Results

**1 - 3**of**3**### New Results for Random Walk Learning

, 2014

"... Abstract In a very strong positive result for passive learning algorithms, Bshouty et al. showed that DNF expressions are efficiently learnable in the uniform random walk model. It is natural to ask whether the more expressive class of thresholds of parities (TOP) can also be learned efficiently in ..."

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Abstract In a very strong positive result for passive learning algorithms, Bshouty et al. showed that DNF expressions are efficiently learnable in the uniform random walk model. It is natural to ask whether the more expressive class of thresholds of parities (TOP) can also be learned efficiently in this model, since both DNF and TOP are efficiently uniform-learnable from queries. However, the time bounds of the algorithms of Bshouty et al. are exponential for TOP. We present a new approach to weak parity learning that leads to quasi-efficient uniform random walk learnability of TOP. We also introduce a more general random walk model and give two positive results in this new model: DNF is efficiently learnable and juntas are efficiently agnostically learnable.

### Arithmetic Cryptography∗

, 2015

"... We study the possibility of computing cryptographic primitives in a fully-black-box arith-metic model over a finite field F. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circ ..."

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We study the possibility of computing cryptographic primitives in a fully-black-box arith-metic model over a finite field F. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits which make only a black-box use of the underlying field, and the ad-versary has a full (non-black-box) access to the field. This model captures many standard information-theoretic constructions. We prove several positive and negative results in this model for various cryptographic tasks. On the positive side, we show that, under reasonable assumptions, computational primitives like commitment schemes, public-key encryption, oblivious transfer, and general secure two-party computation can be implemented in this model. On the negative side, we prove that garbled circuits, multiplicative-homomorphic encryption, and secure computation with low online com-plexity cannot be achieved in this model. Our results reveal a qualitative difference between the standard Boolean model and the arithmetic model, and explain, in retrospect, some of the

### The Fundamental Learning Problem that Genetic Algorithms with Uniform Crossover Solve Efficiently and Repeatedly As Evolution Proceeds

"... This paper establishes theoretical bonafides for implicit concurrent multivariate effect evalu-ation—implicit concurrency1 for short—a broad and versatile computational learning efficiency thought to underlie general-purpose, non-local, noise-tolerant optimization in genetic algorithms with uniform ..."

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This paper establishes theoretical bonafides for implicit concurrent multivariate effect evalu-ation—implicit concurrency1 for short—a broad and versatile computational learning efficiency thought to underlie general-purpose, non-local, noise-tolerant optimization in genetic algorithms with uniform crossover (UGAs). We demonstrate that implicit concurrency is indeed a form of efficient learning by showing that it can be used to obtain close-to-optimal bounds on the time and queries required to approximately correctly solve a constrained version (k = 7, η = 1/5) of a recognizable computational learning problem: learning parities with noisy membership queries. We argue that a UGA that treats the noisy membership query oracle as a fitness function can be straightforwardly used to approximately correctly learn the essential attributes in O(log1.585 n) queries and O(n log1.585 n) time, where n is the total number of attributes. Our proof relies on an accessible symmetry argument and the use of statistical hypothesis testing to reject a global null hypothesis at the 10−100 level of significance. It is, to the best of our knowledge, the first relatively rigorous identification of efficient computational learning in an evolutionary algorithm on a non-trivial learning problem. 1