Marko Beko

Slavisa Tomic, João Pedro Matos-Carvalho, Marko Beko

This work addresses weight optimization problem for fully-connected feed-forward neural networks. Unlike existing approaches that are based on back-propagation (BP) and chain rule gradient-based optimization (which implies iterative execution, potentially burdensome and time-consuming in some cases), the proposed approach offers the solution for weight optimization in closed-form by means of least squares (LS) methodology. In the case where the input-to-output mapping is injective, the new approach optimizes the weights in a back-propagating fashion in a single iteration by jointly optimizing a set of weights in each layer for each neuron. In the case where the input-to-output mapping is not injective (e.g., in classification problems), the proposed solution is easily adapted to obtain its final solution in a few iterations. An important advantage over the existing solutions is that these computations (for all neurons in a layer) are independent from each other; thus, they can be carried out in parallel to optimize all weights in a given layer simultaneously. Furthermore, its running time is deterministic in the sense that one can obtain the exact number of computations necessary to optimize the weights in all network layers (per iteration, in the case of non-injective mapping). Our simulation and empirical results show that the proposed scheme, BPLS, works well and is competitive with existing ones in terms of accuracy, but significantly surpasses them in terms of running time. To summarize, the new method is straightforward to implement, is competitive and computationally more efficient than the existing ones, and is well-tailored for parallel implementation.

Milos S. Stankovic, Marko Beko, Srdjan S. Stankovic

In this paper we propose several novel distributed gradient-based temporal difference algorithms for multi-agent off-policy learning of linear approximation of the value function in Markov decision processes with strict information structure constraints, limiting inter-agent communications to small neighborhoods. The algorithms are composed of: 1) local parameter updates based on single-agent off-policy gradient temporal difference learning algorithms, including eligibility traces with state dependent parameters, and 2) linear stochastic time varying consensus schemes, represented by directed graphs. The proposed algorithms differ by their form, definition of eligibility traces, selection of time scales and the way of incorporating consensus iterations. The main contribution of the paper is a convergence analysis based on the general properties of the underlying Feller-Markov processes and the stochastic time varying consensus model. We prove, under general assumptions, that the parameter estimates generated by all the proposed algorithms weakly converge to the corresponding ordinary differential equations (ODE) with precisely defined invariant sets. It is demonstrated how the adopted methodology can be applied to temporal-difference algorithms under weaker information structure constraints. The variance reduction effect of the proposed algorithms is demonstrated by formulating and analyzing an asymptotic stochastic differential equation. Specific guidelines for communication network design are provided. The algorithms' superior properties are illustrated by characteristic simulation results.