Ads that Stick: Near-Optimal Ad Optimization through Psychological Behavior Models
Authors: Kailash Gopal Darmasubramanian
Akash Pareek
Arindam Khan
Arpit Agarwal
Venue: ICLR 2026 Poster (2026)
Keywords: Mathematical User Modelling
Psychological Model
Ad Optimization
Scheduling
Digital Advertising
Behavioral Modeling
Satiation Effect
Primary area: learning on time series and dynamical systems
TL;DR: We model ad scheduling with psychological effects and design a provably near-optimal algorithm that improves over standard heuristics.

Abstract: Optimizing the timing and frequency of advertisements (ads) is a central problem in digital advertising, with significant economic consequences. Existing scheduling policies rely on simple heuristics, such as uniform spacing and frequency caps, that overlook long-term user interest. However, it is well-known that users' long-term interest and engagement result from the interplay of several psychological effects (Curmei, Haupt, Recht, and Hadfield-Menell, ACM CRS, 2022).

In this work, we model change in user interest upon showing ads based on three key psychological principles: mere exposure, hedonic adaptation, and operant conditioning. The first two effects are modeled using a concave function of user interest with repeated exposure, while the third effect is modeled using a temporal decay function, which explains the decline in user interest due to overexposure.
Under our psychological behavior model, we ask the following question: Given a continuous time interval $T$, how many ads should be shown, and at what times, to maximize the user interest towards the ads?

Towards answering this question, we first show that, if the number of displayed ads is fixed to $n$, then the optimal ad-schedule only depends on the operant conditioning function. Our main result is a quasi-linear time algorithm that, given the number of ads $n$, outputs a near-optimal ad-schedule, i.e., the difference in the performance of our schedule and the optimal schedule is exponentially small.
Our algorithm leads to significant insights about optimal ad placement and shows that simple heuristics such as uniform spacing are sub-optimal under many natural settings. The optimal number of ads to display, which also depends on the mere exposure and hedonistic adaptation functions, can be found through a simple linear search given the above algorithm. We further support our findings with experimental results, demonstrating that our strategy outperforms various baselines.

Published as a conference paper at ICLR 2026
ADS THAT STICK: NEAR-OPTIMAL AD OPTIMIZATION THROUGH PSYCHOLOGICAL BEHAVIOR MODELS
Kailash Gopal Darmasubramanian
Akash Pareek
Department of Computer Science and Engineering Indian Institute of Technology, Madras
Department of Computer Science and Automation Indian Institute of Science Pareek akash. cs cm@gmalll i 11..1........... com
cs22b0 9 8 sma 1. 1itm. ac.
Arindam Khan
Arpit Agarwal
Department of Computer Science and Automation Indian Institute of Science ari ndamkhan sc. ac in
Department Computer Science and Engineering Indian Institute of Technology, Bombay agarpi t out look com
ABSTRACT
Optimizing the timing and frequency of advertisements (ads) a central problem in digital advertising, with significant economic consequences. Existing scheduling policies rely on simple heuristics, such as uniform spacing and frequency caps, that overlook long-term user interest. However, it is well-known that users' long-term interest and engagement result from the interplay of several psychological effects (Curmei, Haupt, Recht, and Hadfield- Hadfield-Menell, ACM CRS, 2022).
In this work, model change in user interest upon showing ads based three key psychological principles: mere exposure, hedonic adaptation, and operant conditioning. The first two effects modeled using a concave function of user interest with repeated exposure, while the third effect is modeled using temporal decay function, which explains the decline in user interest due to overexposure. Under our psychological behavior model, we ask the following question: Given a continuous time interval how many ads should be shown, and at what times, maximize the user interest towards the ads?
Towards answering this question, first show that, if the number of displayed ads is fixed, then the optimal ad-schedule only depends on the operant conditioning function. Our main result is quasi-linear time algorithm that outputs optimal ad-schedule, i.e., the difference in the performance of our schedule and the optimal schedule exponentially small. Our algorithm leads to significant insights about optimal ad placement and shows that simple heuristics such as uniform spacing are sub-optimal under many natural settings. The optimal number of ads to display, which also depends on the mere exposure and hedonistic adaptation functions, can found through a simple linear search given the above algorithm. We further support our findings with experimental results, demonstrating that our strategy outperforms various baselines.
INTRODUCTION
Digital advertising forms the backbone of today's trillion-dollar internet economy, serving as primary channel for both acquiring new customers and sustaining engagement with existing ones. Since user attention is scarce and capturing carries significant economic value, optimizing the timing and frequency of ads becomes critical challenge for advertisers. This issuc arises across many settings, including placing ads in (live) video or audio streams, sending push notifications to app users, embedding sponsored content within across user sessions. In each case, the objective to maximize user engagement and recall while mitigating long-term fatigue or satiation.
long line of empirical work in behavioral psychology has shown that the temporal spacing and frequency of ads have significant effect on the memory retention and fatigue of the customer (Singh
et al. 1994 Sahni 2015: Curmei et al. 2022). For instance, following initial positive neural

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response to repeated stimuli, individuals tend to revert toward baseline level of interest, causing the "thrill" of the same message to fade. This initial boost is known as mere exposure, while the subsequent tapering-off is termed hedonic adaptation in the behavioral psychology literature (Curmei et al. 2022). Moreover, insufficient spacing between two ads can drain attention and affect memory retention in the long run (Singh et al. T994). This effect is referred to as operant conditioning.
Despite the importance of this problem, there has been relatively little work studying the long-term cognitive and psychological effects of ad impressions (Sahni 2015 Aravindakshan & Naik 2015
Rafieian 2023) (see Section I.1 for more details). Most existing approaches for ad placement rely on simple heuristics such as uniform spacing. front-loading, or frequency caps (Aravindakshan &
Naik 2013 Ratician [203]] Despotakis et al. [202T), or on short-sighted policies that treat each ad
impression as independent of the others (Theocharous et al. 2015).
Recent empirical studies have tried to move beyond these simple heuristics. For instance, (Freeman jet al., 2022) ran an experiment with 327 participants to test where ads should be placed in order to reduce negative reactions such anger, irritation, and perceived intrusiveness. Their main hypothesis was that "Mid-roll ads will elicit more anger and be seen as more intrusive than preroll ads", and their experiments confirmed this. They also tested several other hypotheses, all arriving at the same conclusion that placing ads at the beginning is generally more effective than inserting them in the audio
middle. Similarly. (Ritter& & Cho. 2009) conducted an experiment with 129 participants podcasts. Their experiments supported the hypothesis that "Advertising at the beginning of podcasts will generate less intrusiveness, less irritation, more favorable attitudes toward an ad, and less ad avoidance than advertising in the middle of podcasts". Other studies also suggest similar trends. For example, (Goldstein et al., 201 I) argues that a mix of ad placements, at the beginning. at the end, and a small number evenly spaced in the middle, can create a more positive overall experience for users.
These studies motivate us to ask whether there is an underlying psychological reward model that can provide an explanation for these empirical findings:
Question 1.1. Can we design theoretical reward model for ad scheduling that captures the users' psychological behavior?
Ideally, the optimal policy under this reward model should be consistent with the empirical obser- vations cited above. Moreover, this model should have tunable parameters so that it can adapt to different real-world settings. This further motivates an algorithmic question about computing the optimal ad schedule under various parameter settings:
Question 1.2. Under such reward model, can the optimal ad schedule be computed efficiently?
To answer Theorem 1.1
study a dynamic model of user interest under repeated ad exposures. Let n + ads be shown at time i = (lo, 4, 4,..., ),), where each 1 E [0, T] and 4 S 1/+1 for all i E f0, n... Here, 1, represents the time at which the 7th ad is shown, and without loss of generality, we set to = 0 and fn = T. We refer to f interchangeeably as strategy or schedule. Given time horizon 7' and strategy i, the reward obtained from showing the ith ad at time & is denoted by R(E, i). The total reward associated with strategy is then defined as R(f)= = Σ R(E, i).
To capture user psychology in our model, we study R(E,) i) as combination of two functions. The first function captures the (positive) mere exposure and the hedonic adaptation effects and depends only on the number of ads shown previously. We denote this function by B Zzo -> T concave function with respect to the number of ads shown till now, and represents the reward when the 8th ad is shown. The concavity of the function B is justified by the 'diminishing returns' property implied by mere exposure and hedonic adaptation, and is common in recent literature that considers the dynamic effect of actions (Patil et al.) 2023 Blum & Ravichandran 2025). The second function is a temporal exponential decay function with parameter ò E [0, 1 which captures the (negative)
operant conditioning effect so that reward at time t decreases by 51-t' for any ad shown previously at time t'. This temporal decay function is motivated by the classical theory of (Ebbinghaus) 1913) on forgetting curves, which hypothesizes the exponential decline of memory retention over time. This model is similar to the influential goodwill stock model of (Nerlove & Arrow 1962). and exponential discounting models used in control theory (Leqi et al. 2021).
'Throughout, we use the term "reward" to describe the advertiser's 's optimization objective. Depending on the context, this could represent probability of purchase, user satisfaction with product, or engagement with ads.
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The parameter plays key role in capturing different types of user psychological behavior, such as anger, irritation, intrusiveness, or interest. A low value of indicates that past ad impressions have little effect on the user and therefore additional ads do not strongly reduce engagement. In contrast, a high value of 5 implies that ads have more long-term effects on the user (Curmei et al. 2022). This makes user highly sensitive when an ad is shown, which can quickly lead to irritation, loss of
interest, or a perception of intrusiveness. Thus, the parameter d provides a compact way to encode the psychological effects observed in experimental studies such as (Freeman et al. 2022
Ritter &
Cho 2009; Goldstein et al., 2011).
Given these functions, the goal is find the number of ads n + and the strategy f (fo..... tn) [0, 27, such that the total reward R(t) is maximized. Given this model of user reward, we first show that the problem has a special structure: if the number of ads n + is fixed, then the optimal display timing only depends on the function capturing operant conditioning.
To answer Theorem 1.2 our main algorithmic result is a quasi-lineat time algorithm that, given a fixed number of ads 1, outputs a near-optimal ad-schedule. Let ? denote the optimal time to place the i-th ad. We show that each 1 can be approximated by l (produced by our algorithm) with exponentially small error, namely, t 1 S t, S t + 1. Our result is based on several key 2n insights: (I) the operant conditioning function strictly convex, leading to single global optima, (2) the global optima has a special structure such that each f; can be recursively found using ,,,...,,,,- 1,-1, (3) t can be found using binary search with bounded error and, (4) error propagation in the recursive computation can be controlled. Next, to determine the optimal number of ads to display, which depends the mere exposure and hedonic adaptation functions, we perform simple linear search, using the result discussed above.
2nd
We further support our theoretical findings using simulations where we compare the performance of different strategies under our reward model. We demonstrate that our ad placement strategy outperforms other baselines for various values of 5 E 1].
