Optimal testing of discrete distributions with high probability

Diakonikolas, Ilias; Gouleakis, Themis; Kane, Daniel M.; Peebles, John; Price, Eric

doi:10.1145/3406325.3450997

Cited by 14 publications

(26 citation statements)

References 21 publications

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“…Before presenting the sample complexity result, we state the existence of a principled independence testing procedure. Lemma 4.5 (Diakonikolas et al (2021)). There exists an ( , δ)-independence tester I(A, B)…”

Section: Algorithm 1 Causal Transition Model Estimationmentioning

confidence: 94%

Provably Efficient Causal Model-Based Reinforcement Learning for Systematic Generalization

Mutti¹,

Santi²,

Rossi³

et al. 2022

Preprint

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In the sequential decision making setting, an agent aims to achieve systematic generalization over a large, possibly infinite, set of environments. Such environments are modeled as discrete Markov decision processes with both states and actions represented through a feature vector. The underlying structure of the environments allows the transition dynamics to be factored into two components: one that is environment-specific and another one that is shared. Consider a set of environments that share the laws of motion as an illustrative example. In this setting, the agent can take a finite amount of reward-free interactions from a subset of these environments. The agent then must be able to approximately solve any planning task defined over any environment in the original set, relying on the above interactions only. Can we design a provably efficient algorithm that achieves this ambitious goal of systematic generalization? In this paper, we give a partially positive answer to this question. First, we provide the first tractable formulation of systematic generalization by employing a causal viewpoint. Then, under specific structural assumptions, we provide a simple learning algorithm that allows us to guarantee any desired planning error up to an unavoidable sub-optimality term, while showcasing a polynomial sample complexity.

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Section: Algorithm 1 Causal Transition Model Estimationmentioning

confidence: 94%

Provably Efficient Causal Model-Based Reinforcement Learning for Systematic Generalization

Mutti¹,

Santi²,

Rossi³

et al. 2022

Preprint

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“…denotes the total variation distance (statistical distance). This minimum number of samples, n(k, ε, δ), is the sample complexity of closeness testing; and the optimal dependence on all parameters (including δ), up to constant factors, was recently obtained by Diakonikolas, Gouleakis, Kane, Peebles, and Price [DGKPP21] (previous work only focused on, and obtained, the right dependence on k, ε [CDVV14]). 1…”

mentioning

confidence: 90%

“…In [DGKPP21], it was shown that the expectation of Z in the cases p = q and d TV (p, q) > ε differed by a noticeable quantity; a comparatively easy argument then allowed them to prove that Z was with high probability close to its expectation; and suitably thresholding this statistic Z to distinguish between the two cases led to the optimal sample complexity.…”

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confidence: 99%

Optimal Closeness Testing of Discrete Distributions Made (Complex) Simple

Canonne¹,

Sun²

2022

Preprint

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In this note, we revisit the recent work of Diakonikolas, Gouleakis, Kane, Peebles, and Price [DGKPP21], and provide an alternative proof of their main result. Our argument does not rely on any specific property of Poisson random variables (such as stability and divisibility) nor on any "clever trick," but instead on an identity relating the expectation of the absolute value of any random variable to the integral of its characteristic function:Our argument, while not devoid of technical aspects, is arguably conceptually simpler and more general; and we hope this technique can find additional applications in distribution testing.

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“…A number of results have discovered and rediscovered optimal bounds for identity testing [Pan08, VV14, ADK15, DKN15, Gol16, DK16, DGPP18, DKW18, DGPP19], even with optimal dependence on the failure probability δ and on an instance-by-instance basis. The harder problem of equivalence testing was studied in [BFRSW00], and optimal upper and lower bounds were given in [Val11,CDVV14,DKW18,DGKPP21]. Some work has also studied the case where an unequal number of samples are received from the two distributions [AJOS14, BV15,DK16].…”

Section: Related Workmentioning

confidence: 99%

The Price of Tolerance in Distribution Testing

Canonne¹,

Jain²,

Kamath³

et al. 2021

Preprint

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We revisit the problem of tolerant distribution testing. That is, given samples from an unknown distribution p over {1, . . . , n}, is it ε1-close to or ε2-far from a reference distribution q (in total variation distance)? Despite significant interest over the past decade, this problem is well understood only in the extreme cases. In the noiseless setting (i.e., ε1 = 0) the sample complexity is Θ( √ n), strongly sublinear

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Optimal testing of discrete distributions with high probability

Cited by 14 publications

References 21 publications

Provably Efficient Causal Model-Based Reinforcement Learning for Systematic Generalization

Provably Efficient Causal Model-Based Reinforcement Learning for Systematic Generalization

Optimal Closeness Testing of Discrete Distributions Made (Complex) Simple

The Price of Tolerance in Distribution Testing

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