We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current mutation rate and the other half with half the current rate. The mutation rate is then updated to the rate used in that subpopulation which contains the best offspring. We analyze how the (1 + λ) evolutionary algorithm with this self-adjusting mutation rate optimizes the OneMax test function. We prove that this dynamic version of the (1 + λ) EA finds the optimum in an expected optimization time (number of fitness evaluations) of O(nλ log λ + n log n). This time is asymptotically smaller than the optimization time of the classic (1 + λ) EA. Previous work shows that this performance is best-possible among all λ-parallel mutation-based unbiased black-box algorithms. This result shows that the new way of adjusting the mutation rate can find optimal dynamic parameter values on the fly. Since our adjustment mechanism is simpler than the ones previously used for adjusting the mutation rate and does not have parameters itself we are optimistic that it will find other applications.
The first to conduct a rigorous runtime analysis of the were Jansen, De Jong, and Wegener @cite_36 . They proved, among other results, that when optimizing a linear speed-up exists up to a population size of @math , that is, for @math , finding the optimal solution takes an expected number of @math generations, whereas for larger @math at least @math generations are necessary. This picture was completed in @cite_7 with a proof that the expected number of generations taken to find the optimum is @math . The implicit constants were determined in @cite_35 , giving the bound of @math , for any constant @math , as mentioned in the introduction.
@cite_36 @cite_35 @cite_7
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The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival @math and (ii) when it is geometric( @math ) we have @math , where the @math are much smaller than previous estimates. We also study the critical value @math for "prolonged persistence" on graphs with @math vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known @math we give estimates on the rate of convergence. Physicists tell us that @math where @math is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct. Due to its optimality on a single machine for the problem of minimizing average flow time, Shortest-Remaining-Processing-Time ( ) appears to be the most natural algorithm to consider for the problem of minimizing average flow time on multiple identical machines. It is known that @math achieves the best possible competitive ratio on multiple machines up to a constant factor. Using resource augmentation, @math is known to achieve total flow time at most that of the optimal solution when given machines of speed @math . Further, it is known that @math 's competitive ratio improves as the speed increases; @math is @math -speed @math -competitive when @math . However, a gap has persisted in our understanding of @math . Before this work, the performance of @math was not known when @math is given @math -speed when @math . We complement this by showing that @math is @math -speed @math -competitive for the objective of minimizing the @math -norms of flow time on @math identical machines. Both of our results rely on new potential functions that capture the structure of . Our results, combined with previous work, show that @math is the best possible online algorithm in essentially every aspect when migration is permissible. We study the problem of recovering the subspace spanned by the first @math principal components of @math -dimensional data under the streaming setting, with a memory bound of @math . Two families of algorithms are known for this problem. The first family is based on the framework of stochastic gradient descent. Nevertheless, the convergence rate of the family can be seriously affected by the learning rate of the descent steps and deserves more serious study. The second family is based on the power method over blocks of data, but setting the block size for its existing algorithms is not an easy task. In this paper, we analyze the convergence rate of a representative algorithm with decayed learning rate (Oja and Karhunen, 1985) in the first family for the general @math case. Moreover, we propose a novel algorithm for the second family that sets the block sizes automatically and dynamically with faster convergence rate. We then conduct empirical studies that fairly compare the two families on real-world data. The studies reveal the advantages and disadvantages of these two families. Abstract We analyze how the ( 1 + λ ) evolutionary algorithm (EA) optimizes linear pseudo-Boolean functions. We prove that it finds the optimum of any linear function within an expected number of O ( 1 λ n log ⁡ n + n ) iterations. We also show that this bound is sharp for some linear functions, e.g., the binary value function. Since previous works shows an asymptotically smaller runtime for the special case of OneMax , it follows that for the ( 1 + λ ) EA different linear functions may have run-times of different asymptotic order. The proof of our upper bound heavily relies on a number of classic and recent drift analysis methods. In particular, we show how to analyze a process displaying different types of drifts in different phases. Our work corrects a wrongfully claimed better asymptotic runtime in an earlier work [13] . We also use our methods to analyze the runtime of the ( 1 + λ ) EA on the OneMax test function and obtain a new upper bound of O ( n log ⁡ log ⁡ λ log ⁡ λ ) for the case that λ is larger than O ( log ⁡ n log ⁡ log ⁡ n log ⁡ log ⁡ log ⁡ n ) ; this is the cut-off point where a linear speed-up ceases to exist. While our results are mostly spurred from a theory-driven interest, they also show that choosing the right size of the offspring population can be crucial. For both the binary value and the OneMax test function we observe that once a linear speed-up ceases to exist, in fact, the speed-up from a larger λ reduces to sub-logarithmic (still at the price of a linear increase of the cost of each generation).