Are More LLM Calls All You Need? Towards the Scaling Properties of Compound AI Systems
Authors: Lingjiao Chen
Jared Quincy Davis
Boris Hanin
Peter Bailis
Ion Stoica
Matei Zaharia
James Zou
Venue: NeurIPS 2024 poster (2024)
Keywords: Scaling Laws; Compound AI systems; language models
Primary area: generative_models
TL;DR: We study the scaling properties of compound inference systems both theoretically and empirically

Abstract: Many recent state-of-the-art results in language tasks were achieved using compound systems that perform multiple Language Model (LM) calls and aggregate their responses. However, there is little understanding of how the number of LM calls -- e.g., when asking the LM to answer each question multiple times and taking a majority vote -- affects such a compound system's performance. In this paper, we initiate the study of scaling properties of compound inference systems. We analyze, theoretically and empirically, how the number of LM calls affects the performance of Vote and Filter-Vote, two of the simplest compound system designs, which aggregate LM responses via majority voting, optionally applying LM filters. We find, surprisingly, that across multiple language tasks, the performance of both Vote and Filter-Vote can first increase but then decrease as a function of the number of LM calls. Our theoretical results suggest that this non-monotonicity is due to the diversity of query difficulties within a task: more LM calls lead to higher performance on "easy" queries, but lower performance on "hard" queries, and non-monotone behavior can emerge when a task contains both types of queries. This insight then allows us to compute, from a small number of samples, the number of LM calls that maximizes system performance, and define an analytical scaling model for both systems. Experiments show that our scaling model can accurately predict the performance of Vote and Filter-Vote systems and thus find the optimal number of LM calls to make.

Are More LM Calls All You Need? Towards the Scaling Properties of Compound AI Systems
Lingjiao Chen', Jared Davis', Boris Hanin Peter Bailis',
lon Stoica", Matei Zaharia", James Zou?
Stanford University
University of California, Berkeley? Princcton University
Abstract
Many recent state-of-the-art results in language tasks were achieved using com- pound systems that perform multiple Language Model (LM) calls and aggregate their responses. However, there is little understanding of how the number of LM calls - e.g., when asking the LM answer each question multiple times and tak- ing majority vote affects such compound system's 's performance. In this paper, we initiate the study of scaling properties of compound inference systems. We an- alyze, theoretically and empirically, how the number of LM calls affects the per- formance of Vote and Filter- Vote, two of the simplest compound system designs, which aggregate LM responses via majority voting, optionally applying LM filters. We find, surprisingly, that across multiple language tasks, the performance of both Vote and Filter-Vote can first increase but then decrease as a function of the num- ber LM calls. Our theoretical results suggest that this con-monotonicity is due to the diversity of query difficulties within a task: more LM calls lead to higher performance on "easy" queries, but lower performance on "hard" queries, and non- monotone behavior can emerge when task contains both types of queries. This insight then allows us to compute, from a small number of samples, the number of LM calls that maximizes system performance, and define an analytical scaling model for both systems. Experiments show that our scaling model can accurately predict the performance of Vote and Filter- Vote systems and thus find the optimal number of LM calls to make.
1 Introduction
Compound AI systems that perform multiple Language Model (LM) calls and aggregate their re- sponses are increasingly leveraged to solve language tasks 1ZKC#24 23, TAB 23, TWWL+24 22)+ For example, Google's s Gemini Ultra achieved state-of-the-art results on MMLU using a C2 voting strategy: the LM is called 32 times, and then the majority vote of the 32 responses is
used the final response [TAB 23). Other compound systems filter responses using an LM before
selecting one |Alp23]]
natural question is, thus, how does scaling the number of LM calls affect the performance of such compound systems? This question is under-explored in research, but characterizing the scaling dynamics crucial for researchers and practitioners to estimate how many LM calls are needed for their applications and allocate computational resources aptly. Understanding these scaling dynamics also helpful recognizing the limits of compound inference strategies.
As a first step towards answering this question, we study the scaling properties of two popular compound system designs, Vote and Filter-Vote. Vote aggregates multiple proposed answers via majority vote, as in Gemini's CoT@33 Filter-Vote leverages a filter before performing a majority
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Voote Vilte
Performal
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Number of LLM Calls
Number Calls
(a) MMLU PHYSICS
(b) TRUTHFULQA
(c) GPQA
(d) AVERITEC
The scaling behavior of Vote and Fitter-Vote. Interestingly, their performance is often non-monotonic function of the number LM calls. For example, the number LM calls increases, Vote's 's performance initially increases but then decreases, while Filter- Vote's performance initially decreases but then increases on the MMLU PHYSICS dataset.
Figure 1:
vote, similar AlphaCode 2 Alp23.. While the inference system design space is broad, we focus on Vote and Filter-Vote for two reasons. First, they have already been used in real applications, such Gemini's CoT@32 strategy. Second, despite their simplicity, they exhibit nontrivial scaling properties. Specifically, although one might expect their performance to monotonically increase as more LM calls are invoked, we have identified surprising phenomenon, across multiple language tasks, exhibited by these systems: growing the number of LM calls initially improves performance but then degrades it, as shown in Figure
This surprising effect motivates our theoretical study of Vote and Filter- Vote, which explains this non- monotone effect through the diversity of query difficulty in given task (see Figure and Theorem 2). At high level, our results show that more LM calls continuously lead to better performance on "easy" queries and worse performance "easy" and "hard" queries, the non-monotone aggregate behavior emerges. Formally, query is easy compound system with infinitely many LM calls gives correct answer and hard otherwise. We provide mathematical conditions under which query is easy or difficult for Vote or Filter- Vote. We also derive a performance scaling model for both systems that explicitly models query difficulties, and present algorithm that lets users fit the scaling law's parameters using small number samples. In experiments with GPT-3.5, we show that these algorithms can let estimate the scaling behavior for various problems and identify the optimal number of calls to make to the LM maximize accuracy. Our main contributions are:
"hard" queries. When task is some mixture of
Main Contributions
Non-monotonic Scaling Behavior. We find empirically that the performance of Vote and Filter-Vots is often non-monotonic in the number of LM calls.
Query Difficulty-basel Explanation. We propose formal notion of query difficulty. We then argue empirically and show in a simple analytical model that the diversity of query difficulty can explain the scaling behavior of Vote and Filter-Vort Additional LM calls improve performance on easy queries and degrades performance on difficult queries. We provide precise conditions under which the mommooooone scaling behavior emerges datasets containing both easy and difficult queries our analytical model.
Heuristic for Optimal Number of LM Calls. We empirically validate predictions for the optimal number of LM calls from analytical model for Vote and Filter-Vote which can be estimated from a small number queries.
In a nutshell, our work shows that more LM calls do not necessarily improve the performance of compound Al systems and that it is possible, at least some cases, to predict how the number of LM calls affects Al systems' performance and thus decide the optimal number of LM calls for a given task. Our work focuses on tasks with a fairly small number of possible responses (e.g.. multiple-choice questions) that support majority vote, but tasks with many valid outputs, such as