Key insights from our model and relation to previous empirical findings. Recall that experi- mental studies (Freeman al. 2022; Ritter & Cho 2009; Goldstein al.) 2011) suggest that placing ads at the beginning and end is generally more effective than placing them in the middle. While these findings provide useful evidence regarding effective ad placement, they are limited in scope and cannot establish that such strategy is universally optimal across all scenarios. This motivates the need for theoretical framework that can explain and generalize these observations. In particular, our model offers deeper insights into ad placement strategies by explicitly accounting for all values of E [0. 1]. To systematically understand how influences the optimal policy and how ad placement strategies may vary across different values of 6, we analyze the optimal ad policy in our model for all values of E [0, 1). Our analysis demonstrates that the structure of the optimal schedule changes as 5 changes. Building on this, near-optimal ad scheduling algorithm provides not only practical scheduling strategies but also valuable theoretical insights into how ad placement should adapt across the full spectrum of 6. To be precise, we observe that for small 8, the ads in the optimal strategy i are placed almost evenly in [T]. As 5 increases, more and more ads start to concentrate at 0 and T such that the first t2 moves towards O, and the last t, < T moves towards T. The remaining ads are evenly spaced between l; and Ij- All these insights show that the optimality of heuristics such as uniform spacing or placing more ads at the beginning, depend on the value of 5, highlighting the need to adapt schedules based on user behavior (i.e., 8).
1.1 RELATED WORK
Behavioral psychology. While the work on behavioral psychology is vast, in our work we follow (Curmei et al. )222) and focus on three well-studied phenomena from psychology: mere exposure, operant conditioning, and hedonic adaptation. Several works in behavioral psychology has empirically shown the effect of mere exposure and hedonic adaptation under various settings (Cox & Cox| 2002) Fang et al.] 2007| CChuaan et al. 2015) Yang & Galak 2015). The most relevant to our work is the study of (Hekkert et al., 2013) who found that attractiveness of a product increased with the number of times it was shown to a user. Moreover, in a similar context (Nelson & Meyvis 2008) also found evidence of hedonic adaptation as a function of the number of exposures. Similarly, operant conditioning has also been well-studied (See (Cooper et al.) (007) and references therein). While operant conditioning might have additional connotations in the psychology literature, we mainly use it to model the 'annoyance' or 'satiation' effect of repeated exposures (Sahni) 2015). There has been
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some effort on psychology-ave aware 2021), however, incorporating psychological effects in advertising has received less attention.
aware recommendation systems (Curmei al. 2022 Jesse & Jannach
Value of ó in real world. There exists some work related to determining the exact curve to analyze user retention of content. In the work of (Curmei et al.) 2022), the authors note that in real-world scenarios, the value of 5 is 0.98. Several other works (Murre & Dros 2015 Goldstein et al. 2011) provide motivation for setting 5 between 0.7 and 0.99. This is also the regime where uniform spacing a bad strategy as compared the optimal ad schedule SseeeSeciion 7).
Ad scheduling. The problem of optimizing ad schedules has been studied across various communities such as marketing, operations research, and machine learning. One of the earliest work is due to (Nerlove& & Arrow, 1962), who proposed the goodwill-stock model that treats advertising as an investment that builds a "stock" of consumer goodwill (or awareness) which then depreciates over time. Specifically, they considered a dynamical model of change in consumer goodwill given an ad impression, and cast the problem of optimizing the ad schedule as an optimal control problem. However, they do not consider memory/satiation effects, and the optimal policy under their model is to show most of the ads at the start of the time-horizon. (Naik et al. 19988 extended this work to consider memory effects, however, the optimal policy under their dynamical system is not interpretable. (Sahni 2015) conducted field experiments to demonstrate that temporal spacing of ads has a large effect on the memory retention and satiation of the users. More recently, the problem of optimizing ad schedules has been casted as a reinforcement learning problem (Rafieian 2023 Theocharous et al. 2015). While these algorithms tend to be general purpose, their solutions are hard to interpret and can be difficult to execute in practice, given the complexity of ad exchanges (Despotakis et al. (021).
Dynamic rewards and restless bandits. (Leqi et al. 2021 introduced bandit framework that models user satiation using linear dynamical systems and showed that the greedy strategy is optimal when all arms have the same base reward and decay profile. More broadly, several works have explored reward structures where the current payoff depends on historical actions and decays over time (Heidari et al.) 2016 Levine et al.] 2017; Sezzne et al.) [019).. In contrast, (Kleinberg & Immorlica 2018) studies setting where rewards increase with time since the last pull. Related ideas appear in models where rewards evolve based on the number of pulls or the time elapsed since the last interaction (Cella & Cesa-Bianchi 2020 Basu et al., 2019 Warlop et al. 2018 Mintz et al., 2020).
Digital advertising. (Schwartz et al., 2017). the authors investigated customer acquisition through advertisements on online platforms using multi-armed bandits, and designed policy that achieves an 8% improvement in acquisition rate. (Adany et al. 2013) addressed the problem of allocating personalized ads to users, considering each user's 's profile and estimated viewing capacity. (Seshadri al. 2015) explored advertisement scheduling to meet advertisers' campaign goals while maximizing ad-sales revenue. (Dobrita al. 2025) proposed framework that leverages k-nearest neighbors to predict ad positions, enhancing ad scheduling optimization. Aiming to maximize viewership under budget constraints, (Czerniachowska 2019) presented scheduling solution aligned with advertisers' budgets. (Sumita et al. (017) developed a - c-competitiive algorithm for envy-free allocation of video ads where e >> 0 constant. We also note that digital advertising is vast field with many practical considerations - here we summarize work that is closely related to our theoretical modeling.
1.2 ORGANIZATION
In Section 2 we formalize the problem, and in Section 3 we present a near-optimal algorithm for scheduling ads. Next, in [Section 4 we analyze this algorithm for a special case, followed by a
discussion of the broader implications of our work in [Secciion 5) In SSectinn 6) we give an overview of our analysis for the general case. We support our theoretical findings with experiments in Secction 7 and conclude by outlining the key takeaways, limitations, and possible extensions in Section 8 All remaining details and proofs are provided in the Appendix.
2 PROBLEM DESCRIPTION
Let T denote the total time horizon, which is known to in advance. We consider the setting where we have to display n + homogeneous ads in the continuous time interval TJ. We assume for now that displaying these ads is instantaneous, though in Sneetionn E| we will show how extend this to the case when each ad needs the same amount of time. As discussed before, the function R(E, i) is composed of two parts. The first part B(i) is the reward when the 1th ad is shown (note that i could
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also be 0 and corresponds the ad shown at to), and is simply a function of the number of times have shown the ad previously, and not the times at which we show ads. In this work, we consider B(i) to be concave function, for example, the sigmoid function B(i) T+e at but in general, any function would work. The second function depends on both the number of times we show ads and the times & at which ads were shown. Let f = (to, t1........ 1 bn) denote the time and order in which ads were displayed. The second function is Ei= 8t where E [0, 1] and y E RR, is constant that parameterizes the strength of this effect. The term Σ 8 8 E [o, 1] captures the temporal decay in the loss when the 1 ad is shown. Hence, the reward for the 7th ad under strategy & given by R(t, = B(i) - 7 م
84r-1, 8
Let E be an upper bound on the number of ads that can be shown in [0, 7 Our objective is to find the number of ads be shown so that the reward R(t) is maximized.
< n. and the strategy i = (to...., (n) according to which ads should
The reward R(t) can be written
n
1-1
R(t) = Σ R(E,) Σ 1=0
B(I)
Σ B(i) -
==0
Observe that the first term E, B(i is independent of the times at which show the ads, and only depends on the number of ads themselves. The coefficient of the second term, i.e., 7 only plays the role of scaling factor. Hence, maximizing R(()) for given value of is equivalent minimizing the loss L()). where L()) is defined as L(I) = -
Once we can find the optimal value of R8(E) for strategies that involve showing + ads, we can iterate over possible values find the optimal number of ads to be shown. We provide a summary of various notations involved in Table I] Next, we provide an overview of the algorithm to find the optimal number of ads to be shown to maximize the reward, and the optimal strategy / show these ads.
Description
Notation
Number of ads
71 n+1
Decay parameter
Time Horizon
T
Strategy
8= (to:...,tt)
Interest Function
B(i)
Loss Function
L(I) Ejcr(((i-5)
Reward Function
R(I) = (E B(i) - (5
Table 1: Summary of notation
3 NEAR-OPTIMAN ALGORITHM FOR AD SCHEDULING
In this section, we provide an overview of our near-optimal algorithm for ad scheduling. We first consider the case when we know n + 1, the number of ads show. Given the time horizon T and we want to compute the schedule [ = (4o, 4,,..,, (,), such that for all 0 S i S
1, S 4++1-
The first step our algorithm to compute the number of ads show time 0 and T. Note that multiple ads can be shown at the same time, given the instantaneous nature of the ads (we relax this assumption in /eection
Let ta > be the first non-zero time at which an ad is shown, i... 1 ads are shown at time 0. We will define the quantity T; as T; paper we derive our results in terms of T; for E {O..... n). Given the parameters n, T, 8, our first algorithm (Algorithm) I] outputs the value of a and T. 8'=, Later in our analysis, show that all the subsequent ad timings can be found once we know a and T,. We now describe the algorithm to find and T..
8t, and throughout the
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Algorithm 1 Algorithm to obtain a and To
1: Input: n, T.
2: Find the smallest value of a [n/2] s.t.
[a+1)"-2 > 8 and (a5") y
o" 2a
[aB 111-
<
holds.
[a7, 1n-2042
3: For the above value of a, define the function ((Ta)) 2 [1+a7.)
4: Compute the solution for the equation hTT,) = 8 via binary search. For binary search, initialize the search space to [8"', 1]. (see Section for more details) return (a, To)
5:
As Algorithm returns a, we now know the first non-zero time to show an ad. This means that we show -1 ads each to and time t, T. Note that for the corner case when the value of does not exist, and for a detailed algorithm, refer to tofSection D Once and Ta is known, Algorithm shows how to find the near-optimal schedule.
Algorithm 2 Algorithm to obtain near-optimal schedule
1: Input: n, 6, T, a, Ta
Vi (0....,a 1) and f, T, Vjj E (n 2a ,.....,. and in-a- = T to 4: Distribute the remaining lita, Vi E {O,..., ; n 2a} such that t440 bo +i (i--((n---tt))
2: Set f;
3: Set ta = (T.)))
Algorithm 2 provides a near-optimal ad schedule when the number of ads is known in advance. is unknown, we use Algorithm to determine it. Below, we show how to compute the optimal number of ads n, given upper bound ñ on the total ads that can be shown within the time horizon T.
Algorithm 3 Optimal number of ads
1: Input: 8, T,
2: For each n E [īi], compute the near-optimal strategy in using Algorithm)
3: For each near-optimal strategy, compute R(f,)
4: return the value of n that maximizes R((n)
Algorithm returns the optimal number of ads show and the near-optimal strategy to display them. For clarity, we focus our analysis on the case a = 1, i.e., the case where only one ad is shown at time 0 and time respectively. The general case * is discussed in Section D and follows a similar approach. When = 1, our goal is to show that approximating Ti helps us approximate fj.... tn. resulting in a nearly optimal schedule. the next section, we outline our approach for the case, with full proof' provided in [Section 4
ANALYSIS OF OUR ALGORITHM FOR a
In this section, we present our analysis the near-optimal strategy for the case a = 1. For the case of general a, overview of analysis is presented in Section 6 The analysis is divided into three parts: [Section 4. proves that has a unique minimum, [Section 4.2 derives closed-form expressions showing that each T3 can be written in terms of Ti /ecction 4.3 first approximates Ti and then to an additive error of 1 1/2". Once Ti and i are approximated, the remaining T; and 4 values can be computed easily. We now start our analysis by showing that has a unique minima.
