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Advertiser campaign GMV optimization Campaign-level GMV optimization is fundamental for advertisers in TSA. For a campaign, we want to determine each AD’s [MATH] so that GMV is maximized. For a campaign-level GMV, the problem can be formulated as: |
[EQUATION] where [MATH] is the cost regulation parameter and [MATH] is the permissible offset determined from the demands. In practice, [MATH] is set no larger than [MATH] to ensure that the cost does not exceed the keyword bidding cost, i.e., the virtual budget. Meanwhile [MATH] is constrained in ranges determined fro... |
CIA generates fine-grained impression-level bids with coarse-grained AD-level [MATH] and [MATH] for implicit ROI constraints. Note that in basic CIA, [MATH] is used to guarantee a higher ROI while such constraint on [MATH] is eliminated but replaced with the ROI range implied by [MATH] in the GMV optimization. To solve... |
Assume there is a set of [MATH] effective [MATH] , denoted as [MATH] . For each AD, [MATH] is evaluated by the replay module to yield a valuation point [MATH] [MATH] is obtained by replaying with keyword-level bids. Following the discretization, the GMV optimization problem can be reformulated as follows: |
[EQUATION] where [MATH] is an indicator variable that [MATH] means [MATH] is selected for the [MATH] -th AD. The above formulation belongs to a classic dynamic programming problem and can be solved with group Knapsack algorithm ( ). After obtaining the optimal [MATH] , optimal [MATH] can be retrieved accordingly. |
Style comparison For an FMCG store, the launching of new clothing styles now and then is common practice. To reduce the marketing risk of awful designs, advertisers need to be aware of the market response to different styles quickly to adjust future production and inventory. Within CIA, different fashion styles should ... |
[EQUATION] where [MATH] and [MATH] have the same meaning as in GMV optimization and the new variable [MATH] denotes the amount of impressions in all the ADs. |
This problem can also be solved by group Knapsack algorithm following the same logic with GMV optimization. However, for the quadratic form of the objective, the optimal solution would yield uniform impressions among ADs. Based on this, we first calculate the feasible cost range as [MATH] . Then [MATH] and [MATH] are r... |
[EQUATION] The above formulation is a standard quadratic programming problem and can be solved with mature solvers such as OSQP ( ). After obtaining the optimal [MATH] , the replay module combined with the binary search method is utilized to get the optimal [MATH] . Compared with Knapsack algorithms, directly optimizin... |
Ranking with CIA Within keyword bidding, there exists an obvious mismatch: platform tries to optimize [MATH] for revenue while advertisers care more about [MATH] . Meanwhile, consumer experience depending on conversions (consistent with advertisers) is also ignored. To mitigate the mismatch, industrial practice often i... |
Given the new bidding formula, the ranking score in CIA has the following form: [EQUATION] Compared with the original ranking score, CIA takes explicitly [MATH] into bids while the ranking formula is preserved. In essence, budget savings from low-quality impressions are spent on higher-valued impressions, leading to a ... |
We argue that the CIA bidding keeps the GSP auction mechanism unchanged, i.e., the allocation and charging mechanisms have remained the same. Advertisers can still change their keyword-level bid settings to influence [MATH] so that the auction game still continues. For multi-agent behavior, the CIA bidding only takes i... |
Replay based CIA implementation To support optimization in the CIA framework, we need to predict the performance of different [MATH] on an AD. Particularly, we are interested in the following two functions: |
[EQUATION] where [MATH] represents the external bidding environment. For the baseline cost, we have: [EQUATION] A historical auction log contains the contextual information of all the candidate ADs, bids, predicted [MATH] [MATH] and commodity price etc. An asynchronous data collection system based on MaxCompute ( ) is ... |
With the historical daily auction log, the replay works as: 1. Updating the bids with [MATH] , and re-ranking all the ADs in the current auction according to eCPM; |
2. For the top- [MATH] ADs , calculating their cost [MATH] once clicked; 3. Accumulating [MATH] and [MATH] to the cost and GMV, respectively. |
Keyword-level bid settings can also yield rough performance following the same procedure. Reversely, in the optimization process, a mapping from the AD’s daily cost to [MATH] would also be necessary. The inverse estimation of [MATH] can be solved efficiently via binary search due to monotonicity. |
As a black box module, replay encapsulates the functions of winning probability/bid landscape estimation and bidding optimization together (see Fig. ). Currently, it can generate a pair of [MATH] in milliseconds. To alleviate the influence of marketing shifts, CIA can average performance from the auction log of the pas... |
