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167
A competitive test for uniformity of monotone distributions
 in AISTATS, 2013
"... We propose a test that takes random samples drawn from a monotone distribution and decides whether or not the distribution is uniform. The test is nearly optimal in that it uses at most O(n√log n) samples, where n is the number of samples that a genie who knew all but one bit about the underlying ..."
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Cited by 2 (1 self)
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We propose a test that takes random samples drawn from a monotone distribution and decides whether or not the distribution is uniform. The test is nearly optimal in that it uses at most O(n√log n) samples, where n is the number of samples that a genie who knew all but one bit about the underlying distribution would need for the same task. Conversely, we show that any such test would require Ω(n log n) samples for some distributions. 1
TABLE OF CONTENTS
, 2014
"... Copyright Jayadev Acharya, 2014 All rights reserved. The dissertation of Jayadev Acharya is approved, and it is acceptable in quality and form for publication on microfilm and electronically: ..."
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Copyright Jayadev Acharya, 2014 All rights reserved. The dissertation of Jayadev Acharya is approved, and it is acceptable in quality and form for publication on microfilm and electronically:
Tight bounds for universal compression of large alphabets
 In ISIT
, 2013
"... Abstract—Over the past decade, several papers, e.g., [1–7] and references therein, have considered universal compression of sources over large alphabets, often using patterns to avoid infinite redundancy. Improving on previous results, we prove tight bounds on expected and worstcase pattern redund ..."
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Cited by 4 (2 self)
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Abstract—Over the past decade, several papers, e.g., [1–7] and references therein, have considered universal compression of sources over large alphabets, often using patterns to avoid infinite redundancy. Improving on previous results, we prove tight bounds on expected and worstcase pattern redundancy, in particular closing a decadelong gap and showing that the worstcase pattern redundancy of i.i.d. distributions is Θ̃(n1/3)†. I.
Tight Bounds on Profile Redundancy and Distinguishability
"... The minimax KLdivergence of any distribution from all distributions in a collection P has several practical implications. In compression, it is called redundancy and represents the least additional number of bits over the entropy needed to encode the output of any distribution in P. In online estim ..."
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Cited by 7 (3 self)
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The minimax KLdivergence of any distribution from all distributions in a collection P has several practical implications. In compression, it is called redundancy and represents the least additional number of bits over the entropy needed to encode the output of any distribution in P. In online estimation and learning, it is the lowest expected logloss regret when guessing a sequence of random values generated by a distribution in P. In hypothesis testing, it upper bounds the largest number of distinguishable distributions in P. Motivated by problems ranging from population estimation to text classification and speech recognition, several machinelearning and informationtheory researchers have recently considered labelinvariant observations and properties induced by i.i.d. distributions. A sufficient statistic for all these properties is the data’s profile, the multiset of the number of times each data element appears. Improving on a sequence of previous works, we show that the redundancy of the collection of distributions induced over profiles by lengthn i.i.d. sequences is between 0.3 · n 1/3 and n 1/3 log 2 n, in particular, establishing its exact growth power. 1
On Optimal TimerBased Distributed Selection For RateAdaptive Multiuser Diversity Systems
"... Abstract—We develop an optimal, distributed, and low feedback timerbased selection scheme to enable next generation rateadaptive wireless systems to exploit multiuser diversity. In our scheme, each user sets a timer depending on its signal to noise ratio (SNR) and transmits a small packet to ide ..."
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Abstract—We develop an optimal, distributed, and low feedback timerbased selection scheme to enable next generation rateadaptive wireless systems to exploit multiuser diversity. In our scheme, each user sets a timer depending on its signal to noise ratio (SNR) and transmits a small packet to identify itself when its timer expires. When the SNRtotimer mapping is monotone nondecreasing, timers of users with better SNRs expire earlier. Thus, the base station (BS) simply selects the first user whose timer expiry it can detect, and transmits data to it at as high a rate as reliably possible. However, timers that expire too close to one another cannot be detected by the BS due to collisions. We characterize in detail the structure of the SNRtotimer mapping that optimally handles these collisions to maximize the average data rate. We prove that the optimal timer values take only a discrete set of values, and that the rate adaptation policy strongly influences the optimal scheme’s structure. The optimal average rate is very close to that of ideal selection in which the BS always selects highest rate user, and is much higher than that of the popular, but ad hoc, timer schemes considered in the literature. I.
Poissonization and universal compression of envelope classes
"... Abstract—Poisson sampling is a method for eliminating dependence among symbols in a random sequence. It helps improve algorithm design, strengthen bounds, and simplify proofs. We relate the redundancy of fixedlength and Poissonsampled sequences, use this result to derive a simple formula for the ..."
