exist timers τ forwhich sharp restart is beneficial.The threshold form of the CV criteria is induced to the CV-index criteria. Specif-ically, the input’s CV index I CV – the input’s ‘CV inequality’ – is compared to thethreshold level . To determine the existence of timers for which the application of thesharp-restart algorithm is detrimental or beneficial, one needs to check if the input’s‘CV inequality’ I CV is below or above the threshold level , respectively. The CV-index criteria described above expose a connection between sharp restart andthe measurement of socioeconomic inequality. In the following sections we shall showthat this connection extends far beyond the CV-index criteria. To that end we shalluse an additional socioeconomic working tool – the Lorenz curve – which is describedin this subsection. Armed with the Lorenz curve, we shall show that the connectionbetween sharp restart and inequality indices is profound and broad, and that inequalityindices purvey valuable information on whether sharp restart is detrimental or benefi-cial.As in subsection 2.3, consider a human society comprising of members with non-negative wealth values. The distribution of wealth among the society’s members isquantified by the society’s
Lorenz curve y = L ( x ) (0 ≤ x , y ≤
1) [62]-[65]. Specifically,the Lorenz curve y = L ( x ) has the following socioeconomic meaning: the low (poor)100 x % of the society members possess 100 y % of the society’s overall wealth.The Lorenz curve y = L ( x ) resides in the unit square (0 ≤ x , y ≤ L ( ) = L ( ) =
1. (II) It is monotone increasingand concave. (III) It is invariant with respect to the currency via which wealth is mea-sured. The first two properties imply that the Lorenz curve y = L ( x ) is bounded fromabove by the square’s diagonal line y = x (see Fig. 1).As noted in subsection 2.3, the human society under consideration has two socioe-conomic extremes: perfect equality and perfect inequality. Recall that in a perfectlyequal society all members share a common positive wealth value. In the space ofLorenz curves the perfect-equality socioeconomic extreme is characterized by the diag-onal line y = x (0 ≤ x , y ≤ y = L ( x ) from the diagonal line y = x can serve as a geometric gauge of the society’s socioeco-nomic inequality: the deviation of the society from the perfect-equality socioeconomicextreme.Evidently, there are many different ways of measuring the deviation of the Lorenzcurve y = L ( x ) from the diagonal line y = x . In turn, these different ways yield differentinequality indices. An illustrative example is described as follows. For a fixed number7 .0 1.00.01.0 q y=L(x) y = x Figure 1: Lorenz-curve illustration. The Lorenz curve y = L ( x ) is depicted in blue, andthe diagonal line y = x is depicted in orange; the Lorenz curve and the diagonal linereside in the unit square (0 ≤ x , y ≤ x = q , depicted in dashed black – between theLorenz curve and the diagonal line. q (where 0 < q < x = q of the unit square. The verticaldistance – along the vertical line x = q – between the Lorenz curve y = L ( x ) and thediagonal line y = x is: q − L ( q ) (see Fig. 1). This vertical distance takes values inthe range [ , q ] . Consequently, the corresponding ‘normalized’ vertical distance [ q − L ( q )] / q takes values in the unit interval [ , ] . It is straightforward to check that thenormalized vertical distance [ q − L ( q )] / q meets the three inequality-index propertiesthat were postulated in subsection 2.3.Any non-negative valued random variable, with a positive mean, has a correspond-ing Lorenz curve [55, 56]. Indeed, as with inequality indices: deem the random variableunder consideration to represent the wealth of a randomly-sampled member of a vir-tual society. Then, the Lorenz curve of the random variable is that of its correspondingvirtual society. As above, we take the random variable under consideration to be theinput T of the sharp-restart algorithm. Thus, henceforth, y = L ( x ) shall manifest theinput’s Lorenz curve. In this section we establish connections between sharp restart on the one hand, and theGini and Bonferroni inequality indices on the other hand. Specifically, we shall showthat these two inequality indices of the input T yield existence criteria that are similarto the CV-index criteria of subsection 2.3, and that have a similar threshold form.8 .1 Gini-index criteria Perhaps the best-known and most popular socioeconomic inequality index is the
Giniindex [43]-[49]. This subsection addresses the Gini index I Gini of the input T .In terms of the input’s Lorenz curve, the input’s Gini index I Gini is twice the areacaptured between the Lorenz curve y = L ( x ) and the diagonal line y = x [47]. Namely, I Gini = (cid:90) [ q − L ( q )] dq . (4)The integral appearing on the right-hand side of Eq. (4) is the average of the verticaldistances [ q − L ( q )] (0 < q <
1) between the Lorenz curve and the diagonal line (seeFig. 1); these vertical distances were noted in subsection 2.4.The Gini-index representation of Eq. (4) exhibits no clear connection to Eq. (1),the output’s mean. To make the connection more apparent we use the following, alter-native, Gini-index representation [56]: I Gini = − µ E [ min { T , T } ] , (5)where T and T are IID copies of the input T . The Gini-index representation of Eq. (5)is based on the disparity between two means: the mean of the minimum min { T , T } versus the input’s mean µ . The smaller the disparity between the two means – thecloser is the Gini index to its zero lower bound. Conversely, the larger the disparitybetween the two means – the closer is the Gini index to its unit upper bound.In addition to the Gini-index representation of Eq. (5), we also use the followingrepresentation of the output’s mean: M ( τ ) = F ( τ ) E [ min { T , τ } ] . (6)The derivation of Eq. (6) is detailed in the Methods. Eqs. (5) and (6) incorporatesimilar terms – the term E [ min { T , T } ] in the former, and the term E [ min { T , τ } ] in thelatter. This similarity suggests that a connection between the input’s Gini index I Gini and the output’s mean M ( τ ) may exist. We shall now establish such a connection.Denote by f max ( t ) ( t ≥
