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Devroye , Reddad : Discrete minimax estimation with trees
"We propose a simple recursive data-based partitioning scheme which produces piecewise-constant or piecewise-linear density estimates on intervals, and show how this scheme can determine the optimal L1L1 minimax rate for some discrete nonparametric classes."
to:NB  density_estimation  devroye.luc  minimax  nonparametrics  statistics 
12 hours ago by cshalizi
[1908.06130] Average-Case Lower Bounds for Learning Sparse Mixtures, Robust Estimation and Semirandom Adversaries
"This paper develops several average-case reduction techniques to show new hardness results for three central high-dimensional statistics problems, implying a statistical-computational gap induced by robustness, a detection-recovery gap and a universality principle for these gaps. A main feature of our approach is to map to these problems via a common intermediate problem that we introduce, which we call Imbalanced Sparse Gaussian Mixtures. We assume the planted clique conjecture for a version of the planted clique problem where the position of the planted clique is mildly constrained, and from this obtain the following computational lower bounds: (1) a k-to-k2 statistical-computational gap for robust sparse mean estimation, providing the first average-case evidence for a conjecture of Li (2017) and Balakrishnan et al. (2017); (2) a tight lower bound for semirandom planted dense subgraph, which shows that a semirandom adversary shifts the detection threshold in planted dense subgraph to the conjectured recovery threshold; and (3) a universality principle for k-to-k2 gaps in a broad class of sparse mixture problems that includes many natural formulations such as the spiked covariance model.
"Our main approach is to introduce several average-case techniques to produce structured and Gaussianized versions of an input graph problem, and then to rotate these high-dimensional Gaussians by matrices carefully constructed from hyperplanes in 𝔽tr. For our universality result, we introduce a new method to perform an algorithmic change of measure tailored to sparse mixtures. We also provide evidence that the mild promise in our variant of planted clique does not change the complexity of the problem."
to:NB  high-dimensional_statistics  minimax  computational_complexity  statistics  theoretical_computer_science 
yesterday by cshalizi
[1901.00555] An Introductory Guide to Fano's Inequality with Applications in Statistical Estimation
"Information theory plays an indispensable role in the development of algorithm-independent impossibility results, both for communication problems and for seemingly distinct areas such as statistics and machine learning. While numerous information-theoretic tools have been proposed for this purpose, the oldest one remains arguably the most versatile and widespread: Fano's inequality. In this chapter, we provide a survey of Fano's inequality and its variants in the context of statistical estimation, adopting a versatile framework that covers a wide range of specific problems. We present a variety of key tools and techniques used for establishing impossibility results via this approach, and provide representative examples covering group testing, graphical model selection, sparse linear regression, density estimation, and convex optimization."
to:NB  information_theory  minimax  statistics  estimation  to_read 
2 days ago by cshalizi
Kontorovich , Pinelis : Exact lower bounds for the agnostic probably-approximately-correct (PAC) machine learning model
"We provide an exact nonasymptotic lower bound on the minimax expected excess risk (EER) in the agnostic probably-approximately-correct (PAC) machine learning classification model and identify minimax learning algorithms as certain maximally symmetric and minimally randomized “voting” procedures. Based on this result, an exact asymptotic lower bound on the minimax EER is provided. This bound is of the simple form c∞/ν√c∞/ν as ν→∞ν→∞, where c∞=0.16997…c∞=0.16997… is a universal constant, ν=m/dν=m/d, mm is the size of the training sample and dd is the Vapnik–Chervonenkis dimension of the hypothesis class. It is shown that the differences between these asymptotic and nonasymptotic bounds, as well as the differences between these two bounds and the maximum EER of any learning algorithms that minimize the empirical risk, are asymptotically negligible, and all these differences are due to ties in the mentioned “voting” procedures. A few easy to compute nonasymptotic lower bounds on the minimax EER are also obtained, which are shown to be close to the exact asymptotic lower bound c∞/ν√c∞/ν even for rather small values of the ratio ν=m/dν=m/d. As an application of these results, we substantially improve existing lower bounds on the tail probability of the excess risk. Among the tools used are Bayes estimation and apparently new identities and inequalities for binomial distributions."
to:NB  learning_theory  statistics  minimax  kontorovich.aryeh  kith_and_kin  to_read 
18 days ago by cshalizi
JoeDog/byron: A multi-game tic-tac-toe game with running score and a race against the clock. Features a M.E.N.A.C.E. engine
A multi-game tic-tac-toe game with running score and a race against the clock. Features a M.E.N.A.C.E. engine - JoeDog/byron
github  java  tic-tac-toe  minimax  montecarlo 
5 weeks ago by snahor
Tic Tac Toe
cwoebker's blog - insights on technology and science
minimax  python 
may 2016 by Sidon

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