4.1 HAS A UNIQUE MINIMA
In this section, we want to argue about the number of minima for the loss function L. Specifically, want to show that L is strictly convex function, and hence, it has most one minima. This would imply that local minima is also the global minimum (Boyd & Vandenberghe, 2004). This characterization helps find the minima for L.

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For strategy f, to form a feasible solution, it must satisfy t, f,+1 and 0 S t, S T. Then, the set of feasible solutions, which we denote by D, is defined as follows: t, E R.0 S t, S T, S t441 0, t. T. We first show why to 0 and t, T are required to minimize L.
Lemma 4.1. In any optimal solution i that minimizes the loss function L, we have to = 0 and In T.
For L = -Eis 84- let dij = 4 - 1j. where i > j and denote = Ei>, 80s, The feasible solution D' for T,' is given by: dij E R vi j. aij 2 0, ond T, dij + Ojk Ojk-
Using Theorem 4.1, we only consider those f where to and t, T. Since to and en are fixed, all references to VL hereafter are with respect to &1. 22..... fn 1- To show that L has a unique global minima, we divide our analysis into four parts: (1) We establish that there exists exactly one global minima in D' for the function L', by showing that L' is strictly convex and the space D' is compact. (2) We show a bijection between D and D'. (3) We show that, according to the previous bijection, L' indeed models L. (4) We finally show that the minima of L' corresponds exactly to the minima of L. and vice versa. Note that the missing proof Theorem 4.1] the technical details to prove that T. has a unique minimum, and the proof of the following theorem, can be found in (Section A
Theorem 4.2. The function I. admits a unique minima in DD
As have shown that L has a unique minima. To find the minima, we first argue that all T/'s can be expressed in terms of Ti.
4.2 DEPENDENCY OF Ti's ON Ti
Let T; = 8th for E (0. n). In this section, we show that T2, .3.... Tn can be expressed in terms of Tı. We have that to and tn T. Hence, To 1 and Tù 8". Further, ti would be the time at which the second ad (the first after the starting time = 0) is shown. In correspondence to this time 4, Ti would be defined as 5",
As shown in Section 4.1 eventual goal is to find this minima. Let Ho (00) HH ( + ạी similarly H,
we know that L = E;>i 84-- has a unique minima and therefore
30 + 5ア +.... a = Ej= I. In the following lemma, we show how T, can be expressed in ZT. terms of T;-1, Hi-1, H;-2, which will later help us to express T, in terms of Ti
Lemma 4.3. For < S 1, for any strategy i (0, 4, 1..... ,nn-1, T) corresponding which
V = 0, the following relations hold:
-1
AH: (T
T
2H
We now use Theorem 4.3 to show that every T3 can expressed in terms of Ti, which is the main theorem of this section.
Theorem 4.4. For any strategy 7 -= (0, 4, ... 1n 7) corresponding to which V = 0, for all i
in {1, 11. n 1}. Ta can be written terms of Ti as follows:
T
T 11)-111
(2)
Theorem 4.4 follows via proof by induction on the expressions of H; and T,, and using the induction hypothesis Lemma 4.3 Note that Theorem 4.4 we have shown that T2...., Tn-1 can be expressed in terms of T, We now show how to express T, in terms of Ti.
Lemma 4.5, For < i n, for any strategy = (0, t1. (1...,,t-1,177 corresponding to which
VL 0, the following relation holds:
T
(1 + n- T,
To conclude, derived many interesting relations between Ti, Te,..., Tr in this section. The missing details and proofs can be found in Section B In the following section, we use the theorem and lemmas of this section to find an approximate value T. and t; for all i (l...., 1}.
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4.3 APPROXIMATING tz's
In Section 4.1 we showed that a unique minima exists for L, and in Section 4.2 we showed that (Z2...., I') can be expressed in terms of Ti. In this section, we show how to find an approximate solution for :T by finding an approximate solution for T1. Note that as Ti = 5 once we obtain an approximate solution for T1, we also get solution for t1. To this end, we describe the outline of this section before proving every detail in [Section C We first show that Ti has unique solution and then demonstrate how to approximate Ti using binary search. Once we obtain an approximation for T1, we use it to approximate T, i E (2.... ,n] using Theorem 4.4 Next, we approximate & using T1, which further helps in approximating t2....,,
For an optimal Et, 7 5 be the optimal value of T; that satisfies VL = Similarly, define T2 ...... T First, we show the following result for T;
Lemma 4.6. Assuming T -) s Ti & ( + e), then T; can be bounded by T (1 4) S T, < T .1 c)
We now use the above lemma to approximate 4, which is our main result.
Lemma 4.7. For < 1. 4
logg T; is bounded by 1 2 22n(2)
[1g(1 S S
ln(2 Tog(1/6)
log(1/8) we have, E S S t Corollary 4.8. Setting 2 22)
We have thus shown that each t, can be approximated within an exponentially small error of the optimal ty, completing the analysis of our near-optimal strategy for 1. We next discuss several important implications and behaviors of the solution.
5 IMPLICATIONS
Having already established the optimality for the objective function L, we now proceed to analyze its structural properties. These properties reveal how the placement of advertisements varies with 8, offering deeper insights into the behavior our strategy under different values of 5. We summarize here the behavior of our near-optimal solution, which is mathematically proved in Secciiinn E
The near-optimal ad schedule exhibits the following patterns ó varies: (a) As 8 O, ads are placed at evenly spaced intervals, (b) as å increases, more ads concentrate times and T, (c) d increases, the first t >> 0 moves towards 0, and the last tj < T moves towards T. The remaining ads are evenly spaced between t; and tz.
Observation 5.1.
The above properties indicate form of clustering behavior in near-optimal solution. When - O, is optimal to display the advertisements at uniformly spaced intervals. As increases, a greater number of advertisements are placed at the endpoints, and T, while the remaining ones evenly distributed in the interior of the interval. In the limit as 5 + 1, the majority of the ads are concentrated at and T, with only few ads placed uniformly in between.
ANALYSIS OF OUR ALGORITHM FOR GENERAL a
In our analysis far, have assumed that T > n logolo//ooooo (2), i.e., 1. We now study the when T < 1/ (2), i.e., * 1. In this case, the minima do not lie in the interior of space D. However, in Theorem 4.2 have shown that there exists unique minimum in D. Therefore, such a minimum must lie on the boundary of D.
To find the minima when T < log1/8(2), (2), divide analysis into three parts. We first discuss the set of inequalities for which the minima lie at the boundary of 'D. Next, similar to Section 4.2 we describe the relation between the T.'s when S n logi,s(2) (2), which will eventually help us find the minima. We provide an algorithm to iteratively check for the existence of the minimum and to find it.
In this process, we obtain the recursive relation for Tatt
Lemma 6.1. Assuming to f--1 1 O, write Tati in terms of To
(aT,)*+1
Tott = (TT, + 1)2
(4)
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We need to add some boundary checking conditions in order to find the right value of a. Precisely, we need to check if at ta = O, we find the correct value of a, using similar techniques as before, we can solve for Ta and recursively obtain the entire schedule. The final version of the algorithm is given Appendix
--- [a+1)--z 2 8" at la T, we have oi [ad" +1)= I 8 Once
[a8" y=+222
S aDas Algorithm
7 EXPERIMENTS
In this section, we evaluate the performance of our near-optimal strategy through four distinct experiments: (1) how does the near-optimal strategy vary as the value of ó changes? (2) comparison between our strategy and other baseline strategies? (3) how does the loss function change with a change in the number of ads? (4) how to find the optimal number of ads? We provide a brief overview of each experiment below. For detailed setup and additional results, please refer to Sectionn F. The complete code available at
7.1 VARIATION IN NEAR OPTIMAL STRATEGY WITH CHANGE IN 8
In this experiment, we illustrate how strategy evolves the parameter d increases from 0 to 1. Figure la depictss the outcome when the number of ads n + 1 is odd. Please refer to the the ChefSection F for the odd case. Initially, for 5 < 0.4, the ads are placed nearly equidistantly. As ō increases beyond 0.4, the ads gradually bifurcate- half of them shift towards = 0, and the other half move towards t 20. This experiment aligns precisely with the behavior we obtained Theorem 5.1
7.2 OUR STRATEGY VS BASELINE STRATEGIES
We compare near-optimal strategy against three baselines Uniform, Corner, and Random. Uniform:
Ads are placed uniformly over [0, T", with equal spacing.
Corner: Half of the ads placed f 0, half at = T.
Random: The first and last ads are fixed at = 0 and t = T, respectively, while the remaining n 2 ads distributed uniformly random over the interval (0, T))
Our experiment models a video streaming setting, where users engage for 1.5-2 hours and are shown approximately 15 ads. Therefore, we have n
15 and T 100. For this experiment, we focus on high value of ò (greater than 0.9) as suggested in (Curmei et al. 2022). As observed in our experiment (Figure 1b) and in Theorem E.13 when is small, the Uniform strategy performs well. Conversely, when approaches I, the Corner strategy becomes more effective. Our results show that the optrmol strategy adapts to and consistently outperforms all baseline strategies. Specifically, at 6 around 0.98, our strategy outperforms all other baseline strategies at least by 10%.
7.3 CHANGE IN LOSS WITH NUMBER OF ADS
We now conduct another experiment to quantify the loss incurred by our near-optimal strategy due to the effect of operant conditioning. as the number of ads increases. Let # (n + 1) denote the loss associated with showing n ads under our strategy. A natural intuition is that if showing n ads results in a loss of I # (n + 1), then doubling the number of ads would roughly double the loss, i.e., # (2 + 1)) ~ 2 * (n + 1). Our experiments actually support this intuition Figure Ic
Interestingly, for smaller values of (around 0.7), the loss remains relatively stable even as the number of ads increases. However, we observe sharp rise in loss as 5 iinreesess from 0.9 to 0.99, indicating increased sensitivity to operant conditioning in this regime (see Figure lcp.
7.4 OPTIMUM NUMBER OF ADS
In our final experiment, demonstrate that for given user (fixed 8), the optimal number of ads n n changes under different mere exposure and hedonic adaptation functions. We use a sigmoid reward function B(i) k - where captures the overall strength of these effects and e controls sensitivity to the number of ads. For k. > the function is concave and increasing. As shown in Figure Id the reward initially rises with more ads due to the dominance of mere exposure. Beyond a point, the negative impact of hedonic adaptation and operant conditioning becomes significant, causing the reward to decline. The peak of this curve corresponds to the optimal number of ads. For other related experiments, refer to Seection F

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20.0
100
Our Strategy
17.5
Unifomm Strategy
No.
Random Strategy
Ad No.
Strategy
12.5
Ad No.
Time
Ad No.
60
7.5
Ad No. 5
Ad No.
40
5.0
Ad No.
2.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.90
0.92
0.94
0.96
0.98
1.00
6
Optimal strategy for 7 Ads, Time 20
(b) Loss vs 8 for (n
1)
15, T
100
0.6
700
50
600
0.8
k=
500
40
6 0.99
Reward
30
300
20
200
10
10
20
Number of Ads (n+1]
30
40
50
Number Ads (n+1)
(c) Loss vs (n + 1) ads
(d) Reward v/s (n 1) for B(i) 146
Figure (a) Change in near-optimal strategy with 5 for ads and T 20. (b) Loss between near-optimal and baseline strategies for n + = 15, T = 100. (c) Loss increases as the number of ads increases. (d) Gain function used to find the optimal number ads for a user with d = 0.9.