0) the density function of the random variable max { T , T } where, as in Eq. (5), T and T are IID copies of the input T . With this density functionat hand, the following formula is presented: (cid:90) ∞ (cid:20) M ( τ ) − µµ (cid:21) f max ( τ ) d τ = − I Gini . (7)The derivation of Eq. (7) is detailed in the Methods.Eq. (7) manifests the effect of the input’s Gini index I Gini on the output’s mean M ( τ ) . Indeed, Eq. (7) yields the following pair of Gini-index criteria : • If the input’s Gini index is smaller than half, I Gini < , then there exist timers τ for which sharp restart is detrimental. • If the input’s Gini index is larger than half, I Gini > , then there exist timers τ for which sharp restart is beneficial. 9hese Gini-index criteria have a threshold form that is identical to that of the CV-index criteria of subsection 2.3. Specifically, the input’s Gini index I Gini – the input’s‘Gini inequality’ – is compared to the threshold level . To determine the existence oftimers for which the application of the sharp-restart algorithm is detrimental or bene-ficial, one needs to check if the input’s ‘Gini inequality’ I Gini is below or above thethreshold level , respectively.On the one hand, I CV and I Gini are markedly different inequality indices of theinput T . On the other hand, these very different inequality indices yield very similarexistence results regarding timers τ for which sharp restart is detrimental or beneficial.We shall elaborate on the relation between the CV-index criteria and the Gini-indexcriteria in the discussion at the end of this section. Not as popular as the Gini index, yet no less profound, is the
Bonferroni index [66]-[72]. This subsection addresses the Bonferroni index I Bon f of the input T .In terms of the input’s Lorenz curve, the input’s Bonferroni index I Bon f is theaverage of the normalized vertical distances [ q − L ( q )] / q (0 < q <
1) between theLorenz curve and the diagonal line [72]. Namely, I Bon f = (cid:90) q − L ( q ) q dq . (8)As noted in subsection 2.4, for any fixed number q , the normalized vertical distance [ q − L ( q )] / q is an inequality index. Hence, the Bonferroni index I Bon f is an averageof inequality indices.The Bonferroni-index representation of Eq. (8) exhibits no clear connection toEq. (1), the output’s mean. To make the connection more apparent we use φ ( τ ) = E [ T | T ≤ τ ] – the input’s conditional mean, given the information that the input is nolarger than the timer. In terms of the input’s density and distribution functions, thisconditional mean is given by: φ ( τ ) = (cid:90) τ t f ( t ) F ( τ ) dt = F ( τ ) (cid:90) τ t f ( t ) dt . (9)Evidently, this conditional mean is no larger than the input’s mean: φ ( τ ) ≤ µ .In terms of the conditional mean φ ( τ ) , the input’s Bonferroni index I Bon f admitsthe following representation [72]: I Bon f = − µ (cid:90) ∞ φ ( τ ) f ( τ ) d τ . (10)The integral appearing on the right-hand side of Eq. (10) is a weighted average ofthe conditional mean φ ( τ ) , where the averaging is with respect to the input’s densityfunction. The Bonferroni-index representation of Eq. (10) is based on the disparitybetween two terms: the weighted average of the conditional mean (cid:82) ∞ φ ( τ ) f ( τ ) d τ versus the input’s mean µ . The smaller the disparity between the two terms – the closer10s the Bonferroni index to its zero lower bound. Conversely, the larger the disparitybetween the two terms – the closer is the Bonferroni index to its unit upper bound.In terms of the conditional mean φ ( τ ) , the output’s mean of Eq. (1) admits thefollowing representation: M ( τ ) = φ ( τ ) + τ ¯ F ( τ ) F ( τ ) . (11)The derivation of Eq. (11) is detailed in the Methods. Both Eqs. (10) and (11) incorpo-rate the term φ ( τ ) . This commonality suggests that a connection between the input’sBonferroni index I Bon f and the output’s mean M ( τ ) may exist. We shall now establishsuch a connection.Introduce the value ν = (cid:90) ∞ t f ( t ) F ( t ) dt = (cid:90) ∞ ln (cid:20) F ( t ) (cid:21) dt . (12)With the value ν at hand, the following formula is presented: (cid:90) ∞ (cid:20) M ( τ ) − µµ (cid:21) f ( τ ) d τ = ν − µµ − I Bon f , (13)The derivation of Eq. (13) is detailed in the Methods.Eq. (13) manifests the effect of the input’s Bonferroni index I Bon f on the out-put’s mean M ( τ ) . Indeed, setting the threshold level l Bon f = ν − µµ , Eq. (13) yields thefollowing pair of Bonferroni-index criteria : • If the input’s Bonferroni index is smaller than its threshold level, I Bon f < l Bon f ,then there exist timers τ for which sharp restart is detrimental. • If the input’s Bonferroni index is larger than its threshold level, I Bon f > l Bon f ,then there exist timers τ for which sharp restart is beneficial.The Bonferroni-index criteria have a threshold form that is similar to the threshold formof the CV-index criteria of subsection 2.3, and to the threshold form of the Gini-indexcriteria of subsection 3.1. Specifically, the input’s Bonferroni index I Bon f – the input’s‘Bonferroni inequality’ – is compared to the threshold level l Bon f = ν − µµ . To determinethe existence of timers for which the application of the sharp-restart algorithm is detri-mental or beneficial, one needs to check if the input’s ‘Bonferroni inequality’ I Bon f isbelow or above the threshold level l Bon f , respectively.
We conclude this section with remarks regarding the resemblance and the interplaybetween the CV criteria of subsection 2.2, and the Gini-index criteria of subsection3.1. The remarks are based on the deviation T − T between two IID copies, T and T ,of the input. 