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(d) Filter-Vote overall (100%)
(e)
Filter-Vote easy (56%)
(f) Filter-Vote difficult (44%)
Vote Filter-Vote
Figure 2: Performance breakdown on MMLU PYHSICS, As the number of LM calls increases, Vote and Filter-Vote perform increasingly better on easy queries but increasingly worse on difficult ones.
chat, remain under-explored. We have released the code and datasets used in this paper, and hope to excite more research regarding the scaling properties of compound Al systems.
2 Related Work
Neural Scaling Laws. There has been extensive research on how the training parameters affect the performance of neural language models 20 SGS BDK 22I, MLP*23, [4]. Among others, the model loss has been empirically shown to follow a power law as the number of model parameters and training tokens [KMH++0]) Researchers have also proposed theories to explain the empirical scaling laws [BDK+ By contrast, recent work has found that scaling up the model parameters leads to performance decay certain tasks [MLP+23]. To the best of our knowledge, there is no study on how the number of LM calls affects the performance of a compound system, which is complementary to scaling laws training parameters and data set sizes.
Compound Systems using LMs. Many inference strategies that perform multiple model calls have been developed to advance performance various language processing tasks IXCG+23
2 TAB=23 TWL WW
SPW23). For example, Gemini reaches sfute-of-the-arsd art performance on MMLU via its CoT@32 majority voting scheme [TAR 23).
ZKAW23 CZ77X
Self-consistency [WWS 22] boosts the performance of ffain-of-thhouht via majority vote scheme over multiple reasoning paths generated by Pal M-540B. Finally, AlphaCode [Alp23]] matches the 85th percentile of humans in coding contest regime by generating up to million samples per problem via an LM and then filtering the answer set down. While these approaches are empiri- cally compelling, there has been little systematic study of how the number of LM calls affects these systems' performance, and how it should be tuned for a given application.
Compound System Evaluation. Compound AI systems are increasingly evaluated in tradi- tional benchmarks [CKB*2I HBK TVCMIN, NWD as well as domain-specift new datasets H
MMA 24, KCM and the agent environments [LYZ 23, 2
TYW+241 24]. We refer the interested readers to survey for more details (CWWW+2) Existing pa- pers focus on obtaining a single metric of given system, while our goal is to understand how the
compound system's performance is affected by its parameters (e.g., # of LM calls).

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Algorithm 2: Vilter-Vote.
Input: A user query 2;, # gen responses K Output: A response ù)
Algorithm 1: Vote.
Sample 01, 022,................ 0g i.i.d. from θ;
Input: user query v. # gen responses K
Generate 2k, C = (rr, 02)
1, ... K;
Output: A response
Generate Wkk (2x 2bi ek), k: 1,... K;
1 Sample 01, 022,... 0g iid. from O;
Generate 2 G(zz, 02), k = 1, 1,,,-,,,4 KK;
if max; wı = 0 then
L Set ÿK V V(z,, ;
3 Set ÿK = (Z1,'' ;²k);
Return UK
else
L Set OK = V(1z | we = 18);;
Return ýK
3 Inference System Designs
In this paper, we focus on two simple and natural inference system designs: Vote and Filter- Vote. Vote is inspired by and resembles several real-world compound AI systems, such as self- consistency [WWS= Medprompt [NKM**3), and Gemini CoT @ 32 strategy (3),+ while Filter- Vilter-Vote represent many other real-world compound Al systems including AlphaCode 2 [[Al23]] and AlphaGeometry (T4). Note that this paper focuses on tasks with small number of possible answers.
Building Blocks. Vote and Filter- Vote rely on three building blocks, a generator G(.,)). a majority voter V(-), and filter -)) The generator G(,,) -) takes user query x and B E θ as inputs and produces a candidate answer and an explanation. Here, instantiations of are a design choice of users and can encode many generation strategies. For example, even with a single fixed LM, diverse generations may be achieved by using non-zero temperature and different prompt word- ings or few-shot examples for each call to the LM. If @ contains different LMs, then this system definition can also represent LM ensembles. The majority voter V returns the mode of its input. i... (zi, 22, ... zk) arg m Σ -= 12.=0- and breaks ties arbitrarily. Here, A is the space of all possible answers. Finally, the filter ((.-)) takes the user query and multiple candidate answers as input, and only returns the subset that an LM believes is correct.
Vote. Given a user query v, Vote (i) first creates K candidate answers by calling the generator G, and then (ii) uses the majority voter V to choose one as the final response ÿx. The details are given in Algorithm M.
Filter-Vote Given a user query, Filter-Vote (i) first generates multiple candidate answers, (ii) re- moves few candidate answers by the filter 4; and (iii) then uses the majority voter V to choose one from the remaining answers as the final response. If all answers are removed by the filter, then V is applied on the original candidate answers. Algorithm gives the formal description.
4 Analytical Model of Scaling Behavior
Now we present our analytical performance model of Vote and Filter- Vote strategies. Specifically, we are interested in understanding the behavior of F(K; D) - Elix = vl. where the expectation over D and the candidate responses. All notations are summarized in Table MI
4.1 When do more LM calls lead to increase decrease in performance?
Our first key insight is that individual query difficulty is crucial in LM calls' effects. To see this, let us first introduce query difficulty indicator.
Definition 1. Given a user query d(x) is called an query difficulty indicator if
(0
iff d(x) > 0.
lim F(K,x) K
11
iff d(x <0

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Table I: Notations.
Symbol
Meaning
I
an input query
y
the correct answer
K
the number of L.M calls the output by one LM call
2k
the output by an inference system using LM calls
D/Drr
test dataset/train dataset
A
answer space
a
fraction of easy queries
P1
probability of 2 being correct for easy queries
P2
probability of 2 being correct for difficult queries
F(K; D)
Accuracy of an Inference System with K LM calls per query D
G(K; D)
Analytical Performance Model (to approximate F(K; D))
D))
Inverse U shape
Subset Difficuty
U shape
Subset 2 Diffcullyy
Subset 2 Difficulty
Increase
0.6
0.6
0.6
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0.4
0.4
0.2
0.2
0.2
0
%
Subset 1 Difficulty
Subset Difficulty
Subset 1 Difficulty
0.2
0.5
= 0.8
Figure 3: How the query difficulties shape the landscape one-layer Voting Inference System's performance. Informally, if the overall task is "casy" (p1 + P2 1), but the fraction of "hard" queries is large (a < Systems' performance increases first but then decreases. We call such a landscape "inverse U shape". Similarly, if the overall task is "hard" (p1 + P2 < 1), but the fraction of "hard" queries is small (a 1- 7). then enlarging the number of LM calls leads an initial decrease and then increase. Such landscape is called a "U shape". When a is large, the U-shape is less likely to occur while the inverse U-shape becomes more common. Smaller a leads an opposite trend.
4), then as the number of LM calls increases, the Voting Inference
Intuitively, positive query difficulty indicator implies the query is difficult, i.e., infinitely many LM calls lead an incorrect final answer, and negative value implies the query is easy, i.e., infinitely many LM calls eventually give correct final answer. For simplicity, F(K,x) is always either 0 or for Vote and Filter-Vote, i.e., eventually the answer correct or incorrect. Also note that d(x) is scale-invariant, i.e., if d(x)) is item difficulty indicator, then for any positive scalar y > 0, y x d(x)) is also a difficulty indicator. We will call a query x difficult (easy) if d(x)) 0 (d(x) < 0). We give two concrete instantiations follows.
assume that the limit of
Lemma 1. For Vote, (x) A maxaşy Pr[G(z, 0) a) Pr[G(z, 0) = y] is an [G (z. G2(z,))) Then
query difficulty indicator. For Filter-Vote denote G(x, 0) dr(x) A maxa4y (r, G,,,0), (z 0)) is query difficulty indicator.
a,,( G (a, 0), G2(r, 0))
1] - (r, =
5

