Abstract-This paper considers unconstrained convex optimization problems with time-varying objective functions. We propose algorithms with a discrete time-sampling scheme to find and track the solution trajectory based on prediction and correction steps, while sampling the problem data at a constant rate of 1{h, where h is the sampling period. The prediction step is derived by analyzing the iso-residual dynamics of the optimality conditions. The correction step adjusts for the distance between the current prediction and the optimizer at each time step, and consists either of one or multiple gradient steps or Newton steps, which respectively correspond to the gradient trajectory tracking (GTT) or Newton trajectory tracking (NTT) algorithms. Under suitable conditions, we establish that the asymptotic error incurred by both proposed methods behaves as Oph 2 q, and in some cases as Oph 4 q, which outperforms the state-of-theart error bound of Ophq for correction-only methods in the gradient-correction step. Moreover, when the characteristics of the objective function variation are not available, we propose approximate gradient and Newton tracking algorithms (AGT and ANT, respectively) that still attain these asymptotical error bounds. Numerical simulations demonstrate the practical utility of the proposed methods and that they improve upon existing techniques by several orders of magnitude.

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Despite their attractiveness, popular perception is that techniques for nonparametric function approximation do not scale to streaming data due to an intractable growth in the amount of storage they require. To solve this problem in a memory-affordable way, we propose an online technique based on functional stochastic gradient descent in tandem with supervised sparsification based on greedy function subspace projections. The method, called parsimonious online learning with kernels (POLK), provides a controllable tradeoff between its solution accuracy and the amount of memory it requires. We derive conditions under which the generated function sequence converges almost surely to the optimal function, and we establish that the memory requirement remains finite. We evaluate POLK for kernel multi-class logistic regression and kernel hinge-loss classification on three canonical data sets: a synthetic Gaussian mixture model, the MNIST hand-written digits, and the Brodatz texture database. On all three tasks, we observe a favorable trade-off of objective function evaluation, classification performance, and complexity of the nonparametric regressor extracted the proposed method.

Abstract-We develop algorithms that find and track the optimal solution trajectory of time-varying convex optimization problems which consist of local and network-related objectives. The algorithms are derived from the prediction-correction methodology, which corresponds to a strategy where the timevarying problem is sampled at discrete time instances and then a sequence is generated via alternatively executing predictions on how the optimizers at the next time sample are changing and corrections on how they actually have changed. Prediction is based on how the optimality conditions evolve in time, while correction is based on a gradient or Newton method, leading to Decentralized Prediction-Correction Gradient (DPC-G) and Decentralized Prediction-Correction Newton (DPC-N). We extend these methods to cases where the knowledge on how the optimization programs are changing in time is only approximate and propose Decentralized Approximate Prediction-Correction Gradient (DAPC-G) and Decentralized Approximate PredictionCorrection Newton (DAPC-N). Convergence properties of all the proposed methods are studied and empirical performance is shown on an application of a resource allocation problem in a wireless network. We observe that the proposed methods outperform existing running algorithms by orders of magnitude. The numerical results showcase a trade-off between convergence accuracy, sampling period, and network communications.

Reinforcement learning, mathematically described by Markov Decision Problems, may be approached either through dynamic programming or policy search. Actor-critic algorithms combine the merits of both approaches by alternating between steps to estimate the value function and policy gradient updates. Due to the fact that the updates exhibit correlated noise and biased gradient updates, only the asymptotic behavior of actor-critic is known by connecting its behavior to dynamical systems. This work puts forth a new variant of actor-critic that employs Monte Carlo rollouts during the policy search updates, which results in controllable bias that depends on the number of critic evaluations. As a result, we are able to provide for the first time the convergence rate of actor-critic algorithms when the policy search step employs policy gradient, agnostic to the choice of policy evaluation technique. In particular, we establish conditions under which the sample complexity is comparable to stochastic gradient method for non-convex problems or slower as a result of the critic estimation error, which is the main complexity bottleneck. These results hold for in continuous state and action spaces with linear function approximation for the value function. We then specialize these conceptual results to the case where the critic is estimated by Temporal Difference, Gradient Temporal Difference, and Accelerated Gradient Temporal Difference. These learning rates are then corroborated on a navigation problem involving an obstacle, which suggests that learning more slowly may lead to improved limit points, providing insight into the interplay between optimization and generalization in reinforcement learning.