Changing assignment weights with time-based confounders

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by ALEXANDER WAKIM

Ramp-up and multi-armed bandits (MAB) are widespread methods in on-line managed experiments (OCE). These methods contain altering project weights throughout an experiment. Nonetheless, if one modifications project weights when there are time-based confounders, then ignoring this complexity can result in biased inference in an OCE. Within the case of MABs, ignoring this complexity may also result in poor complete reward, making it counterproductive in the direction of its meant objective. On this put up we talk about the issue, an answer, and sensible issues.

Background

On-line managed experiments

An internet managed experiment (OCE) randomly assigns totally different variations of an internet site or app to totally different customers as a way to see which model causes extra of some desired motion. On this put up, these “variations” are known as arms and the specified motion is known as the reward (arms are sometimes known as “remedies” and reward is commonly known as the “dependent variable” in different contexts). Examples of rewards for an OCE embrace the quantity of product purchased, the quantity of people that join a publication, the quantity of people that begin an account, and so on. This put up considers a typical design for an OCE the place a consumer could also be randomly assigned an arm on their first go to through the experiment, with project weights referring to the proportion which can be randomly assigned to every arm. 

There are two widespread causes project weights could change throughout an OCE. The primary is a method known as ramp-up and is suggested by many specialists within the discipline [1]. The second widespread motive is multi-armed bandit algorithms (MAB) that maximize reward by assigning extra customers to a profitable arm sooner as a way to make the most of it sooner.

Ramp-up

Ramp-up is when the experiment initially provides a small project weight to the brand new arm and, because the experiment continues, will increase the brand new arm’s project weight. One motive to do ramp-up is to mitigate the danger of by no means earlier than seen arms. For instance, think about a fantasy soccer website is contemplating displaying superior participant statistics. A ramp-up technique could mitigate the danger of upsetting the positioning’s loyal customers who maybe have robust preferences for the present statistics which can be proven. One more reason to make use of ramp-up is to check if an internet site’s infrastructure can deal with deploying a brand new arm to all of its customers. For instance, think about a smaller web site that’s contemplating including a video internet hosting function to extend engagement on the positioning. The web site desires to ensure they’ve the infrastructure to deal with the function whereas testing if engagement will increase sufficient to justify the infrastructure.

Though there are a lot of methods to do ramp-up, we assume that when a consumer is assigned an arm, they’ll keep assigned to this arm at some point of the experiment. When project weights change in a ramp-up experiment, there are durations of fixed project weights that we outline as epochs. We think about the project weights for every epoch to be the proportion of beforehand unassigned customers which can be assigned to every arm upon first go to throughout that epoch. The project weights inside epochs could or could not add as much as 100%. If they don’t add as much as 100%, we name this a partial site visitors ramp-up and a consumer who visits throughout that epoch could also be unassigned and might be assigned on a future go to in a later epoch. Once they add as much as 100%, we name this a full site visitors ramp-up. 

Task

Weights

Epoch 1

Epoch 2

Present Arm

90%

50%

New Arm 

10%

50%

Fig 1. A easy instance of a full site visitors ramp-up technique the place 90% of customers are assigned the present arm and 10% are assigned the brand new arm in epoch 1. In epoch 2, 50% of beforehand unassigned customers are assigned every the present and new arm.
There may be a whole lot of consideration that ought go into the design and evaluation of a selected ramp-up technique. Describing the exact choice standards for various ramp-up designs is out of scope for this put up. As an alternative, we deal with the case the place an experimenter has determined to run a full site visitors ramp-up experiment and needs to make use of the information from all the epochs within the evaluation. The fantasy soccer and video internet hosting examples, which we are going to talk about in additional element later, spotlight conditions the place this design could be thought of, regardless of potential complexity within the evaluation.

