The OneMax problem is a standard benchmark optimisation problem for a binary search space. Recent work on applying a Bandit-Based Random Mutation Hill-Climbing algorithm to the noisy OneMax Problem showed that it is important to choose a good value for the resampling number to make a careful trade off between taking more samples in order to reduce noise, and taking fewer samples to reduce the total computational cost. This paper extends that observation, by deriving an analytical expression for the running time of the RMHC algorithm with resampling applied to the noisy OneMax problem, and showing both theoretically and empirically that the optimal resampling number increases with the number of dimensions in the search space.
Droste @cite_6 defined a model for the OneMax problem, in which during every fitness evaluation of the bit-string @math , there was exactly one uniformly-chosen random bit mis-read with probability @math . Hence the measured noisy fitness values are equivalent to replacing the Gaussian noise term in by an appropriate discrete random variable taking values from @math . Under this scheme, Droste showed that (1+1)-EA with mutation probability @math could optimise a @math -bit OneMax problem corrupted by , with high probability, in polynomial time if @math is @math .
@cite_6
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We study the well-known black-box optimisation algorithm (1+1)-EA on a novel type of noise model. In our noise model, the fitness function is linear with positive weights, but the absolute values of the weights may fluctuate in each round. Thus in every state, the fitness function indicates that one-bits are preferred over zero-bits. In particular, hillclimbing heuristics should be able to find the optimum fast. We show that the (1+1)-EA indeed finds the optimum in time @math if the mutation parameter is @math for a constant @math However, we also show that for @math the (1+1)-EA needs superpolynomial time to find the optimum. Thus the choice of mutation parameter is critical even for optimisation tasks in which there is a clear path to the the optimum. A similar threshold phenomenon has recently been shown for noise-free monotone fitness functions. In basic genetic algorithm (GA) applications, the fitness of a solution takes a value that is certain and unchanging. This formulation does not work for ongoing searches for better solutions in a nonstationary environment in which expected solution fitness changes with time in unpredictable ways, or for fitness evaluations corrupted by noise. In such cases, the estimated fitness has an associated uncertainty. The uncertainties due to environmental changes (process noise) and to noisy evaluations (observation noise) can be reduced, at least temporarily, by re-evaluating existing solutions. The Kalman formulation provides a formal mechanism for treating uncertainty in GA. It provides the mechanics for determining the estimated fitness and uncertainty when a new solution is generated and evaluated for the first time. It also provides the mechanics for updating the estimated fitness and uncertainty after an existing solution is re-evaluated and for increasing the uncertainty with the passage of time. A Kalman-extended GA (KGA) is developed to determine when to generate a new individual, and when to re-evaluate an existing one and which to re-evaluate. This KGA is applied to the problem of maintaining a network configuration with minimized message loss, with mobile nodes and stochastic transmission. As the nodes move, the optimal network changes, but information contained within the population of solutions allows efficient discovery of better-adapted solutions. The sensitivity of the KGA to several control parameters is explored. In practical applications evaluating a fitness function is frequently subject to noise, i. e., the “true fitness” is disturbed by some random variations. Evolutionary algorithms (EAs) are often successfully applied to noisy problems, where they have turned out to be particularly robust. Theoretical results on the behavior of EAs for noisy functions are comparatively very rare, especially for discrete search spaces. Here we present an analysis of the (1 + 1) EA for a noisy variant of OneMax and compute the maximal noise strength allowing the (1 + 1) EA a polynomial runtime asymptotically exactly. The methods used in the proofs are presented in a general form with clearly stated conditions in order to simplify further applications. Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noise-handling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noise-handling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the Compact Genetic Algorithm can overcome the shortsightedness of mutation-only heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property. This extended abstract summarizes our work "The Benefit of Recombination in Noisy Evolutionary Search," which appeared in Proceedings of International Symposium on Algorithms and Computation (ISAAC), 2015, pp. 140--150. Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem---e.g., in determining the relationship between genetics and the presence or absence of a disease---or they may be a result of extreme quantization. In one-bit compressed sensing it has recently been shown that the number of one-bit measurements required for signal estimation mirrors that of unquantized compressed sensing. Indeed, @math -sparse signals in @math can be estimated (up to normalization) from @math one-bit measurements. Nevertheless, controlling the precise accuracy of the error estimate remains an open challenge. In this paper, we focus on optimizing the decay of the error as a function of the oversampling factor @math , where @math is the number of measurements. It is known that the error in reconstructing sparse signals from standard one-bit measurements is bounded below by @math . Without adjusting the measurement procedure, reducing this polynomial error decay rate is impossible. However, we show that an adaptive choice of the thresholds used for quantization may lower the error rate to @math . This improves upon guarantees for other methods of adaptive thresholding as proposed in Sigma-Delta quantization. We develop a general recursive strategy to achieve this exponential decay and two specific polynomial-time algorithms which fall into this framework, one based on convex programming and one on hard thresholding. This work is inspired by the one-bit compressed sensing model, in which the engineer controls the measurement procedure. Nevertheless, the principle is extendable to signal reconstruction problems