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Here, dy (x) > O indicates that an incorrect answer is more likely to be generated than the correct one, hence the query is difficult. Similarly, dr(z) implies that an incorrect answer more likely to be kept in the filtered answer set.
More LM calls elicit higher performance on easy queries, but lower performance on difficult queries. Therefore, the performance, F(K; D), is more difficult to characterize when the data set D contains both easy and difficult queries. Here, we study Vote on a special case of D to understand how the difficulty impacts the performance function D):
case study on specific dataset. Let us consider a specific dataset Da.pr.p2 with answer space cardinality | Here, [0, 1] queries in D are æ such that Pr[G(zz, 0) = y] = pı > 1.2 and
1 - a queries are 22 such that Pr Pr[ 0) = y] = p2 < 2. The following theorem qualitatively characterizes the performance of Vote on this dataset.
p2(1- (o(1-Pa)( pa) + 1. pf + P2 + 1 and K is odd, then F(K; D Dosp..2)) pi (1--))(pp: 1p)1
Theorem 2. Let L
increases monotonically, pf p2 and a 2 1 1
decreases monotonically, if + P2 < 1 or S -
increases and then decreases, Pf + P2 > and a < 7
decreases and then increases, if pl + p2 < 1 and a > 1-
Theorem @ precisely connects the query difficulty with the performance landscape. Here, t is a constant that only depends on P1 and p2. i.e., the probability of an LM's generation being correct on easy and hard queries, respectively. Intuitively, t quantifies the difficulty similarity between the easy and hard queries: becomes larger if the easy queries are more difficult (p) smaller) or the hard queries are less difficult (p2 is larger). Interestingly, it suggests that, for some query difficulty distribution, non- monotone effect of the number of LM calls is expected. Informally, if the overall task is "easy" (p1 + Ps > 1), but the fraction of "hard" queries is large (a < - 7), then as the number of LM calls increases, the Voting Inference Systems' performance increases first but then decreases. We call such landscape a "inverse U shape". Similarly, if the overall task is "hard" (p1 + P2 < 1), but the fraction of "hard" queries is small (a > 1 7). then enlarging the number of LM calls leads an initial decrease and then increase. Such a landscape is called "U shape". This well explains the U-shape of Inference Systems' performance shown Figure II Figure El visualizes the effects of query difficulty on the performance landscape in more detail.
4.2 What is the analytical scaling model?
Now we derive an analytical scaling model for both Vote and Filter- Vote. Noting that the perfor- mance is the average of F(K, x) for each in the dataset, the key challenge is identifying the shape of F(K,z) z) for easy and difficult queries. Let us first consider the special case |A| = 2, where we can obtain close form result for Vote.
Theorem 3. If |A|
2, then on any query x, the performance of Vote is F(K, x) = ta-1(1 1)b-ldt is the regu- I--dyte) (K+1) K, where La(a,b) & So" t-ta-1 ta-1(1 1)- ta-
larized incomplete beta function.
Thus, for Vote with |A| = 2, F),, = EEnD EewD(Li-syon K+1 K}
How about Filter-Vote and the general answer space? Admittedly, an exact scaling model is chal- lenging to obtain. Instead, we give an approximation model inspired by the special case. We first note that F(K,x) should be treated separately for difficult and easy queries: after all, as a function of a, the incomplete beta function Ig (a, a) monotonically increases/decreases if x > 2 (x < })) Second, (a, a) grows roughly exponentially in and this trend should hold for general answer space and for both Vote and Filter-Vort Hence, we propose the following scaling model
G(K,z) (a(e)K-a(aVVtaalt
fd(()) > 0,
1-e (a-eaaaa-aa ff((z) < 0

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where constants (1(2) > 0, c2(2) > 0, cx(x) do not depend on the number of LM calls K. There- fore, our analytical performance scaling model is G(K, D) [x-D|[((, In practice, one can use a training dataset Dr- to fit the parameters in G(K, D). Note that given query the pa- rameters e; (z) can be different for Vote and Votter-Vote. In particular, the filter is of high quality, then the performance should converge quickly, and thus the constants c;(+) are likely to be larger. Otherwise, the performance should scale slower, and thus the constants c(-) should be smaller. We will show in the experiments that G(K, D) matches the empirical performance F(K, D) accurately.
4.3 How to optimize the number of LM calls?
In general, one can always (i) fit the analytical scaling model G(K, D), and (ii) then use maxk G(K, D) obtain the optimal number of LM calls. Interestingly, we show that for special case, we can derive the optimal number LM calls.
Theorem 4. If pı + P2 > and a < 1. then the number of LM calls K that maximizes Da.pr.p)) for Vote (up to rounding) is
2p1 log T-o 1-2p2
K 2
log P2(1-P2)
p1
The optimal number of LM calls depends on the query difficulty. For example, K** will be larger grows (up to - )). That is, if there are more "easy" queries than "difficult" queries, then more LM calls should be adopted.
Experiments
We compare the empirical performance of Vote and Filter- Vote and the performance predicted by our analytical scaling model. Our goal is three-fold: (i) validate that there are cases where more LM calls do not monotonically improve the performance of these Inference Systems, (ii) justify that the number of LM calls has opposite effects on easy and difficult queries, and (iii) explore whether our analytical scaling model can accurately predict the performance of Vote and Filter-Vote, and thus guide the design of Inference Systems such as optimizing number of LM calls.
Datasets and LM. To understand the scaling properties of Vote and Voter we conduct sys- tematical experiments on both (i) real-world datasets and (ii) synthetic datasets with controlled query difficulties. Specifically, the wol-world datasets include MMLU PHYSICS [HBB*22 20], TRUTH- FULQA [LHE2T] GPQA [RHS*23), and AVERITEC ISGYZE! MMLU PHYSICS contains high school physics questions extracted from the original MMLU dataset. TRUTHFULQA measures whether language model is truthful in generating answers questions. GPQA queries are gener- ated by experts biology, physics, and chemistry. Each query in AVERITEC is a claim and the goal is to verify its correctness. Each query is prompted as multiple- evaluation. The details of these datasets and prompts can be found in the Appendix. The synthetic dataset is Da.p. p2 introduced in Section B, and study the scaling behavior by varying the parameters. We use CPPT-35--turo-O01 on the real-world datasets. All experiments are averaged over 1,000 runs.
question for objective
Non-monotonic Scaling Behavior. We start by understanding how the number of LM calls affects the performance of Vote and Filter-Vote empirically. As shown in Figure II we observe a non- monotonic behavior: more LM calls can sometimes lead to a drop in performance! This underscores the importance of scaling performance modeling.
A case study on AVERITEC. Now let us perform a case study on the AVERITEC dataset to understand the intriguing behavior better. In particular, we use the deployment partition of AVERITEC [SGV24]] which contains 500 fact verification questions. The goal is to determine given claim should (A) refused, (B) supported, or there (C) conflicting evidence or (D) not enough evidence. We evaluate the performance of both Vote and Filter- Vote on AVERITEC. In addition, we fit our analytical scaling model with 2, 5, 10, 20, 50, 100 LM calls. Then we use it to predict the performance of Vote and Filter- using 100 randomly drawn number of LM calls.
7