Complexity arises within the presence of time-based confounders, when taking an unweighted common throughout knowledge that’s aggregated throughout epochs can result in biased estimates (whether or not it’s a partial or full site visitors ramp-up technique). At a excessive stage, it is because altering the project weights causes the distribution of the time-based confounder in every experiment arm to be totally different than the general inhabitants. Utilizing the fantasy soccer instance, we are going to simulate a scenario the place the bias is so robust that an inferior choice on which arm to deploy is made. We’ll supply a weighted common resolution that permits an experimenter to do full site visitors ramp-up and mixture knowledge throughout epochs to get an unbiased estimate of the impact of every arm.

Multi-armed bandits

Multi-armed bandits (MAB) are a category of algorithms that maximize reward by assigning extra customers to higher performing arms sooner as a way to make the most of them sooner. MAB algorithms are fashionable throughout most of the giant internet corporations. Corporations like Google [2], Amazon [3], and Microsoft [4] have all revealed scholarly articles on this matter.

Generally a MAB is run to maximise reward throughout an OCE with the objective of finally ending the MAB, figuring out which arm is best, and deploying the higher arm. Different occasions a MAB is run perpetually to maximise reward and to not essentially discover a profitable arm. Simply as in ramp-up, making inferences whereas ignoring the complexity of time-based confounders which can be current can result in biased estimates. Furthermore, sure MAB algorithms will be counterproductive in the direction of maximizing reward if assumptions aren’t met. This put up will talk about learn how to use knowledge from a MAB to get unbiased estimates. We’ll supply methods, issues, and issues to consider when operating a MAB to maximise reward.

 


F
ig 2. Multi-armed bandit algorithms are sometimes considered by imagining a gambler sitting at a number of slot machines, deciding which machine to play as a way to maximize winnings.

Naive aggregation of information in ramp-up can result in biased estimates

In an OCE, the typical reward for every arm is commonly estimated by the typical of the noticed outcomes in that arm. In an OCE with fixed project weights and a consultant pattern, that is an unbiased estimator.

Nonetheless, when there are altering project weights, then an unweighted common of information throughout the epochs generally is a biased estimate. The reason being that altering project weights introduces a dependence between time of first go to and chance of being assigned to an arm. If time of first go to can also be correlated with reward, this introduces bias. Time of first go to and reward could also be correlated resulting from time-based confounders, which we outline as a variable that’s correlated with project time and reward. To summarize, altering project weights causes the distribution of the time-based confounder for these assigned to every arm to be unrepresentative of the general inhabitants of customers who go to through the experiment. Potential time-based confounders embrace day-of-week, time-of-day, or frequent vs. rare customers.

When there are altering project weights and time-based confounders, this complication should be thought of both within the evaluation or the experimental design. An choice that may generally work is to make use of solely the information from customers assigned within the remaining epoch. This results in legitimate inference as long as the ultimate epoch is run lengthy sufficient to be consultant with respect to the time-based confounder. If the time-based confounder is one thing like day-of-week, then ensuring the ultimate epoch is run for not less than a full 7-day cycle can do the trick. Nonetheless, if the confounder is frequent/rare consumer, this selection could not work if one is unwilling to reassign customers as they ramp-up (there could also be statistical or product causes for not wanting to do that). It’s because customers assigned within the remaining epoch are much less more likely to be frequent customers than the customers who go to at any level within the experiment. Within the case of full site visitors ramp-up, customers assigned within the remaining epoch should not have visited in earlier epochs and therefore are much less more likely to be frequent customers. Within the case of partial site visitors ramp-up, a proportion of customers who visited throughout earlier epochs had been assigned in these earlier epochs. If the proportion is giant, customers assigned within the remaining epoch are rather a lot much less more likely to have visited in earlier epochs. If the proportion is small, customers assigned within the remaining epoch are solely barely much less more likely to have visited in earlier epochs and the bias could have little sensible significance.

Utilizing knowledge from the ultimate epoch is only one technique to analyze experiments that use ramp-up methods within the presence of time-based confounders — different experimental designs or methods to research the information may present legitimate inference. With that being mentioned, full site visitors ramp-up and mixing knowledge throughout epochs would generally be engaging, supplied one might get unbiased estimates. We provide two examples the place this can be the case.

