Abstract
We consider probabilistic circuits working over the real numbers and using arbitrary semialgebraic functions of bounded description complexity as gates. In particular, such circuits can use all arithmetic operations (+, −, ×, ÷), optimization operations (min and max), conditional branching (if-then-else), and many more. We show that probabilistic circuits using any of these operations as gates can be simulated by deterministic circuits with only about a quadratical blowup in size. A slightly larger blowup in circuit size is also shown when derandomizing approximating circuits. The algorithmic consequence, motivating the title, is that randomness cannot substantially speed up dynamic programming algorithms.
- L. M. Adleman. 1978. Two theorems on random polynomial time. In Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science (FOCS). 78--83.Google Scholar
Digital Library
- M. Ajtai and M. Ben-Or. 1984. A theorem on probabilistic constant depth computations. In Proceedings of the 16th Annual ACM Symposium on Theory of Computing (STOC). 471--474.Google Scholar
- M. Ajtai, J. Komlós, and E. Szemerédi. 1983. Sorting in c log n parallel steps. Combinatorica 3, 1 (1983), 1--19.Google Scholar
Digital Library
- N. Alon and E. R. Scheinerman. 1988. Degrees of freedom versus dimension for containment orders. Order 5 (1988), 11--16.Google Scholar
Cross Ref
- S. Basu, R. Pollack, and M.-F. Roy. 1996. On the combinatorial and algebraic complexity of quantifer elimination. J. ACM 43, 6 (1996), 1002--1045.Google Scholar
Digital Library
- S. Ben-David and M. Lindenbaum. 1998. Localization vs. identification of semi-algebraic sets. Machine Learning 32 (1998), 207--224.Google Scholar
Digital Library
- C. H. Bennett and J. Gill. 1981. Relative to a random oracle A, PA ≠ N PA ≠ co-N PA with probability 1. SIAM J. Comput. 10, 1 (1981), 96--113.Google Scholar
Cross Ref
- P. Bürgisser, M. Karpinski, and T. Lickteig. 1993. On randomized semi-algebraic test complexity. J. Complexity 9, 2 (1993), 231--251.Google Scholar
Digital Library
- F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, and K. Werther. 1995. On real Turing machines that toss coins. In Proceedings of the 27th Annual ACM Symposium. on Theory of Computing (STOC). 335--342.Google Scholar
- D. Dubhashi and A. Panconesi. 2009. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press.Google Scholar
- R. M. Dudley. 1984. A Course on Empirical Processes. In Lecture Notes in Mathematics, vol. 1097. Springer, Berlin.Google Scholar
- P. Goldberg and M. Jerrum. 1995. Bounding the Vapnik-Chervonenkis dimension of concept classes parametrized by real numbers. Machine Learning 18 (1995), 131--148.Google Scholar
Digital Library
- O. Goldreich. 2011. In a world of P = BPP. In Studies in Complexity and Cryptography. Lecture Notes in Computer Science, vol. 6650. Springer, 191--232.Google Scholar
- D. Grigoriev. 1999. Complexity lower bounds for randomized computation trees over zero characteristic fields. Computational Complexity 8, 4 (1999), 316--329.Google Scholar
Digital Library
- D. Grigoriev and M. Karpinski. 1997. Randomized Ω(n2) lower bound for knapsack. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC). 76--85.Google Scholar
- D. Grigoriev, M. Karpinski, F. Meyer auf der Heide, and R. Smolensky. 1997. A lower bound for randomized algebraic decision trees. Computational Complexity 6, 4 (1997), 357--375.Google Scholar
Cross Ref
- D. Haussler. 1992. Decision theoretic generalizations of the PAC model for neural nets and other learning applications. Inf. Comput. 100 (1992), 78--150.Google Scholar
Digital Library
- R. Impagliazzo and A. Wigderson. 1997. P = BPP unless E has subexponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th ACM Symposium on Theory of Computing (STOC). 220--229.Google Scholar
- U. Manber and M. Tompa. 1985. The complexity of problems on probabilistic, nondeterministic, and alternating decision trees. J. ACM 32, 3 (1985), 720--732.Google Scholar
Digital Library
- A. A. Markov. 1958. On the inversion complexity of systems of Boolean functions. J. ACM 5, 4 (1958), 331--334.Google Scholar
Digital Library
- F. Meyer auf der Heide. 1985. Simulating probabilistic by deterministic algebraic computation trees. Theor. Comput. Sci. 41 (1985), 325--330.Google Scholar
Cross Ref
- J. Milnor. 1964. On the Betti numbers of real varieties. Proc. Amer. Math. Soc. 15 (1964), 275--280.Google Scholar
Cross Ref
- M. Mitzenmacher and E. Upfal. 2005. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press.Google Scholar
- H. Morizumi. 2012. Limiting Negations in Probabilistic Circuits. New Trends in Algorithms and Theory of Computation, Departmental Bulletin Paper 1799, pages 81--83. Kyoto University Research Information Repository.Google Scholar
- R. Motwani and P. Raghavan. 1995. Randomized Algorithms. Cambridge University Press.Google Scholar
- D. Pollard. 1984. Convergence of Stochastic Processes. Springer-Verlag.Google Scholar
- P. Pudlák and V. Rödl. 1992. A combinatorial approach to complexity. Combinatorica 12, 2 (1992), 221--226.Google Scholar
Digital Library
- N. Saxena. 2009. Progress on polynomial identity testing. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 99 (2009), 49--79.Google Scholar
- J. T. Schwartz. 1980. Fast probabilisitic algorithms for verification of polynomial identities. J. ACM 27, 4 (1980), 701--717.Google Scholar
Digital Library
- A. Seidenberg. 1954. A new decision method for elementary algebra. Ann. of Math. 60 (1954), 365--374.Google Scholar
Cross Ref
- A. Shpilka and A. Yehudayoff. 2010. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science 5, 3--4 (2010), 207--388.Google Scholar
Digital Library
- M. Snir. 1985. Lower bounds on probabilistic linear decision trees. Theor. Comput. Sci. 38 (1985), 69--82.Google Scholar
Cross Ref
- A. Tarski. 1951. A Decision Method for Elementary Algebra and Geometry (2nd ed.). University of California Press, Berkeley and Los Angeles, Calif.Google Scholar
- V. N. Vapnik and A. Ya. Chervonenkis. 1971. On the uniform convergence of relative frequencies of events to their probabilities.Theory Probab. Appl. 16 (1971), 264--280.Google Scholar
Cross Ref
- H. E. Warren. 1968. Lower bounds for approximation by non-linear manifolds. Trans. Amer. Math. Soc. 133 (1968), 167--178.Google Scholar
Cross Ref
- R. Zippel. 1979. Probabilistic algorithms for sparse polynomials. In Lecture Notes in Computer Science, vol. 72. Springer, 216--226.Google Scholar
Index Terms
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