/pdf/sequential-design-of-experiments-3hb9thw5u7.pdf

Sequential Design of Experiments

We provide a complete characterization of the complexity of best-arm identification in one-parameter bandit problems. We prove a new, tight lower bound on the sample complexity. We propose the 'Track-and-Stop' strategy, which is proved to be asymptotically optimal. It consists in a new sampling rule (which tracks the optimal proportions of arm draws highlighted by the lower bound) and in a stopping rule named after Chernoff, for which we give a new analysis.

/pdf/optimal-best-arm-identification-with-fixed-confidence-58mbo93b1d.pdf

Optimal Best Arm Identification with Fixed Confidence

We give a complete characterization of the complexity of best-arm identification in one-parameter bandit problems. We prove a new, tight lower bound on the sample complexity. We propose the `Track-and-Stop' strategy, which we prove to be asymptotically optimal. It consists in a new sampling rule (which tracks the optimal proportions of arm draws highlighted by the lower bound) and in a stopping rule named after Chernoff, for which we give a new analysis.

Sequential Tests of Statistical Hypotheses.

Outlier hypothesis testing is studied in a universal setting. Multiple sequences of observations are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are distributed according to an outlier distribution, distinct from the typical distribution governing the observations in all the other sequences. Nothing is known about the outlier and typical distributions except that they are distinct and have full supports. The goal is to design a universal test to best discern the outlier sequence(s). For models with exactly one outlier sequence, the generalized likelihood test is shown to be universally exponentially consistent. A single-letter characterization of the error exponent achievable by the test is derived, and it is shown that the test achieves the optimal error exponent asymptotically as the number of sequences approaches infinity. When the null hypothesis with no outlier is included, a modification of the generalized likelihood test is shown to achieve the same error exponent under each non-null hypothesis, and also consistency under the null hypothesis. Then, models with more than one outlier are studied in the following settings. For the setting with a known number of distinctly distributed outliers, the achievable error exponent of the generalized likelihood test is characterized. The limiting error exponent achieved by such a test is characterized, and the test is shown to be asymptotically exponentially consistent. For the setting with an unknown number of identically distributed outliers, a modification of the generalized likelihood test is shown to achieve a positive error exponent under each non-null hypothesis, and also consistency under the null hypothesis. When the outlier sequences can be distinctly distributed (with their total number being unknown), it is shown that a universally exponentially consistent test cannot exist, even when the typical distribution is known and the null hypothesis is excluded.

Universal Outlier Hypothesis Testing

We consider the problem of detecting an odd process among a group of Poisson point processes, all having the same rate except the odd process. The actual rates of the odd and non-odd processes are unknown to the decision maker. We consider a time-slotted sequential detection scenario where, at the beginning of each slot, the decision maker can choose which process to observe during that time slot. We are interested in policies that satisfy a given constraint on the probability of false detection. We propose a variation on a sequential policy based on the generalised likelihood ratio statistic. The policy, via suitable thresholding, can be made to satisfy the given constraint on the probability of false detection. Furthermore, we show that the proposed policy is asymptotically optimal in terms of the conditional expected stopping time among all policies that satisfy the constraint on the probability of false detection. The asymptotic is as the probability of false detection is driven to zero. We apply our results to a particular visual search experiment studied recently by neuroscientists. Our model suggests a neuronal dissimilarity index for the visual search task. The neuronal dissimilarity index, when applied to visual search data from the particular experiment, correlates strongly with the behavioural data. However, the new dissimilarity index performs worse than some previously proposed neuronal dissimilarity indices. We explain why this may be attributed to some experiment conditions.

Learning to Detect an Oddball Target

We consider the problem of detecting an odd process among a group of Poisson point processes, all having the same rate except the odd process. The actual rates of the odd and non-odd processes are unknown to the decision maker. We consider a time-slotted sequential detection scenario where, at the beginning of each slot, the decision maker can choose which process to observe during that time slot. We are interested in policies that satisfy a given constraint on the probability of false detection. We propose a generalised likelihood ratio based sequential policy which, via suitable thresholding, can be made to satisfy the given constraint on the probability of false detection. Further, we show that the proposed policy is asymptotically optimal in terms of the conditional expected stopping time among all policies that satisfy the constraint on the probability of false detection. The asymptotic is as the probability of false detection is driven to zero. 
We apply our results to a particular visual search experiment studied recently by neuroscientists. Our model suggests a neuronal dissimilarity index for the visual search task. The neuronal dissimilarity index, when applied to visual search data from the particular experiment, correlates strongly with the behavioural data. However, the new dissimilarity index performs worse than some previously proposed neuronal dissimilarity indices. We explain why this may be attributed to the experiment conditons.

Learning to detect an oddball target

We highlight the usefulness of city-scale agent-based simulators in studying various non-pharmaceutical interventions to manage an evolving pandemic. We ground our studies in the context of the COVID-19 pandemic and demonstrate the power of the simulator via several exploratory case studies in two metropolises, Bengaluru and Mumbai. Such tools may in time become a common-place item in the tool kit of the administrative authorities of large cities.

/pdf/city-scale-agent-based-simulators-for-the-study-of-non-zs4qhsh95x.pdf

City-Scale Agent-Based Simulators for the Study of Non-pharmaceutical Interventions in the Context of the COVID-19 Epidemic: IISc-TIFR COVID-19 City-Scale Simulation Team.

Neuroscientists have recently shown that images that are difficult to find in visual search elicit similar patterns of firing across a population of recorded neurons. The $L^{1}$ distance between firing rate vectors associated with two images was strongly correlated with the inverse of decision time in behavior. But why should decision times be correlated with $L^{1}$ distance? What is the decision-theoretic basis? In our decision theoretic formulation, we model visual search as an active sequential hypothesis testing problem with switching costs. Our analysis suggests an appropriate neuronal dissimilarity index, which correlates equally strongly with the inverse of decision time as the $L^{1}$ distance. We also consider a number of other possibilities, such as the relative entropy (Kullback–Leibler divergence) and the Chernoff entropy of the firing rate distributions. A more stringent test of equality of means, which would have provided a strong backing for our modeling, fails for our proposed as well as the other already discussed dissimilarity indices. However, test statistics from the equality of means test, when used to rank the indices in terms of their ability to explain the observed results, places our proposed dissimilarity index at the top followed by relative entropy, Chernoff entropy, and the $L^{1}$ indices. Computations of the different indices require an estimate of the relative entropy between two Poisson point processes. An estimator is developed and is shown to have near unbiased performance for almost all operating regions.

Neural Dissimilarity Indices That Predict Oddball Detection in Behaviour

Active Sequential Hypothesis Testing (ASHT) is an extension of the classical sequential hypothesis testing problem with controls. Chernoff [1] proposed a policy called Procedure A and showed its asymptotic optimality as the cost of sampling was driven to zero. In this paper we study a further extension where we introduce costs for switching of actions. We show that a modification of Chernoff's Procedure A, one that we call Sluggish Procedure A, is asymptotically optimal even with switching costs. The growth rate of the total cost, as the probability of false detection is driven to zero, and as a switching parameter of the Sluggish Procedure A is driven down to zero, is the same as that without switching costs.

Nidhin Koshy Vaidhiyan

Papers

Learning to Detect an Oddball Target

Learning to detect an oddball target

City-Scale Agent-Based Simulators for the Study of Non-pharmaceutical Interventions in the Context of the COVID-19 Epidemic: IISc-TIFR COVID-19 City-Scale Simulation Team.

Neural Dissimilarity Indices That Predict Oddball Detection in Behaviour

Active search with a cost for switching actions