For the primary instance, think about a small web site that could be a platform for content material on private finance. Think about you run this website and wish to check a brand new video internet hosting function to hopefully improve engagement in your website. How prepared your customers are to have interaction with private finance content material relies on whether or not or not it’s the weekend. Right here, day-of-week is a time-based confounder.

Though growing engagement is the objective of this new video internet hosting function, you additionally wish to make certain your small web site can deal with this. So that you initially give this new function an project weight of 25%, then if all goes properly 50%, then finally 75%. Since so many customers got the function in earlier epochs, there’s a whole lot of data being left on the desk for those who had been to disregard these epochs. So combining knowledge from epochs is engaging.

For the second instance, we are going to go into extra element within the subsequent part.

Instance: fantasy soccer web site

Fantasy soccer is a digital sport the place customers choose skilled soccer gamers to play on their staff firstly of the season. Earlier than every week’s video games, a consumer chooses a subset of those gamers to “play” that week. The higher the picked soccer gamers carry out in an actual soccer sport, the extra factors the consumer will get.

Think about you might be experimenting on a fantasy soccer website and also you wish to present some superior participant statistics as a way to support customers in selecting the gamers to play that week. This alteration provides customers a teaser of what they’d get in the event that they signed up for a premium account. The objective of this modification is to extend the variety of non-premium customers who join a premium account. You resolve to make use of a ramp-up technique to mitigate the danger of your non-premium customers having robust preferences for the standard participant statistics which can be presently proven. Nonetheless, ramp-up provides problems that you simply ought to think about. For instance, a few of your customers go to regularly to tinker with their lineups, verify scores, verify rankings, and so on. and different customers go to sometimes. For the reason that frequent customers go to extra typically, it’s extra seemingly for his or her first go to to be earlier within the experiment. The frequent vs. rare consumer variable is a time-based confounder if, along with being correlated with time of first go to, it has an impact on the chance of signing up for a premium account.

Fantasy soccer follows a weekly schedule based mostly on the actual soccer video games which can be performed on Thursday, Sunday, and Monday. We name a Tuesday via Monday cycle a “soccer week” to tell apart it from an arbitrary 7 day interval that would span two soccer weeks. Because of the weekly schedule, each frequent and rare customers will seemingly go to not less than as soon as throughout a soccer week, with every soccer week’s guests being consultant of the general inhabitants of customers the positioning is curious about. So when operating an experiment on a fantasy soccer website, it’s a good suggestion to think about knowledge from an entire soccer week as a way to have a consultant pattern of the inhabitants.

In the event you needed to mitigate danger and do ramp-up, utilizing full site visitors ramp-up and mixing knowledge throughout epochs is a horny choice partly because of the weekly schedule of fantasy soccer. With this selection, you could possibly do ramp-up and run the experiment for only one soccer week. If as a substitute you modified the ramp-up design and thought of solely the ultimate epoch, earlier epochs would have to be deliberate such that the ultimate epoch ran for a full soccer week. Though not insurmountable, this does add some complexity within the experimental design and will trigger the experiment to take meaningfully longer than every week to complete.

We simulate knowledge utilizing this fantasy soccer instance for example how naive aggregation of information in ramp-up can result in biased estimates. We do the very same simulation twice, however in a single simulation the hypothetical experimenter modifications the project weights half manner via and within the second they don’t. We’ll see that the bias attributable to naive aggregation with altering project weights will be so robust that it modifications the result of the experiment.

On this simulation, all customers who go to through the experiment are randomly assigned an arm on their first go to and keep assigned to that arm at some point of the experiment. The experiment will run for a full soccer week as a way to get a consultant pattern and to decide earlier than subsequent week’s video games. We simulate 2,000,000 customers who go to through the experiment. The variety of simulated customers is giant sufficient in order that the simulation provides almost similar solutions every time it is run. Because of this we don’t report uncertainty measures or statistical significance within the outcomes of the simulation. Time of first go to and time of project for consumer i are represented by the identical variable $Z_i$ and is simulated by a uniform distribution $$

Z_i sim mathrm{Uniform}(0, 1)

$$ the place experiment period is normalized to be between $0$ and $1$.

