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1. Scalable Zonotopic Under-approximation of Backward Reachable Sets for Uncertain Linear Systems

   https://doi.org/10.1109/LCSYS.2021.3123228  

   Yang, Liren; Ozay, Necmiye (January 2022, IEEE control systems letters)

   Zonotopes are widely used for over-approximating forward reachable sets of uncertain linear systems for verification purposes. In this paper, we use zonotopes to achieve more scalable algorithms that under-approximate backward reachable sets of uncertain linear systems for control design. The main difference is that the backward reachability analysis is a twoplayer game and involves Minkowski difference operations, but zonotopes are not closed under such operations. We underapproximate this Minkowski difference with a zonotope, which can be obtained by solving a linear optimization problem. We further develop an efficient zonotope order reduction technique to bound the complexity of the obtained zonotopic underapproximations. The proposed approach is evaluated against existing approaches using randomly generated instances and illustrated with several examples. 

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2. Efficiently Solving Turn-Taking Stochastic Games with Extensive-Form Correlation

   https://doi.org/10.1145/3580507.3597665  

   Zhang, Hanrui; Cheng, Yu; Conitzer, Vincent (July 2023, Proceedings of the 24th ACM Conference on Economics and Computation)

   We study equilibrium computation with extensive-form correlation in two-player turn-taking stochastic games. Our main results are two-fold: (1) We give an algorithm for computing a Stackelberg extensive-form correlated equilibrium (SEFCE), which runs in time polynomial in the size of the game, as well as the number of bits required to encode each input number. (2) We give an efficient algorithm for approximately computing an optimal extensive-form correlated equilibrium (EFCE) up to machine precision, i.e., the algorithm achieves approximation error 𝜀 in time polynomial in the size of the game, as well as log(1/𝜀). Our algorithm for SEFCE is the first polynomial-time algorithm for equilibrium computation with com- mitment in such a general class of stochastic games. Existing algorithms for SEFCE typically make stronger assumptions such as no chance moves, and are designed for extensive-form games in the less succinct tree form. Our algorithm for approximately optimal EFCE is, to our knowledge, the first algorithm that achieves 3 desiderata simultaneously: approximate optimality, polylogarithmic dependency on the approximation error, and compatibility with stochastic games in the more succinct graph form. Existing algorithms achieve at most 2 of these desiderata, often also relying on additional technical assumptions. 

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3. Computing under-approximations of multivalued decision diagrams

   Hosseini, Seyedehzahra; Ciardo, Gianfranco (May 2023, Application and Theory of Petri Nets and Concurrency)

   Gomes, L.; Lorenz, R. (Ed.)

   Efficient manipulation of binary or multi-valueddecision diagrams (BDDs or MDDs) is critical in symbolic verification tools. Despite the applicability of MDDs to real-world tasks such as discovering the reachable states of a model, their large demands on hardware resources, especially memory, limit algorithmic scalability. In this paper, we focus on memory-constrained algorithms that employ a novel O(m log n)-time under-approximation technique for MDDs, where m and n are the number of MDD edges and nodes, respectively. The effectiveness of our approach is demonstrated experimentally by a reduction in peak memory usage for the symbolic reachability computation of a set of Petri nets. 

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4. Differential Privacy on Finite Computers

   https://doi.org/10.29012/jpc.679  

   Balcer, Victor; Vadhan, Salil (October 2019, Journal of Privacy and Confidentiality)

   We consider the problem of designing and analyzing differentially private algorithms that can be implemented on discrete models of computation in strict polynomial time, motivated by known attacks on floating point implementations of real-arithmetic differentially private algorithms (Mironov, CCS 2012) and the potential for timing attacks on expected polynomial-time algorithms. As a case study, we examine the basic problem of approximating the histogram of a categorical dataset over a possibly large data universe X. The classic Laplace Mechanism (Dwork, McSherry, Nissim, Smith, TCC 2006 and J. Privacy \& Confidentiality 2017) does not satisfy our requirements, as it is based on real arithmetic, and natural discrete analogues, such as the Geometric Mechanism (Ghosh, Roughgarden, Sundarajan, STOC 2009 and SICOMP 2012), take time at least linear in |X|, which can be exponential in the bit length of the input.   In this paper, we provide strict polynomial-time discrete algorithms for approximate histograms whose simultaneous accuracy (the maximum error over all bins) matches that of the Laplace Mechanism up to constant factors, while retaining the same (pure) differential privacy guarantee. One of our algorithms produces a sparse histogram as output. Its ``"per-bin accuracy" (the error on individual bins) is worse than that of the Laplace Mechanism by a factor of log|X|, but we prove a lower bound showing that this is necessary for any algorithm that produces a sparse histogram. A second algorithm avoids this lower bound, and matches the per-bin accuracy of the Laplace Mechanism, by producing a compact and efficiently computable representation of a dense histogram; it is based on an (n+1)-wise independent implementation of an appropriately clamped version of the Discrete Geometric Mechanism. 

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5. Efficient sampling from the Bingham distribution

   Ge, Rong; Lee, Holden; Lu, Jianfeng; Risteski, Andrej (April 2021, ALT)

   We give a algorithm for exact sampling from the Bingham distribution p(x)∝exp(x⊤Ax) on the sphere Sd−1 with expected runtime of poly(d,λmax(A)−λmin(A)). The algorithm is based on rejection sampling, where the proposal distribution is a polynomial approximation of the pdf, and can be sampled from by explicitly evaluating integrals of polynomials over the sphere. Our algorithm gives exact samples, assuming exact computation of an inverse function of a polynomial. This is in contrast with Markov Chain Monte Carlo algorithms, which are not known to enjoy rapid mixing on this problem, and only give approximate samples. As a direct application, we use this to sample from the posterior distribution of a rank-1 matrix inference problem in polynomial time. 

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