CONCLUSION
In this paper, consider model that incorporates dynamic psychological effects - mere exposure, hedonic adaptation, and operant conditioning into the problem of ad scheduling. We present a near- optimal strategy schedule ads based into the problem of ad scheduling, for example, equal spacing of ads might not be optimal under many settings, and it might be better show ads in the beginning and at the end compared to the middle of the time-horizon. We also support these theoretical results using simulations.
behavioral model. Our strategy leads to several insights
8.1 LIMITATIONS/XTTENSSOI
Scasonality and non-stationary rewards. While our model can work well for real-world settings where the rewards approximately stationary, such as inserting ads into (live) video stream, our model does not handle scenarios where the rewards are affected by seasonality or time-of-dat effect. For example, unlikely that sending push notification at night will result in user attention. will be interesting to extend our model to account for non non-stationary rewards.
Competition between ads. In our work, we consider the optimization of the ad schedule from the point of view of single advertiser or an ad agency running multiple homogeneous ads. In the future will be interesting to account for externalities in the form of competing ads across various channels.
Incorporating context or side-infee inle-ffformmain.. Our model does not incorporate the context side- information of the user or the ad into the optimization problem. This is motivated by the fact that under many scenarios, advertisers do not have access to user information, such advertising streaming platforms. Another future direction is to incorporate additional context of the user and ad.
Learning the reward function. Our current setup assumes knowledge of the reward function (including the parameter 5). While our methodology is flexible enough to allow various types of reward functions, it will be interesting study problem joint learning and optimization problem in multi-armed bandits setting. Another interesting direction would be to conduct sensitivity analysis and identify reliable indicators of 8 in Ad-Date for designing experiments estimate 8.
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many
ad break?, less- more- han-manywaaaaaa- ads cessed: 2025-09-02.
2023.
Ac-
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A MISSING DETAILS OF SECTION 4.1
Before proving that L has a unique minima, we want to highlight that the proofs of this section work for all values of We now provide the missing details of Section 4.1
A.1 PROOF OF LEMMA 4.1
We first argue that to 0, in any optimal solution f. The argument for tn = ' is quite similar.
For the sake of contradiction, assume that have an optimal solution i where to * 0. Consider solution t' where E2 i for all i >> and 16 0.
We have
L(t) = Σ
8 to +
Σ
84:
j>0
5>130
and
I(() Σ
849-16
Σ
843-4
120
1>i20
Σ
84 +
Σ
84
1
1>130
Σ
84 + Σ
8'
8
j>i>0
Σ
8 80 + Σ
84:-t2
=
(()).
This implies that L(t') < L(t), which is contradiction.
A.2 PROOF THAT L HAS UNIQUE MINIMA
As discussed in Section 4.1 our approach to proving that T. has unique minima is divided into four key steps. We now prove each these steps detail.
Lemma A.1. L' admits a unique minima in D'.
Proof. Observe that 8 is strictly convex for constant 5 E (0, 1), as the double derivative of 8al positive. Since the sum of strictly convex functions is also strictly convex, L' is a strictly convex function.
Observe that ozj + ant + ayo ano T. Hence, az < for i,j. This means that the feasible region of dij is bounded, and since the region is an intersection of linear inequalities, the feasible region of Osj polytope.
Also, note that D' describes polytope and hence D' is convex region. As L' is strictly convex and D' is convex region, L' has at most one minima D' (Boyd & Vandenberghe 2004). Again, we know that polyhedra are compact sets. Therefore, the function L' has a unique minimum in D' (Boyd Vandenberghe 2004).
We now present bijection B between the space D and D' and show that B preserves minima. Define the mapping B DD + D' as follows:
Given fv, we set a:0 t,
Define aij aND aju-
We now show that B is bijection:
BiD
44
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Lemma A.2. The mapping B is bijection from D D'
Proof. Let (() = a, for an element E D. By definition, = 4 - 1 2 since i> Also, note that aij + Ojk t, ij + tj ig - 1 aik. Hence, dij satisfies the constraints associated with D'. Hence, B is feasible in the domain D',
Observe that if + 6. then 0:a + d'00 hence it is injection.
Given dij. let define where 4, = 450- We show that B(1) = a. Let B(() = a', and we will show a' Firstly, by definition of and t,
d'1 t; 0.0
We also have
dej
400
d 4, - 1j 0:0 aj0 aij.
Therefore a' and hence for each a, there exists & such that B(t) Therefore, B is surjection. Since B is both an injection and surjection, it is bijection.
Corollary A.3. As B is bijection, it is also invertible. It can be verified that,
(B (a)); a.20
To this end, we now show that L' indeed models L according to B: Lemma A.4. L(t) LBB(()).
Proof.
I(t) =
Σ
8h1
(2)
Σ
B(())-BB1 B(G))
Σ 80.0-o,0
Σ 80Ls
i>J
= L'((()).
Next, demonstrate that B preserves the minima, i.e., t" is minima of L iff B(t*) is minima of L'. Lemma A.S. t" is minima of T. iff B(t") is minima of T/.
Proof. Let us first prove the forward direction. t' is local minima of L, then by definition, there exists an such that for all t, jt I+1 c, so have that L(t) 2 L(t"). Set e' c. We want to show that in the neighborhood around a* B((L), for all [a(a) T'(aa).
For the sake of contradiction, assume that there exists E D' such that a²1 S and L'(a) < Ta(a") Consider t B (a). First, observe that It -t- [t- < since a has t embedded in it. Also, we have L(t) (a) < L'(a")= = L(t"), which
contradiction to the fact that t" a local minima.
We now prove the other direction. Let a" be local minimum for L'. Hence, by definition, there exists an e such that for all a, a - a < e, we have L(a)= 2 L(a*). Let us set = e/2n².
For the sake of contradiction assume there exists a E D such that /r 1*| < and T(()) < L((t). Observe that (IB(t) B((*)l)= < 2- max, (t, - f) n² s 2n (using the fact that the magnitude of vector is more than the value of each of its components). Also, (L'(B(t) L' LB(((t)) < 0, which a contradiction to the fact that a" was a local minima.
Putting everything together, present the proof of Theorem 4.2
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A.3 PROOF OF THEOREM 4.2
First, we showed that the function L' admits a unique minima D' in Lemma A.1 We then proved that the number minima of in D is the same as the number minima L' D' in Lemma A.5 Hence, L admits unique minima in D.
B MISSING DETAILS OF SECTION 4.2
In this section, and in Appendix C focus the case 1, and defer the general case (a # 1) to Appendix D When a 1, the two conditions Line 2 of Algorithm | simplify to 2n-2 > 5
6² The second condition always holds, while the first condition holds only when 7 n 1o Therefore, in this section and in Appendix C we assume T >> logi/s
and
(6
167 107-1
whenever needed. The case 7 S // (2), where a 1, handled in Appendix D We
provide the missing details of Section 4.2 To this end, show how T, is can be expressed in terms of Ti.... T:-1 and T to Tn.
Lemma B.1. For 1 < S n 1. for any strategy = (0, t1, .tn-1.7) corresponding to which
L 0, we have,
(8%)2- =
(0 +...+
8/1-1
Proof. As know VL 0, therefore the partial derivative with respect can be written
a(L)
0
a(1))
This implies that
8
16 + 8 4... 8'm) =
8%
We rewrite the above equation
(8
(8"2
8t 4.. )
850
Similarly, for any i {,..... n 1}, we have
A(L)
O(t.)
1
84
+
(8 + +... 8
86a
84
84--1
(8+1)2
(8t141 + 8tm).
(5)
8
+
89
5-1-1
This proves the lemma.
B.1 PROOF LEMMA 4.3
As T, 5%, using Lemma B.1 we have
T.
Σ 1z
(6)
H111
1=1+1
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We rewrite the above equation
H = Σ
Tj
1=+1
Similarly, we have
a H
Σ T
(8)
j=1
Subtracting (8) from
we get
H²-IT? H1-2Th-1-
Σ T₃- I
jmit1
jmi
This implies
H₁-1T?+ + HH
-
Observe that we now have a quadratic equation in T,, Also note that Ti > as 8t 0, the quadratic equation above can only have a positive root. Therefore,
4H 2
T
2H:-1
We now want to show that every T. can be expressed in terms of T1. To this end, we first show that T2 can be expressed function of Ti and then formalize it for any T,
Claim B.2. At any time (f1,... ,ta) where VL = 0, T2 can be written in terms of as follows:
T2 ++
(9)
Proof. From Lemma
we have
-1 V
+ 2
T=
2H:-1
For i = 2 we get
T: -1+ V + 4
2Hr
As to 0, we have 8 which implies Hu 1 and H1 (1+ t.). Putting this in the above equation, we get:
-1+ 144. (1+ +1+#111 YIT
T3
2. + ++
+1 / 47 + 4T +
= Ti
2. (T) +1)
-1+ 27' +1
T
2(Ti +1
T
1+T
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B.2 PROOF OF THEOREM 4.4
We prove this theorem by induction. As our base case, we know that the theorem holds for T2 (Claim B.2
Let us assume that the induction hypothesis holds till T:-1. We want to show that the theorem also holds for T. To start with, let us first calculate H2 for
H
Tj
j=0
(1+
=1+
Σ
(Using induction hypothesis)
TY
(+ yh
+++
(Using geometric progression formula)
1+I1
1
[1+7] b
-14
T
Ti
Ti
1 + 7i
=1+
Ti
1 1 Ti
Ti
Using Equation
we have,
-1+ V + +411-2211-1(22)
T
2H,-1
-1 / 1 (1+T)) 1i
(1+T1)
i--1
(1+7))-2))
T--1
Ti
2( 147 )i
-T-1 -1 + V 4(7i + 1)(
2(1 +T)
T -1+ 27' +1
2(1 + TT)-II
Ti
(1 (1+
Corollary B.3. At any time (1,,.., ln) where VL = 0, for all in [n 1}, H3 can be expressed in
terms of as follows:
H, + Ti
(10)
Ti
B.3 PROOF LEMMA 4.5
From Lemma B.I
we have that
T (T2 + T3 + +Tn)
(11)
Which can be written
T (T2 + T3 +... Ta Tn
By using Theorem 4.4 substitute T,
(T (1+71)*
for < i < to get
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(1+Th) + (71)s
(T1)²
+ ... + (T))-1
T2 + T3 +
+ Tn-1
(1+
(1+
(Ti)² 1 1-
Tj n-
11
(1+Ti)
1 7
I+71
T
(1 2
Hence, we have
T (1+ m-) = Tn-
Corollary B.4. For 1 S i S n, any time (1,, (n)) where LL = 0, the following relation holds:
8hn
(1+ 01+ 8
(12)
C MISSING DETAILS SECTION
As discussed Section we first show that Ti has unique solution, then we approximate the Ty's, and lastly we approximate ty 's. We now start by showing that Tj has a unique solution.
C.1 Ti HAS UNIQUE SOLUTION
In Lemma 4.5 we showed relation between Ti and Tn Here we first show that the function T
(1+71) with respect to Ti is strictly monotone and then argue that 7; has unique solution which minimizes L provided that > n (2)
Lemma C.1. Let ((Ti)
T
Then increases strictly monotonically with Ti.
(1+T1)
Proof.
Th
n.
T (1 + Tj nn
(n 2)
dT, (1 + T, m-
(1 + T-)--2
(1 + n-1
> 0
Hence, ((T)) strictly monotonically increasing with Ti,
Note that ((Ti) is strictly monotonically increasing with Ti and T,, = 8f is constant, show that Ti has a unique solution. To show this, for convenience we assume that T > n log1/5(2). We remove this assumption later Appendix D
Lemma C.2. Assuming T n log (2), g(Ti) 8T admits unique solution.
