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cs.DS updates on arXiv.org

PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
The Entropy of Backwards Analysis
Mathias Bæk Tejs Knudsen, Mikkel Thorup · 2017-04-15 · via cs.DS updates on arXiv.org

Backwards analysis, first popularized by Seidel, is often the simplest most elegant way of analyzing a randomized algorithm. It applies to incremental algorithms where elements are added incrementally, following some random permutation, e.g., incremental Delauney triangulation of a pointset, where points are added one by one, and where we always maintain the Delauney triangulation of the points added thus far. For backwards analysis, we think of the permutation as generated backwards, implying that the $i$th point in the permutation is picked uniformly at random from the $i$ points not picked yet in the backwards direction. Backwards analysis has also been applied elegantly by Chan to the randomized linear time minimum spanning tree algorithm of Karger, Klein, and Tarjan. The question considered in this paper is how much randomness we need in order to trust the expected bounds obtained using backwards analysis, exactly and approximately. For the exact case, it turns out that a random permutation works if and only if it is minwise, that is, for any given subset, each element has the same chance of being first. Minwise permutations are known to have $Θ(n)$ entropy, and this is then also what we need for exact backwards analysis. However, when it comes to approximation, the two concepts diverge dramatically. To get backwards analysis to hold within a factor $α$, the random permutation needs entropy $Ω(n/α)$. This contrasts with minwise permutations, where it is known that a $1+\varepsilon$ approximation only needs $Θ(\log (n/\varepsilon))$ entropy. Our negative result for backwards analysis essentially shows that it is as abstract as any analysis based on full randomness.