We describe a slightly subexponential time algorithm for learning parity functions in the presence of random classification noise, a problem closely related to several cryptographic and coding problems. Our algorithm runs in polynomial time for the case of parity functions that depend on only the first O (log n log log n ) bits of input, which provides the first known instance of an efficient noise-tolerant algorithm for a concept class that is not learnable in the Statistical Query model of Kearns [1998]. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model.In coding-theory terms, what we give is a poly( n )-time algorithm for decoding linear k × n codes in the presence of random noise for the case of k = c log n log log n for some c > 0. (The case of k = O (log n ) is trivial since one can just individually check each of the 2 k possible messages and choose the one that yields the closest codeword.)A natural extension of the statistical query model is to allow queries about statistical properties that involve t -tuples of examples, as opposed to just single examples. The second result of this article is to show that any class of functions learnable (strongly or weakly) with t -wise queries for t = O (log n ) is also weakly learnable with standard unary queries. Hence, this natural extension to the statistical query model does not increase the set of weakly learnable functions.

We describe a slightly sub-exponential time algorithm for learning parity functions in the presence of random classification noise. This results in a polynomial-time algorithm for the case of parity functions that depend on only the first O(log n log log n) bits of input. This is the first known instance of an efficient noise-tolerant algorithm for a concept class that is provably not learnable in the Statistical Query model of Kearns [7]. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model. In coding-theory terms, what we give is a poly(n)-time algorithm for decoding linear k × n codes in the presence of random noise for the case of k = clog n log log n for some c > 0. (The case of k --- O(log n) is trivial since one can just individually check each of the 2 k possible messages and choose the one that yields the closest codeword.) A natural extension of the statistical query model is to allow queries about statistical properties that involve t-tuples of examples (as opposed to single examples). The second result of this paper is to show that any class of functions learnable (strongly or weakly) with t-wise queries for t = O(log n) is also weakly learnable with standard unary queries. Hence this natural extension to the statistical query model does not increase the set of weakly learnable functions.

We generalize Sudan's (see J. Compl., vol.13, p.180-93, 1997) results for Reed-Solomon codes to the class of algebraic-geometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error-correction bound (d-1)/2, d being the minimum distance of the code. Our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields. For the latter problem we design a polynomial-time algorithm and show that the resulting overall list-decoding algorithm runs in polynomial time under some mild conditions. Several examples are included.

Generalizing the high-noise decoding methods of [1, 19] to the class of algebraic-geometric codes, we design the first polynomialtime algorithms to decode algebraic-geometric codes significantly beyond the conventional error-correction bound. Applying our results to codes obtained from curves with many rational points, we construct arbitrarily long, constant-rate linear codes over a fixed field F q such that a codeword is efficiently, non-uniquely reconstructible after a majority of its letters have been arbitrarily corrupted. We also construct codes such that a codeword is uniquely and efficiently reconstructible after a majority of its letters have been corrupted by noise which is random in a specified sense. We summarize our results in terms of bounds on asymptotic parameters, giving a new characterization of decoding beyond the error-correction bound. 1 Introduction Error-correcting codes, originally designed to accommodate reliable transmission of information through unreliable ...

We review the field of result-checking, discussing simple checkers and self-correctors. We argue that such checkers could profitably be incorporated in software as an aid to efficient debugging and enhanced reliability. We consider how to modify traditional checking methodologies to make them more appropriate for use in real-time, real-number computer systems. In particular, we suggest that checkers should be allowed to use stored randomness: that is, that they should be allowed to generate, preprocess, and store random bits prior to run-time, and then to use this information repeatedly in a series of run-time checks. In a case study of checking a general real-number linear transformation (e.g., a Fourier Transform), we present a simple checker which uses stored randomness, and a self-corrector which is particularly efficient if stored randomness is employed.

We review the field of result-checking and suggest that it be extended to a methodology for enforcing hardware/software reliability. We thereby formulate a vision for "self-monitoring" hardware/software whose reliability is augmented through embedded suites of run-time correctness checkers. In particular, we suggest that embedded checkers and correctors may be employed to safeguard against arithmetic errors such as that which has bedeviled the Intel Pentium Microprocessor. We specify checkers and correctors suitable for monitoring the multiplication and division functionalities of an arbitrary arithmetic processor and seamlessly correcting erroneous output which may occur for any reason during the lifetime of the chip.

Article Comparing algorithm for dynamic speed-setting of a low-power CPU Share on Authors: Kinshuk Govil Computer Science Division, University of California, Berkeley, CA Computer Science Division, University of California, Berkeley, CAView Profile , Edwin Chan Computer Science Division, University of California, Berkeley, CA Computer Science Division, University of California, Berkeley, CAView Profile , Hal Wasserman Computer Science Division, University of California, Berkeley, CA Computer Science Division, University of California, Berkeley, CAView Profile Authors Info & Claims MobiCom '95: Proceedings of the 1st annual international conference on Mobile computing and networkingDecember 1995 Pages 13–25https://doi.org/10.1145/215530.215546Online:01 December 1995Publication History 309citation1,647DownloadsMetricsTotal Citations309Total Downloads1,647Last 12 Months29Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

We review the field of result-checking, discussing simple checkers and self-correctors. We argue that such checkers could profitably be incorporated in software as an aid to efficient debugging and reliable functionality. We consider how to modify traditional checking methodologies to make them more appropriate for use in real-time, real-number computer systems. In particular, we suggest that checkers should be allowed to use stored randomness: i.e., that they should be allowed to generate, pre-process, and store random bits prior to run-time, and then to use this information repeatedly in a series of run-time checks. In a case study of checking a general real-number linear transformation (for example, a Fourier Transform), we present a simple checker which uses stored randomness, and a self-corrector which is particularly efficient if stored randomness is allowed.>