Experimental Evaluation In this section, we first introduce our experimental settings including baseline algorithms and evaluation metrics. Then we report some offline simulation and online A/B test results with further analysis and discussions. |
6.1 Experimental settings To test the performance of the CIA framework, we decompose evaluations into three levels, i.e., AD level, campaign level and platform level. |
The AD level evaluation bids as equation ( ) (abbreviated as CIA-AD) and is compared with keyword-level bids. The evaluation of CIA-AD is only used to illustrate the effectiveness of CIA over keyword-level bids. For the AD-level evaluation, we compare the performance of GMV, ROI, CVR, and PPC. Formally, CIA-AD is evalu... |
KB: Standard keyword-level bid settings are used to bid each impression. Campaign level experiments constitute the core evaluations. We test the GMV optimization (CIA-CAMP1) and style comparison (CIA-CAMP2) performance in Section 3.2 . The following baselines are used ( [MATH] ): |
CoKB: For campaign GMV optimization, joint optimization of budget allocation and keyword-level bids is implemented ( ), to compare with CIA-CAMP1. |
KB-CAMP: For style comparison, advertisers set the same keyword bids on different ADs to compare their performance, which serves as the baseline of CIA-CAMP2. The normalized standard deviation of cost spent on different ADs in CIA-CAMP2 is compared. |
The platform level experiments mainly focus on evaluating the effects on the platform when massive advertisers switch to CIA bidding. Specifically, advertiser GMV/ROI and platform revenue are evaluated simultaneously. |
To make the results more persuasive, all the experiments are carried out on massive randomly selected ADs/campaigns across two weeks. For the sake of commercial privacy, all performance measures are presented in the form of a relative shifting ratio compared with the baseline. |
6.2 Offline simulation The offline simulation operates on historical auction logs and relies on replay for optimization and evaluation. |
AD-level: We randomly choose [MATH] ADs and test their performance with CIA-AD. The overall average performance and four examples are shown in Tab. |
From Tab. we can see that with an approximately equal cost, the GMV and ROI of CIA-AD has an increase of nearly [MATH] percent. Meanwhile, CVR obtains the maximum increase of [MATH] percent, which means that more impressions with high conversion rates are acquired by CIA bidding. This coincides with our analysis as the... |
Campaign-level: For a campaign-level offline simulation, the results of randomly selected [MATH] campaigns which opt in CIA are shown in Tab. . It shows that with a nearly equal cost, CIA leads to an increase of GMV/ROI by [MATH] percent, which is reasonable since CoKB still bids on the keyword level although co-optimi... |
For the style comparison demand, the performance of CIA-CAMP2 with [MATH] campaigns is listed below in Tab. . Compared with naive equal keyword bids for style comparison, CIA-CAMP2 arrives a more uniform impression allocation, leading to a fair style comparison. |
Platform level: Multiple advertisers to opt in CIA will definitely change the global bidding environment, bringing complex consequences. To further evaluate the effects, we conduct an offline performance test by randomly switching a proportion of all the ADs to bid with CIA. The results of the whole platform (including... |
It shows that with more and more advertisers changing to CIA, the platform revenue (reflected by advertisers’ costs) and advertiser ROI grows together, which means that CIA has a positive effect on a healthy marketing ecosystem. |
6.3 Online experiments The online experiments compare different strategies by randomly allocating online queries to buckets implementing different strategies. |
AD-level: We apply the replay module on the historical log to estimate a proper [MATH] to balance the cost. Then it is used online to evaluate the performance of CIA-AD as displayed in Tab. , where“overall” summarizes the whole [MATH] ADs and A,B,C,D are the same with those in offline evaluations. |
With a [MATH] percent increase in cost, overall GMV and ROI encounters an increase of [MATH] and [MATH] percent, respectively. The PPC also sees an increase of [MATH] percent, which means that platform efficiency also increases. |
Campaign-level: Online evaluation results of CIA-CAMP1 and CIA-CAMP2 with the same set of offline campaigns are listed in Tab. and Tab . , respectively. The fundamental increase of GMV, ROI, CVR and decrease of PPC follows the right logic. We can also see that in online experiments, different campaigns have various het... |
Platform level: A large-scale online experiment with [MATH] ADs changing to CIA bidding is conducted. The overall ROI of the ADs with CIA bidding sees an [MATH] increase with a [MATH] increase in cost, showing that advertiser ROI and platform revenue grow together. For all the ADs, the average increase ratio is smaller... |
