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Let L be a language that can be decided in linear space and let ϵ>0 be any constant. Let A be the exponential hardness assumption that for every n, membership in L for inputs of length n cannot be decided by circuits of size smaller than 2ϵn. We prove that for every function f:{0,1}∗→{0,1}, computable by a randomized logspace algorithm R, there exists a deterministic logspace algorithm D (attempting to compute f), such that on every input x of length n, the algorithm D outputs one of the following:1)The correct value f(x).2)The string: “I am unable to compute f(x) because the hardness assumption A is false”, followed by a (provenly correct) circuit of size smaller than 2ϵn′ for membership in L for inputs of length n′, for some n′=Θ(logn); that is, a circuit that refutes A. Moreover, D is explicitly constructed, given R.We note that previous works on the hardness-versus-randomness paradigm give derandomized algorithms that rely blindly on the hardness assumption. If the hardness assumption is false, the algorithms may output incorrect values, and thus a user cannot trust that an output given by the algorithm is correct. Instead, our algorithm D verifies the computation so that it never outputs an incorrect value. Thus, if D outputs a value for f(x), that value is certified to be correct. Moreover, if D does not output a value for f(x), it alerts that the hardness assumption was found to be false, and refutes the assumption.Our next result is a universal derandomizer for BPL (the class of problems solvable by bounded-error randomized logspace algorithms) 1 : We give a deterministic algorithm U that takes as an input a randomized logspace algorithm R and an input x and simulates the computation of R on x, deteriministically. Under the widely believed assumption BPL=L, the space ...
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Regularization for Shuffled Data Problems via Exponential Family Priors on the Permutation Group
In the analysis of data sets consisting of (X, Y)-pairs, a tacit assumption is that each pair corresponds to the same observational unit. If, however, such pairs are obtained via record linkage of two files, this assumption can be violated as a result of mismatch error rooting, for example, in the lack of reliable identifiers in the two files. Recently, there has been a surge of interest in this setting under the term “Shuffled Data” in which the underlying correct pairing of (X, Y)-pairs is represented via an unknown permutation. Explicit modeling of the permutation tends to be associated with overfitting, prompting the need for suitable methods of regularization. In this paper, we propose an exponential family prior on the permutation group for this purpose that can be used to integrate various structures such as sparse and local shuffling. This prior turns out to be conjugate for canonical shuffled data problems in which the likelihood conditional on a fixed permutation can be expressed as product over the corresponding (X,Y)-pairs. Inference can be based on the EM algorithm in which the E-step is approximated by sampling, e.g., via the Fisher-Yates algorithm. The M-step is shown to admit a reduction from n^2 to n terms if the likelihood of (X,Y)-pairs has exponential family form. Comparisons on synthetic and real data show that the proposed approach compares favorably to competing methods.
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- Award ID(s):
- 2120318
- PAR ID:
- 10446725
- Editor(s):
- Ruiz, Francisco; Dy, Jennifer; van de Meent, Jan-Willem
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 206
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 2939-2959
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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