in a variety of binary statistical models as well as statistical estimation problems like logistic regression. We demonstrate that many detection methods are designed to identify only a sufficently accurate bounding box, rather than the best available one. To address this issue we propose a simple and fast modification to the existing methods called Fitness NMS. This method is tested with the DeNet model and obtains a significantly improved MAP at greater localization accuracies without a loss in evaluation rate, and can be used in conjunction with Soft NMS for additional improvements. Next we derive a novel bounding box regression loss based on a set of IoU upper bounds that better matches the goal of IoU maximization while still providing good convergence properties. Following these novelties we investigate RoI clustering schemes for improving evaluation rates for the DeNet wide model variants and provide an analysis of localization performance at various input image dimensions. We obtain a MAP of 33.6 @79Hz and 41.8 @5Hz for MSCOCO and a Titan X (Maxwell). Source code available from: https: github.com lachlants denet We investigate the clustering performances of the relaxed @math means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decay exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed @math means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed @math means SDP allows to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed @math means. We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random @math -regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for @math the cover time is asymptotic to @math , where @math . In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph @math . For @math independent walks on @math , the cover time @math is asymptotic to @math , where @math is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is @math . If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for @math walks to broadcast a single piece of information to each other is asymptotic to @math as @math . We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of @math explosive particles ( @math even) tends to @math as @math . The case of @math coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to @math . Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with @math predators and @math prey the expected detection time is @math , where @math is the @math th harmonic number. The ( @math 1+ź) EA with mutation probability c n, where @math c>0 is an arbitrary constant, is studied for the classical OneMax function. Its expected optimization time is analyzed exactly (up to lower order terms) as a function of c and @math ź. It turns out that 1 n is the only optimal mutation probability if @math ź=o(lnnlnlnn lnlnlnn), which is the cut-off point for linear speed-up. However, if @math ź is above this cut-off point then the standard mutation probability 1 n is no longer the only optimal choice. Instead, the expected number of generations is (up to lower order terms) independent of c, irrespectively of it being less than 1 or greater. The theoretical results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds. Experimental supplements shed light on the optimal mutation probability for small problem sizes. This paper considers the effect of stochasticity on the quality of convergence of genetic algorithms (GAs). In many problems, the variance of building-block fitness or so-called collateral noise is the major source of variance, and a population-sizing equation is derived to ensure that average signal-to-collateral-noise ratios are favorable to the discrimination of the best building blocks required to solve a problem of bounded deception. The sizing relation is modified to permit the inclusion of other sources of stochasticity, such as the noise of selection, the noise of genetic operators, and the explicit noise or nondeterminism of the objective function. In a test suite of five functions, the sizing relation proves to be a conservative predictor of average correct convergence, as long as all major sources of noise are considered in the sizing calculation. These results suggest how the sizing equation may be viewed as a coarse delineation of a boundary between what a physicist might call two distinct phases of GA behavior. At low population sizes the GA makes many errors of decision, and the quality of convergence is largely left to the vagaries of chance or the serial fixup of flawed results through mutation or other serial injection of diversity. At large population sizes, GAs can reliably discriminate between good and bad building blocks, and parallel processing and recombination of building blocks lead to quick solution of even difficult deceptive problems. Additionally, the paper outlines a number of extensions to this work, including the development of more refined models of the relation between generational average error and ultimate convergence quality, the development of online methods for sizing populations via the estimation of populations via the estimation of population-sizing parameters, and the investigation of populationsizing in the context of niching and other schemes designed for use in problems with high cardinality solution sets. The paper also discusses how these results may one day lead to rigorous proofs of convergence for recombinative G As operating on problems of bounded description. Given @math noisy samples with @math dimensions, where @math , we show that the multi-step thresholding procedure based on the Lasso -- we call it the Thresholded Lasso , can accurately estimate a sparse vector @math in a linear model @math , where @math is a design matrix normalized to have column @math norm @math , and @math . We show that under the restricted eigenvalue (RE) condition (Bickel-Ritov-Tsybakov 09), it is possible to achieve the @math loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle while selecting a sufficiently sparse model -- hence achieving sparse oracle inequalities ; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. In some sense, the Thresholded Lasso recovers the choices that would have been made by the @math penalized least squares estimators, in that it selects a sufficiently sparse model without sacrificing the accuracy in estimating @math and in predicting @math . We also show for the Gauss-Dantzig selector (Cand e s-Tao 07), if @math obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, one will achieve the sparse oracle inequalities as above, while allowing at most @math irrelevant variables in the model in the worst case, where @math is the smallest integer such that for @math , @math . Our simulation results on the Thresholded Lasso match our theoretical analysis excellently. We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the @math workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60-62]. In this model there are @math processors @math , each of which is initially given a private input bit @math . The goal is for @math to learn the value of @math , for some specified function @math , using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176-180] gave a noise-resistant protocol that allows @math to learn the entire input with constant probability in @math broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of @math on the depth of a tree that learns the entire input. While the above lower bound is for an @math -bit function, we also show an @math lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least @math for a constant parameter @math (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of @math on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) @math lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts. We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ= ξ t (x): (t,x)∈ℤν+1 . The random walk transition probabilities are taken as P(X t +1= y|X t = x,ξ t =η) =P0( y−x)+ c(y−x;η(x)). We assume that the variables ξ t (x):(t,x) ∈ℤν+1 are i.i.d., that both P0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X t , and there is a corresponding random correction of order ( ) to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X t and a corresponding correction of order ( ) to the C.L.T.. Proofs are based on some new L p estimates for a class of functionals of the field. We present a theory, based on statistical mechanics, to evaluate analytically the performance of uncoded, fully synchronous, randomly spread code-division multiple-access (CDMA) multiuser detectors with additive white Gaussian noise (AWGN) channel, under perfect power control, and in the large-system limit. Application of the replica method, a tool developed in the literature of statistical mechanics, allows us to derive analytical expressions for the bit-error rate, as well as the multiuser efficiency, of the individually optimum (IO) and jointly optimum (JO) multiuser detectors over the whole range of noise levels. The information-theoretic capacity of the randomly spread CDMA channel and the performance of decorrelating and linear minimum mean-square error (MMSE) detectors are also derived in the same replica formulation, thereby demonstrating validity of the statistical-mechanical approach. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. In addition, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math in a Euclidean ball of known radius. Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the (μ+1) GA using uniform crossover on the fitness function Jumpk. All previous results in this context only hold for unrealistically low crossover probability pc=O(k n), while we give analyses for the setting of constant pc 2 and constant pc, we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of μ: O (nk-1) for duplicate elimination minimisation, O (n2 log n) for maximising the convex hull, O(n log n) for det. crowding (assuming pc = k n), O(n log n) for maximising the Hamming distance, O(n log n) for fitness sharing, O(n log n) for the single-receiver island model. This proves a sizeable advantage of all variants of the (μ+1) GA compared to the (1+1) EA, which requires θ(nk). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally. In this paper, we study a classification problem in which sample labels are randomly corrupted. In this scenario, there is an unobservable sample with noise-free labels. However, before being observed, the true labels are independently flipped with a probability @math , and the random label noise can be class-conditional. Here, we address two fundamental problems raised by this scenario. The first is how to best use the abundant surrogate loss functions designed for the traditional classification problem when there is label noise. We prove that any surrogate loss function can be used for classification with noisy labels by using importance reweighting, with consistency assurance that the label noise does not ultimately hinder the search for the optimal classifier of the noise-free sample. The other is the open problem of how to obtain the noise rate @math . We show that the rate is upper bounded by the conditional probability @math of the noisy sample. Consequently, the rate can be estimated, because the upper bound can be easily reached in classification problems. Experimental results on synthetic and real datasets confirm the efficiency of our methods. Let X 1 ,...., X n be a collection of iid discrete random variables, and Y 1 ,...., Y m a set of noisy observations of such variables. Assume each observation Y a to be a random function of a random subset of the X i s, and consider the conditional distribution of X i given the observations, namely μ i (x i ) = P X i = x i |Y (a posteriori probability). We establish a general decoupling principle among the X i s, as well as a relation between the distribution of μ i , and the fixed points of the associated density evolution operator. These results hold asymptotically in the large system limit, provided the average number of variables an observation depends on is bounded. We discuss the relevance of our result to a number of applications, ranging from sparse graph codes and multi-user detection, to group testing. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods from such measurements typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math within a Euclidean ball of known radius. This paper studies problems of inferring order given noisy information. In these problems there is an unknown order (permutation) @math on @math elements denoted by @math . We assume that information is generated in a way correlated with @math . The goal is to find a maximum likelihood @math given the information observed. We will consider two different types of observations: noisy comparisons and noisy orders. The data in Noisy orders are permutations given from an exponential distribution correlated with (this is also called the Mallow's model). The data in Noisy Comparisons is a signal given for each pair of elements which is correlated with their true ordering. In this paper we present polynomial time algorithms for solving both problems with high probability. As part of our proof we show that for both models the maximum likelihood solution @math is close to the original permutation @math . Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts.