Effective Dimension in Bandit Problems under Censorship
Abstract
In this paper, we study both multi-armed and contextual bandit problems in censored environments.
Our goal is to estimate the performance loss due to censorship in the context of classical algorithms designed for uncensored environments.
Our main contributions include the introduction of a broad class of censorship models and their analysis in terms of the effective dimension of the problem – a natural measure of its underlying statistical complexity and main driver of the regret bound.
In particular, the effective dimension allows us to maintain the structure of the original problem at first order, while embedding it in a bigger space, and thus naturally leads to results analogous to uncensored settings.
Our analysis involves a continuous generalization of the Elliptical Potential Inequality, which we believe is of independent interest.
We also discover an interesting property of decisionmaking under censorship: a transient phase during which initial misspecification of censorship is self-corrected at an extra cost; followed by a stationary phase that reflects the inherent slowdown of learning governed by the effective dimension.
Our results are useful for applications of sequential decision-making models where the feedback received depends on strategic uncertainty (e.g., agents’ willingness to follow a recommendation) and/or random uncertainty (e.g., loss or delay in arrival of information).
Introduction
Bandit problems are prototypical models of sequential decision-making under uncertainty.
They are widely studied due to their applications in recommender systems, online advertising, medical treatment assignment, revenue management, network routing and control [26, 39].
Our work is motivated by settings in which the feedback received by the decision-maker in each round of decision is censored by a stochastic process that depends on the current action as well as past history of feedbacks and actions.
For instance, in typical missing data problems, the decision-maker needs to deal with frequent losses of information (or delays in arrival of information) due to exogeneous failures such as faulty and/or unreliable communication.
Missing observations in dynamical interactions with the environment are a common concern in diverse fields ranging from operations management to health sciences to physical sciences [45, 20, 29].
In other settings, such as AI-driven platforms for health alerts, route guidance, and product recommendations [7, 46], the reception of feedback depends on whether or not the decision (or recommendation) is adopted by strategic agents (e.g. patients, customers or drivers) with private valuations.
Thus, from the platform’s viewpoint, the adoption behavior of heterogeneous agents can be regarded as a stochastic censorship process .
In static environments, the bias induced by the presence of randomly missing information has been thoroughly studied [29, 33].
However, in online settings, the dynamics of learning and acting are inherently coupled: since censorship mediates current information of the environment, it impacts the outcome of data-driven decision process; this in turn conditions the future decisions and future censored feedback, creating a complex and endogenous joint temporal dependency.
Our work contributes to the analysis of such phenomena for a broad classes of decision and censorship models.
Importantly, it is the first normative inquiry of how censorship impacts the statistical complexity of bandit problems.
We develop an analysis approach that is useful for both estimating the performance loss due to censorship and refining the classical algorithms designed for uncensored environments.
1.1 Related Work
Within the extensive bandits literature, well-surveyed in [26, 39], our work is most closely related to stochastic delayed bandits.
Initially, this line of work focused on the joint evolution of actions and information in settings where the reception of the latter is delayed [15].
Of particular interest is the packet loss model recently introduced in [ 24 ] , which provides the regret bound O
( 1 p R T )
where
R T is the uncensored regret and p the censorship probability.
Analogous results have been shown in the context of Combinatorial Multi-Armed Bandits with probabilistically triggered arms; see for example, [11] and [43].
Our work provides a systematic approach to study more general censorship models, and sheds light on how the impact of coupled feedback and censorship realizations on the expected regret can be evaluated in terms of the effective dimension of the problem.
Importantly, we also tackle the contextual bandit problems, where relatively few results are available on the regret under missing or censored feedback.
A notable exception is the work of [42], who focus on a different information structure and obtain a scaling of 1 /p (see Remark 4.4).
A related contribution by [2] provides both a potential-based analysis of the Upper Confidence Bound algorithm (UCB) for multi-armed bandits and an algorithmic variant leveraging the Kaplan-Meier estimator, although their censorship setting is different than ours.
In particular, our results are applicable to settings when delay is significantly large (possibly infinite).
This is in contrast to prior results on bandits with delayed information structure which assume either that the delay is constant , upper bounded , has a finite mean , or simply provide regret guarantees that are linear in the cumulative delays up to time T [15, 22, 30, 47, 34].
Under such assumptions on delay, one usually gets a second order additive dependency of the regret in terms of delay parameters, which practically says that delay is benign for bandits.
On the other hand, we show that censorship leads to a first order multiplicative dependency on regret and we provide a complete characterization of this dependency for a wide range of bandits and censorship models.
Moreover, the abovementioned works primarily focus on modifying well-known bandit algorithms to account for delays, or propose new delay-robust algorithms which may be difficult to implement in practice; a notable exception includes [44] but it focuses on Thompson Sampling.
In our work, we instead focus on estimating the performance loss due to censorship and derive insights on the behavior of well-known UCB class of algorithms [27, 13, 1].
These algorithms are widely used in practice; moreover, their theoretical study has been shown to be useful for analysis of broader class of algorithms (notably Thompson Sampling [3, 36] and Information-Directed Sampling [37, 23]).
On a somewhat related note, the literature on non-stochastic multi-armed bandit problems with delays [32, 9, 21] also tackles multiplicative dependency, although in a different setting than ours.
Another related line of work is Partial Monitoring [6, 25] which deals with generic categorization of learnability, rather than a fine-grained analysis of dimensionality in relation to censorship, which is our current focus.
Our work contributes to the Generalized Linear Contextual Bandits literature [16, 28] in two ways: firstly, through the use of these models in a sequential decision-making framework on which the impact of censorship is assessed in Sec. 4.
Secondly, by showing that our multi-threshold censorship model MT induces, at first order, a non-linear structure that closely mirrors such models.
Our results provide new tools to study this structure.
It is useful to note that the notion of effective dimension has been well-studied in the statistical learning and kernels literature [40, 41]
(where it is defined for a
Gram matrix K n and regularization λ as d n eff ( λ ) =
tr ( K n ( K n + λ I d ) − 1 ) ).
Our work shows that an analogous quantity governs the regret bound of bandit problems in censored settings.
Finally, there is a rich literature on classical missing and censored data problems [29, 33].
Although conditional on the choice of a given action the missing data/censorship process we study is an instance of missing-completely-at-random (MCAR), the online action generating process adds a significant difficulty to the problem: whereas MCAR is typically studied under a well-defined distributional assumption (e.g. i.i.d.
generation of action), our problem needs to deal with adaptive (hence non i.i.d.) data generation process with respect to the filtration of past information.
In particular, the structure of missing data set results from strong endogenous dependencies with past realization of the censorship (see Sec. 2).
1.2 Summary of Results
In Sec. 3, we consider Multi-Armed Bandit (MAB) models and prove that the regret scales as ˜
O ( d eff √ T ) (Thm. 3.1), where d eff is the effective dimension with value (cid:80) a ∈ [ d ] 1 p a .
In doing so, we recover and generalize related results from [24, 11] to more complex regularized settings and noise models.
In particular, we prove that the effective dimension results from characterizing the so-called censored cumulative potential V α .
Interestingly, we also show that the adaptive nature of censorship on V α plays only a second order role (Prop. 3.7), that is, impact of censorship can be treated in an offline manner at first order.
Importantly, our study of MAB under censorship instantiates an analysis framework which extends to Linear Contextual Bandits (LCB) (Sec. 4).
Our main result provides that regret is still governed by the effective dimension, but now with a dependency of ˜ O
( σ (cid:112) d · d eff √ T )
(Thm. 4.1).
To the best of our knowledge, these regret bounds provide the first theoretical characterization in LCB with censorship, and contribute to the literature by evaluating the impact of censorship on the performance of UCB-type algorithms.
Our second main contribution is identifying the effective dimension for a broad class of multi-threshold models MT as well as a precise understanding of the dynamic behavior induced by these models (Thm. 4.6).
In particular, we find that censorship introduces a twophase behavior: a transient phase during which the initial censoring misspecification is self-corrected at an additional cost; followed by a stationary phase that reflects the inherent slowdown of learning governed by the effective dimension.
In extending our analysis from MAB to LCB, we also develop a continuous generalization of the widely used Elliptical Potential Inequality (Prop. 4.3), which we believe is also of independent interest.
Finally, our results (Thm. 3.1 and Prop. 3.2 for MAB and
Thm. 4.1 for LCB) suggest that the UCB class of algorithms is indeed a reliable method for stochastic bandits problems under censorship.
Problem Setup and Background
Bandit Model: We successively consider stochastic multi-armed bandits (Sec. 3) and Linear
Contextual Bandits (LCB) (Sec. 4) in censored environments.
In both settings, at each round t ≤ T , the agent observes an action set A t ⊂ A .
She then selects an action a t ∈
A t (i.e. an arm ) to which a noisy feedback r ( a t ) + ϵ t is associated, where r ( a t ) is a bounded reward and ϵ t is an i.i.d.
sub-Gaussian noise of pseudo-variance σ 2 .
For action a , the sub-optimality gap at time t is denoted ∆ t ( a ) ≜ max ˜ a ∈A t r (˜ a ) − r
( a ) , and the maximal gap ∆ max ≜ max a,t ∆ t ( a ) .
We now recall the specifics of each model:
• MAB : There is a finite number of actions d , enumerated as A ≜ [ d ] , each having a scalar reward θ ⋆a .
Arms are independent : playing one arm gives no information about the others.
• LCB :
The action set A t is a subset of the unit ball B d , possibly infinite.
Unless explicitly mentioned, the reward is assumed to be linear with respect to a latent unknown vector θ ⋆ ∈ R d , i.e. r ( a ) =
⟨ a, θ ⋆ ⟩ .
Non-stochastic contexts are modeled by the fact that A t is drawn by an oblivious adversary.
Here one does not need to rely on the typical
i.i.d assumption on their generating process [47, 16].
Information Structure:
In the classical uncensored setting, the noisy feedback is immediately observed post-decision and utilized to make decisions in the next round.
We introduce the following censorship model: an independent Bernoulli random variable of parameter p ( a t ) denoted as x a t is drawn after each decision a t and the feedback is observed, i.e., realized , if and only if x a t = 1 ; else the feedback is said to be censored .
We recover the uncensored setting when p ( a ) ≡ 1 .
Henceforth, in both finite and linear settings, the Bernoulli parameter corresponding to the censorship probability depends on the action chosen i.e. our model allows the censorship to be heterogeneous across actions.
Given that the action chosen at time t is a random variable, p ( a t ) refers to a random variable as well.
Algorithm
Input: Total time T , Regularization λ , Precision δ for t = 1 , . .
, T do
Provide reward estimator ˜ r λt verifying w.p.
1 − δ : ∀ a ∈
A t , r ( a ) ≤ ˜ r λt ( a ) ;
Play action a t = argmax a ∈A t ˜ r λt ( a ) ; if ( a t , r ( a t ) + ϵ t ) is realized i.e. x a t = 1 then
Update ˜ r λt ; end end
Algorithms: To study the impact of censorship on bandit problems, we consider the class of high-probability index algorithms based on the optimism under uncertainty principle, commonly referred as UCB -algorithms.
Following [23], Algorithm 1 summarizes the generic UCB design framework.
We detail in App.
A the specific instances of UCB for
MAB (resp.
LCB) used in Sec.3 (resp. Sec.4).
Moreover, this family of algorithms strongly relies on regularized reward estimators ˜ r λt , where the regularizer is mostly used to prevent an artificial cold-start exploratory phase.
Performance Criterion: The frequentist performance of the agent is measured by the notion of pseudo regret , i.e., the difference between the algorithm’s cumulative reward and the best total reward.
More formally, we introduce for any policy π ∈ Π :
[Equation]
We aim to provide guarantees on E [ R ( T, π )] with respect to the number of rounds T and quantities that govern the complexity of the problem (for example number of arms, ambient dimension d , parameters of censorship model or smoothness properties of the reward r ).
Here, the expectation is with respect to the noise induced by the feedback, the censorship and a possibly randomized policy.
2.1 Notations
Transpose of a vector u is denoted by u ⊤ , classical Euclidean inner product by ⟨ ., .
⟩ and trace operator by Tr .
For positive semi-definite matrix Σ ∈ R d × d and for any vector u ∈ R d , notation ∥ u ∥ Σ refers to √ u ⊤ Σ u .
We use notation I d to denote the d × d identity matrix.
B d is the unit ball in R d .
[ n ] is the set of integers { 1 , 2 , · · · , n } .
For a given function f , we note f ( i ) the i th derivative of f .
To avoid confusion with the dimension d , we use ∂x instead of dx to denote an infinitesimal increase of x .
We use the asymptotic notations ∼ , O , Θ and ˜ O
( O when log factors are removed).
Finally, for an event H , we use ¬H to denote its complement.
Multi-Armed Bandits 3.1 Effective Dimension and Regret Bounds
The main result of this section is that censorship effectively enlarges the dimension of the problem.
We define the effective dimension as d eff ≜ (cid:80) a ∈ [ d ] 1 p a and our result (Thm. 3.1) shows that, at first order, the regret is guaranteed to be the same as the uncensored problem with d eff arms instead of d .
Theorem 3.1.
Under censorship, the UCB algorithm with regularization λ has an instanceindependent expected regret of:
[Equation]
Furthermore, we obtain analogous regret guarantees for instance-dependent cases where, at first order, the uncensored dimension (cid:80) a ̸ = a ⋆ σ 2 ∆ a enlarges to (cid:80) a ̸ = a ⋆ σ 2 p a ∆ a :
Proposition 3.2.
For a fixed action set A t ≡ [ d ] and for a-priori known action gap ∆ a ≜ max ˜ a θ ⋆ ˜ a − θ ⋆a , the UCB algorithm with regularization λ has the instance-dependent expected regret:
[Equation]
On one hand, a preliminary understanding of censorship posits an increase of the average " regret per information gain " [23] (as it takes longer on average to get the same amount of information) but does not change the underlying complexity of the problem.
One the other hand, our results (Thm. 3.1 and
Prop. 3.2) postulate that the censored problem is equivalent at first order to a higher dimensional problem but explored with the same regret per information gain .
The abovementioned results extends to a-priori known heteroskedasticity (see Rem. 3 and 4 in App.
B).
For this general setting, the effective dimension for instance-independent (resp.
dependent) case is given by (cid:80) a σ 2 a p a (resp.
(cid:80) a ̸ = a ⋆ σ 2 a p a ∆ a ), where σ 2 a is the variance proxy of arm a .
Although the scaling in (cid:80) a 1∆ a p a was already mentioned in [24] for unregularized setting with homogeneous variance σand proven to be optimal, our results generalize these findings.
3.2 Cumulative Censored Potential
We now provide a proof sketch of Thm. 3.1, and in doing so, we instantiate an analysis framework that will be extended in Sec. 4.
This proof consists in the successive elimination of the noise induced by the feedback and censorship.
This leads to regret guarantees on a resulting deterministic quantity by characterizing worst-case learning conditions.
The first step of the proof is a variant of the classical reduction of the UCB regret to another quantity we refer to as the expected cumulative censored potential .
Before stating it, we define at the end of a round t ∈ [ T ] , the random number of times an arm a has been pulled as τ a ( t ) ≜ (cid:80) tl =1 1 { a l = a } .
Similarly, the number of times an action a has been realized at the end of round t is denoted N a ( t ) ≜ (cid:80) tl =1 1 { a l = a, x a l = 1 } .
We then have:
Lemma 3.3.
Given an uniform regularization of λ > 0 , the UCB algorithm verifies:
[Equation]
where, for any α > 0 and π ∈ Π , the cumulative potential under censorship is given by:
[Equation]
Without censorship, the cumulative potential translates the average rate of decay of uncertainty on the reward of different arms and is closely linked to the divergence between the true reward distribution and the empirical distribution of observed rewards [38].
Introducing censorship transforms the classical deterministic decay rate into a stochastic one.
For a typical reward distribution, the rate of decay is proportional to a term in n − α or can be upper bounded by such a term (see for e.g. [38]), where n is the number of observed rewards.
Therefore, a higher α corresponds to faster learning.
In contrast to the classical non-regularized analysis or to the LCB case of Sec. 4, we observe two different orders of α ( 1 / 2 and 1 ) coming from the use of the L ∞ -norm instead of the L 2 -norm.
Taken independently, they lead to respective contributions of O ( d eff log( T )) and O ( (cid:112) d eff T ) .
Note that by working with a general α , our analysis naturally extends beyond sub-Gaussian noise to more general assumptions about the Laplace transform of noise (e.g., lighter or heavier tails), as discussed in Rem.2.
To further study V α , we introduce the following property:
Proposition 3.4.
For all α > 0 , δ ∈ ]0 , 1] and given ψ α a primitive of x (cid:55)→ x − α , we have:
[Equation]
The proof of this proposition involves two steps: firstly, we remove the stochastic dependence induced by the censorship through concentration properties (See App. B), and we then solve the resulting policy maximization problem (Lemma 3.5).
In the first step, we consider for a given δ ∈ ]0 , 1] the event:
[Equation]
where T 0 ( a ) ≜ 24 log( T ) /p
a + 1 and claim that P ( H CEN ( δ )) ≤ 4 d eff δ 2 T − 12 δ 2 , improving a result of [24].
Here H CEN denotes the event where there is a significant gap between the realized and expected number of observed rewards.
We consider its complement in our analysis of the principal order of regret. This allows us to lower bound for each action, the realized number of reward observations by a multiple of the number of times that action was selected, thus eliminating the randomness induced by censoring.
Our second step makes use of the following lemma (also known as a water-filling process in information theory [14]):
Lemma 3.5.
For ψ α a primitive of x (cid:55)→ x − α where α ∈ ]0 , 1] , regularization ( λ a ) a ∈ [ d ] ∈
( R > 0 )
d and censorship vector ( p a )
a ∈ [ d ] , the solution of the optimization problem:
[Equation]
is given by τ ⋆a = 1 p a [ C − λ a ] + , where C ensures the total budget constraint (cid:80) a ∈
[ d ] τ ⋆a = T .
In particular, with λ eff ≜ 1 d eff (cid:80) a ∈
[ d ] λ a p a and λ 0 a ≜ d eff ( λ a − λ eff ) , the optimal solution is given by τ ⋆a ≜ 1 p
[Equation]
d eff ( T − λ 0 a ) for T ≥ max a λ 0 a and the optimal value is d eff
[Equation]
ψ α ( λ a ) .
For unregularized algorithms, this framework can be easily applied to provide instances-dependent guarantees by adding constraints of type τ a ≤ f (∆ a ) within Lemma 3.5.
Optimal guarantees under regularization such as the ones given in Prop.
3.2 require however to consider both orders of V α ( 1 /and 1 ) simultaneously and not independently, leading to slight variations as shown in the proof of
Prop. 3.2.
Next, we further discuss the properties of V α given its importance in our analysis.
3.3 Evaluating Adaptivity Gain
It is well known that adaptivity is a key feature of sequential decision problems: optimal policies use feedback from previous decisions to decide the next action to take based on the data, and in comparison non-adaptive policies can be quite suboptimal.
Somewhat interestingly, the main result of this section is that adaptivity in the context of censoring does not provide a significant advantage to the decision maker.
More precisely, being able to observe which decisions have been censored and adapting to this information does not bring more than a second order gain.
In proving this result, we quantify and gain insight into the expected performance of policies that are adaptive to the realization of the censorship process, in comparison to a class of non-adaptive (i.e., offline) policies.
In fact, through the introduction of H CEN ( δ ) and for any α ∈ [0 , 1] , δ ∈ ]0 , 1] , we showed in Prop.
3.4 the upper bound
[Equation]
for the learning complexity max E [ V
[Equation]
( T, π )] where the maximum is taken over the class of adaptive policies Π adapt , i.e., measurable with respect to the censorship.
Note that the exact value of such maximum is notoriously difficult to study due to the adaptive nature of censorship induced by the decision-making process.
Next, we introduce Π off , the class of policies that are not adaptive with respect to the censorship and we prove that :
Lemma 3.6.
For α ∈ ]0 , 1] and λ > 0 , we have max π ∈ Π off E [ V [Equation] [Equation]
, π )
[Equation]
eff
[Equation]
( T d eff + λ ) .
In other words, restricting attention to offline policies is sufficient to obtain the correct scaling.
The next step to complete our claim is the asymptotic expansion:
Proposition 3.7.
For α ∈ ]0 , 1] , by denoting γ α ( p ) ≜
[Equation]
(cid:88)
[Equation] [Equation] a ̸
[Equation]
a
[Equation]
−
[Equation]
a p
[Equation]
a (cid:17) , we have:
[Equation]
Moreover, if for a given β ∈ ]0 , 1[ , we introduce Π single ( βT ) the policy class whose censorship information set has a single updating at time ⌊ βT ⌋ , we have:
[Equation]
Thus, γ α ( p ) can be viewed as an adaptivity gain resulting from the continuous correction of the cumulative variance induced by the action selection process.
Essentially, it is closely related to the Jensen Gap of an appropriate random variable and the proof involves the study of the Taylor expansion of the potential function ψ α .
( ⋆⋆ ) tells us that a single observation of the censorship realization is sufficient to obtain a near-optimal gain in adaptivity .
We present a proof sketch of Prop.
3.7 in App.
B. This shows that censorship in MAB can be treated in an offline manner at first order.
Contextual Bandit
In this section, we study Linear Contextual Bandits (LCBs) under censorship.
The regret analysis for the generic censorship model in Sec. 2 is significantly more complex for LCB than for MAB.
This is due to the fact that different actions contribute differently to the information acquisition, leading to a non-linear phenomenon governing the trade-off between reward and information gain (see Sec.4.4).
4.1 Multi-threshold Models and Regret Bounds
To address the abovementioned challenge, we now introduce a simple multi-threshold censorship model, which enables a precise regret analysis.
In particular, we consider that feedback is censored according to the following action-dependant probability:
[Equation]
where ( ϕ j ) j ≤ k +1 is an increasing sequence verifying ϕ
= − π 2 , ϕ k +1 =
π 2 and u ∈ R
d is a unit vector.
We assume that ( p j ) j ≤ k is decreasing, i.e. the censorship is increasing with j in direction u .
Henceforth, we refer to the interval [sin( ϕ j ) , sin( ϕ j +1 )
[ as region j .
Note that simple models such as uniform censorship are subsumed by this family (for k equals 0 ).
The two main features of the multi-threshold model are: the radial aspect (the censorship probability depends on the action through a scalar product with a given vector) and the monotonicity (the censorship is monotone in the value of this scalar product).
Note that MT can be seen as a piecewise constant approximation of any Generalized Linear Model (GLM) [31].
Thus, the simplicity of this censorship model is not an inherently limiting factor on the generality of our subsequent results.
Moreover, MT admits a natural behavioral interpretation: Such a distribution can be seen as induced by a population model of heterogeneous random-utility maximizing agents.
A single threshold model (i.e. k equals 1 ) corresponds to a given agent type, and the multithreshold model naturally results from aggregate responses of heterogeneous population [4].
We now state the main result of this section:
Theorem 4.1.
For a given multi-threshold censorship model MT , there exits d eff such that the UCB algorithm with regularization λ has an instance-independent expected regret of:
[Equation]
Importantly, note the mapping from the original dimension d to the enlarged (cid:112) d · d eff , in contrast to the previous dilation d (cid:55)→ d eff for the case of MAB problems.
An extension to Generalized
Linear Contextual Bandits is provided in App.
C.6 where we show that the dimension is governed by (cid:112) d · d eff /κ , with κ corresponding to a minimum of the derivative of the link function (encompassing the smoothness of the GLM at its maximum) [28, 16].
We conjecture that this result still holds if we relax the monotonicity property of MT although it will require some modifications in the proofs of section D.
On the other hand, we believe that the radial property is necessary, considering the related literature on GLMs (further discussed in App.
C.6) where it appears prominently.
4.2 Generalized Cumulative Censored Potential
Analogous to the MAB case, we now introduce for LCB the random matrices corresponding to the effective realization W Ct ≜
λ I d + (cid:80) tn
=1 x a t a t a ⊤ t and the expected realization W t ≜
λ I d + (cid:80) tn
=1 p ( a t ) a t a ⊤ t .
We also introduce the continuous counterpart of W t defined as W ( t ) ≜
λ I d
+ (cid:82) t u =0 p ( a ( u )) a ( u ) a ( u ) ⊤ ∂u , where ( a ( u ))
u ≤ T is an integrable deterministic path.
4 We emphasise that the use of continuous counterpart is key in enabling our next results.
As in the MAB case, we bound the regret although now using a generalization of V α :
Lemma 4.2.
For all δ ∈ ]0 , 1] , there exists a constant ˜ β δ ( T ) =
Θ( [Equation] d log( T )) such that
[Equation]
where, for α > 0 and π ∈ Π , the linear extension of the cumulative censored potential is given by:
[Equation]
The proof idea is analogous (albeit more complex) than in the finite action case (see App. C).
In order to get a handle on V α , we again leverage a two-step approach: first we eliminate the randomness due to censorship (here, we utilize matrix martingale inequalities) and then optimize the resulting deterministic quantity seen through a continuous lens.
The first step requires the following result:
Proposition 4.3.
For any δ ∈ ]0 , 1] , λ > 0 , α > 0 and policy π ∈ Π , we have:
[Equation]
where C ( δ ) ≜ 8( λ + 1) max(log( d/δ )) /λ, 1) /λ .
The key idea of this result is to observe that the telescopic sum on which the classical Elliptical
Potential lemma [1, 35, 8] heavily relies on is, in fact, the discrete approximation of an integral over a matrix path.
This critical methodological contribution is further discussed in Rem. 1 and 5.
Remark 1.
One way to fully appreciate the generality of this result is to consider the simpler case of classical uncensored environment for which we obtain for α > 0 , α ̸ = 1 :
[Equation]
For α = 1 , a similar reasoning is applied using the formula Tr(log
( A )) =
log(det A ) :
T (cid:88)
[Equation]
A deeper study of the eigenvalues of W 1 − α T then yields the worst-case upper bound
d α ( dλ +
T ) 1 − α / (1 − α ) for α < 1 and dλ 1 − α / ( α − 1) for α > 1 , recovering more naturally and extending the results of [8].
Thus, analogous to the water filling process highlighted in the MAB case in
Lemma 3.5, we now consider a spectral water-filling process [14] optimizing over the eigenvalues of ψ α ( W T ) with a slight abuse of notations ( W 1 − α T and log W T in this discussion).
Following Rem.1, for the general censored case the challenge now becomes to identify a suitable matrix operator on which the aforementioned spectral maximization can be performed.
By applying Lemma 4.2, we henceforth focus on the case of α = 1 for which Prop. 4.3 implies that for any policy:
[Equation]
Next, we focus on maximizing this integral over the policy class Π and again recover the notion of effective dimension.
4.3 Effective Dimension in Linear Settings
We now highlight immediate properties of the effective dimension, and then present its general study for the multi-threshold model MT .
Lemma 4.4.
Let us consider an uniform censorship model p : a (cid:55)→ ¯ p .
By leveraging the case of equality in the Arithmetic-Geometric inequality applied to the eigenvalues of W T , we then simply deduce the associated effective dimension d eff ≜ d/ ¯ p :
[Equation]
In fact, the logarithmic scaling of this quantity persists while moving beyond the uniform censorship assumption.
This also highlights the importance of the leading dimension factor, crudely upper bounded by d/p min in the next lemma:
Lemma 4.5.
For any censorship function p , by introducing lower and upper bounds ( p min , p max ) of p , we have:
[Equation]
Related problems in the Generalized Linear Models literature [47, 28, 16] are implicitly solved in the spirit of Lemma 4.5, where a minimum of the derivative of the link function plays the role of p min above.
However, when the function p varies with action a , a more careful analysis is required to derive useful dimensional bounds.
Our next major result addresses this gap in the literature by improving the bounds provided in Lemma 4.5:
Theorem 4.6.
For a multi-threshold censorship model MT , we have:
[Equation]
where d eff is the effective dimension.
Furthermore, d eff is characterized by two cases:
• Case 1: Single region j effective dimension d eff = dp j . • Case 2: Bi-region
( i, j ) effective dimension, with i < j :
[Equation]
where l ( i
j ) ≜ sin 2 ( ϕ i ) sin 2 ( ϕ j ) and u ( i, j ) ≜ cos 2 ( ϕ i ) cos 2 ( ϕ
j
The implications of these cases are further discussed in Fig.2 in App.
D. Notice that a necessary condition for the bi-region ( i, j ) effective dimension to arise is the constraint on p i p
j :
[Equation]
In the limit p i p j
→ r ⋆ ( i, j ) , d eff goes again to d/p j .
We interpret this limiting case as locally hard in the sense that censorship in region j is sufficiently important in comparison to all other regions to impose a maximal effective dimension to the problem, irrespective of the values of p i , matching
Lemma 4.5.
On the other hand, for the other limiting case (under additional mild assumptions), we find that d eff also goes to d/p j , but now for a uniformly hard reason: that is, censorship is approximately constant and equal to p j , recovering the Lemma 4.4.
Finally, in between these two extremes lies the minimum effective dimension for a given value of p i p j .
4.4 Temporal dynamics of W ( t )
The proof of Thm.
4.6 requires the characterization of the dynamics of the optimal policy of ( P ).
Importantly, we discover that the evolution of W ( t ) is described by two qualitatively different regimes as outlined next.
It turns out that our continuous approach to analyzing cumulative censored potential is an important tool to obtaining this result.
Transient Regime: There exists a decreasing sequence of censorship regions { i 1 = k, . .
, i l } of length l ∈ [ k +1] and associated time sequence { t 0 ≜ 0 , t 1 , . .
. , t l } such that whenever t j ≤ t ≤ t j + for a given index j ≤ l − 1 , the evolution of W ( t ) is given by:
[Equation]
where W i denotes the d × d diagonal matrix diag ( cos 2 ( ϕ i ) d − 1 , . .
, cos 2 ( ϕ i ) d − 1 , sin 2 ( ϕ i )) .
Interestingly, the initial misspecification of censorship is self-corrected during this transient step but at an extra cost.
This characterization of transient regime highlights an important consequence of using classical algorithms in censored environments.
Steady State Regime: Post-transient regime, the dynamics of W ( t ) enter a steady state regime, where one of the two cases necessarily arise: 5 .
• Case 1: Single region i l .
This case arises when the last element of the time sequence t l is equal to + ∞ and we have the single region evolution for all t ≥ t l − 1 :
[Equation]
The effective dimension corresponding to this dynamics is d/p i l , with the following equality for T ≥ t l − 1 :
[Equation]
• Case 2: Bi-region ( i l +1 , i l ) .
This case arises when the steady-state dynamics of W ( t ) span the two regions ( i l +1 , i l ) with i
l +1 <
i l .
For all t ≥ t l , we have the evolution: W
[Equation]
where λ ⋆ and the proportionality factor are specified in SI.
The corresponding effective dimension is given by ( D ) and the following equality holds for all T ≥ t l :
[Equation]
For further discussions on transient and steady state regimes
[Equation]
we refer to Fig.3, 4 and 5. in App.
D.
Concluding Remarks
In this work
[Equation]
we demonstrate that the complexity of bandit learning under censorship is governed by the notion of effective dimension.
To do so, we developed a novel analysis framework which enables us to precisely estimate this quantity for a broad class of multi-threshold censorship models.
An important future work would be to extend our model and approach to Bayesian settings, which will likely provide us with useful insights on the cumulative censored potential V α , as initiated by [18].
Future work also includes relaxing the Missing Completely at Random (MCAR) property in favor of time-dependent censorship models such as Markov Decision Processes (MDPs).
We believe that tools similar to those developed in our potential-based analysis can be applied in this case.
Finally, the contributions of our work may be of interest to the recent value alignment literature, where the question of learnability under humain-AI interactions is central.
We do not envision any negative societal impacts of our work other than that of bandits algorithms deployed in AI-driven platforms.
Acknowledgments and Disclosure of Funding
This research project is supported by the AFOSR FA9550-19-1-0263 “Building attack resilience into complex networks” Grant. The authors would like to thank Prem Talwai and the anonymous reviewers for providing insightful comments and suggestions
A Preliminaries
In this section
[Equation]
we first provide the instances of the UCB algorithm used in Sec.3 and Sec.4.
We also indicate in Tab.1 the notations used throughout the paper to help the reader.
A.1 UCB algorithms
• UCB-MAB: Following [26], the UCB algorithms for the MAB case with homogeneous regularization λ > 0 uses the following optimistic reward estimator at time t :
[Equation]
It is based on the use of the regularized empirical mean to estimate the reward of action a at the end of
round t :
[Equation]
The high-confidence property of this algorithm is proven in Lemma B.1.