11t is straightforward to observe that the mean square deviation (MSD) between thetwo IID copies is twice the input’s variance: E [ | T − T | ] = σ . Consequently, interms of the MSD, the input’s squared CV admits the following representation: σ µ = E [ | T − T | ] µ . (14)An alternative to the aforementioned MSD is the mean absolute deviation (MAD)between the two IID copies, E [ | T − T | ] [73]. From a planar-geometry perspective,the MSD manifests aerial distance, and the MAD manifests walking distance [55]. Interms of the MAD, the input’s Gini index admits the following representation [56]: I Gini = E [ | T − T | ] µ . (15)Eqs. (14) and (15) highlight the resemblance between the input’s squared CV, σ / µ , and the input’s Gini index, I Gini . Jensen’s inequality implies that the squaredMAD is no larger than the MSD, E [ | T − T | ] ≤ E [ | T − T | ] . Consequently, Eqs. (14)and (15) imply the following relation between the input’s Gini index and the input’sCV: √ · I Gini ≤ σµ . (16)The interplay between the CV criteria and the Gini-index criteria comprises fourdifferent scenarios (see Fig. 2): “DD”, “BB”, “DB” and “BD”. In the “DD” and “BB”scenarios the Gini and CV criteria are in accord – both asserting the existence of timersfor which sharp restart is either detrimental (“DD”) or beneficial (“BB”). In the “DB”and “BD” scenarios the Gini and CV criteria are in dis-accord – one criterion assertsthe existence of timers for which sharp restart is detrimental, and another criterionasserts the existence of timers for which sharp restart is beneficial. The “DB” and “BD”scenarios underscore the fact that sharp restart can be detrimental for some timers, andbeneficial for other timers [1]. An illustrative example that demonstrates the “DB”scenario is described in Fig. 3. The inequality-indices criteria established so far, in sections 2 and 3, were existenceresults. Namely, these criteria determined the existence of timers τ for which the ap-plication of sharp restart is detrimental or beneficial. However, these criteria do notaddress particular timers τ . Indeed, these criteria are incapable of determining if sharprestart with a particular timer τ is detrimental or beneficial. This section begins to ad-dress – via socioeconomic inequality indices – particular timers τ . Specifically, thissection shall address sharp restart with two specific and ‘natural’ timers: a timer whosevalue is the input’s mean, and a timer whose value is the input’s median.Along this section the following representation of the output’s mean of Eq. (1) shallbe used: M ( τ ) = µ + τ − E [ | T − τ | ] F ( τ ) . (17)12 DD C V Not feasibleBBDB BDGini
Figure 2: A Gini-CV ‘phase diagram’. The horizontal axis represents the value of theinput’s Gini index I Gini , and the vertical axis represents the value of the input’s CV σµ . The line √ I Gini = σµ is depicted in blue; according to Eq. (16), feasible pairs ofGini-CV values ( I Gini , σµ ) do not reside below this line. The Gini-CV ‘phase diagram’comprises four different regions – each manifesting a different scenario. The trapezoidregion “DD” (0 ≤ I Gini < and √ I Gini ≤ σµ < < I Gini ≤ √ I Gini ≤ σµ < ∞ ): inthis region both the CV criteria and the Gini-index criteria assert the existence of timersfor which sharp restart is beneficial. The rectangular region “DB” (0 ≤ I Gini < and1 < σµ < ∞ ): in this region a Gini-index criterion asserts the existence of timers forwhich sharp restart is detrimental, and a CV criterion asserts the existence of timersfor which sharp restart is beneficial. The triangular region “BD” ( < I Gini < √ and √ I Gini ≤ σµ < E [ | T − τ | ] appearingin Eq. (17) is the mean absolute deviation (MAD) [73] between the input T and thetimer τ . In this subsection we consider sharp restart with the particular timer τ = µ , the input’smean. Namely, a timer that equals the mean of the task’s completion time T . Asshall be shown below, sharp restart with this particular timer is intimately related to aninequality index of the input that is termed Pietra index [83]-[88].From a functional-analysis perspective, Eq. (4) implies that the input’s Gini index I Gini is based on the “ L distance” between the Lorenz curve y = L ( x ) and the diagonalline y = x . Switching from the L distance to the “ L ∞ distance” yields the input’s Pietra13 .0 0.2 0.4 0.6 0.8 1.0 1.20.00.51.01.5 d . G i n i C V DB Figure 3: Lognormal example of the “DB” scenario. Lognormal random variables areof major importance across the sciences [74]-[78]. In particular, Lognormal servicetimes are widely observed in call centers [79]-[81]. In this example we consider theinput T to be Lognormal: ln ( T ) is a normal random variable with an arbitrary mean,and with standard deviation d . The squared CV of this input is exp ( d ) −
1, and theGini index of this input is erf ( d / ) [82] (here erf ( z ) denotes the Gauss error function).As functions of the positive standard-deviation parameter d : the squared CV is depictedin blue, and twice the Gini index is depicted in orange. The CV is larger than one if andonly if d > (cid:112) ln ( ) , and the Gini index is smaller than half if and only if d < − ( ) .Hence, the scenario “DB” holds when the standard-deviation parameter is in the range (cid:112) ln ( ) < d < − ( ) , which is indicated in the figure.index [86]: I Pietra is the maximal vertical distance between the input’s Lorenz curve y = L ( x ) and the diagonal line y = x . Namely, I Pietra = max ≤ q ≤ [ q − L ( q )] . (18)The Pietra-index representation of Eq. (18) exhibits no clear connection to Eq. (1),the output’s mean. To make the connection more apparent we use the following, alter-native, Pietra-index representation [56]: I Pietra = µ E [ | T − µ | ] . (19)The Pietra-index representation of Eq. (19) is based on the term E [ | T − µ | ] , the MADbetween the input T and its mean µ [89]. The smaller this MAD – the closer is thePietra index to its zero lower bound. Conversely, the larger this MAD – the closer isthe Pietra index to its unit upper bound.Setting τ = µ in Eq. (17), and using Eq. (19), yields the formula M ( µ ) − µµ = ¯ F ( µ ) − I Pietra F ( µ ) . (20)14q. (20) manifests, for the particular timer τ = µ , the effect of the input’s Pietra index I Pietra on the output’s mean M ( µ ) . Indeed, set the threshold level l Pietra = ¯ F ( µ ) , theprobability that the input exceeds its mean. Then, Eq. (20) yields the following pair of Pietra-index criteria : • Sharp restart with timer τ = µ is detrimental if and only if the input’s Pietraindex is smaller than its threshold level, I Pietra < l Pietra . • Sharp restart with timer τ = µ is beneficial if and only if the input’s Pietra indexis larger than its threshold level, I Pietra > l Pietra .Jensen’s inequality implies that the squared Pietra-index MAD is no larger than theinput’s variance, i.e. E [ | T − µ | ] ≤ E [ | T − µ | ] = σ . Consequently, twice the input’sPietra index is no larger than the input’s CV, 2 · I Pietra ≤ σµ . Also, the input’s Pietraindex is no larger than the input’s Gini index, I Pietra ≤ I Gini [90].These Pietra-CV and Pietra-Gini relations – combined, respectively, with the CVcriteria of subsection 2.2, and with the Gini-index criteria of subsection 3.1 – yieldthe following