Let $A_i =1$ if consumer $i$ is a “frequent consumer” and $A_i =0$ in any other case. A “frequent consumer” is somebody who regularly visits the web site, so their first go to is extra more likely to come earlier throughout the soccer week than “rare customers”. Therefore, the frequent customers usually tend to be assigned earlier. The connection between these variables is simulated by $$

A_i sim mathrm{Bernoulli}(1- Z_i)

$$A consumer can join a premium account solely as soon as. Whether or not or not consumer $i$ converts through the experiment is simulated by the chances in Fig 3.

Fig 3. Conversion charges used within the simulation for frequent and rare customers for the present and new arm. 

Discover that the brand new arm performs higher on frequent customers and the present arm performs higher on rare customers. This isn’t an instance of Simpson’s paradox since one arm does not uniformly carry out higher on each frequent and rare customers.

The experimenter on this simulation desires to ensure the brand new arm isn’t going to alienate customers who maybe have robust preferences for the present participant statistics which can be proven. So as to mitigate danger, the experimenter decides to “play it protected” and never assign too many customers to the brand new arm. So the experimenter decides to initially assign 90% of customers to the present arm and 10% to the brand new arm.

Half-way via the experiment, early knowledge means that the brand new arm isn’t an entire catastrophe. Think about two hypothetical experimenters: one which ramps-up the project weights and one that doesn’t. These two experimenters will use the simulated knowledge as described above.

Experimenter 1: No ramp-up

Experimenter 1 retains the project weights fixed all through the period of the experiment. That is simulated by $$
T_i sim mathrm{Bernoulli}(0.1)quadmbox{for all $Z_i$}
$$ the place $T_i=0$ means consumer i is assigned to the present arm and $T_i=1$ if assigned to the brand new arm. On this situation, the next are pattern conversion charges on the finish of the experiment:

Present Arm:  0.0390 

New Arm:  0.0371 

Right here the experimenter decides to maintain the present arm.

Experimenter 2: Ramp-up

Experimenter 2 decides to ramp-up the project weights to the brand new arm and assigns 50% of customers to the present arm and 50% to the brand new arm within the second epoch. That is simulated by $$

T_i sim mathrm{Bernoulli}(0.1)quadmbox{if $Z_i<0.5$}
T_i sim mathrm{Bernoulli}(0.5)quadmbox{if $Z_i>0.5$}

$$ the place $T_i=0$ means consumer $i$ is assigned to the present arm and $T_i=1$ if assigned to the brand new arm. Discover that there’s a dependence between which arm consumer $i$ is assigned to and when they’re assigned. Nonetheless, when they’re assigned can also be correlated with which arm they like. Thus the ensuing pattern conversion charges can be biased. Certainly, in our simulations we get the next pattern conversion charges on the finish of the experiment:

Present Arm:  0.0362 

New Arm:  0.0395 

Right here, the experimenter finally ends up deploying an inferior arm. The bias attributable to altering project weights and never contemplating this within the evaluation modifications the experimenter’s choice.

Ramp-up resolution: measure epoch and situation on its impact

If one desires to do full site visitors ramp-up and use knowledge from all epochs, they need to use an adjusted estimator to get an unbiased estimate of the typical reward in every arm. By conditioning on a consumer’s epoch, we are able to get an unbiased estimate without having to know and observe all doable time-based confounders. Let’s formally outline a consumer’s epoch.

Epoch: If project weights are modified at occasions $Z^*_1, …, Z^*_J$ then the project weights are fixed throughout $[Z^*_j, Z^*_{j+1})$. The period of constant assignment weights $[Z^*_j, Z^*_{j+1})$ will be called epoch $j$. The epoch of user $i$ is determined by their assignment time and will be denoted by $E_i$.