6 Under a-priori known heteroskedasticity, the reward estimator can be expressed as:
[Equation]
• UCB for LCB
Following [1, 26], the UCB algorithms for the LCB case with homogeneous regularization λ > 0 uses the following optimistic reward estimator at time t :
[Equation]
where we introduced the random quantity:
[Equation]
It is based on the use of the regularized least square estimator to estimate the vector θ ⋆ at the end of round t :
[Equation]
The high-confidence property of this estimator is proven in Lemma C.1.
B Proof of Sec. 3 - Multi-Armed Bandits
In this section, we prove the results in Sec.3 on the MAB case.
We start by proving Lemmas 3.3, B.1,
B.2, 3.5 and Prop. 3.4.
Thanks to those results, we then tackle Thm. 3.1 and Prop. 3.2.
To conclude the section, we further study the properties of the adaptivity gain, by proving Lemma 3.6 and Prop.
Recall that effective dimension d eff is referring to (cid:80) a ∈ [ d ] 1 p a in this section.
[Table]
Stochastic feedback noise a time t .
Sub-Gaussian with pseudo-variance parameter σ 2 .
If σ 2 depends on the action selected (heteroskedasticity), we use σ 2 a instead.
[Table]
Unknown
[Table]
function,
[Table]
to scalar reward.
Parameterized by unknown latent state θ ⋆ .
[Table]
If ∆ t ( a ) is independent of t , we use ∆ a ≡ ∆ t ( a ) .
∆ max is an upper bound of ∆ t ( a ) for all actions a and time t .
[Table]
Probability that action a is censored if selected, used in Sec. 3.
Notation p ( a ) is used in Sec.4 to emphasize the dependency of p on action a .
[Table]
≜
Parameters of the multi-threshold censorship model.
Vector u defines the direction of censorship, ( ϕ j ) j ≤ k +1 define the censorship regions with fixed censorship probability and ( p j ) j ≤ k define the probability of being censored for each region j .
[Table]
≜
Random variable indicating if feedback is censored as round t .
Follows i.i.d
Bernoulli distribution of parameter p ( a t ) .
[Table]
≜
Regularization tuning parameter.
λ a is used if heterogeneous actionbased regularization.
[Table]
≜
High-probability upper bound on the sub-optimality gap, used in UCB algorithms.
[Table]
≜
Random cumulative censored potential, seen as a function of policy π and number of rounds T .
First introduced in Sec.3 and extended in
Sec.4.
ψ
[Table]
≜
Primitive of the function x
(cid:55)→ x − α , for a given α > 0 .
[Table]
Total number of time action a is realized at the end of round t by policy π .
Used in Sec.3.
[Table]
≜
Total number of time action a is played at the end of round t by policy π .
Used in Sec.3.
W
[Equation]
≜
Censored Design Matrix.
Linear generalization of ( N a ( t ))
a ∈ [ d ] .
Used in Sec.4.
W
[Table]
≜
Expected Design Matrix.
Linear generalization of ( p a τ a ( t ))
a ∈ [ d ] .
Used in Sec.4.
W
[Equation]
[Table]
≜
Continuous generalization of the expected design matrix W t .
B.1 Proof of Lemma 3.
Lemma 3.3.
Given an uniform regularization of λ > 0 , the UCB algorithm verifies:
[Equation]
where, for any α > 0 and π ∈ Π , the cumulative potential under censorship is given by:
[Equation]
Proof.
At a given round t ∈ [ T ]
[Equation]
we have under the event ¬H λ UCB introduced in Lemma B.1:
[Equation]
where the inequality comes from the definition of the UCB algorithm and the conditioning on ¬H λ UCB .
We find there the origin of the two different orders of N a ( 1 / 2 and 1 ).
Taken independently, those lead to a contribution of respectively O ( d eff log( T )) and O ( (cid:112) d eff T ) .
More precisely, we have:
[Equation]
Therefore, thanks to Lemma B.1, we deduce that:
[Equation]
Finally
[Equation]
we conclude that:
[Equation]
B.2 Statement and Proof of Lemma B.
The main step in this reduction from regret to cumulative censored potential is the study of the failure of optimism event thanks to the following result:
Lemma B.1.
For a regularization λ > 0 and δ ∈ ]0 , 1]
[Equation]
we introduce the event:
[Equation]
We then have P ( H λ UCB ) ≤ 2 dT 2 .
Proof.
Although this event is similar to the one introduced in the classical UCB proof idea
[Equation]
the subtlety comes from the randomness induced by the censorship as well as the impact of regularization
[Equation]
The main idea is adopt a worst-case agnostic approach.
First
[Equation]
let’s note that for a given t ∈ [ T ] , a ∈ [ d ] , we have:
[Equation]
Therefore
[Equation]
for a given a ∈ [ d ]
[Equation]
t ∈ [ T ]
[Equation]
by introducing the event B ( t,a ) ≜
[Equation]
|
[Equation]
|
[Equation]
log
[Equation]
, we deduce:
[Equation]
Then, we have:
[Equation]
=
[Equation]
a
[Equation] [Equation]
where we successively used union bounds over the action set and number of realizations and conditioned over number of realizations k .
We re-indexed the random sub-Gaussian variables ( ϵ t ) for last expression thanks to the i.i.d property.
Then, for a given k , using Hoeffding inequality for sub-Gaussian variables, we have:
[Equation]
where the used that fact that (cid:80) kl
=1 ϵ l is sub-Gaussian of pseudo-variance parameter kσ
Therefore, this yields:
[Equation]
Finally, we conclude that P ( H λ UCB ) ≤
2 d T 2 .
Remark 2.
We note that assuming tails distribution for the reward noise ϵ of the form:
[Equation]
for a given q > 0 , as suggested for instance in [47], would lead the use of the confidence interval:
[Equation]
Indeed, the same reasoning as above would then yield:
[Equation]
and therefore P ( H λ,q UCB ) ≤ 2 dT 2 .
For q = 1 , we recover the sub-Gaussian case, which in turns lead to the study of V 1 / 2 , as done in Lemma 3.3.
For general q > 0 , we would would then consider V q/ (1+ q ) , which lead to the upper bound O ( d q/ (1+ q ) eff
T 1 / (1+ q ) ) through the use of Prop. 3.4.
B.3 Statement and Proof of Lemma B.
Lemma B.2.
For any δ ∈ ]0 , 1] , λ > 0 and censorship model, let’s introduce the event:
H
[Equation]
CEN ( δ )
[Equation] [Equation]
a
[Equation]
[ d ] , t ∈ [ T ] , N a ( t ) < (1 − δ ) p a τ a ( t ) and τ a ( t ) ≥ T 0 ( a ) } , where T 0 ( a ) ≜ 24 log( T ) /p
a + 1 .
We then have P ( H
I CEN
[Equation]
Proof.
First
[Equation]
we apply successively two unions bounds over the action set and the number of realizations, mirroring the analysis of [24]:
[Equation]
We then use a multiplicative Chernoff inequality for Binomial Distribution to deduce:
[Equation]
The novelty of our proof is to leverage a integral comparison to deduce the improved control:
[Equation]
a ∈
[Equation]
d
[Equation] [Equation]
Picking for instance δ = 12 yields P
( H I CEN ( 12 )) ≤ 16 d eff T 3 .
B.4 Proof of Lemma 3.
Lemma 3.5.
For ψ α a primitive of x (cid:55)→ x − α where α ∈ ]0 , 1] , regularization ( λ a ) a ∈ [ d ] ∈
( R > 0 )
d and censorship vector ( p a )
a ∈ [ d ] , the solution of the optimization problem:
[Equation]
is given by τ ⋆a = 1 p a [ C − λ a ] + , where C ensures the total budget constraint (cid:80) a ∈
[ d ] τ ⋆a = T .
In particular, with λ eff ≜ 1 d eff (cid:80) a ∈
[ d ] λ a p a and λ 0 a ≜ d eff ( λ a − λ eff ) , the optimal solution is given by τ ⋆a ≜ 1 p a d eff ( T − λ 0 a ) for T ≥ max a λ 0 a and the optimal value is d eff ψ α
( Td eff + λ eff ) − (cid:80) a ∈ [ d ] 1 p a ψ α ( λ a ) .
Proof.
We first introduce the Lagrangian of the problem L ( τ 1 , . . .
, τ d , µ ) :
= (cid:80) a ∈ [ d ] 1 p a (cid:16) ψ α ( p a τ a + λ a ) − ψ α ( λ a ) (cid:17) + µ ( T − (cid:80)
a ∈ [ d ] τ a ) .
Differentiating with respect to τ a for all a ∈ [ d ] yields the equations:
[Equation]
We then write it equivalently as:
[Equation]
However, since ( τ a ) must be nonnegative, it may not always be possible to find a solution of this form.
We then verify using KKT conditions that the solution:
[Equation]
where C ensures the total budget constraint (cid:80)
a ∈ [ d ] τ ⋆a = T , is optimal.
In particular, whenever
T ≥ max a λ 0 a , we recover the solution provided in the second part the Lemma.
B.5 Proof of Prop. 3.
Proposition 3.4.
For all α > 0 , δ ∈ ]0 , 1] and given ψ α a primitive of x (cid:55)→ x − α , we have:
[Equation]
Proof.
For a given α ∈ ]0 , 1] , we condition on the event H
I CEN ( δ ) introduced in Lemma B.2 and consider the cases τ a ( t ) ≥ T 0 ( a ) and τ a ( t ) <
T 0 ( a ) .
This yields for any policy π ∈ Π :
[Equation]
We then apply the Lemma 3.5 with constant ˜ λ ≜ λ/ (1 − δ ) to deduce: max
[Equation]
Then
[Equation]
we conclude thanks to Lemma B
[Equation]
2 that:
[Equation]
In particular
[Equation]
for α = 1 and δ =
[Equation]
this involves:
[Equation]
and for α = 12 and δ = 12 , this yields:
[Equation]
B.6 Proof of Thm. 3.
Theorem 3.1.
Under censorship
[Equation]
the UCB algorithm with regularization λ has an instanceindependent expected regret of:
[Equation]
Proof.
We first apply Lemma 3.3 to deduce:
[Equation]
We then apply proposition 3.4, with δ = 1 / 2 in order to deduce:
[Equation]
By taking λ = o (log( T )) and considering only the leading order
[Equation]
we conclude that:
[Equation]
Note that our proof easily allows to get high-probability bounds on regret instead of bounds on its expected value.
Remark 3.
We now extend Thm.
3.1 to heteroskedastic MAB.
In this model, the pseudo-variance of the sub-Gaussian noisy reward is arm-dependent and denoted σ a .
Moreover, the value of σ a is known to the designer of the algorithm, that is, it can be used as a parameter for the UCB algorithm.
We first apply a slightly modified version of Lemma 3.3 to deduce:
[Equation]
where for α > 0 and π ∈ Π , we introduced the variance-based cumulative potential:
[Equation]
Thus, heteroskedasticity induces the mapping ˘ p a ≡ p a /σ 1 /αa and ˘ λ a ≡ λ/σ 1 /αa .