Pietra-index corollary : If the input’s Pietra index is larger than half, I Pietra > , then there exist timers τ for which sharp restart is beneficial. In casethe input’s median is no-larger than the input’s mean, m ≤ µ , the Pietra-index criteria‘upgrade’ the Pietra-index corollary as follows: If the input’s Pietra index is larger thanhalf, I Pietra > , then sharp restart with timer τ = µ is beneficial.Last, with regard to being larger than the level , we note the following resem-blances between the CV, Gini, and Pietra indices. CV index: I CV > is equivalentto E [ | T − µ | ] > µ . Gini index: I Gini > is equivalent to E [ | T − T | ] > µ . Pietraindex: I Pietra > is equivalent to E [ | T − µ | ] > µ . In this subsection we consider sharp restart with the particular timer τ = m , the input’smedian. Namely, a timer that equals the median of the task’s completion time T . Asshall be shown below, sharp restart with this particular timer is intimately related toan inequality index of the input that is termed vertical-diameter index (Vdiam index)[90]-[91].The description of the Vdiam index uses the Lorenz curve y = L ( x ) and its corre-sponding complementary Lorenz curve, y = ¯ L ( x ) (0 ≤ x , y ≤
1) [56]. The socioeco-nomic meaning of the complementary Lorenz curve is: the top (rich) 100 x % of thesociety members possess 100 y % of the society’s overall wealth. The complementaryLorenz curve y = ¯ L ( x ) has the same properties as the Lorenz curve y = L ( x ) , albeit one:it is concave rather than convex. Consequently, in the unit square, the complementaryLorenz curve y = ¯ L ( x ) is bounded from below by the diagonal line y = x .As described above, the input’s Pietra index I Pietra is the “ L ∞ distance” betweenthe input’s Lorenz curve y = L ( x ) and the diagonal line y = x . Switching from the “ L ∞ Specifically, the median m is the unique positive value at which the input’s distribution function andsurvival function intersect: F ( m ) = = ¯ F ( m ) . As the input’s density function is considered to be positive-valued over the positive half-line, the input’s median is well defined indeed. L ∞ distance” betweenthe Lorenz curve and the complementary Lorenz curve, yields the input’s Vdiam index[90]-[91]: I Vdiam is the maximal vertical distance between the input’s Lorenz curve y = L ( x ) and the complementary Lorenz curve y = ¯ L ( x ) . It turns out that I Vdiam isthe normalized vertical distance – along the vertical line x = – between the input’sLorenz curve y = L ( x ) and the diagonal line y = x [90]-[91]. Namely, I Vdiam = − L (cid:18) (cid:19) . (21)The Vdiam-index representation of Eq. (21) exhibits no clear connection to Eq.(1), the output’s mean. To make the connection more apparent we use the following,alternative, Vdiam index representation [56]: I Vdiam = µ E [ | T − m | ] , (22)where m is the input’s median. The Vdiam-index representation of Eq. (22) is based onthe term E [ | T − m | ] , the MAD between the input T and its median m . The smaller thisMAD – the closer is the Vdiam index to its zero lower bound. Conversely, the largerthis MAD – the closer is the Vdiam index to its unit upper bound.Setting τ = m in Eq. (17), and using Eq. (22), yields the formula M ( m ) − µµ = m µ − I Vdiam . (23)Eq. (23) manifests, for the particular timer τ = m , the effect of the input’s Vdiam index I Vdiam on the output’s mean M ( m ) . Indeed, set the threshold level l Vdiam = m µ , theratio of the input’s median to the input’s mean. Then, Eq. (23) yields the followingpair of Vdiam-index criteria : • Sharp restart with timer τ = m is detrimental if and only if the input’s Vdiamindex is smaller than its threshold level, I Vdiam < l Vdiam . • Sharp restart with timer τ = m is beneficial if and only if the input’s Vdiam indexis larger than its threshold level, I Vdiam > l Vdiam . We conclude this section with remarks regarding the input’s mean and median, the CVcriteria of subsection 2.2, and the Vdiam-index criteria of subsection 4.2. The remarksare based on an optimization-problem perspective.The “ L p distance” between the input T and the timer τ is D p ( τ ) = { E [ | T − τ | p ] } / p . (24)The parameter p of the L p distance takes values in the range p ≥
1, and its most notablevalues are p = p =
2. Specifically, the L distance is the aforementioned mean16bsolute deviation (MAD) E [ | T − τ | ] between the input T and the timer τ . And, the L distance is the square root of the mean square deviation (MSD) E [ | T − τ | ] betweenthe input T and the timer τ .The inputs’s mean and median are intimately related, respectively, to the L and L distances. Indeed, consider the optimization problem min < τ < ∞ D p ( τ ) . This optimiza-tion problem seeks the timer τ that is closest – in the L p distance – to the input T . Forthe L distance the minimum is attained at the input’s mean, and the minimum valueis the input’s standard deviation: arg min < τ < ∞ D ( τ ) = µ and min < τ < ∞ D ( τ ) = σ .For the L distance the minimum is attained at the input’s median, and the minimumvalue is the MAD between the input and its median: arg min < τ < ∞ D ( τ ) = m andmin < τ < ∞ D ( τ ) = E [ | T − m | ] .The CV criteria of subsection 2.2 assert that – in order to determine the existenceof timers τ for which the sharp restart algorithm is detrimental or beneficial – one hasto compare the input’s standard deviation σ to the input’s mean µ . In other words,the CV criteria compare the minimal value min < τ < ∞ D ( τ ) to the minimizing pointarg min < τ < ∞ D ( τ ) .Eq. (23) can be re-written as follows: M ( m ) − µ = m − E [ | T − m | ] . (25)Consequently, the Vdiam-index criteria of subsection 4.2 – regarding restart with theparticular timer τ = m – can be re-formulated as follows: sharp restart is detrimentalif and only if E [ | T − m | ] < m , and is beneficial if and only if E [ | T − m | ] > m . In otherwords, the Vdiam-index criteria compare the minimal value min < τ < ∞ D ( τ ) to theminimizing point arg min < τ < ∞ D ( τ ) .Thus, from the perspective of the optimization problem min < τ < ∞ D p ( τ ) , there isan analogy between the CV criteria of subsection 2.2 and the Vdiam-index criteria ofsubsection 4.2. Indeed, as pointed out above, both these criteria compare the minimalvalue min < τ < ∞ D p ( τ ) to the minimizing point arg min < τ < ∞ D p ( τ ) . Notably, the CVcriteria and the Vdiam-index criteria are based on rather ‘neat’ formulae – Eq. (2) andEq. (25), respectively. The previous section addressed two particular timers: τ = µ , where µ is the input’smean; and τ = m , where m is the input’s median. To each of these timers a specificinequality index of the input was matched, and using these inequality indices it wasdetermined when sharp restart (with these timers) is detrimental or beneficial. Thissection goes beyond the particular mean and median timers, and it addresses generaltimers 0 < τ < ∞ . As noted in subsection 2.3, in order to measure the inherent ‘socioeconomic inequal-ity’ of the input T , this random variable was deemed to be the wealth of a randomly-sampled member of a virtual society. Now, with respect to this virtual society, consider17lso a randomly-sampled Dollar. Namely, sample at random a single Dollar from thesociety’s overall wealth, and set T dol to be the wealth of the society member to whomthe randomly-sampled Dollar belongs [56].A temporal description of the random variable T dol is as follows. Perform, repeat-edly and independently, n rounds of the task under consideration (whose completiontime is the random variable T ). Specifically: starting at time t =
0, perform the task forthe first round; then, upon the first completion, start performing the task for the secondround; then, upon the second completion, start performing the task for the third round;and continue so on and so forth for n consecutive rounds. Denoting by { T , · · · , T n } the durations of the tasks – these durations being IID copies of the input T – we obtainthat: T is the completion time of the first round, T + T is the completion time of thesecond round, ... , and T + · · · + T n is the completion time of the last round.Now, place an observer at a random time epoch along the temporal interval [ , T + · · · + T n ] . This random placement of the observer implies that: the observer samplesthe task that is performed at round k with probability T k / ( T + · · · + T n ) . In the limit n → ∞ , the duration of the task that is sampled by the observer converges, in law, to therandom variable T dol .As the input T , also T dol is a positive-valued random variable. In terms of theinput’s density function and mean, the density function of the random variable T dol is f dol = µ t f ( t ) ( t >
0) [56]. In turn, the distribution and survival functions of the randomvariable T dol are, respectively: F dol ( t ) = µ (cid:90) t s f ( s ) ds (26)( t ≥ F dol ( t ) = µ (cid:90) ∞ t s f ( s ) ds (27)( t ≥ y = L ( x ) couples together the input’s distribution func-tion F ( t ) , and the distribution function F dol ( t ) of the random variable T dol . Indeed,the socioeconomic definitions of the Lorenz curve y = L ( x ) , and of the distributionfunctions F ( t ) and F dol ( t ) , implies that [90]-[91]: L [ F ( t )] = F dol ( t ) (28)( t ≥ F ( t ) = L − [ F dol ( t )] (29)( t ≥ x = L − ( y ) denotes the inverse function of the input’s Lorenz curve y = L ( x ) . Subsection 2.4 introduced, for a fixed number q (where 0 < q < [ q − L ( q )] / q : the ‘normalized’ vertical distance – along the vertical line x = q
18 between the Lorenz curve y = L ( x ) and the diagonal line y = x . With respect to theinput’s distribution function F ( t ) , set q to be the quantile corresponding to the timer τ ,i.e. q = F ( τ ) . Eq. (28) implies that q − L ( q ) q = − F dol ( τ ) F ( τ ) . (30)The quantity appearing in Eq. (30) is an inequality index with an underpinning ver-tical Lorenz-curve geometric meaning. This quantity is henceforth termed the input’s vertical index , and is denoted I V ( τ ) .Introduce the threshold level l V ( τ ) = τµ ¯ F ( τ ) F ( τ ) . (31)With the vertical index I V ( τ ) of Eq. (30) at hand, as well as the threshold level l V ( τ ) of Eq. (31), the following formula is presented: M ( τ ) − µµ = l V ( τ ) − I V ( τ ) . (32)The derivation of Eq. (32) is detailed in the Methods.Eq. (32) manifests the effect of the input’s vertical index I V ( τ ) on the output’smean M ( τ ) . Indeed, Eq. (32) yields the following pair of vertical-index criteria : • Sharp restart with timer τ is detrimental if and only if the input’s vertical indexis smaller than its threshold level, I V ( τ ) < l V ( τ ) . • Sharp restart with timer τ is beneficial if and only if the input’s vertical index islarger than its threshold level, I V ( τ ) > l V ( τ ) .In particular, setting τ = µ in the vertical-index criteria yields the Pietra-indexcriteria of subsection 4.1. And, setting τ = m in the vertical-index criteria yields theVdiam-index criteria of subsection 4.2. The vertical-index criteria of the previous subsection are based on the vertical distancebetween the Lorenz curve y = L ( x ) and the diagonal line y = x . This subsection shiftsfrom the vertical-distance perspective to a horizontal-distance perspective.For a fixed number q (where 0 < q < y = q of the unitsquare. The horizontal distance – along the horizontal line y = q – between the Lorenzcurve y = L ( x ) and the diagonal line y = x is: L − ( q ) − q . This horizontal distancetakes values in the range [ , − q ] . Consequently, the ‘normalized’ horizontal distance [ L − ( q ) − q ] / ( − q ) takes values in the unit interval [ , ] . It is straightforward tocheck that the normalized horizontal distance [ L − ( q ) − q ] / ( − q ) meets the threeinequality-index properties that were postulated in subsection 2.3. As in Eq. (29) above, x = L − ( y ) denotes the inverse function of the input’s Lorenz curve y = L ( x ) . F dol ( t ) of the random variable T dol , set q to be the quantile corresponding to the timer τ , i.e. q = F dol ( τ ) . Eq. (29) implies that L − ( q ) − q − q = − ¯ F ( τ ) ¯ F dol ( τ ) . (33)The quantity appearing in Eq. (33) is an inequality index with an underpinning hori-zontal Lorenz-curve geometric meaning. This quantity is henceforth termed the input’s horizontal index , and is denoted I H ( τ ) .Introduce the threshold level l H ( τ ) = ττ + µ . (34)With the horizontal index I H ( τ ) of Eq. (33) at hand, as well as the threshold level l H ( τ ) of Eq. (34), we present the following formula: M ( τ ) − µµ = ( τ + µ ) ¯ F dol ( τ ) µ F ( τ ) · [ l H ( τ ) − I H ( τ )] . (35)The derivation of Eq. (35) is detailed in the Methods.Eq. (35) manifests the effect of the