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36%
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dr(#r) (a) < 0
dx(z) > 0
(a) Overall performance
(b) Performance breakdown
(c) Scaling model prediction
Claim:
Forty percent of lowa's energy resources are from renewables
One LLM call
(D)
More LLM calls
29%
An easy query
Question: The above claim is?
(B)
(A) Refuse (B) Support (C) Conflicting
(C)
67%
Evidence (D) Enough Evidence
4%
Claim: All USA Ballots Arriving After Election Day Will Be Thrown Out Question: The above claim is? (A) Refuse (B) Support (C) Conflicting
One LLM call
(A)
More LLM calls
34%
(D)
A difficult query
56%
Evidence (D) Not Enough Evidence
10%
(d) Examples of easy and hard queries
Figure 4: A case study on the AVERITEC dataset. (a) more LM calls are invoked, the overall performance of Vote and Filter- Vote both initially increases but then decreases. (b) This U-shape can be perfected explained by the opposite effects on easy and difficult queries: More LM calls lead to higher performance on easy queries, but lower performance on difficult ones. (c) Our analytical
scaling model accurately predicts the empirical performance. (d) Examples an easy query and a difficult one. One LM call gives the correct answer with probability higher than any other answers (67%), ), and thus Vote with more calls eventually gives the correct answer. For the difficult query, probability of the correct answer (34%) is lower than that of an incorrect answer (56%). Thus, Vote with more LM calls eventually always generates a wrong answer.
As shown in Figure A. there are several intriguing behaviors. First, more LM calls do not always lead fact, the performance of both Vote and Filter- Vote increases first but then decreases as the number of LM calls increases from 2 to 1000 (Figure (())). The performance break- down (shown in Figure [b)) gives a natural explanation. More LM calls lead to higher performance when a query is relatively difficult (dr(x) performance when a query is relatively easy (dr(x) < 0 for Filter- Vote and dv(x) < for Vote). We also observe high correlation between the performance predicted by our proposed analytical scaling model and the empirical performance, as depicted in Figure (c). This implies the optimal number of LM calls can also be accurately predicted by our scaling model. Finally, we also give examples of one easy query and one difficult query in Figures @ (d). On the easy example, one LM call gives the correct answer with probability higher than any other answer (67%). Thus, more LM calls eventually give the correct answer. On the difficult query, the probability of the correct answer (34%) is lower than that of an incorrect answer (56%). Thus, Vote with more LM calls eventually always generates a wrong answer.
to better performance.
for Filter- Vote and (x) > 0 for Vote), but lower
Scaling model performance on real-world datasets. Next study how our analytical scaling model generalizes to other real-world datasets. In particular, we our analytical model with 2, 5, 10, 50, and 100 LM calls, and then use the scaling model to predict the performance 100 randomly drawn number of LM calls.
Figure shows the correlation between the predicted and empirical performance three real-world datasets, namely, MMLU PHYSICS, TRUTHFUL-QA and GPQA. We first note that the best per- formance of the Filter- Vote system is not necessarily better than that of the Vote system. Indeed,

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Predicted Performane no
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0.30 0.31 0.32 0.33 0.35 Empirical Performance
Empirical Performance
Empirical Performance
(a) MMLU PYHSICS
(c) TRUTHFULQA
(b) GPQA
Figure 5: How the proposed analytical scaling model performs other real-world datasets, namely, (a) MMLU PHYSICS, (b) TRUTHFULQA, and (c) GPQA. Here, we the scaling model with 5. 10, 50, and 100 LM calls, and then use the scaling model to predict the performance on 100 randomly drawn number of LM calls. Overall, we observe that our analytical model can predict the performance of the two compound systems accurately. Interestingly, the prediction accuracy on the Vote system is relatively higher than that on the Filter- Vote system. This is expected as the Filter- Vote system generalizes the Vote system and thus its scaling behavior can be more complex.
on TRUTHFULQA, the Vote system's 's best performance is higher than that of the Filter- Vote sys- tem. This further highlights the importance of performance prediction, even if there is no budget constraint. Second, we observe that our scaling model can predict the performance of both Vote and Filter- Vote systems accurately across all these real-worle datasets. This is because our scaling model carefully takes into account both difficult and easy queries and thus reflects the monoooo behavior. Interestingly, the prediction accuracy on the Vote system is higher than that on the Filter- Vote system. This is perhaps because the Filter- Vote system generalizes the Vote system and thus its scaling behavior can be more complex.
Difficulty distribution determines whether more LM calls help. To systematically understand how the query difficulty affects Inference Systems' performance landscape, we synthesize bi-level difficult dataset Da.pi p2. vary (i) the fraction of the easy subset and (ii) the query difficulty P1 P2. and then study the scaling performance of Vote. When it is clear from the context, we may also call (PI,P2 P2) query difficulty.
As shown in Figure B. observe that query diffieulty plays important role in the number of LM calls' effects. For example, when the difficulty parameter is (0.85, 0.1) and the fraction of easy queries is 0.6. Vote's performance is monotonically increasing as the number of LM calls grows. However, adding more hard queries by changing the fraction a = 0.4 changes the trend: the performance goes down first for small call numbers and then goes up for larger numbers of calls. is also interesting to notice an inverse "U""shape. For example, when query difficulty is (0.85, 0.4). there is clear U-shape performance. Overall, this justifies that (i) there are cases where more LM calls is not beneficial, and (ii) the diversity of query difficulty critically determines these cases.
Analytical scaling model predicts the optimal number of LM calls. Identifying the optimal number of calls is an important implication of the scaling properties. Here, we compare the optimal number of calls for Vote predicted by our analytical model and the optimal numbers empirically observed on the bi-level difficult dataset Da.pr.P2. As summarized in Table D (in the Appendix), the predicted optimal number is exactly the observed optimal number of LM calls, for all query difficulties evaluated. This validates the assumptions made by Theorem !!
6 Conclusion
In this paper, we systematically study how the number of LM calls affects the performance two natural inference strategy designs: majority voting (Vote) and majority voting after filtering results with an LM (Filter- Vote). We find that increasing the number of LMM calls can lead non-monotone behavior, egg. first increasing performance and then decreasing it. We offer theoretical analysis