Proof. Observe that 4 E {0..... (0....,T), T). so Ti E 10th 1).
When Ti 8T
T
g(T1)
(1 + Ti n-2
Ti
T
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By our assumption that T > n nlogi/5(2), let show that 8² 2n-
T > (n 2) log1/8 (2)
T (log(1/0) > (n 2) (og(2)
8T 1
2n-2
When 7, =
9(T;) = 1/2" > T
(Assuming T n nlog1/6
As g(T;) Tn when Ti = 5 and g(T;) Tn when Tj 1, using intermediate value theorem, there exists a point p E [6" 1] such that gp)) = Tn Since is a continuous and strictly monotonically increasing function, the point p unique. Hence, g(Ti) admits a unique solution.
Lemma C.3. A solution to g(Ti) = - corresponds to a solution to V = 0.
Proof. We began with set of n independent equations (Equation 5) and, through algebraic manipu- lation, derived another set of independent equations (Equations B.4 B.4 4.4) Therefore, solving for g(Ti) = 5 and computing the corresponding values of T; yields a solution that also satisfies the original equations. Since Equation plimlliss
immpsss V. O, this solution corresponds minimum of T..
Lemma C.2 states that when T > n log (2). i.e.. T much larger compared n, there exists unique interior point solution to our problem. In section [ we remove this assumption and show how to find the solution of (Ti).. But for simplicity, we proceed with the assumption now.
C.2 APPROXIMATING Ti USING BINARY SEARCH
As we know that (Ti) has unique solution and g is monotonically increasing with Ti, we perform binary search and estimate Ti to an accuracy of 1/8".
Lemma C.4. Assuming T n (2)), estimate the value of T1 to accuracy of 1/8", in polynomial time.
T is strictly monotonically increasing with Ti. Let Proof. We know, (T1) (1+T))"
h(Ti) g(Ti) T
As T, is constant, h(T;) also strictly monotonically increasing with Tı.
Also, as Tj E 16T 1] and T n log1/6 it can be verified that at Ti 87, h(TT)) = 1(87) < O. and at T, = 1, 1(7:) = M(1) > 0.
We perform binary search deseribed in Algorithm to obtain Tj. We later show that Algorithm 4 takes O(3n + Toog, (1/0)) iterations to obtain an approximation of 1/8". Since we only do constant-time operations and function evaluations each iteration, we need O(n(nT)) time per iteration. Therefore, the overall time to obtain an approximation of to factor of 1/8" would be ((3n + 7 log, (1/5)) In(nT))
Note that the size of our search domain is initially 1- < 1 and in each round, the size of our domain decreases by factor of 2. Let be the total number of iterations, then after r 3n log, (1/8) iterations, our domain size decreases to factor of less than
23n
loga (1/8)
1
8T
8
1
<
Ti
(Ti 2 8*).
g
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Algorithm 4 Binary Search
1: low t- 8
2: high 4-1 1
3: Orounde f- 3n + T log (1/6)
4: mid <- (low + high)/2
5: curr 4
6: while cury Prounds do
7: mid t (low + h
8: if h((mid)= = 0 then
9: return mid
else if h(mid) then
11: low + mid
else
13: high + mid
end if
15:
curr +- curr +1
16:
end while
17:
return mid
For an optimal E let 7 = 5 be the optimal value of Ti that satisfies VL = 0. Similarly, define T2.... Tx the following lemma, we bound the error in Tt's with respect to the estimation we make for Ti.
C.2.1
PROOF OF LEMMA 4.6
Using Theorem we know
T (1 Ti)-1
T.
Let us assume that we can bound Ti
(TI-))STSST((L
Then we get,
Ti
T- (1+ Ti)
(1+))
< (1 (1+7)(1-))=-
(1)))
< (1+T7)(1 O)-I
(TT) (1 ()
(1+ (1-
(1+c)*
T (1 (1-6)--1
Using known inequalities like (1 + x) < (2r 1)x and 1_ 5 1 + 2x for 1/2.
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We have,
(1+))*
T (1-c)
(1)
ST. (1++ (2 -77 1)2c)
S ST. (1 (2 - 1)e)
Similarly, we get
Ti T- (1+ TT)
(1)+
(1+ (1+
(T
(1+77)(1++)) (1+))(+)+))) +))
(TT) (1-c)*
(1 + +1++)-111
(1 (1+c)--1
(1 (1-c)*
= T (1+c)
Again, we know that (1 (1-z)² 2 1 - (2" 1) and
1 145
2 1 X.
We have,
2 T. (1-c)*
(1+
2 (1))))(((11
= =T, (1-6) 2i-1
2 T. (1 11-(221-1 1)c)
Putting everything together, we get
T: We can rewrite the above equation as
S - 1)e)
T - (1 4'c) < T: STST (1+4'e)
Since i <
-1- < ST (1+44)
Hence, we have proved the lemma.
Corollary C.5. Setting 1/8", we get,
T (1 - 5 5T: T: - (1+ 1/2").
The above corollary is obtained by putting 6 = 1/8 because we have already shown in Lemma C.4 that Ti can be estimated to factor of 1/8 using Algorithm
C.3 APPROXIMATING 4 USING TІ
As we have shown that we can estimate TI.... T, in terms of multiplicative error, and since T, = 8%; bound I in terms of additive error.
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C.3.1
PROOF OF LEMMA 4.7
As Ti satisfies
We have,
T (1++)
8 (*+0((1+)))
424 logs +e)
424 log 1/8
log(1 + E)
424 log 1/6 (c)
Similarly, we have
2 (1 ()
8t. 2 8 ()
45 4 = logs - 6)
1
45 4 log(1/5)
log(1 c)
4st (og(1/0)
(-2 (22cc)
(2)c
4 S t
log(1/5)
Hence, G
2m(2)e
24 S
2ln(2)c
log(1/8)
log(1/8]
Corollary C.6. Setting z"
log(1/6) 2ln(2)
have,
1
f
<4
<
2
2
D ANALYSIS WHEN a + 1
In our analysis so far, we have assumed that > n (/). i.e., a 1. We now study the case when 7' < n 1og1/5 (2), i.e., a + 1. In this case, the minima do not lie in the interior of space D. However, Theorem 4.2 we have shown that there exists unique minimum in D. Therefore, such minimum must lie the boundary of DD
To find the minima when < n logi/s (2), we divide our analysis into three parts.
1. We first discuss the set of inequalities for which the minima lie at the boundary of D.
2. Next, similar to Section 4.2 we describe the relation between the 7t's when T' & n (2), which will eventually help us find the minima.
3. We then provide an algorithm to iteratively check the existence of the minima, if it exists, and how to find it.
D.I FINDING THE RIGHT SUBSPACE TO SEARCH FOR THE MINIMA
This section focuses determining the value of In algorithm, perform linear search possible values of a, and in each iteration, we check whether the conditions
2a
1 (as (o8T)n+2-0 2 2n
(a + na 2 8T and
(a8 + 1)n S 8
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satisfied. Theorem D.3 shows that this linear search correctly identifies the value of a, i.e., if satisfies these conditions, then the optimal schedule, we have t; and in- = T for all i a.
From the definition of space D in Section 4.1 note that the boundaries of D are t, 2 0, I and & S 1++1- But the set equations obtained so far indicates that for the optimal ł, t; < 1,+1. We therefore show that any solution with the added constraint 1; f 1 also corresponds to solution to our initial set of equations for D. To this end, with the additional constraint that & = f4+1 we obtain the following two of equations.
1. Substituting li-1 for i, in L and writing VL 0 where is taken with respect to (4,,..., 44-1,111111 1,,,, 1n-1)-
2. Substituting 1, for 441 in and writing VL = where taken with respect to (fi...., t1 ,,, 1n-1
We denote {t1....,te .......,..1111 1,1,,, ................................ in-1) } as first equation and {t1, t4, fn-1) as second equation. We now show the following lemma.
Lemma D.1. Any solution to the set of equations obtained by solving VL = O, under the additional assumption 1 = li+1, is also a solution to the original set of equations obtained by solving V T. 0.
Proof. We here present a proof sketch as the proof is similar to the proof of Lemma B.1
Let denote original set of equations obtained by ((L)/tt in Lemma B.1 by Eg- When perform / in our first set of equations, it looks like we have substituted i, with 1,41 in our set of original equations, Denote by E] the equation obtained by ((L)/tt, in our first set of equations. For example, when 0L/0t, = 0, we get
d(L) 8th 01
16 + 8
28 +.... 8t").
8to
Similarly, our second set of equations looks like we have substituted 4++ with I; in our set of original equations. Denote by E3 the equation obtained by ((L)/Ot, in our second set of equations. For example, when 0L/öt = 0, we get
a(L) = 8t 800 0t:
+
16 + ..... + 28th + ... + 8*')-
561
Observe that since there is unique minima for L, solving the above two equations must give the same solution (or can be shown that both equations have no solution). We can now construct where 7' = (l,,..., .......,,,, (n-1) 1) is the solution obtained by the first set of equations by calculating VL = O, once substituting f++1 by t,.
=== ........ L4 44, .., (n-1), (......,,n-))))
Let first talk about the constraints that are obtained by equation E) where j i 1. In our solution 7', we have that = 1+ But 8 is identical to what obtain when set 4 14+ in Ej. Hence, if we start with a solution F' of E). it will also be solution to Ez.
Let F" (f1.... 1++1, f..1..... .., 1n- where (fI...., tn-1) is the solution of the second equation. We observe that the solutions that we get by solving the first and second sets of equations must correspond to the minima, and we can easily show that this maps to the minima of by setting +++ t2 or t 1++1. Since there is unique minima of L, E =
Let us now see the constraints that are obtained by equation E7, where j i 1. our solution t', we have that I, = 1++1. But E is identical to what we obtain when we set I, = 1,41 in & Hence, we start with solution t" of E7. it will also be solution to Ey.
Putting these together, we obtain that t' satisfies all the equations for our original set of equations.
We now show that an analogous result to Lemma D.I holds when we set f, = and t, = T, where set either i 0 or t,-1 = T. Observe that this is the most
j > i. Note that at the beginning general thing to do, since we set t; = 0, where > 1, then by the constraints, t, = 0 Vjj < i, we will have 4 0. Thus, the added constraint still encompasses this solution. We now show that as L.
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has unique optimum, setting = 0 leads to fn-1 T. We capture this in general by the following lemma.
Lemma D.2. the optimal solution exists in the subspace A of obtained by setting 11, lzy.... bi to zero and In-js.... ln-1 1 to zero. Then, there is an optimal solution in the subspace A' of by setting 1 12, .. zero and 1n-k:... to zero, where k max(i,j)-
Proof. We show this by contradiction. Let us assume without loss of generality that i < j. Assume that a solution exists in the subspace A, but not in A'. Then this solution has a non-zero entry among (44,'.. (x), say at the x position. Let us call this solution ! = (41, (2,,2...... (--1nn)..... 1), where t1 t2 = and in-j =
-- = O.
For constant T, let T - t= -T ... t..) We call this solution 8 (T t1 T tx).
Observe that L(P)= = L(t). Therefore, if f is a global minimum to L, then 8' is also a global minimum to L. But this contradicts the existence of unique global minima.