Following the proof of Prop. 3.4, we deduce for any α > 0 and time allocation ( τ a ( T ))
a ∈ [ d ] :
[Equation]
In order to apply Lemma 3.5, we introduce the notation:
[Equation]
1 − δ d eff ˘ d eff and
[Equation]
a
[Equation]
eff
[Equation] [Equation]
eff ) .
and we deduce that whenever T ≥ max a ˘ λ 0 a , we have:
[Equation]
In particular, by considering the case α = 1 / 2 and only the leading order, we deduce that:
[Equation]
, where as affirmed ˘ d eff = (cid:80) a ∈ [ d ] σ 2 a p a .
B.7 Proof of Prop.
Proposition 3.2.
For a fixed action set A t ≡ [ d ] and for a-priori known action gap ∆ a ≜ max ˜ a θ ⋆ ˜ a − θ ⋆a , the UCB algorithm with regularization λ has the instance-dependent expected regret:
[Equation]
Proof.
As in the proof of Lemma 3.4, for a given round t ∈ [ T ] , we have under the event ¬H λ UCB
[Equation]
It is as an inequality of the second degree and thus for any t ∈
[ T ] , a ∈ [ d ] :
[Equation]
where x 1 = 2 λ ∥ θ ⋆ ∥ ∞ and x 2 = 2 (cid:112) 6 σ 2 log( T ) .
Solving it yields:
[Equation]
or equivalently:
[Equation]
where we used the notation Θ( T ) to simplify the presentation.
Therefore, under ¬H I CEN ( 12 )
[Equation]
we have:
[Equation]
This yields a conditional regret of:
[Equation]
where a ⋆ ≜ argmax ˜ a ∈A θ ⋆ ˜ a and an expected regret of:
[Equation]
In particular
[Equation]
for the regularization λ
= o (log( T ))
[Equation]
we have the asymptotic:
[Equation]
And thus, we conclude that:
[Equation]
Again, note that our proof easily allows to get high-probability bounds on regret instead of bounds on its expected value.
Remark 4.
As in the instance-independent case, previous reasoning immediately extends to a-priori known heteroskedasticity and yields the upper bound:
[Equation]
Next, we provide additional insights to the main result of this section.
In particular, we seek to gain intuition about how the policies that are adaptive to the realization of censorship process would perform in expectation against a class of non-adaptive (i.e. offline ) policies.
In order to precisely derive asymptotic behavior of such policies, we introduce and study a continuous counterpart of the discrete original policy maximization problem
max π ∈ Π E
[ V α ( T, π )] .
Lemma 3.6 provides the basis for continuous approach in the case of offline policies by leveraging concentration inequalities for inverse Binomial distribution.
We then extend this approach in the proof of Prop. 3.7.
This extension enables us to provide an exact expression for the asymptotic gain of a policy class that monitors the censorship at a single point in time, as well as estimate the gain from fully adaptive policies.
B.8 Proof of Lemma 3.
Lemma 3.6.
For α ∈ ]0 , 1] and λ > 0 , we have max
[Equation]
Proof.
Given the offline nature of the policy class, we have:
[Equation]
where we have re-indexed ( N a ( t )) by actions, where ( τ a ) a ∈
[ d ] is a time allocation such that (cid:80) a ∈
[ d ] τ a = T and where for a given action a , ( X an ) n ≤ τ a are dependent random variables verifying X n +1 =
X an + B ( p a ) and X an ∼ B ( n, p a ) .
To lower bound this quantity, we fix a time allocation ( τ a ) a ∈ [ d ] and use the fact that x (cid:55)→ x − α is convex with Jensen’s inequality to deduce:
[Equation]
We then leverage the fact that ( px + λ ) − α ≥ (cid:82) x x − 1 ( pu + λ ) − α ∂u to deduce:
[Equation]
and therefore, for any time allocation ( τ a ) a ∈ [ d ] , we have:
[Equation]
Although the maximum over time allocation is given by Lemma 3.5, we simply use the allocation ( T p a d eff ) a ∈ [ d ] to deduce:
[Equation]
By making the distinction between α = 1 and α < 1 to obtain the explicit expression of ψ α , we then show that the LHS is equivalent to d eff ψ α ( Td eff + λ ) .
The proof of the upper bound is more involved.
In proving it, let’s first assume that:
Claim B.3.
For all a ∈ [ d ] , n ≥ 1 , there exists a constant C a 2 such that:
[Equation]
Given this result
[Equation]
we deduce:
[Equation]
Therefore
[Equation]
we have:
[Equation]
We first consider the second maximization problem and note that:
[Equation]
where we used an integral comparison and Lemma 3.5 to deduce the O scaling and the fact that α ∈ ]0 , 1] the deduce the o scaling.
Similarly, we know that the first maximization problem scales as d eff ψ α ( Td eff + λ ) through another integral comparison and use of Lemma 3.5.
Given this, we conclude that the upper bound is equivalent to d eff ψ α ( Td eff + λ ) .
Thanks to those two results, we finally affirm that:
[Equation]
The last step needed is to prove Claim.
B.3.
In doing so, we extend Lemma 5 .
3 of [5] to more general inverse power function (i.e. α ̸ = 1 ) with regularization λ > 0 .
We first introduce a tuning parameter u ≥ 0 and write:
[Equation]
Using a Berstein inequality for Binomiale variable, we have for all θ > 0 and n ∈ N :
[Equation]
Thus, for all 0 < θ ≤ 3( √ 5 −√ 3) 2, by setting u ≡ (1 − √ 2 θ − θ 3 ) ≥ 0 , we obtain that:
[Equation]
By taking θ =
A log( np a + λ ) np a for another tunable parameter A and for n large enough to ensure
A log( np a + λ ) np a ≤ 3( √ 5 −√ 3) 2, this yields:
[Equation]
an ]
[Equation]
For n sufficiently large to ensure
[Equation]
≤ 1 / 2 and given that α ∈ ]0 , 1] , we then have:
[Equation]
To conclude, we take take A ≡ ( α + 1) and this ensure that for n sufficiently large, we obtain:
[Equation]
[ X an ] + λ ) α E
[Equation]
The leading order of this quantity is (cid:113) log
[Equation]
n ) n = o ( n − 1 / 4 ) and therefore, we conclude that there exists a constant C a 2 , depending on λ, α and p a such that for all n ≥ 1 :
[Equation]
where C a 2 is artificially increased to remove the two lower bounds conditions on n .
B.9 Proof of Prop. 3.
Proposition 3.7.
For α ∈ ]0 , 1] , by denoting γ α ( p ) ≜
[Equation]
, we have:
[Equation]
Moreover, if for a given β ∈ ]0 , 1[ , we introduce Π single ( βT ) the policy class whose censorship information set has a single updating at time ⌊ βT ⌋ , we have:
[Equation]
Thus, we find that the power of a single monitoring is sufficient to ensure almost the same gain as adaptivity i.e. constant monitoring.
The linear dependency in T 0 (due to the linear increase of variance in Binomial models) is also surprising.
In non-asymptotic regime, it is still true but for β verifying 0 < β − ≤ β ≤ β + < 1 for given ( β − , β + ) .
We also observe a more general concave property of the single monitoring gain seen as a function of T 0 , with limits equals to 0 on the borders on the interval.
We conjecture that this concavity is likely to turn in a submodular dependency for several monitoring shots.
Proof.
Single Monitoring: We first prove a slightly extended version of ( ⋆⋆ ) by considering a monitoring at time T 0 and we recover the results of Prop. 3.7 by setting T 0 ≡ βT , for a given β ∈ ]0 , 1[ .
For the first step of the proof, we consider the continuous approximation of
[Equation]
] given by the optimization problem over continuous variables:
[Equation]
In SM , the single monitoring max player initially commits to an allocation of the T 0 first rounds through the policy ( τ a ( T 0 ))
a ∈ [ d ] , with resulting gain expressed as the second term of the maximization problem.
The player then observes the realization N a ( T 0 ) ∼ B ( τ a ( T 0 ) , p a ) and allocates the rest of the T − T 0 budget through the allocation ( τ a ( T )) a ∈ [ d ] , with resulting gain expressed as the first term of the maximization problem. .
Therefore, the single monitoring gain assesses the value of observing the deviation of N a ( T 0 ) from its expectation τ a ( T 0 ) p a .
In an analogous way, we then introduce the continuous approximation of max π
[Equation]
Π off E
[ V α ( T, π )] given by:
[Equation]
In OFF , ( N a ( T 0 )) a ∈ [ d ] is not observed and thus can not be leveraged by the offline player to adapt the second part of the allocation.
On what follows, we use E [ N ] and V [ N ] to denote respectively the mean and variance of N , the empirical discrete distribution over ( N a ( T 0 )) a ∈ [ d ] with associated weights (1 /p a d eff ) a ∈ [ d ] .
Given a realization of ( N a ( T 0 )) a ∈ [ d ] , we use Lemma 3.5 to deduce the optimal choice of ( τ a ( T )) a ∈ [ d ] in SM and resulting expected conditional gain:
[Equation]
where the formula is valid under the assumption ∀
a ∈
[ d ] , T − T 0 ≥ d eff ( N a ( T 0 ) − E [ N ]) .
Such assumption encompass the fact that the remaining budget T − T 0 should be sufficient to correct the deviation observed.
Logically, we know that on expectation E [ N a ( T 0 ) − E
[ N ]] = 0 , that is no systematic deviation is expected.
For for all realization of randomness, the following deterministic crude upper bound hold:
[Equation]
this in turn imposes:
[Equation]
For instance, in the uniform censorship model, this yields condition T 0 T ≤ dp dp + d − 1 .
Nevertheless, this is overly conservative and we can get considerably stronger results by considering high-probability concentration results on N a ( T 0 ) .
Indeed, thanks to Chernoff Bounds for Binomial distribution, we have for δ ≡ T − 1 / 4, that with probability at least 1 − 2 d exp {− δ 2 T 0 / 3 d eff } , for all a , (1 − δ ) T 0 /d eff ≤ N a ( T 0 ) ≤ (1 + δ ) T 0 /d eff .
In particular, this yields (1 − δ ) T 0 /d eff ≤ E [ N ] ≤ (1 + δ ) T 0 /d eff .
Under this event, we have d eff ( N a ( T 0 ) − E [ N ]) ≤ 2 δT 0 , which imposes T ≥ (1 + 2 δ ) T
= T 0 + 2 T 3 / 4 0 .
In particular, for T 0 ≡ βT , where β ∈ ]0 , 1[ , such condition will always be verified for T large enough.
On the other hand, still using Lemma 3.5, we write the conditional expected gain of the offline policy on this same realization of ( N a ( T )) a ∈ [ d ] for OFF as:
[Equation]
where ψ α ( N a ( T 0 )) is artificially introduced.
The difference between the two comes from the possibility for the monitoring policy to homogenize the realized N a ( T 0 ) into a uniform E [ N ] .