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a (easy fraction) =0.4
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Figure 6: Overall performance of one-layer Voting Inference Systems on synthetic datasets with bi-level difficultyy. Overall, observe that increasing the number of LM calls does not necessar- ily lead to performance improvements. For example, when the query difficulty is (0.85, 0.4), the performance increases first and then decreases as the number of LM calls grows. When the query difficulty is (0.65, 0.1), there is reverse trend: the performance goes down first and then goes up. This validates our empirical observation that a larger number of LM calls and thus more resources may not necessarily result in better performance.
that attributes this phenomenon to the diversity query difficulties within a task, and conduct experiments validate our analysis. Furthermore, show how estimate the optimal number of LM calls to make for a given task using small number of queries to fit the parameters of our analytical model, thus helping practitioners optimize their system designs. Overall, our study shows that more LM calls are not necessarily better and underscores the importance of compound system design. To stimulate further research, we have released our code and datasets, available at Attps: //github. com com/Lchee001/Compooooo Iaap://///Coooonnnasaaaaaaaaaei
We hope our findings and analysis will inspire more research into maximizing the effectiveness inference time compute.
Acknowledgement
This work was supported in part by Google PhD Fellowship, Sloan Fellowship, NSF CCF 1763191, NSF CAREER AWARD 1651570 and 1942926, NIH P30AG059307, NIH U0IMH098953, grants from the Chan-Zuckerberg Initiative. Sutherland, and affiliate members and other supporters of the Stanford DAWN project and UC Berkeley SKY Lab, including Google, IBM, Intel, Lacework, Meta, Microsoft, Mohamed Bin Zayed University of Artificial Intelligence, Nexla, Samsung SDS, Uber, and VMware. BH is supported by a 2024 Sloan Fellowship in Mathematics, NSF CAREER grant DMS-2 143754, and NSF grants DMS-188 1855684 and RMMS-213390 213886. We also thank anonymous reviewers for helpful discussion and feedback.
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A Broader Impacts
Many inference strategies are often computationally and financially expensive deploy due to mul- tiple calls of language models. Our study suggests not blindly scaling up, but determining the resources allocated these inference strategies carefully. More broadly, our paper paves the way for data-centric inference strategy designs.
B Limitations
In this paper, we focus our analysis and experiments on the scaling behaviors for two instances of compound Al systems, but there are many other types of compound Al systems. For ease of evaluation, our experiments are conducted on relatively objective language tasks, but it remains open to study how the performance scales subjective tasks such as generating poems and writing essays. Another interesting open problem how to predict query difficult without querying the LMs. Note that in this work, we do not discuss the cost of LM calls; this is important dimension to weigh in practice, and in future work we will investigate how to balance cost, performance, and latency scaling.
C Missing Proofs
We give all missing proofs here.
C.1 Useful Lemmas
Lemma 5. Suppose D is 2-level difficult with (a,P1, P2) and ||||| = 2, and W.L.O.G. K is odd.
Then F(K; D) = alpp [K+1, K + (1 T)T pa (
( K where I (a, & fe ---1
// fo (a-1(1 1)0-1 is the regularized incomplete beta function.
Proof. To analyze how the performance scales as the network size increases, let us first expand the performance of Inference System by the law of total expectation
F(K: = E = y) = Σ Ely - y/r())r)))))))) = pi Pr((((( = p.l
(1)
Now consider Fl = y/r(x)) = p) for any p. Note that there are total possible generations (by
||-||| 2), so the estimated generation is correct if and only if more than half of the generations
correct. That is say, Ei-- ()) = p] Pr
I / 1- Pr(E 1 1, S
2.1 For ease of notation, denote Z Σ k-1 1 1z.=y. Note that 1 1z,=y is Bernoulli variable with parameter p and thus Z is a Binomial variable with parameter (K,p). Therefore, have
K
Pr(Z S w) = T TK w., w 1), where (a, b) A So0 (-= 6- di is the incomplete beta function. Thus, the above gives us
K
K
K
K 2
Ebi w/r(2) p| 1 I-
1+
+11
1p[l ا
If K odd, we can write it as IE'g = --/(()))))))))))))))>>))) = p] = Ip( K++ K+1) We can now plug this back into equation (II) to obtain
K +1
K+1
F(K; D)
Σ Elu = vlr(x) = p:] Pr(r(x)= pi] = Σ (
Pr(r(1)) = pil
2
2
i=1
1=1
K+1
K
K K+1
K+1
a pt
+ (1 T
2
2
2
2
which completes the proof.
Lemma 6. (MM 1, M + 1) L,M M) + M (1-p)
(2p 1), where Bla, b) 2 So ta-1(1
MBCMM
-d is the beta function.
19

---

Proof. Noting that 12(a + 1, Iz(a,b) p"(1-p)" aB(a.)) we have
pd1 p) 1
(M + 1, M +1 - (,M,
MBMM,M++1
4) (a + p1(1-p)" bB(a.)) we have
PM (1 p)
L(M,M+1)- (),,M) + MB(M,
Thus, 1p(M + 1, 1) (1-)) (-
MBMM M) M MCM. The Pascal's identity implies that B(a, 6
1) c+b ((a,b), b) and thus BMM, M +1 1)- = B(M, (MM,,)//2 M)/2. Thus, we have
MM (1))
p) MB(M, M
pM1
( - D)M
(1- A)
-
I,M + 1, 1) = MB(M, M)
MB(M,M)
MB(M,
Extracting the common factors gives - (1))
MB(M, MB(M,M) MB(M, [1 - 2(1 -p]] p (1 - PM
(1 (1)))
(2p 1)
MB(M, M)
(1-))))
That is, + 1,MM+,,M+)== + MB(M,M) (2p - 1), which completes the proof.
C.2 Proof of Lemma
Proof. Let us first consider Vote.
= a). We want to first note that lim Pr(ux = y"] = 1, where 3² 2 arg maxag Pr[G([r 0)
To see this, we can first write PríeK 1²] Pr[maxx44y Σ 1e,=a < Σ 1 -
Pr
1= K I = K
1 ==-"" <0. Va # y²). Subtracting from both sides, we have 1 - Pr(ůk =
0'1 -11-Prin PrrUu Di= 1 1m00 D 1 m > 0]. Now by union bound, we can bound the right hand side by Easy Pr Pr I, 1, > 0). Now, note that each term within the inner summation i.i.d. Furthermore, observe that the expectation is E[1 - 1 y - Pr]
a] - Pr/G 0) y"1 < O. Therefore, by law of large numbers, we have limx Pr|E 1,=== -.,=, 0] = Since there are only finite number of a E A, the sum of this quantity over all should also be O. Therefore, we have
K
K
lim - Pr(ux 1²1 lim Pr[Uase K-00
Σ
1
m > 0 = 0
K-000
1=1
Thus, limx Pr(ůk y²1=1. =1
Now, dr((x) implies y/ = y² and thus lim F(K, 1. dv(x)) implies y + y and thus lim F(K, D) 0.
Now let us consider Votter-Vote.
Abusing the notation a little, we claim that again lomg-+00 Prlpk = yr] = 1, where 4" A arg argmaNaes (r(G((x,)) 0) = al0(x, (z( 0), (z(x,0)) = 1), where recall that G1 0), G2(1,
G(I, 0). To see this, we can first write Pr(ix = y41 Pr|maxapy D/= 1; < ΣK
2i=1 w;] = Pr DK
K Pr(ůx - 1 = - 1x < Va + y²] Pr[Uust0 ΣK 1.,a
V*) Subtracting from both sides, we have
+= w; > 0]. Now by union bound, we can bound the right hand side by Easy Pr|E, 1,, 1. W, O.. Now, note that each term within the inner summa- tion i.i.d. Furthermore, observe that the expectation IF 2:= - 1 [ = Pr[Gi (z (z 0), G2(z,)) = |rPrGG(z, 0) = y", 2(x, G (z, G2(, = 1] < O. Therefore,
K
20