Conclusion In this paper, we propose a unified advertiser demands optimization framework called CIA in the e-commerce sponsored search. CIA formulates formally various demands with budget and unit conversion cost constraints. To maintain the advertiser knob on ROI, CIA constructs an impression-level bidding equation wi... |
# Source: arxiv 1806.05843 # Title: Bayesian inversion of a diffusion evolution equation with application to Biology # Sections: all # Downloaded: 2026-03-03T05:16:41.055357+00:00 |
Bayesian inversion of a diffusion evolution equation with application to Biology. Abstract A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modeling). When complex dynamical systems are consider... |
Introduction The problem of diffusion in a porous media, which is ubiquitous in physics, engineering and biology, is usually represented by the following partial differential equation: |
[EQUATION] where the spatial domain is [MATH] [MATH] ) and the final time is [MATH] (other initial and boundary conditions are possible). In real world applications, the quantity of interest [MATH] (hereafter called the solution of equation ) is typically the concentration of some chemical and evolves from a null initi... |
This problem has already been solved by different approaches, under distinct sets of hypotheses. In , the authors use a system of ordinary differential equations instead of equation , and minimizes a discrete version of a least-square type functional, while confidence intervals on parameters are given by bootstrapping.... |
. These approaches assume that unknown physical quantities can be modelled with Gaussian Processes . In particular, if [MATH] is Gaussian and if the decay and the diffusion are constant, [MATH] is Gaussian as well (linear transformation of a Gaussian process). The two constant parameters [MATH] and [MATH] can then be e... |
In this work, we apply a more general methodology, based on the recent advent of Bayesian Inverse Problems for infinite dimensional spaces. In a sense, it has the advantage of dealing with the ill-posedness while fully integrating the quantification of uncertainties. Moreover, the possibility of taking naturally physic... |
Bayesian inversion As previously announced, our goal in this work is to infer a source term [MATH] (mRNA concentration) jointly with rates of diffusion [MATH] and decay [MATH] (i.e. the parameter [MATH] ) from noisy and partial measurements of the solution [MATH] (gap protein concentration). This problem is ill-posed f... |
. Doing so, the regularized problem’s solution will be compatible with the dataset, but possibly very different from the real parameter (if there is such thing). Additionally, a particularly valuable information is a representation of all parameters [MATH] that would lead to similar data, giving precise statement on ho... |
. One approach consists in treating these 2 objectives sequentially, first regularizing then quantifying uncertainty. However, the Bayesian methodology for inverse problems ( |
and more recently ) is precisely tailored to complete both tasks at once in an elegant manner. One particularity of these recent contributions is to tackle inverse problems directly in function spaces, postponing discretization at the very end for implementation purposes, leading to algorithms robust to the discretizat... |
In essence, instead of searching for one particular parameter value that would solve the regularized problem, this approach considers the probability distribution of the parameter given the data. Namely, given a prior distribution (see subsection 2.2 ) and few technical conditions on the forward problem (parameter to d... |
2.1 Forward model analysis The first step is to detail precisely the required regularity of the solution map from equation . Using common variational techniques from PDE theory (see |
or ), one can show that this equation has a unique weak solution (see theorem 2.1 which proof is in the appendix) given [MATH] in a domain [MATH] that will be precised later on. Moreover, this solution evolves smoothly when the parameter varies. Without loss of generality and keeping in mind the Biological application,... |
Theorem 2.1 Let [MATH] , then for all [MATH] , equation has a unique weak solution, defining a map: [EQUATION] Moreover, this map has the following properties: |
1. it satisfies the following estimate [MATH] [EQUATION] with [MATH] constant independent of [MATH] 2. it is locally Lipschitz, [MATH] [MATH] such that [MATH] [MATH] [MATH] |
[EQUATION] 3. it is twice Fréchet differentiable on [MATH] The proof is given in the appendix. Let us now justify why the properties given in theorem 2.1 are important for the Bayesian inversion (most of them used in section 2.3 ): |
1. The energy estimate will be critical to establish the continuity of the posterior w.r.t. data, because it relates sufficient integrability conditions on [MATH] w.r.t. [MATH] |
2. Continuity (implied by Fréchet differentiability or local Lipschitz behaviour) will be used to show that the solution map is measurable, |
3. Second order Fréchet differentiability will be necessary for geometric methods in optimization (research of modes) and sampling (Markov-Chain Monte-Carlo), |
