Adaptivity Is Exponentially Powerful for Testing Monotonicity of Halfspaces

We give a poly(log(n),1/epsilon)-query adaptive algorithm for testing whether an unknown Boolean function f:{-1, 1}^n -> {-1, 1}, which is promised to be a halfspace, is monotone versus epsilon-far from monotone. Since non-adaptive algorithms are known to require almost Omega(n^{1/2}) queries to test whether an unknown halfspace is monotone versus far from monotone, this shows that adaptivity enables an exponential improvement in the query complexity of monotonicity testing for halfspaces

Adaptivity Helps for Testing Juntas

We give a new lower bound on the query complexity of any non-adaptive algorithm for testing whether an unknown Boolean function is a k-junta versus epsilon-far from every k-junta. Our lower bound is that any non-adaptive algorithm must make Omega(( k * log*(k)) / ( epsilon^c * log(log(k)/epsilon^c))) queries for this testing problem, where c is any absolute constant <1. For suitable values of epsilon this is asymptotically larger than the O(k * log(k) + k/epsilon) query complexity of the best known adaptive algorithm [Blais,STOC\u2709] for testing juntas, and thus the new lower bound shows that adaptive algorithms are more powerful than non-adaptive algorithms for the junta testing problem

An adaptivity hierarchy theorem for property testing

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adaptivity" it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in k+1 rounds, where the queries in the i'th round may depend on the answers obtained in the previous i-1 rounds. Then, we ask the following question: Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity? We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every n in N and 0 <= k <= n^{0.99} there exists a property Pi_{n,k} of functions for which (1) there exists a k-adaptive tester for Pi_{n,k} with query complexity tilde O(k), yet (2) any (k-1)-adaptive tester for Pi_{n,k} must make Omega(n) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs

An Adaptivity Hierarchy Theorem for Property Testing

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adaptivity" it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in k+1 rounds, where the queries in the i\u27th round may depend on the answers obtained in the previous i-1 rounds. Then, we ask the following question: Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity? We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every n in N and 0 <= k <= n^{0.99} there exists a property Pi_{n,k} of functions for which (1) there exists a k-adaptive tester for Pi_{n,k} with query complexity tilde O(k), yet (2) any (k-1)-adaptive tester for Pi_{n,k} must make Omega(n) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs

An adaptivity hierarchy theorem for property testing

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adaptivity" it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in k+1 rounds, where the queries in the i'th round may depend on the answers obtained in the previous i-1 rounds. Then, we ask the following question: Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity? We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every n in N and 0 <= k <= n^{0.99} there exists a property Pi_{n,k} of functions for which (1) there exists a k-adaptive tester for Pi_{n,k} with query complexity tilde O(k), yet (2) any (k-1)-adaptive tester for Pi_{n,k} must make Omega(n) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs

How social learning strategies boost or undermine decision making in groups

Social interactions resulting in emergent collective behaviour play a key role in almost all layers of society, from local, small-scale interactions, such as people crossing the street, to global, large-scale interactions, such as the spread of fake news on online platforms. In our digital and interconnected world, it is increasingly important to understand the emergence of beneficial or detrimental collective dynamics. The characteristics of such dynamics are expected to depend greatly on the nature of information individuals have personally acquired and how they learn from others. Yet, how the decision-making processes shape the resulting collective dynamics remains poorly understood. When do individuals seek more information from social sources? How do individuals reap the benefits when navigating in social environments, and when do they fail to do so? This dissertation aims to answer these questions extending established theories and frameworks from individual decision-making into the social realm. This approach allows for the operationalization of personal and social information in a theory-driven manner, thereby achieving a deeper understanding of the individual-level decision process. The first chapter provides a introductory overview of the interplay between personal information use, social learning strategies and collective dynamics, and introduces the key theories and models I will expand on in this dissertation. In Chapter 2, inspired by Brunswick's lens model, I investigate how individuals form beliefs about the meaning of ecological structures (i.e., cues). Here, participants had to categorize images based on multiple cues, the meaning of which had to be learned over trials. I showed that participants observing the same cues formed different beliefs about the cue meanings. This diversity in cue beliefs is, in turn, an important process governing the quality of social information. The greater this diversity, the more independent personal information is, and the stronger the potential for social information use. Participants, however, failed to realize the full potential of this diversity because they only changed their personal decisions if a large majority disagreed with them. Simulating different strategies of social information use, I show that this reliance on strongly agreeing majorities impedes individuals from benefiting from diversity. This chapter thus identifies diversity in cue beliefs as an important factor allowing individuals in groups to benefit from the wisdom of each other, while simultaneously highlighting the importance of the individuals' social learning strategies to exploit this diversity. Chapter 3 dives deeper into the social learning strategies individuals use. By carefully controlling the social information displayed to participants, the study in this chapter provides an in-depth analysis of social learning strategies. Participants were confronted with an estimation task. They first provided an independent estimate, after which they observed estimates of others. Using Bayesian modelling techniques, I show that the incorporation of others' opinions strongly depends on how consistent those opinions are with an individual's own opinion and the degree of agreement among others. Individuals also strongly differ in the social learning strategies they use. These results elucidate what aspects are conducive for people to change their minds and contribute to the understanding of how individuals’ social information use shapes opinion and information dynamics in our interconnected society. In Chapter 4, I embed individuals a in temporal dynamic system which allows the investigation of the use of information in interaction with the emergent collective dynamic. Here, my focus is on social interactions where multiple people make decisions sequentially and thereby are simultaneously emitters and receivers of social information. To shed light on the unfolding dynamic in such settings, I will introduce the social drift-diffusion model (DDM). The model allows the investigation of the cognitive processes underlying the integration of personal and social information dynamically over time, and the subsequent collective dynamic. Analysis of the data shows that correct information spreads when the participants’ confidence reflects accuracy and when more confident participants decide faster. Under these conditions, later-deciding participants are likely to adopt social information and thereby to amplify the correct signal provided by early-deciding participants. The social DDM successfully captures all the key dynamics observed in the social system, revealing the cognitive underpinnings of information cascades in social systems. The general principles of personal and social information use that emerge from Chapter 4 allow to investigate the optimal behaviour when deciding sequentially. In Chapter 5, I develop an agent-based version of the social DDM and embed it in evolutionary algorithms, allowing the identification of evolutionarily advantageous strategies. I show that the individuals' decision time should depend on the quality of information, with the most accurate individuals deciding first. For all later.deciding individuals it is evolutionary advantageous to imitate the (often accurate) first decision. When introducing asymmetric error costs, single individuals should develop response biases to avoid the more costly error. In groups, however, such response biases can have dramatic consequences, as these biases are likely to be amplified in the group. As a result, individuals in large groups should use much weaker response biases to benefit from social information. I conclude that individuals facing asymmetric error costs in social environments need to carefully trade off the expressed response bias and sensitivity to social information to avoid the more costly error but simultaneously benefit from the collective. Overall, this thesis deepens our understanding of social dynamics by accounting for individual-level decision-making processes across various choice problems. I show that the behaviour of individuals in social environments can significantly differ depending on the personal information individuals possess and the strategies individuals use. Furthermore, I highlight the importance of accounting for such differences to predict the emergence of beneficial or detrimental dynamics in social environments