A Statistical Learning Theory Approach for Uncertain Linear and Bilinear Matrix Inequalities

In this paper, we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities, which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results in statistical learning theory, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chervonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of these problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality. The effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters.Comment: 19 pages, 2 figures, Accepted for Publication in Automatic

Reinforcement Learning Based Minimum State-flipped Control for the Reachability of Boolean Control Networks

To realize reachability as well as reduce control costs of Boolean Control Networks (BCNs) with state-flipped control, a reinforcement learning based method is proposed to obtain flip kernels and the optimal policy with minimal flipping actions to realize reachability. The method proposed is model-free and of low computational complexity. In particular, Q-learning (QL), fast QL, and small memory QL are proposed to find flip kernels. Fast QL and small memory QL are two novel algorithms. Specifically, fast QL, namely, QL combined with transfer-learning and special initial states, is of higher efficiency, and small memory QL is applicable to large-scale systems. Meanwhile, we present a novel reward setting, under which the optimal policy with minimal flipping actions to realize reachability is the one of the highest returns. Then, to obtain the optimal policy, we propose QL, and fast small memory QL for large-scale systems. Specifically, on the basis of the small memory QL mentioned before, the fast small memory QL uses a changeable reward setting to speed up the learning efficiency while ensuring the optimality of the policy. For parameter settings, we give some system properties for reference. Finally, two examples, which are a small-scale system and a large-scale one, are considered to verify the proposed method

Memory lower bounds for deterministic self-stabilization

In the context of self-stabilization, a \emph{silent} algorithm guarantees that the register of every node does not change once the algorithm has stabilized. At the end of the 90's, Dolev et al. [Acta Inf. '99] showed that, for finding the centers of a graph, for electing a leader, or for constructing a spanning tree, every silent algorithm must use a memory of $\Omega(\log n)$ bits per register in $n$-node networks. Similarly, Korman et al. [Dist. Comp. '07] proved, using the notion of proof-labeling-scheme, that, for constructing a minimum-weight spanning trees (MST), every silent algorithm must use a memory of $\Omega(\log^2n)$ bits per register. It follows that requiring the algorithm to be silent has a cost in terms of memory space, while, in the context of self-stabilization, where every node constantly checks the states of its neighbors, the silence property can be of limited practical interest. In fact, it is known that relaxing this requirement results in algorithms with smaller space-complexity. In this paper, we are aiming at measuring how much gain in terms of memory can be expected by using arbitrary self-stabilizing algorithms, not necessarily silent. To our knowledge, the only known lower bound on the memory requirement for general algorithms, also established at the end of the 90's, is due to Beauquier et al.~[PODC '99] who proved that registers of constant size are not sufficient for leader election algorithms. We improve this result by establishing a tight lower bound of $\Theta(\log \Delta+\log \log n)$ bits per register for self-stabilizing algorithms solving $(\Delta+1)$-coloring or constructing a spanning tree in networks of maximum degree~$\Delta$. The lower bound $\Omega(\log \log n)$ bits per register also holds for leader election