To get an unbiased estimate for the average reward in arm $t$, we can infer the arm’s reward in each epoch and take a weighted average across the epochs with equal weights in the arms $$

hat{theta}_{t, mathrm{adjusted}} = sum_j hat{E}[Y_i | T_i=t, E_i=j] P(E_i=j)

$$Right here $Y_i$ is the noticed reward for consumer $i$ and $hat{E}$, $hat{P}$ point out estimates of the corresponding anticipated worth and chance. Inside an epoch, project weights are fixed and so a pattern imply utilizing knowledge from the suitable epoch and arm is an unbiased estimate of $E[Y_i | T_i=t, E_i=j]$. Discover that the weights $P(E_i=j)$ don’t depend upon the assigned arm, so must be estimated from all customers no matter arm project. Intuitively the adjusted estimator reweights every epoch estimate in proportion to the proportion of the general inhabitants that arrived in that epoch. Distinction this with an unadjusted estimate the place every arm’s pattern imply is just consultant of the inhabitants of customers assigned to the given arm. These populations will not be the identical throughout arms when there are altering project weights and time-based confounders.

Certainly, after we use this resolution within the simulation, we get again the identical end result as if we hadn’t modified project weights. To make use of the answer, let’s first take a look at the noticed conversion charges in every epoch

Fig 4. Noticed conversion charges for every epoch/arm mixture. 

Subsequent discover that 50% of all customers within the experiment had been assigned in Epoch 1 and 50% had been assigned in Epoch 2 so $hat{P}(E_i=1 )=0.5$ and $hat{P}(E_i=2 )=0.5$ (first arrival time is uniform and the epochs are equal lengths). Combining all these collectively as described above,

Present Arm: $hat{theta}_{0, mathrm{adjusted}} = 0.0295 occasions 0.5 + 0.0483 occasions 0.5 =0.0389$

New Arm: $hat{theta}_{1, mathrm{adjusted}} = 0.0340 occasions 0.5 + 0.0406 occasions 0.5=0.0373$

We see with this resolution that the altering project weights downside disappears. An experimenter who modifications project weights will get the identical reply because the experimenter who doesn’t change project weights (modulo some rounding errors) as long as they use the adjusted estimator.

Utilizing pattern proportions, as we do right here, is a straightforward technique to estimate $hat{E}[Y_i | T_i=1,E_i=j]$ and $hat{P}(E_i=j )$. In observe, one could wish to use extra complicated fashions to make these estimates. For instance, one could wish to use a mannequin that may pool the epoch estimates with one another by way of hierarchical modeling (a.okay.a. partial pooling) if one has many quick epochs.

Justification of the answer

Within the above simulation we see that when a hypothetical experimenter modifications project weights and makes use of the proposed resolution, they get the identical reply as in the event that they hadn’t modified project weights. Right here we formally justify that when project weights are modified in an experiment, this resolution is an unbiased estimator of what’s known as the typical therapy impact (ATE). To formally outline ATE, we first introduce potential outcomes,

Potential final result: Let $Y_i (t)$ denote the realized reward for consumer $i$ if they’d been given arm $t$. Potential outcomes are counterfactual conditional statements about what would have occurred if the consumer was assigned a specific arm.

The typical therapy impact (ATE) is formally outlined because the anticipated distinction in potential outcomes between customers assigned to totally different arms $$

ATE =E[Y_i(1)] – E[Y_i(0)]

$$ATE is vital in experiments as a result of it tells us concerning the causal impact of every arm. In a altering project weights situation, we should depend on two assumptions to indicate that our adjusted estimate is unbiased for ATE [5]. The primary is known as consistency,

Consistency: the noticed final result if assigned to arm $t$ is similar because the potential final result if assigned to arm $t$, i.e $Y_i=Y_i(t)$ if $T_i=t$.