Let Dj be the subspace obtained by setting L; = O and 4n-i= = T. Timilarry, define D,+1. Then, we have the following theorem:
Theorem D.3. Consider the subspaces D2, Dz.... Dn/2- For each i E [l....,n/22) one of the following hold:
1. The solution lies in the interior of some Dj. where j S
2. The solution lies the boundary of Di,.... ......,Dr, Dj and in Di+1.
Proof. Let us prove the theorem by induction on i. For i = 1, we have demonstrated in Section 4. that the solution lies in Dj. Hence, if it does not lie in the interior of D1 then it lies at the boundary of Dt. In Lemma D.I we have shown that if the minima of L only satisfy equations of the form = 14+1, then our original set of equations also has solution. But this cannot be the case as we have seen that E (t; I 1;+1) has a solution. Hence, the minima for L must also satisfy some equations of the form t, = 0 or/and to T for some j 1. However, it satisfies some equation of the form 1 = 0 for some > i, then by Lemma D.2 it also satisfies l2 = 0, and Ln-2 T and so on.
Now, let us assume that the theorem holds up to hypothesis, we know that either the solution lies in the interior of D Di.... D;-1 or it lies in D, Let us consider the case where lies in D,. boundary of D, If the solution lies at the boundary of D. then it must satisfy some of the equations that form the boundary. As the minima lies in D,, it satisfies 1, = 0 and in- = O, using Lemma D.2
and demonstrate for i. By the induction
does not lie in the interior of D;, then it lies on some
Again, by Lemma D.I. we know that if the minima to L only satisfy equations of the form 4, = 4++1 and not any other equation, then our original set of equations has a solution. Hence, the minima of L must satisfy some equations of the form t, 0 or/and tę T for some j. k > i. However, if it satisfies the equation of the form t, = 0 for some j >> i, it also satisfies f = 0. Also using Lemma D.2] it must satisfy f T. Similarly by Lemma if for some j > i, our equation satisfies by 1 = T, then it also satisfies, 4+1 = O.
Therefore, by these inequalities, the minima must satisfy 4++1 = 0 and in-i-1 T. This proves our lemma.
In the next section, we show how the T''s depend on one another, similar Section 4.2 expressed each T; in terms of Ti.
where we
D.2 DEPENDENCY OF T
In this section, we show how the Ti's relate to one another under the new set of equations. While many of the proofs mirror those in Section 4.2 now extend them to more general setting by incorporating the constraint t₂ = 0 for different values of a.
As have already seen, if the solution of L does not lie in the interior of D. then it must lie in the subspace with the added constraints t2 = 0 and t, = T for some a E {1, .......... n.. To this end, show how T; can be expressed in terms of Ta-
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Lemma D.4. Assuming to t--1
write Tati in terms of Ta
(aT,) (aT. + 1)
Tatt
(13)
Proof. Using Lemma 4.3
we have
1 + v + - T;
2H
(aT.)) Ta (aT. +1j
As base case, when i = 0 we have,
As the first terms are zero, using the induction hypothesis, now calculate Ha which is the sum of the first + k + terms.
H Hatk-
Tj
j=0
(1 a
0
(4T)
(Using induction hypothesis)
1=0
(1 + 5-1
{a
L+aTi yk+1
0
(Using geometric progression formula)
Ti
1+0Ti
1
aTi
[ItaT] al
j++1
a
T1
afT
+
aTa
a
aT'
+
aTa
k+1
a
aT
Now, we have
-14 + 1+ 1+4H. HHati-2 (aooii2HHatiill (T2
Tati
HHati-1
-1+ 1+ 4. al
(1tal1) aTi
)i-1 al (1tal1))
(2 (02i)) )2 (1+oT;) --
oTi
2a
looT )i
aTi
(aTi) -1 + 1 + 4(a7i + ()(fi)
20(1 + o'Ti)'
-1 2aTi +1
(aTi),
2a(1 + aTi)²
(4T))(+1
(1 + oTi)²"
Next, similar to Lemma 4.5
show how to express Ta-a-1 in terms of T...
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Lemma D.5. Assuming to t--1 have
1 (aTa)
n--a+1 a² (1 + aTa) -2
(14)
Proof. Similar to how we obtained Equation 11 l--1- = and in-a-1 _... in T. we get the following
0L/Ota = O, and the fact that to
T (T + +............................... +Tn-a +
a
Hence,
aT Tat1 +... Tn-a = a5
Now, using Lemma D.4 we have
(aTaa²
(aTa)s
(aTa ( n 20
Tat1 + Ta-2 + ................................ + Tn-a
+
+.. +
(1 + aTa)
(1 + aTa)
(1 aT )ma))=22 n
(aTa)²
- ( oTa
)n-20 2a
a
(1 + aTaa
1
of
Ttata
(aT i 22-2
= aTa)a
(1 + aT )n
Thus, putting everything together, have
(aTa) nn (oTa)n22ar4) 20+2
a²6
(15)
(1 aTa) n- 2n
Corollary D.6. Assuming to = -.== l--1 = 0, we have
(ad'a n-
8T a² (1 adta )nta)n=2a"
(16)
Having derived the relation between Ta4i and T,, next them to approximate T, and provide our near-optimal algorithm.
D.3 CHECKING MINIMA AND APPROXIMATION
In this section, we derive the two conditions used in Algorithm D to find the value a. We then provide the fleshed-out version of Algorithm in Algorithm 5 and derive our main theorem for the general setting (a * 1).
Similar to Section 4.3
we first define
1 (aTa) ) 20+2 2 (1 aT n-2a
g(T.)
(17)
Lemma D.7. g(Ta) is strictly monotonically increasing with Ta E 16"h 1].
Proof.
(uT,)m-20-2
(n - 2a + ()TT. (n 20tt1
(aTTa)n-240
- (n - 2a)
dT, a² (1+ T2 m-
(1 + aT aTT')n-2a nn
+1+ aT) )n-201
(n - 2a + (aTT)) 1n n 2a+1 22(aTr)) n2202
(1 aT =1
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Corollary D.8. g(Ta) 8 admits most one solution.
We have shown in Lemma thht that any solution that satisfies Equation 13 the optimum solution. To check if Equation 15 admits a solution, suffices to check the value (Ta) at la = and la T.
Precisely, we need to check if
at to =0
a n2 2a
(a + 1)m-2a 2 8"
at l/
1 (aδT) (n67))-2-220 a² (adT + 11)--2a S 8
The two above inequalities can be verified in (In(nn))) time. According to Theorem D.3 domain reduction, which involves checking whether the inequalities hold and incrementing a if they do not, needs to be performed at most n times. If we find that feasible solution to these inequalities appears after performing domain reduction up to 1 times (for some a S n2/2), then, by Theo- rem D.3] this solution lies in the interior of the domain D.. If we have to do domain reduction n/22 times, then there only one point in the feasible region i.e., 4 for i z n/2 and 1, = for i >> n/2 (if n is odd, then fn/² = T/2).
Once we find value of that satisfies both the conditions, we perform binary search using Algo- rithm 5] which runs in O ((3n + T logz (1/8)) In(nZ") time. This yields approximate solution to To with accuracy 1/8", following an analysis similar to that Lemma C.4 As Lemma C.3 we
can show that solution to g(Ta) 5t corresponds solution to VL U, and thus minimum to L.
After computing Ta Applying similar analysis as in Lemma we obtain all T, with accuracy 1/4". Finally, using these approximate values of T,, we compute the optimal schedule f; in additional (nnTT( In(1/5) time, with an additive error of at most zn
recover the value of Ti's using Lemma D.4 in O(nlognnI)) time.
Note that Algorithm is the complete description of Algorithm We now prove one of our main theorems.
Theorem D.9. From Algorithm we can derive a 1/2"-aaproximaat schedule to the optimal schedule using Algorithm
Proof. As shown in Theorem 4.2 o Section A there exists a unique optimal schedule. In Section B
Lemmas 4.3 and 4.5 establish system of equations whose solution yields this optimal schedule.
Lemma 4.7 in Section 4.3 shows that our algorithm provides 1/22 apprxxipoonnaaioon to the solution of these equations. For the general case * Lemmas D.4 and D.5 in Section D present analogous expressions, and we show that the algorithm achieves the same r/2" appproximmatoonn in this broader context. Putting everything together, conclude that our algorithm yields 1/2" approximmaioon to the optimal schedule.
E MISSING DETAILS OF SECTION 5
In Section we discussed many behaviors of our near-optimal solution. this section, we prove them in detail. To begin, we first present the following property of our solution.
Lemma E.1. For the optimal solution we have t to = tn to
Proof. optimum, we have T, (147))=-Z
T and Ta-1
7 (1+7)
Hence,
Tn
Ti
-i
T Tk-1
To
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Algorithm 5 Binary Search with checking
1: low 8t"
2: high +- 1
3: Brounds t- 3n T log (1/5)
4: mid t (low + high)
5: curr 1
6: 1
7: dr 1
8: while ctr S. n12 do
9: if (a+1)a-1a
8 then
+
11: else if a²
(0s +1ja-z > 8 then
(a5" p+2-2a
12: ctr
+
13: else
14:
ctr
break 16: end if
end while
18: if clr 2 n1/2 then
return (n/2, 1.
20: end if
21: h(Ta) = + (aT.))-22-4
(I+aI.)" 22: while curr Drounds do
23: mid- < (low + h
if h(mid))= then
25:
return mid else if h(mid) < 0 then
27: low & mid
else
29:
high +- mid
30: end if
31:
curr +- curr +
32:
end while
33:
return (a,mid)
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The above equation can be written
8t 8n-1 = =
Therefore, get
tn 1n-1 t1 to
Next, we show that the ratio between T, and T--1 is always fixed.
Lemma E.2. For E (2....,m 1}, we have
T Ti
(18)
T
(1 Ti)
Proof. We have,
T
T
(1+7))
Using Theorem 4.4
T:-1
T--1
(1+2)-2
Ti
(1 + Ti)
Corollary E.3. f2 f1 = 13 12
t- 2n
Now, in the following two lemmas, we show that as we increase the value of ó, the value of f decreases. We first show that tı cannot be greater than T/n.
Lemma E.4. Given 6, T and n, in any optimal solution we have ta < T/n.
Proof. From Corollary B.4 we know that when VL = 0, we have
8tin
8
(19)
(1 + 8t n-2
Let us define h(t,)
8t1m (1481)"
At optimum, we would like h(tt) = 8T.
For fixed 8,
8 -
h(T/n)
(1 + 8 n-
8
T"" For fixed 6, and n, we know that g(Ti) = -1+7))-- strictly monotonically decreasing with Ti (using Lemma C.I).
We have that h(t)) 9(8 Since 8th monotonically decreases with tı, obtain that h(t1) monotonically decreases with t
We know that h(T/n) 8", Since in the optimal solution, require h(t1) 8T. we obtain l < T/n.
Lemma F.S. Fix n and T, then as d increases, the value of l decreases.