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Cited by 2 (0 self)
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Abstract—Poisson sampling is a method for eliminating dependence among symbols in a random sequence. It helps improve algorithm design, strengthen bounds, and simplify proofs. We relate the redundancy of fixedlength and Poissonsampled sequences, use this result to derive a simple formula for the redundancy of general envelope classes, and apply this formula to obtain simple and tight bounds on the redundancy of powerlaw and exponential envelope classes, in particular answering a question posed in [1]. I.
Sorting with adversarial comparators and application to density estimation
"... Abstract—We consider the problems of sorting and maximumselection of n elements using adversarial comparators. We derive a maximumselection algorithm that uses 8n comparisons in expectation, and a sorting algorithm that uses 4n log2 n comparisons in expectation. Both are tight up to a constant fa ..."
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Cited by 1 (1 self)
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Abstract—We consider the problems of sorting and maximumselection of n elements using adversarial comparators. We derive a maximumselection algorithm that uses 8n comparisons in expectation, and a sorting algorithm that uses 4n log2 n comparisons in expectation. Both are tight up to a constant factor. Our adversarialcomparator model was motivated by the practically important problem of densityestimation, where we observe samples from an unknown distribution, and try to determine which of n known distributions is closest to it. Existing algorithms run in Ω(n2) time. Applying the adversarial comparator results, we derive a densityestimation algorithm that runs in only O(n) time. I.
Sublinear algorithms for outlier detection and generalized closeness testing
"... Abstract—Outlier detection is the problem of finding a few different distributions in a set of mostly identical ones. Closeness testing is the problem of deciding whether two distributions are identical or different. We relate the two problems, construct a sublinear generalized closeness test for u ..."
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Abstract—Outlier detection is the problem of finding a few different distributions in a set of mostly identical ones. Closeness testing is the problem of deciding whether two distributions are identical or different. We relate the two problems, construct a sublinear generalized closeness test for unequal sample lengths, and use this result to derive a sublinear universal outlier detector. We also lower bound the sample complexity of both problems. I.
Competitive Distribution Estimation: Why is GoodTuring Good
"... Estimating distributions over large alphabets is a fundamental machinelearning tenet. Yet no method is known to estimate all distributions well. For example, addconstant estimators are nearly minmax optimal but often perform poorly in practice, and practical estimators such as absolute discountin ..."
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Estimating distributions over large alphabets is a fundamental machinelearning tenet. Yet no method is known to estimate all distributions well. For example, addconstant estimators are nearly minmax optimal but often perform poorly in practice, and practical estimators such as absolute discounting, JelinekMercer, and GoodTuring are not known to be near optimal for essentially any distribution. We describe the first universally nearoptimal probability estimators. For every discrete distribution, they are provably nearly the best in the following two competitive ways. First they estimate every distribution nearly as well as the best estimator designed with prior knowledge of the distribution up to a permutation. Second, they estimate every distribution nearly as well as the best estimator designed with prior knowledge of the exact distribution, but as all natural estimators, restricted to assign the same probability to all symbols appearing the same number of times. Specifically, for distributions over k symbols and n samples, we show that for both comparisons, a simple variant of GoodTuring estimator is always within KL divergence of (3 + o(1))/n1/3 from the best estimator, and that a more involved estimator is within Õ(min(k/n, 1/√n)). Conversely, we show that any estimator must have a KL divergence ≥ Ω̃(min(k/n, 1/n2/3)) over the best estimator for the first comparison, and ≥ Ω̃(min(k/n, 1/√n)) for the second.
On the Query Computation and Verification of Functions
"... Abstract—In the query model of multivariate function computation, the values of the variables are queried sequentially, in an order that may depend on previously revealed values, until the function’s value can be determined. The function’s computation query complexity is the lowest expected number ..."
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Abstract—In the query model of multivariate function computation, the values of the variables are queried sequentially, in an order that may depend on previously revealed values, until the function’s value can be determined. The function’s computation query complexity is the lowest expected number of queries required by any query order. Instead of computation, it is often easier to consider verification, where the value of the function is given and the queries aim to verify it. The lowest expected number of queries necessary is the function’s verification query complexity. We show that for all symmetric functions of independent binary random variables, the computation and verification complexities coincide. This provides a simple method for finding the query complexity and the optimal query order for computing many functions. We also show that if the symmetry condition is removed, there are functions whose verification complexity is strictly lower than their computation complexity, and mention that the same holds when the independence or binary conditions are removed. I.
Results 1  10
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167