Consistency typically signifies that customers assigned to an arm will expertise the assigned arm and no different arm. Within the simulation we’re assuming that customers can solely see the model of the web site to which they’re assigned – therefore, consistency. The second assumption is known as conditional ignorability

Conditional ignorability: The potential outcomes of consumer $i$ if they’d been assigned to arm $t$ are conditionally unbiased of which arm is definitely assigned given another variable, i.e $Y_i(t)$ is conditionally unbiased of $T_i$ given another variable.

Within the simulation, the potential final result is conditionally unbiased of arm project given epoch, since project weights are fixed inside an epoch. Thus we now have conditional ignorability no matter what number of time-based confounders are current. With each consistency and conditional ignorability
start{align*}

E[Y_i(t)] &= sum_j E[Y_i(t) | E_i=j] P(E_i=j) quad mbox{(legislation of complete chance)}
&= sum_j E[Y_i(t)| E_i=j] P(E_i=j)quad (Y perp T | E)
&= sum_j E[Y_i| E_i=j, T_i=t]P(E_i=j)quadmbox{(consistency)}

finish{align*} So to get an unbiased estimate of the anticipated potential final result one can mix unbiased estimates for $E[Y_i | T_i=t, E_i=j]$ and $P(E_i=j)$ as we do in $hat{theta}_{t, mathrm{adjusted}}$. From a Bayesian perspective, one can mix joint posterior samples for $E[Y_i | T_i=t, E_i=j]$ and $P(E_i=j)$, which supplies a measure of uncertainty across the estimate. 

This put up discusses the answer within the context of an experiment the place random project and reward each occur on the consumer stage. The answer generalizes trivially if each random project and reward occur at another unit of research. In a extra sophisticated experiment the place random project is on the consumer stage and reward can occur a number of occasions for a consumer, the answer additionally generalizes.

Naive implementation of multi-armed bandits can result in biased estimates and inferior reward

Along with ramp-up, multi-armed bandits (MABs) are a typical motive for altering project weights. MABs are a category of algorithms that maximize reward (conversion charge, gross sales, and so on) by assigning extra customers to higher performing arms sooner as a way to make the most of them sooner. One such MAB algorithm is Thompson sampling which assigns a consumer to an arm in line with the chance that the arm is finest given the information seen up to now [6], [7].

We are able to formally outline Thompson sampling within the case of a two-arm experiment. Let $T_i=1$ if consumer $i$ is assigned to arm 1 and $T_i=0$ if assigned to arm 0. With Thompson sampling, $T_i$ is set by $$

T_i sim mathrm{Bernoulli}(Phi_i)
Phi_i = P(theta_1> theta_0 | Y_1 cdots Y_{i-1})

$$the place $theta_0$ and $theta_1$ are the typical rewards for arm Zero and arm 1 respectively and $Phi_i$ is the posterior chance that arm 1 has higher reward than arm Zero given all the information noticed up to now. Discover that this chance will change with every extra pattern, so project weights will change many occasions all through the MAB.

Though there are all kinds of MAB algorithms, many MAB algorithms are like Thompson sampling and require fashions to provide chances of every arm being one of the best. These MAB algorithms are nice at maximizing reward when the fashions are completely specified and chances are correct. Nonetheless, these chances will be delicate to mannequin misspecification. Inaccurate chances could cause the MAB to do a poor job of maximizing future reward. It’s because the MAB maximizes future reward by utilizing earlier customers to deduce the reward of future customers as in the event that they had been assigned to every arm. If there are time-based confounders, then customers who go to earlier won’t be consultant of later customers. If the mannequin is lacking these time-based confounders then the inference can be poor and the MAB will do a poor job of maximizing future reward.