The random difference G single ( T 0 ) , seen as a function of the realization of ( N a ( T 0 )) is then equal to:
[Equation]
which is exactly the Jensen’s gap of the concave function
¯ ψ α : x (cid:55)→ ψ
[Equation] [Equation]
+ x ) .
The main insight is that this gap is then asymptotically equivalent to:
[Equation]
where the RHS is positive, given that ¯ ψ (2) α
( E [ N ]) is negative.
To show this, we use the original proof of Jensen’s inequality and introduce the interval I ≜ [min a N a ( T 0 ) , max a N a ( T 0 )] to leverage the mean value theorem.
This then yields:
[Equation]
Whenever T 0 is a constant independent of T
[Equation]
for T → + ∞ by explicitly writing the definition of the upper and lower bounds
[Equation]
we have almost surely:
[Equation]
Difficulties arises when T 0 is a function of T , as in the statement of the result where T 0 ≡ βT .
By considering the same concentration event as the one introduced above
[Equation]
we have:
[Equation]
and similarly:
[Equation]
Thus, thanks to the exponential concentration
[Equation]
we conclude that:
[Equation]
Next
[Equation]
we affirm that:
[Equation]
where a ) leverages the previous bounds and where we use for b ) similar concentration results on inverse of Binomial as done for the proof of Lemma 3.
.. We then use the fact that E [ E [ N ]] =
T 0 /d eff to conclude:
[Equation]
We consider this result to be one of the main insight for adaptivity, as it involves that at first order, the gain grows linearly in the expected value of the empirical variance of the arm allocation process.
In opposition to the single monitoring policy, the adaptive policy continuously exploits such variance.
Yet, in doing so, it creates a second order induced variance but we then show that this phenomena is negligible at first order.
To reach a result with explicit dependency on the censorship probability ( p a ) a ∈ [ d ] , we note that:
[Equation]
and therefore:
[Equation]
In particular, for T 0 = βT , this yields E (cid:104) G
[Equation]
The second step closely mirrors the proof of Lemma 3.6 and consists in justifying the use of the continuous approximation for the two optimization problems ( OFF ) and ( SM ).
As in Lemma 3.6, we show that the difference between the continuous and discrete optimization results at most in a second order gain of o ( 1 T α ) , even when maximized as a decoupled quantity.
By combining those two results, we finally deduce as announced:
[Equation]
Complete Adaptivity
We next tackle the proof of ( ⋆ ), where the main idea is to show that a formula analogous to ( V ) holds for the variance of a suited random process.
First, using the same proof technique as in Sec. 4 of [19] thanks to the decaying property of the reward in function of the number of realization, we show that the optimal adaptive policy is the greedy policy, that is the policy that picks at time t the action:
[Equation]
with arbitrary but consistent tie-breaking.
In particular, this ensures that for all actions a, b and time t , we have | N a ( t ) − N b ( t ) | ≤ 1 .
We then introduce the offline and adaptive allocations:
[Equation]
where 1 p a + ξ ai is the total random number of allocation it takes for action a to be realized in the i th selection, ξ ai being equal the centered deviation with respect to the expected value 1 p a .
Of key importance in our proof is S a ( N a ( T )) , the cumulative deviation defined as (cid:80) N a ( T ) i = ξ ai .
Note that it is well approximated in large T regime as a random sum of N a ( T ) i.i.d.
geometric centered variable of parameter p a .
Given this and the total budget constraint, we have the simple relation
[Equation]
b [
[Equation]
N
[Equation]
T )
[Equation]
− S b ( N b ( T )
[Equation]
A relevant quantity to introduce is the random allocation difference ∆ τ a,b between actions a and b defined by:
[Equation]
Using this notation, we simply have τ ona
=
τ offa + (cid:80) b ∆ τ a,b .
We then introduce the random sets
I + ≜ { a : τ ona ≥ τ offa } = { a : (cid:80) b ∆ τ a,b ≥ 0 } and I − ≜ { a : τ ona < τ offa } =
{ a : (cid:80) b ∆ τ a,b <
On the one hand, I + represents the set of actions that are more sampled by the adaptive policy than by the offline policy i.e. that leads to a gain thanks to the greedy property.
One the other hand,
I − is the set of actions under-selected by the adaptive policy, leading to a loss although inferior in absolute value to the resulting gain of I + .
As for the proof of the single monitoring case, we condition on the realization ( N a ( T )) and use a continuous approximation given this conditioning to study the difference of gain.
Thus, we have the action gain for a ∈ I + ,:
[Equation]
On the other hand, for a ∈ I − , we have the action loss still under the continuous approximation:
[Equation]
By introducing G adapt ≜ (cid:80) a ∈
I
+ g a − (cid:80) a ∈
I − l a , the adaptive equivalent of G single and combining previous two results, we deduce:
[Equation]
We then leverage the Taylor expansion ψ (1) α ( N a ( T ))
= ψ (1) α
( ¯ N ( T ))
+ ψ (2)
α ( ¯ N ( T ))
( ¯ N ( T ) − N a ( T )) , where ¯ N ( T ) ≜ (cid:80) a ∈ [ d ] N a ( t )
/d .
We know that the second term is asymptotically negligible given that α > 0 and that the difference between ¯ N ( T ) and N a ( T ) is constant with exponential probability, thanks to the greedy policy property.
We combine this result with the fact that by definition (cid:80)
[Equation]
∈ I
+
[Equation]
−
[Equation]
b ∆ τ a,b = 0 to deduce that at first order:
[Equation]
We see formula ( L ) as the adaptive analogous of ( V ).
Indeed
[Equation]
it involves the product of the second derivative 12 ψ (2)
α
( ¯ N ( T )) , evaluated on a quantity concentrating at T/d eff with a variance term associated to the adaptive action allocation process.
We remark that for any a ∈ [ d ] :
[Equation]
and therefore, by summing:
[Equation]
To obtain the leading order of V [ S a ( N a ( T ))]
[Equation]
we use Wald’s second equation and the fact that
S a ( N a ( T )) is approximated by a sum of geometric random variable of parameter p a , modulo a asymptotically negligible summing constraint due to the fixed total budget T .
This yields V [ S
[Equation]
N
[Equation]
( T ))]
[Equation]
E [ N a ( T )] ∼
[Equation]
p
[Equation]
p
[Equation]
E [ ¯ N ( T )] , where the last results leverages again the fact that the difference between the two quantities is constant with exponential probability.
We conclude with further algebraic calculation that:
[Equation]
By justifying again that the continuous gain approximation leads to terms of order o ( 1 T α ) , as done in the proof of Lemma 3.6, we conclude that:
[Equation]
C Proof of Sec. 4 - Contextual Bandits
In this section, we prove Thm.
4.1 of Sec.4, extending the results of MAB to LCB.
To do so, we prove Lemmas 4.2, C.1 and Prop. 4.3.
Note that the proof of Thm.
4.6 is differed to next section.
We conclude the section by discussing the extension of our analysis to Generalized Linear Contextual
Bandits.
C.1 Proof of Lemma 4.
Lemma 4.2.
For all δ ∈ ]0 , 1] , there exists a constant ˜ β δ ( T ) =
Θ( (cid:112) d log( T )) such that
[Equation]
where, for α > 0 and π ∈ Π , the linear extension of the cumulative censored potential is given by:
[Equation]
Proof.
We have under the event ¬H II UCB ( δ ) introduced in Lemma C.1 and thanks to Holder inequality:
[Equation]
Therefore, the conditional regret is upper-bounded by:
[Equation]
where we introduced ˜ V 12 ( T, π ) ≜ (cid:80)
Tt =1 ∥ a t ∥ W C ( t − 1) − 1 .
Cauchy Schwartz inequality then allows to make the junction ˜
V
( T, π ) ≤ √ T (cid:112) V 1 ( T, π ) .
We then introduce ˜ β
δ ( T ) a deterministic upper bound on β δ ( T ) :
[Equation]
Using the concavity of square root and Jensen’s inequality, we have E [ (cid:112) V 1 ( T, π )] ≤ (cid:112) E [ V 1 ( T, π )] .
Finally, thanks to Lemma C.1, we conclude that:
[Equation]
C.2 Statement and Proof of Lemma C.
Analogous to Lemma B.1 for the MAB case, one key step in the proof is introduction of the failure of optimism event.
Nevertheless, note the difference with the choice of norm.
Lemma C.1.
For any δ ∈ ]0 , 1] , uniform regularization λ > 0 and censored action generating process ( W Ct ) t ≤ T , let’s introduce the event:
[Equation]
We then have P ( H II UCB ( δ ))
≤ δ .
Proof.
The proof closely mirrors the self-normalized bound for vector-valued martingales of Thm.from [1].
The main subtlety is to apply the results to the censored measurable vectors ( x a t a t ) instead of classically ( a t ) .
This yields that with probability 1 − δ , for all t ≥ 0 :
∥
[Equation]
Thus, still on this event, for any t ≥ 0 and action a ∈ R d , we have by definition of ˆ θ λt (Sec.A.1):
[Equation]
and therefore, thanks to Cauchy-Schwartz inequality:
[Equation]
Using previous result, for all a ∈ B d , t ≥ 0 , with probability 1 − δ , we have:
[Equation]
To conclude, we classically plug-in the value a = W Ct
(ˆ θ λt − θ ⋆ ) and divide both sides by ∥ ˆ θ λt − θ ⋆ ∥ W Ct to get that for all t ≥ 0 , with probability 1 − δ , we have:
[Equation]
and therefore, by definition P ( H II UCB ( δ )) ≤ δ .
C.3 Proof of Prop. 4.
Proposition 4.3.
For any δ ∈ ]0 , 1] , λ > 0 , α > 0 and policy π ∈ Π , we have:
[Equation]
where C ( δ ) ≜ 8( λ + 1) max(log( d/δ )) /λ, 1) /λ .
Proof.
First, we use Lemma C.2 to deduce that under H II CEN ( δ ) :
[Equation]
For all t ≥ 1 , we then use the fact that W t ⪯ (1 + 1 λ ) W t − 1 to deduce Tr( W − α t − 1 a t a ⊤ t ) ≤ (1 +λ ) α
Tr( W − α t a t a ⊤ t ) .
The last and most important step is the integral comparison:
[Equation]
In the previous result, the continuous extension ( a ( t ) , W ( t )) t ≤ T of ( a t , W t ) t ∈ [ T ] for a given policy π is defined for any time t ≥ 1 as: a ( t ) ≜ a ⌊ t ⌋ and W ( t ) ≜ (cid:90) t
[Equation]
This yields the result:
[Equation]
Finally
[Equation]
we conclude thanks to Lemma C.2 that:
[Equation]
Remark 5.
The main tour de force of the continuous approximation we employ is to relax the maximization problem by considering the class of continuous deterministic integrable policies, which is considerably more tractable from an analysis perspective.
On the one hand, it allows to get closed-form solution for the maximization problem whereas the discrete approach can only deal with approximations and upper bounds.
On the other hand, it clearly reveals the underlying matrix function the discrete approach is approximating and henceforth allows to leverage powerful integration results.
We leverage again this idea in the context of Sec.4 to tackle impact of censorship.