input’s horizontal index I H ( τ ) on the output’smean M ( τ ) . Indeed, Eq. (35) yields the following pair of horizontal-index criteria : • Sharp restart with timer τ is detrimental if and only if the input’s horizontal indexis smaller than its threshold level, I H ( τ ) < l H ( τ ) . • Sharp restart with timer τ is beneficial if and only if the input’s horizontal indexis larger than its threshold level, I H ( τ ) > l H ( τ ) .In particular, setting τ = µ in the horizontal-index criteria yields the Pietra-indexcriteria of subsection 4.1. We conclude this section with remarks regarding the vertical-index criteria and thehorizontal-index criteria. Also, using the horizontal-index criteria, we shall elaborateon a particular timer: τ = m dol , the median of the random variable T dol .In this section, given a positive timer τ , two inequality indices of the input T werematched to the timer: the vertical index I V ( τ ) of Eq. (30), and the horizontal index I H ( τ ) of Eq. (33). The vertical index I V ( τ ) quantifies the disparity, at the time point t = τ , between the distribution functions of the random variables T and T dol . Similarly,the horizontal index I H ( τ ) quantifies the disparity, at the time point t = τ , betweenthe survival functions of the random variables T and T dol .Accompanying the input’s vertical and horizontal indices, I V ( τ ) and I H ( τ ) , arecorresponding threshold levels: l V ( τ ) of Eq. (31), and l H ( τ ) of Eq. (34). The com-parisons between the vertical and horizontal indices and their corresponding thresholdlevels determines if sharp restart with the timer τ is detrimental or beneficial. While20hese comparisons are equivalent, they provide two different geometric Lorenz-curveperspectives – vertical and horizontal.The Pietra index of subsection 4.1 emanated from maximizing the vertical distancebetween the Lorenz curve y = L ( x ) and the diagonal line y = x . Analogously, the Vdiamindex of subsection 4.1 emanated from maximizing the vertical distance between theLorenz curve y = L ( x ) and the complementary Lorenz curve y = ¯ L ( x ) . Shifting fromthe vertical perspective to the horizontal perspective has the following effects. As in thevertical perspective, maximizing the horizontal distance between the Lorenz curve y = L ( x ) and the diagonal line y = x yields the Pietra index [90]-[91]. However, maximizingthe horizontal distance between the Lorenz curve y = L ( x ) and the complementaryLorenz curve y = ¯ L ( x ) yields an inequality index of the input that is termed horizontal-diameter index [90]-[91].It turns out that the input’s Hdiam index I Hdiam is the normalized horizontal dis-tance – along the horizontal line y = – between the input’s Lorenz curve y = L ( x ) and the diagonal line y = x [90]-[91]. Namely, I Hdiam = L − (cid:18) (cid:19) − . (36)This index is intimately related to the median m dol of the random variable T dol . Indeed,Eqs. (33) and (36) imply that I Hdiam = I H ( m dol ) .The Hdiam index has several representations. One of the input’s Hdiam-index rep-resentations is [56]: I Hdiam = m dol E [ | T − m dol | ] . (37)The Hdiam-index representation of Eq. (37) is based on the term E [ | T − m dol | ] , theMAD between the input T and the median m dol (which is the median of the randomvariable T dol , not of the input T ). The smaller this MAD – the closer is the Vdiamindex to its zero lower bound. Conversely, the larger this MAD – the closer is theVdiam index to its unit upper bound.Setting τ = m dol in Eq. (35), and using Eq. (34), yields the formula M ( m dol ) − µµ = m dol m dol + µ − I Hdiam . (38)Eq. (38) manifests, for the particular timer τ = m dol , the effect of the input’s Hdiamindex I Hdiam on the output’s mean M ( m dol ) . Indeed, set the threshold level l Hdiam = m dol m dol + µ . Then, Eq. (38) yields the following pair of Hdiam-index criteria : • Sharp restart with timer τ = m dol is detrimental if and only if the input’s Hdiamindex is smaller than its threshold level, I Hdiam < l Hdiam . • Sharp restart with timer τ = m dol is beneficial if and only if the input’s Hdiamindex is larger than its threshold level, I Hdiam > l Hdiam .21
Conclusion
Following up on [1], in this paper we continued exploring the effect of sharp restarton mean performance. Using a positive deterministic timer τ , sharp restart is appliedto a general task whose completion time is a positive random variable T . Namely,initiating the task at time t =
0, the task is restarted – as long as it is not accomplished– at the fixed time epochs t = τ , τ , τ , · · · . The mean completion time of the task‘under restart’, M ( τ ) , is compared to the task’s mean completion time, E [ T ] = µ . Thiscomparison determines if sharp restart improves mean performance, M ( τ ) < µ , or if itworsens mean performance, M ( τ ) > µ .The analysis presented in this paper established that inequality indices of the ran-dom variable T hold a treasure trove of information regarding the effect of sharprestart on mean performance. We showed that three inequality indices – CV, Gini,and Bonferroni – determine the very existence of timers with which sharp restart im-proves/worsens mean performance. Given a specific timer τ , we further showed thatthere are inequality indices that relate to this timer, and that these inequality indices de-termine if sharp restart with the specific timer τ improves/worsens mean performance.The novel results established in this paper provide a detailed ‘inequality roadmap’for sharp restart: eight pairs of universal inequality criteria that determine the effectof sharp restart on mean performance. The underpinning inequality indices are sum-marized in Table 1, and the criteria are summarized in Table 2. Each pair of crite-ria comprises a specific inequality index I , and a corresponding threshold level l .All eight pairs of criteria share in common the following threshold pattern. If the in-dex is larger than its corresponding threshold level then mean performance improves: I > l ⇒ M ( τ ) < µ . And, if the index is smaller than its corresponding threshold levelthen mean performance worsens: I < l ⇒ M ( τ ) > µ .On the one hand, inequality indices of the random variable T measure the sta-tistical heterogeneity of the task’s completion time. On the other hand, it is knownthat sharp restart can match the mean-performance of