Typically a MAB can be run to maximise reward throughout an experiment. Right here the MAB finally ends, a winner is said, and the profitable arm is deployed. If one does inference to find out the profitable arm whereas ignoring the complexity of time-based confounders which can be current, then the inference can be biased. The reason being a lot the identical as in ramp-up. Altering project weights causes the distribution of time-based confounders to be systematically totally different throughout arms. Since time-based confounders have an effect on reward, not contemplating time-based confounders which can be current will produce biased outcomes on the impact of every arm.

We proceed with the fantasy soccer instance and see that if a mannequin omits a time-based confounder then MAB algorithms like Thompson sampling will be counterproductive in the direction of maximizing reward. We will even use the instance for example how the naive mannequin can result in inferior selections on which arm to deploy on the finish of the MAB.

Instance: fantasy soccer web site

Let’s return to the fantasy soccer instance and use the identical simulation as described above. However now think about one hypothetical experimenter who runs a MAB to maximise reward through the experiment and the opposite who runs a conventional A/B experiment with fixed and equal project weights. For the experimenter who runs a MAB, there’s extra sampling variation than within the simulation for ramp-up due to the trail dependence between what occurs at first and finish of the experiment. Because of this, we run the simulation with 2,000,000 customers 300 occasions. 

 
The MAB experimenter will use Thompson sampling. Since Thompson sampling requires Bayesian posteriors, the hypothetical experimenters will infer the conversion charge for every arm j with the next easy Bayesian modelbegin{align*}

Y_i &sim mathrm{Bernoulli}(theta_j)
theta_j &sim mathrm{Beta}(1, 1)

finish{align*}Though the mannequin is probably overly easy in the actual world, keep in mind that the information producing course of within the simulation can also be fairly easy. The info producing course of differs from the mannequin solely by the omission of a single variable (frequent vs. rare consumer) that’s correlated with each project time and reward. Omitting confounding variables isn’t often a priority in a conventional A/B check as a result of the legislation of huge numbers and random project typically make these variables equally distributed among the many arms. As we’ll see within the outcomes, this instinct does not apply with time-based confounders in MABs, because the project relies on time.

Simply as within the earlier ramp-up simulation, altering project weights causes some severe points. Fig 5 reveals the estimated ATE for every of the 300 simulations. We see that the MAB experimenter incorrectly picks the brand new arm because the winner in 267 out of 300 simulations. Actually, even when the MAB experimenter accurately picks the present arm because the winner, it dramatically misestimates the ATE relative to the experimenter who runs a conventional A/B check.

Fig 5. Comparability of estimated Common Therapy Impact (ATE) for every of the 300 simulations from a MAB and in opposition to a conventional A/B check with fixed project weights. Right here we see how poor the estimates from a MAB will be.*

Not solely does the MAB experimenter often choose the incorrect arm, we see in Fig 6 that the MAB experimenter additionally has fewer complete conversions than the standard A/B experimenter in 267 out of 300 simulations.

Fig 6. Comparability of complete reward (complete conversions no matter arm) through the experiment for every of the 300 simulations from a MAB in opposition to a conventional A/B check with fixed project weights. Right here we see that MAB results in inferior reward more often than not.* 

*The bimodality in Fig 5 and Fig 6 is attributable to whether or not or not the MAB overconfidently goes into full “exploit” mode by sending almost all customers to the brand new arm (the popular arm amongst frequent customers). Sometimes the MAB “explores” simply sufficient to ship customers to the present arm and finally turns into assured that it’s the profitable arm (the popular arm amongst rare customers). 

On this simulation, altering project weights with a MAB induced biased outcomes and often modified the result of the experiment. Moreover, the MAB experimenter often had inferior reward than the standard A/B experimenter who wasn’t even involved about maximizing reward through the experiment!