To illustrate the abovementioned points
[Equation]
we remark that for the simpler case of classical uncensored environment, we obtain for α > 0 , α ̸ = 1 :
T (cid:88)
[Equation]
For α <
[Equation]
we then have thanks to Lemma 3.5 the worst case bound Tr( W 1 − α T
[Equation]
and henceforth:
[Equation]
On the other hand
[Equation]
for α >
[Equation]
we deduce:
[Equation]
Finally, for α = 1 , we use the formula Tr(log
( A )) =
log(det A ) to deduce:
T (cid:88)
[Equation]
where we used again Lemma 3.5 to obtain the last (worst-case) upper bound.
In doing so, we recover and extend the recent results of [8] in a more natural way.
7 Note that the rank 1 assumption is not needed in the continuous relaxation and therefore our results still hold whenever a ( t ) a ( t ) T is replaced by any positive semi-definite matrix H ( t ) .
C.4 Statement of Lemma C.
In order to prove previous property on V α , a key step mirroring the MAB case is the use of high confidence lower bound on the censorship process, proven using anytime matrix martingale inequalities:
Lemma C.2.
For any δ ∈ ]0 , 1] , λ > 0 and policy π , let’s introduce the event:
[Equation]
where c δ ≜ 8 max( log( d/δ )) λ , 1) .
We then have P ( H II CEN ( δ ))
≤ δ .
Note that picking as in the MAB case δ ∼ d/T 2 would lead to a constant c δ = Θ(log( T )) , that is a worsening confidence interval, except if we manage to control the initialization.
One interesting technical question for future work would be to allow an initialization condition as in Lemma B. ensuring W ( T 0 ) counterbalance log( d/δ ) .
C.5 Proof of Thm. 4.
Theorem 4.1.
For a given multi-threshold censorship model MT , there exits d eff such that the UCB algorithm with regularization λ has an instance-independent expected regret of:
[Equation]
Proof.
Analogous to the MAB case
[Equation]
we use Lemma 4.2 to deduce:
[Equation]
where we have:
[Equation]
We then pick δ = dT 2 , which yields:
[Equation]
We then apply Lemma 4.3 with α = 1 and δ = dT 2 to deduce:
[Equation]
By applying Thm. 4.
[Equation]
we deduce the two possibilities:
• Case 1: Single region i l .
The
effective dimension corresponding to this dynamics is d/p i l , with the following equality for T ≥ t l − 1 :
[Equation]
where we have for T ≥ t l − 1 W ( T )
= p i l ( T − t l − 1 ) W i l + W ( t l − 1 ) .
Explicit formula of ( t n , W ( t n )) are given for all n ≤ l in Cor.
D.1.1.
We then note that:
[Equation]
For T ≥ t l − 1 , we then write this in the form:
[Equation]
where f ( T ) =
o (log( T )) .
• Case 2: Bi-region ( i l +1 , i l ) .
Similarly, for T ≥ t l , we have:
[Equation]
where f ( T ) =
o (log( T )) .
Therefore, for given d eff , f and t 0 , we know that the following holds for all T ≥ t 0 :
[Equation]
Putting the pieces together yields for T ≥ t 0 :
[Equation]
By imposing regularization of order λ
= o (log( T ))
only considering the leading order
[Equation]
this yields:
[Equation]
Finally, by working in large d regime
[Equation]
we finally conclude that:
[Equation]
Again, we note that our proof easily allows to get high-probability bounds on regret instead of bounds
C.6 Extension to Generalized Linear Contextual Bandits
On what follows, we provide a sketch of the extension our results to Generalized Linear Contextual
Bandits (GLCB) but differ the complete treatment to future work.
In this model, the reward of a given action a is assumed to be of the form:
[Equation]
for a given function µ strictly increasing, continuously differentiable and real-valued.
Notable instances of such a problem include the Logistic bandit and the Poisson bandit.
Of particular importance in the dimensionality study of the problem are the constants:
[Equation]
An important requirement of GLCB is the assumption κ > 0 needed to ensure identifiability of θ ⋆ and asymptotic normality.
Given this, the suited definition of pseudo-regret considered is:
[Equation]
Note that this regret can be easily mapped to the one studied above thanks to the fact that L µ is a Lipschitz constant for µ : for all a, ˜ a ∈ ∪A t , | µ ( ⟨ a, θ ⋆ ⟩ ) − µ (
[Equation]
a
[Equation]
⋆ ⟩
[Equation]
| ≤ L
[Equation]
|⟨ a
[Equation]
θ ⋆ ⟩ − ⟨ ˜ a
[Equation]
θ ⋆ ⟩| .
Mirroring the proof of [28], we use a Maximum Likelihood Estimator (MLE) instead of a LeastSquare Estimator for θ ⋆ .
More precisely, we define ˆ θ MLE t as the solution of the equation: t (cid:88)
[Equation]
A minor difference between the approach of [28] and what precedes is the use of a period of initial random sampling (e.g. exploration ) instead of the regularization to ensure inversibility of the design matrix W Ct .
More precisely, the initial sampling ensures that with high-probability, λ min ( W Ct ) > 0 in a finite time T init .
To be possible, this requires the assumption that there exists σ 20 > 0 such that for all t ≥ 1 , we have λ min (cid:0) E a ∈A t (cid:2) aa ⊤ (cid:3)(cid:1) ≥ σ 20 , where the expectation E is associated with an uniform sampling of actions.
Under the same assumption, the impact of censorship on this initialization step is at worst an increase of the sampling time to ˜ T init ≜ T init /p
min , which is still constant.
Following
Lemma 9 of [28], we then consider the censored high-probability confidence set for any δ ∈ [ 1 T , 1] :
[Equation]
and a direct extension of their results allows us to conclude P ( H III UCB ( δ ))
≤ δ .
Note that the constant κ appears when upper bounding in the Loewner order the Fischer Information Matrix of the MLE by the matrix W Ct .
Post-initialization, the conditional regret is then upper bounded by:
[Equation]
Combining these elements and taking δ = 1 T , we conclude that:
[Equation]
where we used Thm.
4.6 to control E
[ V 1 ( π UCB , T )] as done in the proof of Th,4.1.
D Effective Dimension and Temporal Dynamics for Multi-Threshold Models
In this section, we prove Thm. 4.6 and discuss its implications.
In doing so, we introduce and prove Lemmas D.1, D.2, D.3 and Cor.
D.1.1.
We conclude the section by illustrating results for the single-threshold model, through Cor.
D.3.1.
D.1 Supplementary Notations
Without loss of generality (i.e. up to an orthogonal transformation), we can consider that u ≡ e d , the d th basis vector.
Given this, for two regions i < j , we introduce the notations:
l ( i, j ) ≜ sin 2 ( ρ i ) sin 2 ( ρ j ) and u ( i, j ) ≜ cos 2 ( ρ i ) cos 2 ( ρ j ) r ⋆ ( i, j ) ≜ ( d − 1)
u
( i, j ) + l ( i, j ) d and r † ( i, j ) ≜
r ⋆ ( j, i ) =
dl ( i, j ) u ( i, j ) u
( i, j )
+ ( d − 1) l ( i, j )
W i ≜ (cid:32)
cos 2 ( ρ i ) d − 1 I d −(0) (0) sin 2 ( ρ i ) (cid:33) and W ( i, j )
= (cid:32) cos 2 ( ρ j )( u ( i, j )
− p i p
j )
I d −(0) (0) sin 2 ( ρ j )( p i p j − l ( i, j ))
(cid:33) .
Whenever i and j are clear from context, we use in u (resp.
l ) as abbreviation for u ( i, j ) (resp.
l ( i, j ) ).
D.2 Proof of Thm. 4.
Theorem 4.6.
For a multi-threshold censorship model MT , we have:
[Equation]
where d eff is the effective dimension.
Furthermore, d eff is characterized by two cases:
• Case 1: Single region j effective dimension d eff = dp j . • Case 2: Bi-region
( i, j ) effective dimension, with i < j :
[Equation]
( d − 1) 1 − l ( i, j ) p i p
j − l ( i, j ) + u ( i, j ) −u ( i, j ) − p
i p j (cid:35) <
d p j .
( D ) where l ( i, j ) ≜ sin 2 ( ϕ i ) sin 2 ( ϕ j ) and u ( i, j ) ≜ cos 2 ( ϕ i ) cos 2 ( ϕ j ) .
Initalization: Set current region S ← k while a region is reachable from region S do /* Lemma D.1,Fig.3 */ play region S optimal policy until first reachable region i ⋆ is reached; if region i ⋆ is dual reachable from region S then /
* Lemma D.2, Fig.4 */
Bi-region ( i ⋆ , S ) effective dimension (case 2); /* Lemma D.3 */ play Bi-region ( i ⋆ , S ) optimal policy;
End ; else
Update current region S ← i ⋆ ; /* Lemma D.2, Fig.5 */ end end
Single region S effective dimension (case 1); /* Lemma D.1 */ play region S optimal policy;
Proof.
We first summarize the dynamics of the optimal policy of ( P ) through an algorithmic description in Alg. 2.
Two key notions of our analysis are the concepts of reachability and dual reachability of a region i from a base region j , as described in Lemmas D.1 and D.2 and schematized in Fig.3, 4 and 5.
Formally, they can be written as two independent necessary constraints on the ratio p i /p
j : p i /p j < r ⋆ ( i, j ) for reachability and p i /p j > r † ( i, j ) for dual reachability.
The categorization result provided in the statement of Thm.
4.6 follows from the two possible termination condition of the algorithm.
We use as algorithmic invariant to ensure the termination the fact that the set of reachable regions is strictly decreasing for inclusion and finite.
Hence, the while loop will terminate either because a dual reachable region is reached or because no more regions are reachable.
In order to not overload the presentation, time aspect is not present in the algorithmic description but is extensively covered in Lemmas D.1, D.2, D.3 and Cor.
D.1.1, as well as in what follows.
One of our main finding is that the dynamics of the optimal policy of ( P ) are described through W ( t ) by two qualitatively different regimes.
We emphasize that our continuous approach to analyzing cumulative censored potential is key to obtaining these results.
Transient Regime:
From the while loop in the algorithmic description results a so-called transient regime.
More precisely, there exists a decreasing sequence of censorship regions { i 1 = k, . .
, i l } of length l ∈ [ k +1] and associated time sequence { t 0 ≜ 0 , t 1 , . .
. , t l } such that whenever t j ≤ t ≤ t j + for a given index j ≤ l − 1 , the evolution of W ( t ) is given by:
[Equation]
This result follows from a simple induction with repeated use of Lemma D.1, giving the exact sequence of censorship regions, Moreover, closed-formed formula for the time sequence is provided in Cor.
D.1.1.
We interpret this transient step as an adversarial self-correction of the initial misspecification of censorship at an extra cost.
This characterization of transient regime highlights an important consequence of using classical algorithms in censored environments.
Steady State Regime: Post-transient regime, the dynamics of W ( t ) enter a steady state regime, where one of the two cases necessarily arise:
• Case 1: Single region i l .
This case arises when the while loop ends because no other regions are reachable.