any other restart protocol [34,57]. Hence, combining these two facts together, we obtain the following take-home-message: restart improves mean performance when the underlying statistical hetero-geneity is high; and it worsens mean performance when the underlying statistical het-erogeneity is low. The universal inequality criteria established here articulate the take-home-message with unprecedented mathematical precision and resolution.The first two pairs of inequality criteria – CV and Gini – have a fixed threshold level( l = ) that is independent of the task’s completion time. Conversely, the latter six pairsof inequality criteria have threshold levels that depend on the task’s completion time.For the latter six pairs of inequality criteria, the common threshold pattern admits analternative interpretation that is described as follows. Given the value of the inequalityindex I , there is a critical ‘mean value’ µ c that is based on I . If the task’s meancompletion time is larger than the critical value then mean performance improves: µ > µ c ⇒ M ( τ ) < µ . And, if the task’s mean completion time is smaller than the criticalvalue then mean performance worsens: µ < µ c ⇒ M ( τ ) > µ . The critical values µ c are specified in the right column of Table 2.The inequality indices that underpin the inequality criteria have various equivalentrepresentations [56]. As specified in Table 1, one such representation is based on mean22quare/absolute deviations (MSD/MAD) of the random variable T , the task’s comple-tion time; these deviations can be easily and reliably estimated from data. Thus, theapplication of the inequality criteria whose underpinning inequality indices admit aMSD/MAD representation is very practical, and can be done even when the distribu-tion of the task’s completion time is not known in full detail. Indeed, for example:scientists from various disciplines are well accustomed to estimating the CV, and es-timating the Gini index is common practice in economics and in the social sciences;consequently, the use of the CV and Gini criteria should be straightforward to all.Given a task, a ‘natural’ sharp-restart timer to consider is the mean of the task’scompletion time, τ = µ . The Pietra criteria relate this natural timer, determining ifsharp restart with this timer is beneficial or detrimental. In turn, building on relationsbetween the CV, Gini, and Pietra inequality indices, the two following universal Pietracorollaries were also established. If the Pietra index is larger than the level then thereexist timers with which sharp restart is beneficial. Moreover, if – in addition to thePietra index being larger than the level – the input’s median is no-larger than theinput’s mean, m ≤ µ , then: sharp restart with the specific timer τ = µ is beneficial.As the Pietra index admits a MAD representation, the Pietra criteria and corollaries arevery practical and are straightforward to use.This paper unveiled profound connections between two seemingly unrelated topics:the measurement of socioeconomic inequality on the one hand, and the mean perfor-mance of sharp restart on the other hand. These connections provide researchers –theoreticians and practitioners alike – a whole new inequality vantage point, as wellas a whole new inequality roadmap and toolbox, to work with in the interdisciplinaryfield of restart. 23 able 1 Inequalityindex MSD/MADrepresentation Lorenzrepresentation CV I CV − I CV = µ E (cid:104) | T − T | (cid:105) —Gini I Gini = µ E [ | T − T | ] (cid:82) [ q − L ( q )] dq Bonferroni I Bon f = — (cid:82) q − L ( q ) q dq Pietra I Pietra = µ E [ | T − µ | ] max ≤ q ≤ [ q − L ( q )] Vdiam I Vdiam = µ E [ | T − m | ] − L (cid:0) (cid:1) Hdiam I Hdiam = m dol E [ | T − m dol | ] L − (cid:0) (cid:1) − I V ( τ ) = — q − L ( q ) q Horizontal I H ( τ ) = — L − ( q ) − q − q Table 1 : The eight inequality indices that underpin, respectively, the eight pairsof universal inequality criteria for sharp restart. These inequality indices have variousequivalent representations [56], two of which are presented in the table: representationsbased on mean square/absolute deviations (MSD/MAD) of the task’s completion time T ; and representations based on the Lorenz curve L ( · ) of the task’s completion time T (see section 2.4 for the details). Remarks regarding specific inequality indices arethe following. CV and Gini: T and T are two IID copies of the random variable T .Vdiam: m is the median of the random variable T . Hdiam: m dol is the median of therandom variable T dol (see section 5.1 for the details). Vertical: q = F ( τ ) , where F ( · ) is the distribution function of the random variable T . Horizontal: q = F dol ( τ ) , where F dol ( · ) is the distribution function of the random variable T dol (see section 5.1 for thedetails). 24 able 2 Inequalityindex I = Thresholdlevel l = Timer Criticalvalue µ c = I CV Existence —– I Gini Existence —– I Bon f ν − µµ Existence ν + I Bonf I Pietra ¯ F ( µ ) τ = µ ¯ F − ( I Pietra ) I Vdiam m µ τ = m m I Vdiam I Hdiam m dol m dol + µ τ = m dol m dol − I Hdiam I Hdiam I V ( τ ) τµ ¯ F ( τ ) F ( τ ) < τ < ∞ τ I V ( τ ) ¯ F ( τ ) F ( τ ) I H ( τ ) ττ + µ < τ < ∞ τ − I H ( τ ) I H ( τ ) Table 2 : Eight pairs of universal inequality criteria that determine the mean per-formance of sharp restart. For each pair of criteria, the table’s columns specify: theinequality index I on which the criteria are based; the threshold level l that corre-sponds to the inequality index; and the timer parameters τ for which the criteria apply.Comparing the inequality index I to the threshold level l , the criteria assert that: if I > l then mean performance improves; and if I < l then mean performance wors-ens. For each pair of criteria – excluding the CV and Gini criteria – the table’s rightcolumn further specifies the critical ‘mean value’ µ c that is based on the inequality-index value I . Comparing the task’s mean completion time µ to the critical ‘meanvalue’ µ c , the criteria assert that: if µ > µ c then mean performance improves; and if µ < µ c then mean performance worsens. Remarks regarding specific inequality indicesare the following. Bonferroni: the value ν is given by Eq. (12). Pietra and Vertical: F ( · ) and ¯ F ( · ) are, respectively, the distribution function and the survival function ofthe task’s completion time T . 25 cknowledgments . Shlomi Reuveni acknowledges support from the Azrieli Foun-dation, from the Raymond and Beverly Sackler Center for Computational Molecularand Materials Science at Tel Aviv University, and from the Israel Science Foundation(grant No. 394/19). Introduce the function f res ( t ) = µ ¯ F ( t ) (39)( t ≥ F ( t ) is the input’s mean, µ = (cid:82) ∞ ¯ F ( t ) dt , the function f res ( t ) is a probability density: it is non-negative, f res ( t ) ≥ (cid:82) ∞ f res ( t ) dt =