4. The local Lipschitz behaviour is used in the characterization of posterior modes (Maximum a Posteriori). In the rest of this work, we will restrict ourselves to the subset |
[EQUATION] which is implicitly equipped with the norm [EQUATION] Since [MATH] , the solution map is well defined on this subset and keeps all its smoothness properties. Moreover, one can show that a weak solution of equation for [MATH] is also a strong solution |
, but we don’t need these regularity results here. 2.2 Choice of prior distribution The second step is to choose a prior probability distribution on [MATH] , encoding all knowledge on the physics at hand, while being simple enough to keep analysis tractable. Here are the constraints given by the biological application: |
[MATH] must be positive (which imposes decay, not production), [MATH] must be positive and continuous at all time and position (it is a concentration). |
Starting with the depletion and diffusion parameters [MATH] , we choose respectively Borel prior distributions [MATH] and [MATH] on [MATH] with densities w.r.t. Lebesgue’s measure. Now, since [MATH] must be positive, we re-parametrize the problem with |
[EQUATION] where [MATH] . By selecting a Borel probability measure [MATH] on the Banach space [MATH] , both continuity and positivity of the new source [MATH] will be ensured almost-surely. In this work, we choose [MATH] as a Gaussian distribution with covariance operator [MATH] with continuous realizations (see |
for a presentation of infinite dimensional Gaussian measures). Finally, we assume independence between the three components, leading to the following prior distribution: |
[EQUATION] These choices clearly ensure positivity of [MATH] (in the previous sense) and [MATH] . The exponential map in equation can be replaced with any sufficiently differentiable function from [MATH] to [MATH] (to keep the second order Fréchet differentiability of the solution map). Alternative distributions are po... |
or more general convex measures from ) for the regularization. In practice however, our choice is also motivated by the fact that one can find a Gaussian reference measure [MATH] in the form |
[EQUATION] where [MATH] and [EQUATION] such that [MATH] . Indeed, choose [MATH] and [MATH] then [MATH] with [EQUATION] This reference will be critical for modes analysis (section 2.4 ) and MCMC sampling (section 3.2 ). |
2.3 Posterior distribution The third and last step in the theoretical analysis of Bayesian inversion is to show that this particular setting (forward model and prior distribution) leads to a well defined posterior measure using Bayes theorem. This is the purpose of theorem 2.2 which is a direct application of the theor... |
. Consider a dataset [MATH] which corresponds to observations at different times and locations [MATH] and assume they are produced from the following model (in vector notations): |
[EQUATION] where [MATH] [MATH] being the identity matrix of dimension [MATH] ) and [MATH] is the observation operator, mapping directly the PDE parameter [MATH] to the value of the associated solution [MATH] at measurement locations (composition of solution map [MATH] with Diracs). The following theorem, which is prove... |
Theorem 2.2 Let [MATH] be the observation operator defined in equation and [MATH] the probability measure defined in equation (satisfying [MATH] ), then there exists a unique posterior measure [MATH] for [MATH] . It is characterized by the following Radon-Nikodym density w.r.t. [MATH] |
[EQUATION] with [EQUATION] and [EQUATION] Furthermore, the two following integrability conditions: [MATH] [MATH] imply the continuity of [MATH] in the data w.r.t. Hellinger distance. |
Theorem 2.2 gives two distinct results: a) the existence and uniqueness of a posterior (as long as [MATH] is Radon and [MATH] ), b) well-posedness of the Bayesian inverse problem under additional integrability conditions of certain functions. In particular, the need for an exponential moment under [MATH] comes from the... |
2.4 Maximum a posteriori In the previous section, we proved the well-posedness of the Bayesian inverse problem under specific integrability conditions. However, the posterior distribution is only known up to a multiplicative constant, through its density w.r.t. [MATH] . In our application, we will need to summarize [MA... |
) but its optimality properties (under the frequentist paradigm) are not yet well-understood in infinite dimension to the best of our knowledge. This is why posterior modes (or Maximum a Posteriori) are more and more considered instead. Indeed, they provide a clear link with Tikhonov-Philips regularization (see |
) and a practical optimization problem (in case of Gaussian or Besov priors) which can be solved numerically, see theorem 2.3 (which proof is given in the appendix). |
Theorem 2.3 Let [MATH] be the prior probability measure defined in equation and [MATH] the Gaussian reference measure from equation 2.2 . Suppose additionally that |
[EQUATION] is locally Lipschitz, then the modes of [MATH] are exactly the minimizers of the following (generalized) Onsager-Mashlup functional: |
[EQUATION] where [MATH] and [MATH] are respectively the norm of the Cameron-Martin space and the mean of [MATH] . A minimizer will be noted [MATH] |
