Time lower bounds for nonadaptive turnstile streaming algorithms

We say a turnstile streaming algorithm is "non-adaptive" if, during updates, the memory cells written and read depend only on the index being updated and random coins tossed at the beginning of the stream (and not on the memory contents of the algorithm). Memory cells read during queries may be decided upon adaptively. All known turnstile streaming algorithms in the literature are non-adaptive. We prove the first non-trivial update time lower bounds for both randomized and deterministic turnstile streaming algorithms, which hold when the algorithms are non-adaptive. While there has been abundant success in proving space lower bounds, there have been no non-trivial update time lower bounds in the turnstile model. Our lower bounds hold against classically studied problems such as heavy hitters, point query, entropy estimation, and moment estimation. In some cases of deterministic algorithms, our lower bounds nearly match known upper bounds

Linear Convergence of Comparison-based Step-size Adaptive Randomized Search via Stability of Markov Chains

In this paper, we consider comparison-based adaptive stochastic algorithms for solving numerical optimisation problems. We consider a specific subclass of algorithms that we call comparison-based step-size adaptive randomized search (CB-SARS), where the state variables at a given iteration are a vector of the search space and a positive parameter, the step-size, typically controlling the overall standard deviation of the underlying search distribution.We investigate the linear convergence of CB-SARS on\emph{scaling-invariant} objective functions. Scaling-invariantfunctions preserve the ordering of points with respect to their functionvalue when the points are scaled with the same positive parameter (thescaling is done w.r.t. a fixed reference point). This class offunctions includes norms composed with strictly increasing functions aswell as many non quasi-convex and non-continuousfunctions. On scaling-invariant functions, we show the existence of ahomogeneous Markov chain, as a consequence of natural invarianceproperties of CB-SARS (essentially scale-invariance and invariance tostrictly increasing transformation of the objective function). We thenderive sufficient conditions for \emph{global linear convergence} ofCB-SARS, expressed in terms of different stability conditions of thenormalised homogeneous Markov chain (irreducibility, positivity, Harrisrecurrence, geometric ergodicity) and thus define a general methodologyfor proving global linear convergence of CB-SARS algorithms onscaling-invariant functions. As a by-product we provide aconnexion between comparison-based adaptive stochasticalgorithms and Markov chain Monte Carlo algorithms.Comment: SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 201

Runtime Analysis for Self-adaptive Mutation Rates

We propose and analyze a self-adaptive version of the $(1,\lambda)$ evolutionary algorithm in which the current mutation rate is part of the individual and thus also subject to mutation. A rigorous runtime analysis on the OneMax benchmark function reveals that a simple local mutation scheme for the rate leads to an expected optimization time (number of fitness evaluations) of $O(n\lambda/\log\lambda+n\log n)$ when $\lambda$ is at least $C \ln n$ for some constant $C > 0$. For all values of $\lambda \ge C \ln n$, this performance is asymptotically best possible among all $\lambda$-parallel mutation-based unbiased black-box algorithms. Our result shows that self-adaptation in evolutionary computation can find complex optimal parameter settings on the fly. At the same time, it proves that a relatively complicated self-adjusting scheme for the mutation rate proposed by Doerr, Gie{\ss}en, Witt, and Yang~(GECCO~2017) can be replaced by our simple endogenous scheme. On the technical side, the paper contributes new tools for the analysis of two-dimensional drift processes arising in the analysis of dynamic parameter choices in EAs, including bounds on occupation probabilities in processes with non-constant drift

Constraining the Number of Positive Responses in Adaptive, Non-Adaptive, and Two-Stage Group Testing

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of "yes" responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of "yes" responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of "yes" responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p,d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts

Oracle Complexity Classes and Local Measurements on Physical Hamiltonians

The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of estimating ground state energies of local Hamiltonians. Perhaps surprisingly, [Ambainis, CCC 2014] showed that the related, but arguably more natural, problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^QMA[log]-complete even when restricted to more physical Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results. We first give a sequence of results which together yield P^QMA[log]-hardness for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. These equalities simplify the proofs of our subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians employed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians, including physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APX-SIM remains P^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.Comment: 38 pages, 3 figure

Optimal Parameter Choices Through Self-Adjustment: Applying the 1/5-th Rule in Discrete Settings

While evolutionary algorithms are known to be very successful for a broad range of applications, the algorithm designer is often left with many algorithmic choices, for example, the size of the population, the mutation rates, and the crossover rates of the algorithm. These parameters are known to have a crucial influence on the optimization time, and thus need to be chosen carefully, a task that often requires substantial efforts. Moreover, the optimal parameters can change during the optimization process. It is therefore of great interest to design mechanisms that dynamically choose best-possible parameters. An example for such an update mechanism is the one-fifth success rule for step-size adaption in evolutionary strategies. While in continuous domains this principle is well understood also from a mathematical point of view, no comparable theory is available for problems in discrete domains. In this work we show that the one-fifth success rule can be effective also in discrete settings. We regard the $(1+(\lambda,\lambda))$~GA proposed in [Doerr/Doerr/Ebel: From black-box complexity to designing new genetic algorithms, TCS 2015]. We prove that if its population size is chosen according to the one-fifth success rule then the expected optimization time on \textsc{OneMax} is linear. This is better than what \emph{any} static population size $\lambda$ can achieve and is asymptotically optimal also among all adaptive parameter choices.Comment: This is the full version of a paper that is to appear at GECCO 201