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This work addresses a version of the two-armed Bernoulli bandit problem where the sum of the means of the arms is one (the symmetric two-armed Bernoulli bandit). In a regime where the gap between these means goes to zero and the number of prediction periods approaches infinity, we obtain the leading order terms of the minmax optimal regret and pseudoregret for this problem by associating each of them with a solution of a linear heat equation. Our results improve upon the previously known results; specifically, we explicitly compute these leading order terms in three different scaling regimes for the gap. Additionally, we obtain new non-asymptotic bounds for any given time horizon.

We prove that any $n$-qubit unitary transformation can be implemented (i) approximately in time $\tilde O\big(2^{n/2}\big)$ with query access to an appropriate classical oracle, and also (ii) exactly by a circuit of depth $\tilde O\big(2^{n/2}\big)$ with one- and two-qubit gates and $2^{O(n)}$ ancillae. The proofs involve similar reductions to Grover search. The proof of (ii) also involves a linear-depth construction of arbitrary quantum states using one- and two-qubit gates (in fact, this can be improved to constant depth with the addition of fanout and generalized Toffoli gates) which may be of independent interest. We also prove a matching $\Omega\big(2^{n/2}\big)$ lower bound for (i) and (ii) for a certain class of implementations.

This work, for the first time, introduces two constant factor approximation algorithms with linear query complexity for non-monotone submodular maximization over a ground set of size $n$ subject to a knapsack constraint, $\mathsf{DLA}$ and $\mathsf{RLA}$. $\mathsf{DLA}$ is a deterministic algorithm that provides an approximation factor of $6+\epsilon$ while $\mathsf{RLA}$ is a randomized algorithm with an approximation factor of $4+\epsilon$. Both run in $O(n \log(1/\epsilon)/\epsilon)$ query complexity. The key idea to obtain a constant approximation ratio with linear query lies in: (1) dividing the ground set into two appropriate subsets to find the near-optimal solution over these subsets with linear queries, and (2) combining a threshold greedy with properties of two disjoint sets or a random selection process to improve solution quality. In addition to the theoretical analysis, we have evaluated our proposed solutions with three applications: Revenue Maximization, Image Summarization, and Maximum Weighted Cut, showing that our algorithms not only return comparative results to state-of-the-art algorithms but also require significantly fewer queries.

Stochastic gradient MCMC (SGMCMC) offers a scalable alternative to traditional MCMC, by constructing an unbiased estimate of the gradient of the log-posterior with a small, uniformly-weighted subsample of the data. While efficient to compute, the resulting gradient estimator may exhibit a high variance and impact sampler performance. The problem of variance control has been traditionally addressed by constructing a better stochastic gradient estimator, often using control variates. We propose to use a discrete, non-uniform probability distribution to preferentially subsample data points that have a greater impact on the stochastic gradient. In addition, we present a method of adaptively adjusting the subsample size at each iteration of the algorithm, so that we increase the subsample size in areas of the sample space where the gradient is harder to estimate. We demonstrate that such an approach can maintain the same level of accuracy while substantially reducing the average subsample size that is used.

We study the logit evolutionary dynamics in population games. For general population games, we prove that, on the one hand strict Nash equilibria are locally asymptotically stable under the logit dynamics in the low noise regime, on the other hand a globally exponentially stable fixed point exists in the high noise regime. This suggests the emergence of bifurcations in population games admitting multiple strict Nash equilibria, as verified numerically in previous publications. We then prove sufficient conditions on the game structure for global asymptotic stability of the logit dynamics in every noise regime. Our results find application in particular to heterogeneous routing games, a class of non-potential population games modelling strategic decision-making of users having heterogeneous preferences in transportation networks. In this setting, preference heterogeneities are due, e.g., to access to different sources of information or to different trade-offs between time and money. We show that if the transportation network has parallel routes, then the unique equilibrium of the game is globally asymptotically stable.

Time is a crucial factor in modelling dynamic behaviours of intelligent agents: activities have a determined temporal duration in a real-world environment, and previous actions influence agents' behaviour. In this paper, we propose a language for modelling concurrent interaction between agents that also allows the specification of temporal intervals in which particular actions occur. Such a language exploits a timed version of Abstract Argumentation Frameworks to realise a shared memory used by the agents to communicate and reason on the acceptability of their beliefs with respect to a given time interval. An interleaving model on a single processor is used for basic computation steps, with maximum parallelism for time elapsing. Following this approach, only one of the enabled agents is executed at each moment. To demonstrate the capabilities of language, we also show how it can be used to model interactions such as debates and dialogue games taking place between intelligent agents. Lastly, we present an implementation of the language that can be accessed via a web interface. Under consideration in Theory and Practice of Logic Programming (TPLP).

This paper investigates the intelligent computing task-oriented computing offloading and semantic compression in mobile edge computing (MEC) systems. With the popularity of intelligent applications in various industries, terminals increasingly need to offload intelligent computing tasks with complex demands to MEC servers for computing, which is a great challenge for bandwidth and computing capacity allocation in MEC systems. Considering the accuracy requirement of intelligent computing tasks, we formulate an optimization problem of computing offloading and semantic compression. We jointly optimize the system utility which are represented as computing accuracy and task delay respectively to acquire the optimized system utility. To solve the proposed optimization problem, we decompose it into computing capacity allocation subproblem and compression offloading subproblem and obtain solutions through convex optimization and successive convex approximation. After that, the offloading decisions, computing capacity and compressed ratio are obtained in closed forms. We design the computing offloading and semantic compression algorithm for intelligent computing tasks in MEC systems then. Simulation results represent that our algorithm converges quickly and acquires better performance and resource utilization efficiency through the trend with total number of users and computing capacity compared with benchmarks.

Gaussian variational inference and the Laplace approximation are popular alternatives to Markov chain Monte Carlo that formulate Bayesian posterior inference as an optimization problem, enabling the use of simple and scalable stochastic optimization algorithms. However, a key limitation of both methods is that the solution to the optimization problem is typically not tractable to compute; even in simple settings the problem is nonconvex. Thus, recently developed statistical guarantees -- which all involve the (data) asymptotic properties of the global optimum -- are not reliably obtained in practice. In this work, we provide two major contributions: a theoretical analysis of the asymptotic convexity properties of variational inference with a Gaussian family and the maximum a posteriori (MAP) problem required by the Laplace approximation; and two algorithms -- consistent Laplace approximation (CLA) and consistent stochastic variational inference (CSVI) -- that exploit these properties to find the optimal approximation in the asymptotic regime. Both CLA and CSVI involve a tractable initialization procedure that finds the local basin of the optimum, and CSVI further includes a scaled gradient descent algorithm that provably stays locally confined to that basin. Experiments on nonconvex synthetic and real-data examples show that compared with standard variational and Laplace approximations, both CSVI and CLA improve the likelihood of obtaining the global optimum of their respective optimization problems.

Many food products involve mixtures of ingredients, where the mixtures can be expressed as combinations of ingredient proportions. In many cases, the quality and the consumer preference may also depend on the way in which the mixtures are processed. The processing is generally defined by the settings of one or more process variables. Experimental designs studying the joint impact of the mixture ingredient proportions and the settings of the process variables are called mixture-process variable experiments. In this article, we show how to combine mixture-process variable experiments and discrete choice experiments, to quantify and model consumer preferences for food products that can be viewed as processed mixtures. First, we describe the modeling of data from such combined experiments. Next, we describe how to generate D- and I-optimal designs for choice experiments involving mixtures and process variables, and we compare the two kinds of designs using two examples.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.