The precise application of this theorem to our biological setting is done in section 4.1 Metropolis-Hastings algorithm As it was previously announced, our motivation for the Bayesian methodology is the quantification of uncertainty, which will be done by simulation. Among the vast catalogue of methods for probability d... |
) and well defined on function spaces even though ergodicity analysis of such algorithm is still in its infancy . After a short presentation of the functional Metropolis-Hastings algorithm (section 3.1 ), we will focus on a state-of-the-art Markov kernel designed to sample from Gaussian measures (section 3.2 ) and adap... |
3.1 Metropolis-Hastings on function spaces The Metropolis-Hastings algorithm (MH) is a very general method to design Markov chains to sample from a given probability measure. It is based on a two-step process on each iteration: |
1. Given a current state [MATH] , propose a new candidate [MATH] according to a proposal Markov kernel [MATH] (it is a probability distribution on [MATH] for almost any [MATH] ), |
2. Accept the new state [MATH] with probability [MATH] or remain at [MATH] This algorithm provides a sample distributed under a predefined probability measure [MATH] , if one selects [MATH] and [MATH] in a specific way (see |
for a discussion in general state spaces). For instance, let [MATH] and [MATH] , the Metropolis-Hastings algorithm typically considers the following acceptance probability: |
[EQUATION] which, in particular, requires the absolute continuity of [MATH] w.r.t. [MATH] (detailed balance of the Markov chain). Contrary to finite dimensional situations, this condition may be difficult to satisfy and a common way to overcome this situation in Bayesian Inverse problems (see |
) is to select [MATH] revertible w.r.t. [MATH] . Indeed, in this case (with [MATH] ): [EQUATION] In theory, the MH algorithm may be implemented with a large family of proposal kernels [MATH] . In practice however, they need to be as efficient as possible and thus adapted to the problem at hand. Two common desirable pro... |
Adjust the proposal to locally mimic the target distribution [MATH] Include a step size to tune acceptance probability. These two properties may be used to trade-off self-correlation, acceptance rates and convergence speed to high interest areas of the parameter space. The next section presents an algorithm with both p... |
3.2 Geometric MCMC under Gaussian reference We are now going to detail a specific Markov proposal kernel [MATH] , tailored to sample distributions having a density w.r.t. a Gaussian measure [MATH] . Most of recent work on infinite dimensional MCMC methods are based on the following Langevin stochastic differential equa... |
[EQUATION] where [MATH] is a (possibly position-dependent) preconditioner, [MATH] a cylindrical Brownian motion and [MATH] the gradient in [MATH] of the negative log-likelihood. According to |
, a semi-implicit discretization of equation leads to a Markov chain with the following kernel: [EQUATION] where [MATH] is a step-size parameter, [MATH] and: |
[EQUATION] This dynamic explores the parameter space with a balance between Newton-type descent to zones of high density and Gaussian exploration. The philosophy behind this kernel is to use alternative Gaussian reference measures locally adapted to the posterior distribution, since it has been recently showed that hig... |
and later generalized in and ). Indeed, highly informative datasets may result in a posterior measure significantly different from the prior in likelihood-informed directions and non-geometric kernels (such as Independent sampler or preconditioned Crank-Nicholson) are ineffective in this case. However, the infinite dim... |
[EQUATION] where [MATH] is the Gauss-Newton Hessian matrix of [MATH] , which locally adapts to the posterior. This kernel does not preserve the distribution [MATH] but is shown to be absolutely continuous w.r.t. the reference measure [MATH] , almost-surely in [MATH] (under technical assumptions regarding [MATH] ) and t... |
[EQUATION] and noting [MATH] as it is done in , it finally comes: [EQUATION] Finally, the acceptance probability associated to the Markov kernel from equation is |
[EQUATION] This algorithm is well-defined on function spaces (reversibility is ensured w.r.t. [MATH] ), thus it is robust to discretization as required. The [MATH] -mMALA proposal may be computationally expensive, as it requires to compute both gradient [MATH] , Gauss-Newton Hessian [MATH] and the Cholesky decompositio... |
or likelihood-informed in ) to reduce the computational burden. A second alternative is to choose a constant preconditioner, located at a posterior mode for instance (similar to HMALA in |
and gpCN in ). Numerical application We now turn to the practical implementation of the previous methodology on the problem of reverse-engineering for post-transcriptional gap-gene in Drosophila Melanogaster. First, we precise our choice of distributions for the parameters compatible with previous assumptions, and give... |
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