---

by law of large numbers, we have limg Pr[E only finite number of a E A, the sum of this quantity over all should also be Therefore, we have
= - 1 =, > 0] = 0. Since there are
K Σ
lim 1 Pr(ex - v²1 K00
lim Pr[Uapy K-000
1 12 wi 0] 0
km
Thus, lim Príůx = y²] = 1.
Now, dr(x) > implies y" and thus limg F(K, D) = 1. dr(x) > 0 implies + y² and thus limx- F(K, D) 0.
Hence, we have shown that dy (x) and (x) are difficulty indicator for Vote and Filter- Vote, re- spectively.
C.3 Proof of Theorem
Proof. We construct recurrent relation F(K;)) by Applying Lemma S
F(K + D) D)
K 33
K
+3
K
+
K +3
K 1
K +
K ++1
++
-alp
+ (1
4p
-alpp
+ (1-1a)
m)
2
2
2
2
2
2
2
K3
K
+ 3
K +1
K +
1
K+3
3
K +
3
K +1
K +
=a(Tp/
- Ipp
1) + (Tpp
pa
))
2
2
2
2
2
2
2
2
For ease of notation, let us denote M
K+1 2
Applying Lemma B leads to
F(K + D) F(K; D)
P (1- - M
(2p1 1)] + (1 ao) a)) 12 -
(2p2 1)]
MB(M,M)
MB(M, M)
MB(M, M) (a: (1 pi) (1- M1 1) +1 (1-
D (1 p2) 1)]
M
Now rearranging the terms gives
F(K + 2; - D)
a(2p1 1)
P1(1 P1) MM 1]
MB(M, (1 - a1- M (1 - -22) P2) (M (1-2p2) 11 - a)( 2p2) P2(1 - p2)
al2p pi(1-p1) M - 1. Note that AF(M) is mono- For ease of notation, denote (F [1 - 2p3) p=(1-p3)
tonically increasing or decreasing, depending on the parameters P1, It is also easy to show that
a 2 1- 1 if and only AF 2 Now, we are ready to derive the main results by analyzing A AF(M).
P2 > 1 and a 2 1- Now pr(l-p)) > 1, and thus AF(M) is monotonically in- pall-ps]
creasing. Furthermore, it is easy to see lima--xo AF(M) Furthermore, AF(() 2 0. That is, for any given M, A AF(M) is non-negative and thus F(K; D) must be monotoni- cally increasing
Pi P2 1 and < 1 :: Now pi(1-Pt) 1, and thus AF(M) is monotonically
p2(1-P2)
increasing. Furthermore, is easy to see lim (F(M) xc. Furthermore, AF(1) < O. That is, as K and thus M increases, AF(M) is negative and then becomes positive. Therefore, F(K; D) must decrease and then increase
P1+ < 1 and a > 1 1- Now (1(L-p1) p2(1-p²) < L, and thus AF(M) is monotonically decreas- ing. Furthermore, it is easy to see lim AF(M) = -1 Furthermore, AF(1) > 0. That is, as K and thus M increases, A (FM) positive and then becomes negative. There- fore, F F(K; D) must increase and then decrease
21

---

Pi P2 < 1 Laadd a I 1-17: Now pn(l-pi) < 1, and thus AF( is monotonically decreas-
pz(l-Pa)
ing. Furthermore, That is, for any given M, (F(M) is non-positive and thus F(K; D) must be monotoni- cally decreasing
easy see lim AF(M) = -1. Furthermore, AF(1) < 0.
Thus, we complete the proof.
C.4 Proof of Theorem E
Proof. Let us first consider F(K, x) = Pr(ux = y]. Note that there are in total 2 possible genera-
tions (by |||]| = 2), so the estimated generation i is correct if and only if more than half of the gener-
ations is correct. That is to say, Flux = y) Pr|E mu= > / = 1 1u-=y S 2 K For ease of notation, denote 7 k= z. Note that 1l =y is Bernoulli variable with param- eter p and thus Z is Binomial variable with parameter (K, Therefore, we have Pr[Z S w/ =
I1-p(K w, w +11,, 1), where (a, b) _ _o² -- (1 - +) fo t- (1 11-th 11-1)==11 dt is the incomplete beta function. Specifically, Noting that by definition, (z) 1 2p. we can rewrite this as F(K,x) = I-- Taking the integral over completes the proof.
K and get F(K,2) Pr(ůk y) h1-p( K+1 K+1
set w 2
C.5 Proof of Theorem B
Proof. Note that the optimal number of LM calls must ensure that the incremental component
AF(M) O. That is, the optimal value M must satisfy AF(M*) (1-o](1-2pa) P2 P2-
a(2p)-1) p1 M
1 0. Solving the above equation gives us a unique solution
log I-a 1-2ps
2p
M
p=(1-pa) log p1 (1-Pt)
Noting that K* [2M]] completes the proof.
D Additional Experiment Details
D.1 Datasets
We evaluate Vote and Filter-Vote four real-world datasets, namely. GPQA IRHS 23], AVERITEC [SGV244), TRUTHFULQA [LHE21]] and MMLU PHYSICS [HBB 20). AVERITEC fact verification dataset, where each query is claim (e.g.. "All USA ballots Arriving after elec- tion day will be thrown out"), and the goal is to determine if this claim should be refused, supported, or there is conflicting evidence not enough confidence. We use the "development" partition offered in the original paper, which contains 500 queries. GPQA is dataset consisting of multiple- choice questions written by domain experts in biology, physics, and chemistry. We the diamond partition, which contains 198 queries, and as reported by the original paper, enjoys the highest ex- pert accuracy. TRUTHFULQA is another widely used multiple-choice question-ansee answering dataset. We uses the validation parititon, which contains 817 questions spanning 38 categories, including health, law, finance, and politics. MMLU PHYSICS is the high school physics subset of the MMLU dataset, which contains 151 questions. All queries are prompted as multiple-choie questions for ease of evaluation.
Liscense. GPQA is released under the MIT License. AVERITEC uses Creative Commons AtributtionNonCommeer 4.0 International License. TRUSTFULQA and PHYSICS (as part of MMLU) both adopt the Apache License 2.0
D.2 Prompting for generators and filters
We provide the prompts used for the generators and filters in the following boxes.
22

---

Table 2: Optimal number of LM calls prediction. For each data distribution with specific 2-level difficulty parameters P. P2, we sample 100 data points, employ a simulated Voting Inference System with 1000 number of LM calls, estimate the parameters by their empirical mean and adopt the analytical optimal number of LM calls predictor. Across evaluated query difficulties, our pre- dicted optimal number of I.M calls exactly matches the optimal number empirically observed.
Optimal Number LM Calls
α
P1
P2
P1
P2
Analytical/Empirical
0.4
0.85
0.84
0.40
0.4
0.85
0.3
0.38
0.85
0.30
0.4
0.75
0.4
0.41
0.74
0.40
0.4
0.75
0.3
0.44
0.75
0.30
0.4
0.65
0.4
0.41
0.65
0.40
0.5
0.85
0.4
0.50
0.84
0.40
0.5
0.85
0.3
0.50
0.85
0.30
0.5
0.75
0.4
0.51
0.75
0.40
7
0.5
0.75
0.75
0.30
0.5
0.65
0.4
0.48
0.65
0.40
15
0.6
0.85
0.4
0.59
0.84
0.39
5
0.6
0.85
0.3
0.63
0.85
0.30
0.6
0.75
0.4
0.61
0.75
0.40
11
0.6
0.75
0.3
0.59
0.75
0.30
11
0.6
0.65
0.4
0.62
0.65
0.40
30
Prompt for the generator G()
Please answer the following question. You should first analyze it step by step. Then generate your final answer by the answer is (X).
Q: [Query]
A:
Prompt for the filter &()
[User Question]: (query)
[Answer]:
(answer [nswer)
Instruction: Review your previous answer and find problems with your answer. Finally, conclude with either [[correct]] if the above answer is correct or [[wrong]] if it is incorrect.
Think step by step.
Verdict:
D.3 Generation Setup
We set up the temperature as 0.1 since all the evaluation tasks are objective and have clear correct answers. For each question, query GPT-3.5-turbo-012 400 times. Then for each F(K, D), we randomly sample K answers from the 400 answers with replace to simulate the performance once, and report the average over 1000 runs.
D.4 The optimal number of LM calls predicted by the scaling model
Table @ compares the optimal number of LM calls predicted by the scaling model and the ground- truth value the bi-level dataset D Papi Overall, We observe that they match very well.
23