It is equivalent to have the last element
of the time sequence t l is equal to + ∞ and we have the single region evolution for all t ≥ t l − 1 thanks to Lemma D.1:
[Equation]
The effective dimension corresponding to this dynamic is d/p i l , with the following equality for T ≥ t l − 1 :
[Equation]
T
[Equation]
where the closed-form formula for W ( t n ) is provided in Cor.
D.1.1 for all n ≤ l − 1 .
• Case 2: Bi-region ( i l +1 , i l ) .
This case arises when the while loop ends
because
dual reachable region i l +1 is reached from region i l , with i l +1 < i l .
For all t ≥ t l , Lemma D.
yields the evolution:
[Equation]
where λ ⋆ and the proportionality factor are specified in the proof.
The corresponding effective dimension is given by ( D ) and the following equality holds for all T ≥ t l thanks to
Lemma D.3:
[Equation]
where the closed-form formula for W ( t n ) is provided in Cor.
D.1.1 for all n ≤ l .
Remark 6.
Fig.3 and 5 provide further insights on formula ( D ) for d eff .
Throughout the proof and as illustrated on Fig.3, we see that for ( D ) to arise, p i p j must belong to a certain interval
J ≜ ] max(1 , r † ( i, j )) , r ⋆ ( i, j )[ .
As r ⋆ ( i, j ) < u ( i, j ) and r † ( i, j ) >
l ( i, j ) , we see ( D ) as a weighted average of the relative distance of p i p j to u ( i, j ) and l ( i, j ) .
Fig.2 provides a sketch of the variations of d eff as p i
p
j evolves in this interval.
D.3 Statement and Proof of Lemma D.
Lemma D.1 (Reachability Analysis) .
Let’s assume we start at a given time t 1 in transient censored region j , with a matrix
W ( t 1 ) =
(cid:18) λ a I d −(0) (0) λ b (cid:19) , where λ a ≥ λ b .
We introduce I j ≜ { i ; i < j and p i p
j < r ⋆ ( i, j ) } , the set of reachable regions from region j and affirm that we have the two possible cases:
• If I j = ∅ ,
i
e. no region is reachable from region j , we switch to a steady state regime with single region j effective dimension d eff = d/p j
Otherwise, next region added to the transient sequence is
i ⋆ ≜ argmin i ∈ I
j µ ⋆ ( i, j, λ a , λ b ) , at time t 2 ≜ t 1 + 1 p j µ
⋆ ( i ⋆ , j, λ a , λ b ) and we have:
[Equation]
Proof.
First, we note that the initial starting point is recovered for t 1 = 0 , base censored state k and λ a = λ b
= λ
but this Lemma allows to go beyond the first step in the study of the behavior of the system.
We know the temporal evolution for normalized budget µ ≜ p 1 ( t − t 1 ) is of the form:
[Equation]
We recall that the set of actions associated with region j is { a ∈ B d , sin( ϕ j ) ≤ ⟨ a, e d ⟩ < sin( ϕ j +1 ) } .
Therefore, the use of Kiefer-Wolfowitz theorem [26] combined with the fact λ a ≥ λ b yields that the optimal policy while evolving in region j only plays unit action vector v j ≡ (cos( ϕ j ) /
( d − 1) 1 / 2 , . . .
, cos( ϕ j ) / ( d − 1) 1 / 2 , sin( ϕ j )) .
By noting that v j v ⊤ j = W j , we obtain the formula announced.
Reachability of a given state i < j from state j after time t 1 is then defined as:
[Equation]
We interpret this as a classical a first-order optimally condition for convex maximization problems, where the matrix W j is weighted by the censorship probability representing the speed of increase in region j .
We then rewrite this condition as: ∃
[Equation]
p i p j where f ( µ ) ≜
µ sin 2 ( ϕ j ) + λ b µ cos 2 ( ϕ j ) d −+ λ a − 1 .
We know that f is increasing in µ and the LHS of the equation above is decreasing in f ( µ ) as i < j .
Hence, the reachability condition than be stated by looking at the limit of f in + ∞ .
By using the fact that lim µ → + ∞ f ( µ ) =
d sin 2 ( ϕ j ) − 1 cos 2 ( ϕ j ) , we deduce that the reachability condition is equivalent to looking at the position of p i p
j with respect to:
[Equation]
On the one hand, if p i p j ≥ r ⋆ ( i, j ) , the state in never reachable in a finite time.
On the other hand, whenever p i p j < r ⋆ ( i, j ) , the state is reachable by investing a budget µ ⋆
( i, j, λ a , λ b ) such that:
[Equation]
which in turn involves:
[Equation]
In particular, at t 1 = 0 whenever λ b = λ a = λ and j = k , this gives:
[Equation]
The first reachable region from region j is then defined as i ⋆ ≜ argmin i ∈
I µ ⋆ ( i, j, λ a , λ b ) , where
I ≜ { i ; i < j and p
[Equation]
i, j ) } .
Note that at the moment t 2 ≜ t 1 + 1 p j µ ⋆ ( i ⋆ , j, λ a , λ b ) when this region is reached, we have:
[Equation]
On the other hand, whenever the set I is empty, by definition, the process reaches case 1 steady-state regime and only plays optimal policy of region j for remaining budget.
To be fully general, we note that two or more regions can be reached simultaneously.
In this case, the optimal policy tie-breaks by taking the region with maximal index i.e. higher censorship, as further described in Lemma D.2.
D.4 Statement and Proof of Cor. D.1.
More generally, this allows us to deduce the next technical corollary:
Corollary D.1.1.
For a sequence of censored regions { i 1 = k, . . .
, i l , i l +1 , . . .
} , we have for the l th region of the transient sequence, with starting time t l − 1 and ending time t l :
[Equation]
where t l is characterized by:
[Equation]
and where λ W ( t n ) a and λ W ( t n ) b refer respectively to the upper and lower coefficient of the diagonal matrix W ( t n ) .
Proof.
We leverage a simple induction reasoning using for l ≥ 1 the formula given within the proof of lemma D.1:
[Equation]
and the initialization conditions t 0 = 0 and W (0)
=
λ I d .
D.
Statement and Proof of Lemma D.
Lemma D.2 (Dual Reachability Analysis) .
Let’s assume we are currently playing transient region j
and we reach the region i at time t l .
We then have the following two possible cases:
• If p i p j > r † ( i, j ) , we say that regions i is dual reachable from region j , leading to a steady state regime with bi-region ( i, j ) effective dimension.
In such case, for t ≥ t l , the potential increase is of the form:
[Equation]
• Otherwise, we switch from base region j to base region i and continue in the transient regime.
Proof.
Using previous section, we know that W ( t l ) ∝ W ( i, j ) where we recall that the matrix W ( i, j ) has the strong property that the gains in regions i and j are equal i.e.:
[Equation]
One of the main result we show in the multi-threshold censorship model is that for t ≥ t l , we have:
[Equation]
which involves in particular that for t ≥ t l , W ( t ) ∝ W ( i, j )
This is possible thanks to the fact that the optimal policy produces a combination of p i W i and p j W j proportional to W ( i, j ) so that optimally of both regions i and j is maintained while maximal first-order gain is simultaneously ensured.
The proportionality condition is then written as the existence of µ i , µ j > 0 such that p i µ i W i + p j µ j W j ∝ W ( i, j ) or equivalently as:
[Equation]
where µ i and µ j are the infinitesimal time increase in regions i and j .
It leads in turn to the ratio equality:
[Equation]
Thus
[Equation]
we see that bi-region stationarity is possible if and only if p i p j > r †
[Equation]
j ) where we introduced the dual reachability condition:
[Equation]
Hence, the use of the term dual reachability comes from the fact that region i is dual reachable from region j if and only if region j is reachable from region j .
In such case, further algebraic calculation then lead to the instantaneous potential increase ∂W for infinitesimal time ∂t ≜ µ j + µ j :
[Equation]
We then note that:
[Equation]
Therefore
[Equation]
we conclude that:
[Equation]
We then introduce λ ⋆ defined such that:
[Equation]
Given the previous two results
[Equation]
we conclude that for all t ≥ t l :
[Equation]
Note that entering the bi-region stationary regime impedes new regions to be reachable.
Indeed, going back to the initial definition of reachability, region n is said to be reachable from region j after time t l if and only if:
[Equation]
Yet, using previous result on the evolution of W ( t ) , we know that the ratio of those two quantities remain equal for any t ≥ t l i.e. no new regions can be reached.
Moreover, using the optimality criterion of Lemma D.1, when several regions are reached simultaneously, the tie-breaking is performed by considering the most censored region, i.e. the one with the highest i index.
If the chosen region is not dual reachable, then the next one is considered.
In the case where none of them is dual reachable, the base region becomes the maximally censored region and we immediately reiterate the procedure described in Lemma D.2.
D.6 Statement and Proof of Lemma D.
Lemma D.
(Bi-Region Effective Dimension) .
Let’s assume we reach a bi-region ( i, j ) steady state regime at time t l ≤ T .
Then, we have:
[Equation]
where d eff = 1 p j (cid:104) ( d − 1) 1 − l ( i,j ) p i /p
j − l ( i,j ) + u
( i,j ) − 1 u ( i,j ) − p
i /p
j (cid:105) and λ ⋆ is given in the proof of Lemma D.2.
Moreover, we have the cumulative transient potential:
[Equation]
t l
[Equation]
Proof.
For t ≥ t l
[Equation]
we have the infinitesimal two-step increase ∂G during the infinitesimal time ∂t ≜ µ i + µ j :
[Equation]
where we used the property of W ( i, j )
[Equation]
Invoking lemma D.
[Equation]
we know the evolution of W ( t ) for t ≥ t l :
[Equation]
as well as the relations between µ i and µ j :
[Equation]
We invoke the fact that Tr( W ( i, j ) − 1 W j ) =
[Equation]
j + 1 p
i
[Equation]
p j − l to conclude that:
[Equation]
p
[Equation]
Given that ∂t is an infinitesimal time increase, we have in the steady state regime:
[Equation]
We finally note that the cumulative potential coming from the transient period is equal to:
[Equation]
where the closed-form expression of W ( t n ) is given in Corollary D.1.1.
D.7 Special case: Single-threshold model
Corollary D.3.1.
For the single threshold model with two regions 0 and 1 and associated censorship probabilities p 0 < p 1 , our main theorem yields:
• If p 0 p 1 < d − 1 d cos 2 ( ϕ 1 ) , then we reach bi-region steady state regime and have the effective dimension:
[Equation]
• Otherwise, we are from t = 0 in single-region steady state regime and have the effective dimension d eff =
d/p 1 .
Proof.
Using Lemma D.2 in the case of the single threshold model, we note that if region 0 is reachable, it is necessarily dual reachable given that r † (0 , 1) = 0 and henceforth, we always have p 0 /p 1 > r † (0 , 1) .
Thanks to the results of Lemma D.1, we also note that r ⋆ (0 , 1) =
p 0 p 1 <
[Equation]
and that if region 0 is reachable, it is done in a time:
[Equation]