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Proof. We know from Corollary B.4 that at optimal value of t, have
8
(1+811)m~
Differentiating the above equation from both sides, we get
T
8mt
(1 + 8t1 yn-2 (nt, 8 + no In(d) 0c
(1+84)m-
(1 + 8t1 )
8 ((n 2) + 8th )r 2100 5,801-1 1 + (n - 2) + +811 (01))~88+ ln())
24
(1 (1+061)21) )2n-
This implies that
0t, T(1 ((1+8"1)= = (1 + 8' (nt) no In())
8t
8((n 2) (n - 2)0¹ (n())
06
08
Hence, we have
0t1
8 05 ((1 + 8 8 Ind - (n- - 22) In(5)) + ntt + +2t18"
Therefore,
0t a6
(6 = T(1+ - (n + 28
Rearranging the terms, we get
0l (0 20th) -5'(T -2(1 + (T ml1)
Finally, obtain
01
8t1 (T 21,) + (T nll)) ô 6(n + 28²t)
05
As and 1 T/n, the numerator in the above equation is strictly positive. Note that In(ō) and therefore the denominator is strictly negative. Hence, the derivative must also be strictly less than 0. This implies that t is monotonically decreasing with 8.
Next, we show that as 5 + 0, tt's are equally spaced in T. Lemma E.6. As
have 4 Ti/n.
Proof. As T, 5T, for the optimum solution, the corresponding value of Tj satisfies
T
8T
(1+T))m-2
But if the solution is not optimal, the above equation might not be true. Hence, we try estimate the value of the quantity:
1
TT
87
(1+T))-
We would like to show that as 8 + 0, the value of the above equation 7; 8T/n converges to 1,
which will imply that (1+T.)" 5". We now show that this is true.
7
T
87
lim
lim
8T + -i 5-10
8T (1 + 8
lim
6 (1 + 8
As we have already seen that t.'s 's are equally spaced between t and t,-1 (Corollary E.3 and since T/n and fn-1 T T/n (using Lemma R.I as 5 0, this gives I iT iT/n. I
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The properties that we have shown far in this section holds when T > /logi/5(2) (i.e., 1). We now show that similar properties hold when T (5 (i.e., 1). From Section D observe that ò increases, f--1 and ln-a+1 T for a E [n/2 + 1] hold. We now, in the next three lemmas, argue that as 6 increases, la decreases for a + 1.
Firstly, from the analysis in Section D observe that increases, the constraint la holds for larger and larger a. When f--1 0 and ta * 0, from Corollary 16 we can write 8 (ad'a) ) 2a+2
a² (1 + a0'==)"= )n- We now try to find the ranges of for which 6T (ad"n) jm 2--2a+2
a² (1 + ad' )n holds.
Lemma E.7. The equation
8T 1 (aot )m jm 2
(20)
a² (1 + adta )n-2a
1/T
1/T
1
holds in the domain E
n 2a+2
(1+ 2)m-20
Proof. Equation 20 holds when
Solving Equation 20 gives positive solution for tz.
Solving the equation
((a 1)8%, jn-20+1
8 (a 1)2 (1 (a 1)5e 1) 1
(21)
gives a negative solution for ta-1, i.e., the equation is invalid.
(a5%a yn-2+2
Let us define ha (ta) a² (1 +1+ adta )n-2a
(aTa)n nn-20+2 20+2
Using Lemma D.7 we know that g g(Ta)
a² (1 + aTa)" n-2n is monotonically increasing. with
We know that (4.) g(ō"x). Since 8th monotonically decreases with las have that (Ia) monotonically decreases with ta.
Substituting ta O, we get
1 n-2++2 A(0) = a² (1 + m2a
We know that htt.)) & M() since ta 2 by monotonicity. We also know that if Equation 20 has solution typ would have ht,,)) 8T. Using these, obtain that:
a 20-22-2
8T < a² (1 + aa)--aa
1
1/T
8 (1 1/a)m-20
1/T
Hence, Equation 20 is valid only for 8
(1+1/ay==a
1/T
By similar argument, show that Equation
21 is valid only for 8 < (1+
(1+
1/T
For (1+
we would obtain f and Equation 21 would no longer hold.
1/T
Therefore, becomes valid to now write Equation 20 as long as 8 < (1+1/0)=-2)
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1/r
1/7
the domain &
(1+ )
(144)m=00
we have that:
((a - 1)04k) " 2
The equation 8"
(a 1)2 (1 + (a 1) - 1 jn-2a+2 22+2 invalid.
(a6*= ym 2 2a+2
The equation 8
a² (1 + ad'a nn- is valid.
Hence,
the
domain
in
which
Equation
20
holds
is
8
1/T
1/T
1
a-1
)n
20+2
y 20
1/T
Lemma E.8. When 8
have to TT/n - -2a) the only solution
(1 e-1 )- )
1 (ad"n jm
Equation 20 i.e. 8"
a (1 + addaaa un-2a
1/T
Proof. We can verify the statement of the lemma by substituting to Tn((n- - 2a) into both sides of the equation.
and
The LHS becomes
1/T
T
1
8T
nn
2a
nn 20+2
a
2
The RHS becomes:
(udtu n (o-fu)n--aa+) 2a+2
1 n-2022 n-22022
a
a
(1 + to )
a²
(1 + a- 1)m-20
(a 1)
2a+2 n2222
a²
a nn2
n- 20+2
(a8 (a-taan-2e- 20=2
As we know that (ta)) T/(n 2a) is the only solution to Equation 20
is monotonically decreasing with far Hence, to
( 2012
1/T 26
Lemma E.9. For & E
(1+2)²
the value of to which satisfies
Equation 20 strictly monotonically decreases with from
T
to 0.
Proof. We know that when Equation 20 holds, we have.
(a jm 22+2
a² (I + at 2n
33

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Published as a conference paper at ICLR 2026
Differentiating with respect to 6, we obtain that
T (ao-=a)n-20+2
aδ ) ) 2u ) In
Ste 56
+ 11.61-11
a²8 (1 + a8ta )
a²
(1 + ad'n, )
1 (a6" ) 2at2 jn-2at2(1 - -2a)(1 + ad' ) a(6' In 8 + t2
2a
a²
(1 + aB'n )2n-4a
Hence,
T / (1 + a (a8")) = [[1 a8th (n 2a + 2)a (a8** 2a)a 8 16
Ot
+ +14)
06
Rearranging the terms, get
Ot.
T(1 + ad"n) = [(n 1(n-2a) 2(1 + az [][6 (8 In
+ ta]
36
Therefore, separating variables, get
Qta
In
= [ (n (n--ac 2nn-aa+00
+ 2)tal + (ad"")[T' 2t
08
Finally, we can write / 00 as:
0t
[T (n 2a 2)tal + (ao*">[T 26.a]
08
8
1/T
When (1+ -- n
we have for the optimal by Lemma E.8 have that Observe that at this point, the derivative of T with respect to is nega- tive, and it remains negative as long ta I T/(n 2a + 2). Hence, in the domain E 1/T
12
to n-2a+2-
1/T
we have that la decreases strictly monotonically with
respect 8.
We now prove lemma similar to Corollary E.3
Lemma E.10. At the minima, for the first non-zero and last non-T time we have that the ads are equispaced, i.e., ta - la-1 La - an-a11 bn
Proof. From equation 14 we obtain that
(aTa (nTa))-2aa4) n2-242
T Tn-at1= = 8T
a (1 aTaa) 20
From equation 13 we obtain that
1 (aTa)*+
Toti a (aT, + 1)'
Substituting = 2a, get
(aT, 1n 1
T (TT + 1)n-22
a)
Taking the ratio, we obtain
Tn Tn-a+1 To =
To
T,
T
This implics
8ta-at1 th-a -ta-1
Using the fact that tx-a411 T and 1 0, we have
TT -0
Taking log on both sides, we get
T t to
This proves lemma.
34

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Published as a conference paper at ICLR 2026
Now, similar to Lemma E.2 have the following lemma.
Lemma E.11.
T: aT,
T:-1 aT1 +1
Proof. From equation 13 we obtain that
1 (aTa)
Toti a (aTa + 1)r
Tatitl
1 (aTa)t2
(aTa 1) (aTa)
a (aTa + 1)it1 x
Tati
aTa
aTa +1
Hence, the ratio between successive terms is some constant. Now we prove a lemma similar to Lemma B.5
Lemma E.12. The value of ta decreases increases.
Proof. From equation 14, obtain that
(aT,)² " n222+2
Ta-a+1 a² (1 aTTaaTTa n-
We have already established that the RHS is monotonically increasing function in Lemma D.7 Observe that T2-a+1- 8², therefore as ô increases, Ta-a+1 increases. This means that the LHS increases with delta. Since the RHS is monotonie, T. also increases with 6. As Ta 8 ta decreases as 8 increases.
have now established all the behaviors of our near-optimal solution as described below. Observation E.13. The near-optimal ad schedule exhibits the following patterns as å varies:
As 8 + ads are placed evenly spaced intervals.
As 8 increases, more ads concentrate times 0 and T.
As å increases, the first t, moves towards 0, and the last t. T moves towards T. The remaining ads are evenly spaced between t4 and tj.
So far, our analysis has assumed that ads are instantaneous. setting where all ads have equal size and show that optrmmll schedule can still be achieved.
We now extend our approach to the
E. ADVERTISEMENTS
EQUAL SIZES
Let us dive into the case where the ads are of equal size, say s. We want that at any point in time, there is only one ad schedule. For this problem, we consider a new loss function which defined as
(.(t) = Σ 8ti-t--
(22)
HZL
The above loss function can be written as
(t
8ti
(23)
Note that the loss function of Equation 23 is similar to L(t). Hence, the minima obtained for L(f) will also be minima for (f). However, for feasibility, need that the ads at least time
apart. Observe that for Equation 23 we need an interior point solution, and hence we work under the assumption that 7 > n logi/s(22 (2() (i.e., 1).
35

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Published as a conference paper at ICLR 2026
Lemma E.14. When all ads are equal in length, if the second ad be scheduled as per near-optimal schedule, without overlapping with the first ad, then all ads can be scheduled as per our near-optimal schedule.
Proof. To construct the schedule, we place our first ad at to = 0 and the last ad at t, = T We then solve the problem for T
We now want to show that if the second ad can be scheduled optimally, then all ads can be scheduled optimally. To schedule the second ad, we need that 4 to 2 8, which implies 4 2 8 and hence, Ti 80i < 8 This immediately ensures that the second last ad i.e., fn-1 can be scheduled as t - to = bx - 1 (using Lemma E.I.
We also know that for i E {2, .......1 13
T,
T: I+
$ Ti
(Using Lemma E.2
Therefore,
84 < Ti
I 8
Taking log to the base d both sides, obtain
4 4-4441
Hence, there enough space for ads to be scheduled optimally.
So far, we have assumed that the number of ads n is known. In the next section, we show how to determine the optimal number of ads for a user.
E.2 OPTIMIZING THE NUMBER OF ADS TO DISPLAY
Given that the relative strengths of mere exposure and operant conditioning are known, a natural question arises: how many ads should be shown to maximize the overall reward? To answer this, observe that for any fixed number of ads n, the loss due to operant conditioning can be computed exactly. In addition, Σ B(i) can be calculated since it is just function of the number of pulls, allowing us to calculate the total reward R(I)) precisely.
To find the optimal number of ads, evaluate R(T) for each maximum number of ads that can fit within the time horizon T. The value of that maximizes R(()) gives the optimal number of ads. Algorithm 3 performs this procedure and returns both the optimal number of ads and corresponding near-optimal schedule in quaur-quaaaatic time with respect to n. Note that in most real-world scenarios, the maximum number of ads is either known in advance or can be learned efficiently. Moreover, prior work (RedCircle) 2023 Pandora) suggests that the maximum number ads typically not very large. For example, platforms like YouTube and Spotify generally display around 10 and ads per hour, respectively. Hence, can conclude that Algorithm performs in quasi-lince linear time for all practical purposes.