---

3.00335
1.0be00
2.30
2:30m04
2:30a=05
2:31e-05
2.79-004
3.00e0h
3.0 3.09--05
0.0004
3.00225
2.41a-05
8.36w-25
1.04e-04
0.0003
2
1.506-04
b.21e:05
2:14w-04
3.95w-03
2.00320
3.00015
0.0001
0.0001
6
2.00005
(a) a = 0.4
(b) 0.5
(c) 0.6
Figure 7: Mean square error of the performance predicted by our proposed scaling law on synthe- sized datasets with varying bi-level difficulties. Here, we fit the scaling law by performance evalu- ated K 1, 2, 3, 4, 5, and evaluate performance for K from 100. Overall, we observe that the predicted performance accurately matches the empirical evaluation.
D.5 Performance Estimation via Our Proposed Scaling Law
Can proposed scaling model predict the empirical performance of Vote accurately? To answer this, we apply it to each dataset considered Figure f In particular, we feed our estimator with and then
the Inference Systems' performance with the number of LM calls being 1, 2, it to predict the performance for LM calls within the range from to 100. Note that this is quite challenging. the estimator needs to extrapolate, i.e., predict the performance of Voting Inference System whose number of I.M calls is much larger than any number seen the training data.
As shown in Figure A the performance predicted by our estimator accurately matches the empirical observation. Across all query difficulty parameters, the mean square error ranges from 6 to le 4. This suggests that predicting how the number of LM calls affects Voting Inference System feasible and also indicates that our scaling law captures the key performance trend effectively.
24


