1,348 research outputs found
Polynomial-Time Pseudodeterministic Construction of Primes
A randomized algorithm for a search problem is *pseudodeterministic* if it
produces a fixed canonical solution to the search problem with high
probability. In their seminal work on the topic, Gat and Goldwasser posed as
their main open problem whether prime numbers can be pseudodeterministically
constructed in polynomial time.
We provide a positive solution to this question in the infinitely-often
regime. In more detail, we give an *unconditional* polynomial-time randomized
algorithm such that, for infinitely many values of , outputs a
canonical -bit prime with high probability. More generally, we prove
that for every dense property of strings that can be decided in polynomial
time, there is an infinitely-often pseudodeterministic polynomial-time
construction of strings satisfying . This improves upon a
subexponential-time construction of Oliveira and Santhanam.
Our construction uses several new ideas, including a novel bootstrapping
technique for pseudodeterministic constructions, and a quantitative
optimization of the uniform hardness-randomness framework of Chen and Tell,
using a variant of the Shaltiel--Umans generator
Memory-Sample Lower Bounds for Learning Parity with Noise
In this work, we show, for the well-studied problem of learning parity under
noise, where a learner tries to learn from a
stream of random linear equations over that are correct with
probability and flipped with probability
, that any learning algorithm requires either a memory
of size or an exponential number of samples.
In fact, we study memory-sample lower bounds for a large class of learning
problems, as characterized by [GRT'18], when the samples are noisy. A matrix
corresponds to the following learning
problem with error parameter : an unknown element is
chosen uniformly at random. A learner tries to learn from a stream of
samples, , where for every , is
chosen uniformly at random and with probability
and with probability
(). Assume that are such that any
submatrix of of at least rows and at least columns, has a bias of at most . We show that any learning
algorithm for the learning problem corresponding to , with error, requires
either a memory of size at least , or at least samples. In particular, this shows that
for a large class of learning problems, same as those in [GRT'18], any learning
algorithm requires either a memory of size at least or an exponential number of noisy
samples.
Our proof is based on adapting the arguments in [Raz'17,GRT'18] to the noisy
case.Comment: 19 pages. To appear in RANDOM 2021. arXiv admin note: substantial
text overlap with arXiv:1708.0263
Extractor-Based Time-Space Lower Bounds for Learning
A matrix corresponds to the following
learning problem: An unknown element is chosen uniformly at random. A
learner tries to learn from a stream of samples, , where for every , is chosen uniformly at random and
.
Assume that are such that any submatrix of of at least
rows and at least columns, has a bias
of at most . We show that any learning algorithm for the learning
problem corresponding to requires either a memory of size at least
, or at least samples. The
result holds even if the learner has an exponentially small success probability
(of ).
In particular, this shows that for a large class of learning problems, any
learning algorithm requires either a memory of size at least or an exponential number of samples, achieving a
tight lower bound on the size
of the memory, rather than a bound of obtained in previous works [R17,MM17b].
Moreover, our result implies all previous memory-samples lower bounds, as
well as a number of new applications.
Our proof builds on [R17] that gave a general technique for proving
memory-samples lower bounds
GRADIENT-BASED STOCHASTIC OPTIMIZATION METHODS IN BAYESIAN EXPERIMENTAL DESIGN
Optimal experimental design (OED) seeks experiments expected to yield the most useful data for some purpose. In practical circumstances where experiments are time-consuming or resource-intensive, OED can yield enormous savings. We pursue OED for nonlinear systems from a Bayesian perspective, with the goal of choosing experiments that are optimal for parameter inference. Our objective in this context is the expected information gain in model parameters, which in general can only be estimated using Monte Carlo methods. Maximizing this objective thus becomes a stochastic optimization problem. This paper develops gradient-based stochastic optimization methods for the design of experiments on a continuous parameter space. Given a Monte Carlo estimator of expected information gain, we use infinitesimal perturbation analysis to derive gradients of this estimator.We are then able to formulate two gradient-based stochastic optimization approaches: (i) Robbins-Monro stochastic approximation, and (ii) sample average approximation combined with a deterministic quasi-Newton method. A polynomial chaos approximation of the forward model accelerates objective and gradient evaluations in both cases.We discuss the implementation of these optimization methods, then conduct an empirical comparison of their performance. To demonstrate design in a nonlinear setting with partial differential equation forward models, we use the problem of sensor placement for source inversion. Numerical results yield useful guidelines on the choice of algorithm and sample sizes, assess the impact of estimator bias, and quantify tradeoffs of computational cost versus solution quality and robustness.United States. Air Force Office of Scientific Research (Computational Mathematics Program)National Science Foundation (U.S.) (Award ECCS-1128147
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