(1,..., n). where ñ is the
F MISSING EXPERIMENTS FROM SECTION 7
In this section, we provide additional experiments omitted from Section 7 ments, we used an Ilth-generation Core i5 laptop with 8GB of RAM. The experiments took few seconds to run. Since the experiments are designed to be very lightweight, running them any personal computer should not be much of problem. The complete code is available at https//amonymous..uu wwwrouuus.ooenssse www.e...ciccce///AA......... wcwence///eens-steeeeeeee Steisii//..////EDMMM....... We now provide a few more
For the experi-
experiments for the four experiments described in Section
36

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Published as a conference paper at ICLR 2026
F.1 VARIATION IN NEAR OPTIMAL STRATEGY WITH CHANGE
We present our experimental results to illustrate how our strategy evolves as the parameter increases from 0 to 1 for an even n +++1 1.
We observe that the pattern is largely similar to the odd case, with one key difference: for an odd number of ads, the = ad remains fixed 2
T/2, regardless of the value of 5 (See Figure lap. But for an even number of ads (See Figure 2a). the optimal times to show the ads in the first half shift towards while the optimal time to show ads in the second half shifts towards 7 as the value of increases. Hence, it supports the theoretical behavior described in Observation 5.1
F.2 OUR STRATEGY VS BASELINE STRATEGIES
We have presented the performance of our near-optimal strategy compared to three baseline strategies, namely Uniform, Corner, and Random. We now describe them in more detail.
Uniform: Ads are placed uniformly over [0, TJ. with equal spacing.
Corner: Half of the ads are placed at 0, half at
Random: The first and last ads are fixed at & 0 and t T, respectively, while the remaining n ads are distributed uniformly at random over the interval (0, T))
we presented an experiment with n + 15 and 100, modeling a video streaming scenario starting with 8 0.9. We now present a similar experiment for music streaming setting (FigurePb) where a typical session lasts around 60 minutes with approximately ads. This is emulate what popular music streaming services offer in their free plan. It can be observed that our strategy performs better than all the baseline strategies.
In Section 7.2
Strategy
Uniform Strategy
17.5
Random Strategy
15.0
Ad No.
Strategy
Ad No.
10-
No.
10.0
No.
7.5
5.0
4-
2.5
0.0
2
0.6
0.90
0.92
0.94
0.96
0.98
1.00
(a) Our strategy for n + 1 = 6, T
20
(b) Loss with 8 for n + 1 = 6, = 60
60
5
k 25
k=3 3
log[Loss]
0
40
Roward
-5
0.6
30
0.7
6
0.8
-10
0.9
10-
0.999
-15
10
20
30
40
50
60
70
0:
10
20
40
50
(n+1)
Number of Ads (n+1)
(c) log(loss) vs n ads
(d) Reward v/s + for B(i) = k -
1/10
Figure 2: Additional experiments to evaluate the performance of our near-optimal strategy.
37

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Published as a conference paper at ICLR 2026
F.3 CHANGE LOSS WITH NUMBER OF ADS
In Section 7.3 we examined how the loss changes with the number of ads for different values of 5. We extended that experiment by plotting log(loss) versus the number of ads (Figure 2 and observed trends similar to those in Section 7.3
F.4 OPTIMUM NUMBER OF ADS
In Section 7.4 we used a sigmoid reward function to determine the optimal number of ads for various values of k and where k captures the combined strength of hedonistic adaptation and mere exposure, and e controls the sensitivity to the number ads. We now perform a similar experiment using B(i) k(1 -r) (See Figure 2d and show that the peak of the curve corresponds to the optimal number ads.
38


Reviews:
[{"review_id": "xG33socCXE", "reviewer": "Reviewer_Jqk7", "summary": "This paper proposes a principled algorithmic framework for scheduling advertisements by modeling three psychological effects: mere exposure, hedonic adaptation, and operant conditioning. The authors formulate ad scheduling as an optimization problem, specifically minimizing a loss function $L(\\overline{t})$ that depends on the temporal decay of user interest.\n\n1. A novel psychological model for ad scheduling.\n2. A quasi-linear time algorithm (Algorithm 1 / Algorithm 5) that finds a near-optimal schedule with provable, exponentially small approximation error.\n3. Theoretical analysis proving the uniqueness of the optimal solution (via strict convexity) and deriving a novel recursive structure for the optimal ad timings.\n4. Experimental validation showing that the proposed schedule outperforms heuristics like uniform spacing by over 10% in realistic parameter regimes (e.g., high $\\delta$).\n\nHaving thoroughly reviewed the main paper and all appendices (A-F), I find the work to be technically sound, novel, and practically relevant. The theoretical claims are correct. My main concerns are related to presentation, as much of the core theoretical justification is relegated to the appendices, making the main paper difficult to follow and appear incomplete.", "questions": "My initial Questions were largely resolved by the appendix in the supplementary material.\n\n- **Q: How sensitive are the results to misspecification of $\\delta$?**\n    - This question remains open. A sensitivity analysis would be a valuable addition, as $\\delta$ is unlikely to be known perfectly in practice.\n\nThis paper provides solid theoretical contributions and a practical, efficient algorithm. With the appendices, it is clear the technical work is sound. The required revisions are purely focused on improving clarity and presentation to ensure the paper's contributions are properly understood by the reader.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. **Well-Motivated Problem:** The paper addresses a practically important problem with clear economic implications. The grounding in established psychological principles (mere exposure, hedonic adaptation, operant conditioning) is a significant strength and is well-modeled.\n2. **Solid and Complete Theoretical Contributions:** The main theoretical results are non-trivial and, upon review of the appendices, are fully proven.\n    - **Uniqueness (Thm 4.2):** The proof of strict convexity and a unique minimum is correct and detailed in Appendix A.\n    - **Recursive Structure (Thm 4.4, Lemma 4.5):** The recursive formulas for $T_i$ are proven in Appendix B. I have numerically verified these formulas.\n    - **General $a \\neq 1$ Case (Appendix D):** My initial concern about the analysis being limited to $a=1$ was unfounded. Appendix D provides a complete and correct analysis for the general (boundary) case $a \\neq 1$, deriving the analogous recursive relations (Lemma D.4) and the general solution condition (Lemma D.5).\n    - **Approximation Guarantees (Cor 4.8, Thm D.9):** The exponentially small approximation error $|t_i^* - t_i| \\le 1/2^n$ is proven for the $a=1$ case in Appendix C and generalized for all $a$ in Theorem D.9.\n3. **Practical and Efficient Algorithm:** The proposed algorithm (detailed in Algorithm 1 and Algorithm 5) is efficient (quasi-linear time) and implementable. The code provided was functional and reproduced the experimental claims.\n4. **Validated Experimental Claims:** All experimental claims in Section 6 are verified. The results showing uniform spacing is optimal for small $\\delta$ and endpoint-concentration is optimal for large $\\delta$ align with the theory and provide valuable, actionable insights.", "weaknesses": "The paper's primary weaknesses are not technical but related to presentation and clarity. The main paper, read in isolation, is confusing and appears incomplete.\n\n1. **Critical analysis deferred to Appendix:** The paper is not self-contained. The analysis for the general $a \\neq 1$ case (Appendix D) and the proofs for all key theorems (Thm 4.2, Thm 4.4) and approximation bounds are entirely in the appendix. While this is common, the $a \\neq 1$ analysis is a _core_ part of the algorithmic contribution (Algorithm 1) and its omission from the main text makes the paper's claims seem unsupported.\n    - I would suggest to add theorem statements for the general $a \\neq 1$ case in the main paper. Also, add 1-2 paragraph proof sketches for Theorems 4.2 and 4.4 to the main text.\n2. **Algorithm presentation is opaque:** The conditions in Algorithm 1 are opaque and unmotivated. The appendix (Algorithm 5 and Appendix D.3) reveals these are the boundary-checking conditions for finding the correct $a$.\n    - I would recommend to add a remark after Algorithm 1 explaining the purpose of these conditions, as detailed in Appendix D.3.\n3. **Issues pertaining to confusing notation:** The use of \"$n+1$ ads\" with indices 0 to $n$ is non-standard and causes confusion. The relationship between the paper's $n$ and the code's $n$ is also implicit.\n    - Add a notation table at the start of Section 2. Explicitly define $T_1$ in Section 4.2 and its relationship to $T_a$.\n4. **Limited experimental scope:** The experiments are entirely synthetic. While they effectively validate the algorithm's properties, the paper would be stronger if it addressed how the model parameters ($\\delta$, $B(\\cdot)$) could be calibrated using real-world user data.\n    - Having a discussion on parameter estimation and acknowledge the limitation of not having real ad-serving data would help improve the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BOOntl3296", "reviewer": "Reviewer_CFrE", "summary": "A reward model for advertising is derived from the marketing literature. \nIn the context where we have to expose a user during a known interval, an optimization algorithm is derived. \nSome qualitative insights are provided on the behavior of the solution.", "questions": "* Can you provide an updated version of figure 1?\n* could you provide a summary of the proof that would clearly delineate what is easy and what is hard in the derivations?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The authors did an excellent job at introducing a reward model by bridging results from other fields. \nI found the article very informative and inspiring.", "weaknesses": "Main weakness : the technical sections of the paper should be improved\n* the graphs cannot be read properly (no color)\n* discussion and explanation on the baselines in the main text would be welcome\n* the proof of the statements are sometime two hand waving, the mathematical writing should be improved\n* I am not sure the algorithm itself is a very technical contribution (mostly basic maths). While this is not a reason for rejection or anything, I believe the proof sketch should be better articulated", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3zQ2lR7qAp", "reviewer": "Reviewer_tYzy", "summary": "The paper performs a theoretical quantative study of a model of ad fatigue, which is important in advertising, by considering the number of ads shown to a user and the effect of ad recency on the user's inclination to positively interact with an ad. A bi-level algorithmic framework is established for computing an optimal ad schedule given a fixed number of ads as the 'inner' problem, and computing the optimal number of ads as the 'outer' problem.\n\nDespite a few weaknesses, the work is enlightening and interesting.", "questions": "1. Will the code to reproduce the results be published?\n2. Why do you believe that a concave and increasing function on the entire domain is a reasonable model to gain insights?\n3. How do you propose fitting the parameters of such a model if we want to actually use it in an advertising platform?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The phenomenon is interesting and important.\n2. A fresh deviation from all this \"attention is all you need\" trend, where a phenomenon is actually studied in depth.\n3. The framework is simple, without many 'moving parts'. \n4. The bi-level idea is nice and practical, if indeed we aim to plan an ad schedule for each user separately.\n5. The results are insightful. For example, it's not trivial that the optimal schedule tends to \"show a lot only at the beginning and the end of the time-frame\" only when the fatigue parameter is very close to 1, and otherwise close-to-uniform strategy appears optimal (Figure 1 (a))", "weaknesses": "1. In practice, privacy concerns don't let you, in many cases, know when you're showing an ad to the same user. So such an idea is practical mainly in \"closed gardens\" where all users are always logged in and we know who they are. Most advertising marketplaces are not like that.\n2. Code to reproduce the experiments is important, and any claim to publish it is missing from the paper.\n3. The function B(i) is concave and increasing, but we don't know if it's the right model. There are many ways to model a concave and not necessarily increasing function. It may eventually asymptotically be concave and increasing, but it might be the case that the effect of its shape before asymptotic behavior kick in is not trivial.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)