MAB resolution: measure project time and situation on its impact

If one desires legitimate inference after utilizing a MAB, one might use the same technique because the one described for ramp-up. Simply as within the case of ramp-up, one can get legitimate inference without having to know and observe all doable time-based confounders by conditioning on steady project time and utilizing a weighted common over project time as follows$$

P(Y_i(t) | theta )= int P(Y_i | Z_i=z, T_i=t, theta) P(Z_i=z | theta) dz

$$the place are mannequin parameters. Conditioning on steady project time Zi, quite than discrete project epoch, is critical as a result of the project weights can change with every extra pattern in a MAB. Discover that this implies a mannequin that circumstances on steady project time is required. Conditioning on steady project time is far more troublesome in observe than conditioning on discrete epoch. For instance, what occurs if one doesn’t have a superb understanding of how steady project time impacts reward? Will the inference nonetheless be strong?

This solves the bias downside however doesn’t repair the issue of maximizing reward through the experiment when the mannequin omits time-based confounders. Since maximizing reward is the rationale to make use of a MAB, fixing the bias downside isn’t motive sufficient to run a MAB if it may’t additionally maximize reward. For maximizing reward through the experiment when there are time-based confounders, one ought to think about a category of MABs known as stressed multi-armed bandits which don’t assume reward is fixed over time [8]. Nonetheless, stressed multi-armed bandits add new assumptions about how reward modifications over time. Contemplating how counterproductive issues will be when assumptions will not be met in Thompson sampling, one ought to totally perceive and check these assumptions earlier than implementing.

When utilizing MABs, an experimenter must critically assess whether or not or not the mannequin is an effective match to the information at hand. Typically easy fashions gained’t work. Even when the mannequin is generally appropriate, however is lacking only one time-based confounder, the MAB can result in each unhealthy inference and inferior reward. Furthermore, if a MAB ends and one of many arms in consideration is deployed, then the MAB is getting used like an A/B check. Given all this, maybe one ought to deal with legitimate inference quite than taking up the complexities required to additionally maximize reward.

Conclusion

Altering project weights throughout an experiment is a typical observe. Specialists within the discipline typically suggest ramping up project to never-before-seen arms [1] and MAB algorithms are fashionable in business [2], [3], [4]. This put up reveals that naive implementations of ramp-up and MABs can break the inferential guarantees of randomized experiments and, within the case of MABs, can result in inferior reward relative to a easy A/B testing design with fixed project weights.

We provide an answer for an unbiased estimator for the typical therapy impact (ATE) when an experimenter does full site visitors ramp-up and aggregates knowledge throughout epochs within the presence of fixing project weights and time-based confounders. The answer is an easy weighted common of reward throughout epochs that permits the experimenter to make use of knowledge throughout your complete experiment. For MABs, the technique of conditioning on project time is a bit trickier and doesn’t repair the issue of maximizing reward. With this in thoughts, one wants to think about why they’re utilizing a MAB. If the objective is de facto to maximise reward, one ought to totally perceive and critically check the assumptions of the MAB algorithm and any fashions the algorithm could require earlier than implementing. If the objective is to finally deploy one of many arms in consideration, maybe one shouldn’t be taking up the complexities required for utilizing a MAB.

Though this weblog put up makes some particular factors about altering project weights in an A/B experiment, there’s a extra common takeaway as properly. A/B testing isn’t easy simply because knowledge is massive — the legislation of huge numbers doesn’t care for all the pieces! Even with massive knowledge, A/B assessments require considering deeply and critically about whether or not or not the assumptions made match the information. Merely trusting strategies which have been used previously, both by the experimenter or the business, can typically lead experimenters astray. 

References

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[3] Hill, Daniel N., et al. “An environment friendly bandit algorithm for realtime multivariate optimization.” Proceedings of the 23rd ACM SIGKDD Worldwide Convention on Information Discovery and Knowledge Mining. ACM, 2017.

[4] Audibert, Jean-Yves, and Sébastien Bubeck. “Greatest arm identification in multi-armed bandits.” 2010.

[5] Imbens, Guido W., and Donald B. Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge College Press, 2015.

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[8] Whittle, P. “Stressed Bandits: Exercise Allocation in a Altering World.” Journal of Utilized Chance, vol. 25, no. A, 1988, pp. 287–298., doi:10.2307/3214163.

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