1. In effect, f res ( t ) is the density function of theinput’s “residual lifetime” [92]. The corresponding distribution and survival functionsare F res ( t ) = (cid:82) t f res ( s ) ds and ¯ F res ( t ) = (cid:82) ∞ t f res ( s ) ds . Moreover, the correspondingmean is µ res = (cid:82) ∞ ¯ F res ( t ) dt = (cid:82) ∞ t f res ( t ) dt = µ (cid:82) ∞ t ¯ F ( t ) dt = µ (cid:82) ∞ t f ( t ) dt = µ E (cid:2) T (cid:3) = µ (cid:0) σ + µ (cid:1) . (40)In the second line of Eq. (40) we used integration by parts, and in the third line of Eq.(40) we used the following representation of the input’s variance: σ = E (cid:2) T (cid:3) − E [ T ] .Dividing both sides of Eq. (1) by µ , and using the distribution function F res ( t ) , wehave M ( τ ) µ = F res ( τ ) F ( τ ) . (41)Eq. (41), together with the coupling between distribution and survival functions, im-plies that M ( τ ) − µµ = M ( τ ) µ − = F res ( τ ) F ( τ ) − = F res ( τ ) − F ( τ ) F ( τ ) = ¯ F ( τ ) − ¯ F res ( τ ) F ( τ ) . (42)In turn, Eq. (42) implies that [ M ( τ ) − µ ] F ( τ ) = µ [ ¯ F ( τ ) − ¯ F res ( τ )] . (43)Integrating both sides of Eq. (43), and using Eq. (40), yields Eq. (2): (cid:82) ∞ [ M ( τ ) − µ ] F ( τ ) d τ = µ (cid:82) ∞ [ ¯ F ( τ ) − ¯ F res ( τ )] d τ = µ [ (cid:82) ∞ ¯ F ( τ ) d τ − (cid:82) ∞ ¯ F res ( τ ) d τ ] = µ ( µ − µ res )= µ − (cid:0) σ + µ (cid:1) = (cid:0) µ − σ (cid:1) . (44)26 .2 Derivation of Eqs. (6) and (7) The output of the sharp-restart algorithm admits the following stochastic representation[1]: T R = min { T , τ } + T (cid:48) R · I { T > τ } , (45)where T (cid:48) R is an IID copy of the output T R . Applying expectation to both sides of Eq.(45) yields E [ T R ] = E [ min { T , τ } ] + E (cid:2) T (cid:48) R · I { T > τ } (cid:3) . (46)As T (cid:48) R is an IID copy of T R , we have E [ T (cid:48) R · I { T > τ } ] = E [ T (cid:48) R ] · E [ I { T > τ } ]= E [ T (cid:48) R ] Pr ( T > τ ) = E [ T R ] ¯ F ( τ ) . (47)And, substituting Eq. (47) in Eq. (46) we have E [ T R ] = E [ min { T , τ } ] + E [ T R ] ¯ F ( τ ) . (48)In turn, Eq. (48) implies that E [ T R ] F ( τ ) = E [ min { T , τ } ] , (49)and Eq. (49) yields Eq. (6).In what follows T and T are IID copies of the input T . The distribution function ofthe random variable max { T , T } is F ( t ) ( t ≥ f max ( t ) = F ( t ) f ( t ) ( t > f ( τ ) ,and using the density function f max ( t ) , we have M ( τ ) f max ( τ ) = E [ min { T , τ } ] f ( τ ) . (50)Integrating Eq. (50) yields (cid:82) ∞ M ( τ ) f max ( τ ) d τ = (cid:82) ∞ E [ min { T , τ } ] f ( τ ) d τ = (cid:82) ∞ [ (cid:82) ∞ min { t , τ } f ( t ) dt ] f ( τ ) d τ = (cid:82) ∞ (cid:82) ∞ min { t , τ } f ( t ) f ( τ ) dtd τ = E [ min { T , T } ] . (51)As f max ( t ) is a density function, Eqs. (51) and (5) yield Eq. (7): (cid:82) ∞ (cid:104) M ( τ ) − µµ (cid:105) f max ( τ ) d τ = (cid:82) ∞ (cid:104) M ( τ ) µ − (cid:105) f max ( τ ) d τ = µ (cid:82) ∞ M ( τ ) f max ( τ ) d τ − (cid:82) ∞ f max ( τ ) d τ = µ E [ min { T , T } ] − = ( − I Gini ) − = − I Gini . (52)27 .3 Derivation of Eqs. (11) and (13) Integration by parts implies that (cid:90) τ ¯ F ( t ) dt = (cid:90) τ t f ( t ) dt + τ ¯ F ( τ ) . (53)Combined together, Eq. (1), Eq. (53) and Eq. (9) yields Eq. (11): M ( τ ) = F ( τ ) (cid:82) τ ¯ F ( t ) dt = F ( τ ) (cid:82) τ t f ( t ) dt + τ ¯ F ( τ ) F ( τ ) = φ ( τ ) + τ ¯ F ( τ ) F ( τ ) . (54)Note that we can write Eq. (54) in the following form M ( τ ) = φ ( τ ) + τ F ( τ ) − τ . (55)Multiplying both sides of Eq. (55) by the term f ( τ ) yields M ( τ ) f ( τ ) = φ ( τ ) f ( τ ) + τ f ( τ ) F ( τ ) − τ f ( τ ) . (56)Integrating Eq. (56), and using Eq. (12), we have (cid:82) ∞ M ( τ ) f ( τ ) d τ = (cid:82) ∞ φ ( τ ) f ( τ ) d τ + (cid:82) ∞ τ f ( τ ) F ( τ ) d τ − (cid:82) ∞ τ f ( τ ) d τ = (cid:82) ∞ φ ( τ ) f ( τ ) d τ + ν − µ . (57)As f ( t ) is a density function, Eqs. (57) and (10) yield Eq. (13): (cid:82) ∞ (cid:104) M ( τ ) − µµ (cid:105) f ( τ ) d τ = (cid:82) ∞ (cid:104) M ( τ ) µ − (cid:105) f ( τ ) d τ = µ (cid:82) ∞ M ( τ ) f ( τ ) d τ − (cid:82) ∞ f ( τ ) d τ = µ [ (cid:82) ∞ φ ( τ ) f ( τ ) d τ + ν − µ ] − = µ (cid:82) ∞ φ ( τ ) f ( τ ) d τ + νµ − = (cid:0) − I Bon f (cid:1) + νµ − = ν − µµ − I Bon f . (58)28 .4 Derivation of Eq. (17) Note that max { T , τ } + min { T , τ } = T + τ , (59)and max { T , τ } − min { T , τ } = | T − τ | . (60)Subtracting Eq. (60) from Eq. (59) yields2 min { T , τ } = T + τ − | T − τ | . (61)In turn, applying expectation to both sides of Eq. (61) we have2 E [ min { T , τ } ] = E [ T + τ − | T − τ | ]= E [ T ] + E [ τ ] − E [ | T − τ | ]= µ + τ − E [ | T − τ | ] . (62)Combining together Eq. (6) and Eq. (62) yields Eq. (17 ): M ( τ ) = E [ min { T , τ } ] F ( τ ) = µ + τ − E [ | T − τ | ] F ( τ ) . (63) Combining together Eq. (54) and Eq. (26) we have M ( τ ) = F ( τ ) (cid:82) τ t f ( t ) dt + τ ¯ F ( τ ) F ( τ ) = µ F dol ( τ ) F ( τ ) + τ ¯ F ( τ ) F ( τ ) . (64)Eq. (64) yields Eq. (32): M ( τ ) − µµ = µ M ( τ ) − = µ (cid:104) µ F dol ( τ ) F ( τ ) + τ ¯ F ( τ ) F ( τ ) (cid:105) − = τµ ¯ F ( τ ) F ( τ ) − (cid:104) − F dol ( τ ) F ( τ ) (cid:105) . (65)Eq. (65) implies that [ M ( τ ) − µ ] F ( τ ) = τ [ − F ( τ )] − µ [ F ( τ ) − F dol ( τ )]= τ ¯ F ( τ ) − µ [ ¯ F dol ( τ ) − ¯ F ( τ )] = ( τ + µ ) ¯ F ( τ ) − µ ¯ F dol ( τ ) . (66)29q. (66) yields M ( τ ) − µµ (cid:104) µτ + µ F ( τ ) ¯ F dol ( τ ) (cid:105) = M ( τ ) − µτ + µ F ( τ ) ¯ F dol ( τ ) = τ + µ ( τ + µ ) ¯ F ( τ ) − µ ¯ F dol ( τ ) ¯ F dol ( τ ) = ¯ F ( τ ) ¯ F dol ( τ ) − µτ + µ = (cid:104) − µτ + µ (cid:105) − (cid:104) − ¯ F ( τ ) ¯ F dol ( τ ) (cid:105) = ττ + µ − (cid:104) − ¯ F ( τ ) ¯ F dol ( τ ) (cid:105) . (67)In turn, Eq. (67) yields Eq. (35). References [1] Eliazar, I. and Reuveni, S., 2020. Mean-performance of sharp restart I: Statisticalroadmap. J. Phys. A: Math. Theor. 53 405004.[2] C. W. Gardiner,
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