Reviews:
[{"review_id": "yhz19lTOPu", "reviewer": "Reviewer_sDbu", "summary": "Recent state-of-the-art results in language tasks have been achieved using compound systems that make multiple calls to Language Models (LMs) and aggregate their responses. However, there is limited understanding of how the number of LM calls affects the performance of these compound systems. This paper studies the scaling properties of compound inference systems, specifically analyzing how the number of LM calls impacts the performance of two simple compound system designs, Vote and Filter-Vote, which use majority voting to aggregate LM responses, with or without applying filters. The analysis, both theoretical and empirical, reveals that the performance of Vote and Filter-Vote systems can initially improve but then decline as the number of LM calls increases. This non-monotonic behavior is attributed to the varying difficulty of queries within a task: more LM calls enhance performance on \"easy\" queries but reduce it on \"hard\" queries.", "questions": "Are there methods beyond Vote and Vote-Filter that can be applied here? And can this approach generalize if difficulties are probabilistic to begin with?", "limitations": "The paper focuses on analyzing and experimenting with the scaling behaviors of two specific instances of compound AI systems, though there are many other types. The experiments are conducted on relatively objective language tasks for ease of evaluation. It remains an open question how performance scales on more subjective tasks, such as generating poems and writing essays. Another open problem is how to predict the difficulty of queries without actually querying the language models.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The findings of the authors are of definite interest, and the phenomenon that they identify is thought provoking, although mathematically intuitive. I think the paper has potential for good to high impact. The problem itself is conveyed very clearly, and the authors made the right choice in including the 'main contributions' box and Figure 1 on the first page of the paper itself. \n\nThe theory and substance of the paper is strong, and worth the status of the conference. \n\nThe study also provides a method to determine the optimal number of LM calls that maximizes system performance, based on a small number of samples, and develops an analytical scaling model for both systems. Experiments confirm that the scaling model can accurately predict the performance of Vote and Filter-Vote systems, identifying the optimal number of LM calls.", "weaknesses": "One weakness is in the insistence on using multiple choice benchmarks, while not being more expansive about using more benchmarks (for example, there are many common sense benchmarks that are also multiple choice). Another weakness is that I still don't quite buy the way in which difficulty was determined experimentally. However, the question itself is a provocative one, and I'm not sure that any answer is really perfect. The paper does a good job at starting the conversation and showing one way in which all of this can be formalized.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the helpful feedback and support of our paper! We have addressed the questions as follows.\n\n***Are there methods beyond Vote and Vote-Filter that can be applied here? Can this approach generalize if difficulties are probabilistic to begin with?***: Thanks for bringing up this point. We indeed believe that our methods can apply beyond just Vote and Vote-Filter. We chose to focus on those two systems because … [paste] … We have in mind to do some followup work on more complex systems. One example is Ranking-Vote: Each LLM call provides a ranking of possible answers, and then we perform ranking aggregation (e.g., Borda count) to produce a final answer. Here it’s indeed important to consider probabilistic query difficulty. The effect will be significant because while in Vote and Vote-Filter we have a law of large numbers that governs the resulting prediction (polarizing the difficulty to a binary easy or had), in more complex models, we expect that the full distribution of query difficulty over a dataset will impact model performance. \n\n***Can this approach generalize if difficulties are probabilistic to begin with?***: Yes. For example, one might take a Bayesian view: difficulties can be initially quantified as a prior distribution, and invoking more LLM calls offers a more accurate estimation of the posterior.", "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": "rGrpZcm9cE", "reviewer": "Reviewer_A9Tt", "summary": "In this paper, the authors studied the scaling properties of compound inference systems. Theoretically and empirically, the authors answered the properties of multiple LM calls.", "questions": "1. Is the optimal number of LM calls model-specific or not?", "limitations": "1. According to definition 1, a query can only be categorized into easy and difficult according to the difficulty indicator. The is a simplified case of the real-world scenarios. Besides, the  difficulty indicator is not model-agnostic, which may also reduce the significant of findings.\n\n2. The optimal LM calls depends on query difficult while the difficult level is simply either difficult or easy. A more rational way is to measure the least number of calls for a query to get correct answer.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. Answered several important questions in multiple LM calls, which may benefit the compound inference systems.\n\n\n2. Heuristic for Optimal Number of LM Calls.", "weaknesses": "1. The authors only studied two simple  natural inference system designs, i.e., vote and filter vote, which might reduce the significant of findings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the helpful feedback and support of our paper! Please see below for our response.\n\n***Simple System***: We indeed focus on simple systems. There are  two reasons. First, they both represent real-world systems. For example, the Cot@32 approach by Google Gemini is indeed a simple vote method. Second,  the observed phenomenon is pretty surprising, and we want to understand it empirically and analytically in the simplest place it appears. We believe it is always important to find a minimal working example. Of course, it will be valuable to investigate in challening settings in future work. \n\n***Is the optimal number of LM calls model-specific or not?***: Thanks for this great question. We actually predict the optimal number of calls in two ways (K* for a special case and the scaling law function G in general). They are both model-specific. This is an important point and we will emphasize it in the revision.\n\n***the least number of calls for a query to get correct answer***: Thank you so much for pointing this out. We think this is a great intuitive way to explain how to characterize query difficulty. It is very closely connected, in fact, to the characterization we currently use. It is a kind of soft version of our difficulty level. Indeed, let’s take Vote as an example.\n\n$$\nd_V(x) = \\max_a \\Pr[G(x, \\theta) = a] - \\Pr[G(x, \\theta) = y]\n$$\n\nHere, the inverse of $Pr[G(x,\\theta)=y]$ is the expected number of calls to obtain at least one correct answer, which is one formalization of the reviewer’s soft difficulty. We focus on the binary difficulty because it offers a natural explanation of the non-monotonic behavior. We appreciate the suggestion for providing this nice intuition and will put a sentence about it into the revision.", "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": "R9OdWIZwSC", "reviewer": "Reviewer_DYM5", "summary": "This paper evaluates the LM calls and the task performance based on two compound system designs: Vote and Filter-Vote, for performing multiple-choice selection tasks. The authors conduct theoretical analysis for the system designs and the scaling behavior by proposing formal notion of query difficulty and model behavior. They suggest that adding LM calls does not always lead to monotonic task performance. Basically, additional LM calls increase performance on easy queries and decrease performance on difficult queries; increasing the number of LM calls can lead to non-monotone behavior depending on task difficulty. The authors also conduct empirical analysis on different datasets including one synthetic dataset to validate the claims, and notice that the analytical scaling model can accurately predict the performance of Vote and Filter-Vote.", "questions": "1. Since all your datasets are multiple choice tasks, how did you extract the results of the LLM outputs for answer mapping? Perhaps the token logits? If that is the case, did you check the robustness of the results of the token logits comparing to the real textual outputs?\n2. Did you (Or do you think whether it is worthwhile to) conduct additional robustness tests (e.g. by shuffling the option orders, or adding input perturbations) to validate the LLM outputs?\n\nRecent evaluation papers that could be relevant, they more or less discuss that the difficulty of the tasks could affect model robustness: https://aclanthology.org/2024.naacl-long.295/\nhttps://aclanthology.org/2024.findings-naacl.130/", "limitations": "The authors already talk about the Limitations in Appendix B.", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper offers a novel insights to the relation of task difficulty, LM calls, model behavior. \n2. The overall writing of the paper is very easy to follow. Throughout the writing, the paper has made the main scope clear.\n3. Both the theoretical analysis and empirical experimentation are well conducted to support the main claim.", "weaknesses": "Although I found that the authors have made solid empirical experimentation to support the claim, the empirical results are not well presented. The authors experimented with several different datasets, but only showed a few case studies on a certain dataset in the main body of the paper. I would at least present the overall results of all datasets in the experiment section if the page limit permits.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the helpful feedback and support of our paper! Please see below for our response.\n\n***present the overall results of all datasets in the experiment section if the page limit permits***: Thanks for your suggestion! We will move more empirical results to the experiment section in the revised paper.\n\n\n***answer extraction***: We prompted the LLM to generate its final output in a specific format (i.e., “the answer is (X)”) and then used regular expression to extract the output.\n\n\n***Did you (Or do you think whether it is worthwhile to) conduct additional robustness tests (e.g. by shuffling the option orders, or adding input perturbations) to validate the LLM outputs?***: Thank you very much for pointing this out! Robustness studies are orthogonal but complementary to the focus of this paper. For example, it would be interesting to study how input perturbation affects query difficulty distribution. We will add a discussion in the revision.\n\n***Recent evaluation papers***: Thanks for the reference. We will add a discussion to them in the revised paper.", "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": "F5iwPaj1C8", "reviewer": "Reviewer_NgrA", "summary": "LLM inference systems often generate multiple answers (function calls) to a query and then aggregate the answers with rules like vote, filter-vote. This paper investigates how the number of function calls influence the influence of performance of the compound system. More concretely, this paper found that there often exist a U-shape overall performance curve with an increasing number of function calls where easy queries benefit from more function calls and hard queries can be harmed by the function calls. Some theoretical results are given for toy settings and an optimal number of function calls can be derived accordingly. Empirical results are given for simple settings.", "questions": "As stated in the weakness.", "limitations": "Not applicable.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- Clarity: The paper is mostly clear and easy to understand.\n- Originality: The problem is less studied in the literature.", "weaknesses": "**1. The results are not surprising.**\n\nThe problem is quite simple and straightforward if we drop the LLM background. Basically, we are drawing samples uniformly at random from a distribution (function calls) and then apply majority vote to them, then with more samples, the uniform + majority vote leads to select the one sample with highest probability (if majority exists). If the majority is the correct answer, then the sample is \"easy\", otherwise it is \"hard\". Mixing easy and hard majority naturally leads to U-shape in the performance curve.\n\n\n**2. The model prediction may not outperform simple hyperparameter-search.**\n\nThe \"optimal\" function calls prediction is based on strong assumptions which may not hold true and estimate accurately in practice. Instead, simple hyperparameter search is usually much more robust performance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the feedback. We answer the questions as follows.\n\n***The results are not surprising***: We appreciate your comment but we would like to respectfully argue that our results are interesting. First, many practitioners and researchers believe that they can consistently improve performance with more LLM calls (e.g. https://arxiv.org/abs/2402.05120). In this context, our finding that more LLM calls can hurt performance in specific settings is unexpected and a useful contribution to the literature.  Furthermore, we provide an explanation of this counterintuitive phenomenon based on query difficulty and provide both mathematical and empirical justifications. The prediction based on our theory matches the empirical performance accurately. These results have not been not studied in existing work and opened the door to understanding and optimizing the design of compound LLM systems. \n\n\n***model prediction is based on strong assumption***: Thank you for this question. We would like to clarify that the algebraic formula (K*) shown in Theorem 4 is used for a special case. On real-world datasets, we use the scaling law function empirically fitted on a small dataset for prediction purposes, which does not require strong assumptions. The scaling law function’s predictions are empirically accurate (see Figure 4(c)). We will highlight this in the revised paper.\n\n***Comparison with hyper-parameter search algorithms***: Thank you very much for pointing this out! We conducted additional experiments to compare our proposed scaling laws with two popular hyper-parameter search algorithms, bayesian optimization and tree structured parzen estimator, with the same number of performance evaluation budgets. Specifically, we evaluate these hyper-parameter search algorithms on GPQA, AVERITEC, and MMLU PHYSICS, with 5 calls to the performance oracle (the same input to our scaling law method). As shown in the following table, our scaling law reaches the most accurate prediction and overall accuracy. We will add this discussion in the revised version. Notably, both the Parzan and Bayesian optimizers would have recommended the users to use a large number of LLM calls (>100), which would have incurred much more cost while performing worse than our estimator.  \n\n| Method                            | GPQA        |         | AVERITEC    |         | MMLU PHYSICS |         |\n|:---------------------------------:|:-----------:|:-------:|:-----------:|:-------:|:------------:|:-------:|\n|                                   | LLM Calls   | Acc     | LLM Calls   | Acc     | LLM Calls    | Acc     |\n| **Our scaling law**               | **19**      | **0.317** | **5**       | **0.367** | **13**       | **0.542** |\n| Tree-structured parzen estimator  | 444         | 0.307   | 114         | 0.361   | 418          | 0.530   |\n| Bayesian optimization             | 430         | 0.307   | 377         | 0.360   | 47           | 0.536   |\n| Ground truth                      | 13          | 0.320   | 4           | 0.368   | 10           | 0.543   |", "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:
We thank all reviewers for their feedback! Please find our answers and clarifications in the individual responses.

Meta-review: Increasingly, systems designed to perform natural language tasks aggregate outputs from multiple calls to LLMs. This work asks an incisive question about such approaches: What is the relationship between the number of LM calls and systems performance? Intriguingly, in the straightforward case of multiple choice QA, this relationship is non-monotonic. In addition to this empirical finding, the authors provide a formalism to analyze LLM calls and offer a heuristic for the optimal number of calls to make for a task (estimated on the basis of a small number of queries). 

The biggest weakness of the work is that the empirical evaluation is somewhat limited here, as it is restricted to multiple choice QA datasets. Nonetheless, there was a consensus amongst reviewers that the question being raised and the initial empirical evaluation (as well as formalism) on offer is likely of broad interest to the community, and may provoke discussion and follow-up work.
Decision: Accept (poster)