Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSublinear Least-Squares Value Iteration via Locality Sensitive Hashing
115
0
0.0
(
0
)
Added by
Zhaozhuo Xu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We present the first provable Least-Squares Value Iteration (LSVI) algorithms that have runtime complexity sublinear in the number of actions. We formulate the value function estimation procedure in value iteration as an approximate maximum inner product search problem and propose a locality sensitive hashing (LSH) [Indyk and Motwani STOC98, Andoni and Razenshteyn STOC15, Andoni, Laarhoven, Razenshteyn and Waingarten SODA17] type data structure to solve this problem with sublinear time complexity. Moreover, we build the connections between the theory of approximate maximum inner product search and the regret analysis of reinforcement learning. We prove that, with our choice of approximation factor, our Sublinear LSVI algorithms maintain the same regret as the original LSVI algorithms while reducing the runtime complexity to sublinear in the number of actions. To the best of our knowledge, this is the first work that combines LSH with reinforcement learning resulting in provable improvements. We hope that our novel way of combining data-structures and iterative algorithm will open the door for further study into cost reduction in optimization.
rate research
Read More
Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval. Arkin et al. represent polygons by their turning function - a function which follows the angle between the polygons tangent and the $ x $-axis while traversing the perimeter of the polygon. They define the distance between polygons to be variations of the $ L_p $ (for $p=1,2$) distance between their turning functions. This metric is invariant under translation, rotation and scaling (and the selection of the initial point on the perimeter) and therefore models well the intuitive notion of shape resemblance. We develop and analyze LSH near neighbor data structures for several variations of the $ L_p $ distance for functions (for $p=1,2$). By applying our schemes to the turning functions of a collection of polygons we obtain efficient near neighbor LSH-based structures for polygons. To tune our structures to turning functions of polygons, we prove some new properties of these turning functions that may be of independent interest. As part of our analysis, we address the following problem which is of independent interest. Find the vertical translation of a function $ f $ that is closest in $ L_1 $ distance to a function $ g $. We prove tight bounds on the approximation guarantee obtained by the translation which is equal to the difference between the averages of $ g $ and $ f $.
Extended differential privacy, a generalization of standard differential privacy (DP) using a general metric, has been widely studied to provide rigorous privacy guarantees while keeping high utility. However, existing works on extended DP are limited to few metrics, such as the Euclidean metric. Consequently, they have only a small number of applications, such as location-based services and document processing. In this paper, we propose a couple of mechanisms providing extended DP with a different metric: angular distance (or cosine distance). Our mechanisms are based on locality sensitive hashing (LSH), which can be applied to the angular distance and work well for personal data in a high-dimensional space. We theoretically analyze the privacy properties of our mechanisms, and prove extended DP for input data by taking into account that LSH preserves the original metric only approximately. We apply our mechanisms to friend matching based on high-dimensional personal data with angular distance in the local model, and evaluate our mechanisms using two real datasets. We show that LDP requires a very large privacy budget and that RAPPOR does not work in this application. Then we show that our mechanisms enable friend matching with high utility and rigorous privacy guarantees based on extended DP.
We develop efficient algorithms for estimating low-degree moments of unknown distributions in the presence of adversarial outliers. The guarantees of our algorithms improve in many cases significantly over the best previous ones, obtained in recent works of Diakonikolas et al, Lai et al, and Charikar et al. We also show that the guarantees of our algorithms match information-theoretic lower-bounds for the class of distributions we consider. These improved guarantees allow us to give improved algorithms for independent component analysis and learning mixtures of Gaussians in the presence of outliers. Our algorithms are based on a standard sum-of-squares relaxation of the following conceptually-simple optimization problem: Among all distributions whose moments are bounded in the same way as for the unknown distribution, find the one that is closest in statistical distance to the empirical distribution of the adversarially-corrupted sample.
Metagenomic binning is an essential task in analyzing metagenomic sequence datasets. To analyze structure or function of microbial communities from environmental samples, metagenomic sequence fragments are assigned to their taxonomic origins. Although sequence alignment algorithms can readily be used and usually provide high-resolution alignments and accurate binning results, the computational cost of such alignment-based methods becomes prohibitive as metagenomic datasets continue to grow. Alternative compositional-based methods, which exploit sequence composition by profiling local short k-mers in fragments, are often faster but less accurate than alignment-based methods. Inspired by the success of linear error correcting codes in noisy channel communication, we introduce Opal, a fast and accurate novel compositional-based binning method. It incorporates ideas from Gallagers low-density parity-check code to design a family of compact and discriminative locality-sensitive hashing functions that encode long-range compositional dependencies in long fragments. By incorporating the Gallager LSH functions as features in a simple linear SVM, Opal provides fast, accurate and robust binning for datasets consisting of a large number of species, even with mutations and sequencing errors. Opal not only performs up to two orders of magnitude faster than BWA, an alignment-based binning method, but also achieves improved binning accuracy and robustness to sequencing errors. Opal also outperforms models built on traditional k-mer profiles in terms of robustness and accuracy. Finally, we demonstrate that we can effectively use Opal in the coarse search stage of a compressive genomics pipeline to identify a much smaller candidate set of taxonomic origins for a subsequent alignment-based method to analyze, thus providing metagenomic binning with high scalability, high accuracy and high resolution.
We give a new approach to the dictionary learning (also known as sparse coding) problem of recovering an unknown $ntimes m$ matrix $A$ (for $m geq n$) from examples of the form [ y = Ax + e, ] where $x$ is a random vector in $mathbb R^m$ with at most $tau m$ nonzero coordinates, and $e$ is a random noise vector in $mathbb R^n$ with bounded magnitude. For the case $m=O(n)$, our algorithm recovers every column of $A$ within arbitrarily good constant accuracy in time $m^{O(log m/log(tau^{-1}))}$, in particular achieving polynomial time if $tau = m^{-delta}$ for any $delta>0$, and time $m^{O(log m)}$ if $tau$ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser---at most $sqrt{n}$ nonzero coordinates---and there were intrinsic barriers preventing these algorithms from applying for denser $x$. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor $T$, given access to a tensor $T$ that is $tau$-close to $T$ in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of $T$ and $T$ have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
