On characterizing optimal learning trajectories in a class of learning problems

In this brief paper, we provide a mathematical framework that exploits the relationship between the maximum principle and dynamic programming for characterizing optimal learning trajectories in a class of learning problem, which is related to point estimations for modeling of high-dimensional nonlinear functions. Here, such characterization for the optimal learning trajectories is associated with the solution of an optimal control problem for a weakly-controlled gradient system with small parameters, whose time-evolution is guided by a model training dataset and its perturbed version, while the optimization problem consists of a cost functional that summarizes how to gauge the quality/performance of the estimated model parameters at a certain fixed final time w.r.t. a model validating dataset. Moreover, using a successive Galerkin approximation method, we provide an algorithmic recipe how to construct the corresponding optimal learning trajectories leading to the optimal estimated model parameters for such a class of learning problem.

A decision-theoretic model for a principal-agent collaborative learning problem

In this technical note, we consider a collaborative learning framework with principal-agent setting, in which the principal at each time-step determines a set of appropriate aggregation coefficients based on how the current parameter estimates from a group of $K$ agents effectively performed in connection with a separate test dataset, which is not part of the agents' training model datasets. Whereas, the agents, who act together as a team, then update their parameter estimates using a discrete-time version of Langevin dynamics with mean-field-like interaction term, but guided by their respective different training model datasets. Here, we propose a decision-theoretic framework that explicitly describes how the principal progressively determines a set of nonnegative and sum to one aggregation coefficients used by the agents in their mean-field-like interaction term, that eventually leading them to reach a consensus optimal parameter estimate. Interestingly, due to the inherent feedbacks and cooperative behavior among the agents, the proposed framework offers some advantages in terms of stability and generalization, despite that both the principal and the agents do not necessarily need to have any knowledge of the sample distributions or the quality of each others' datasets.

A naive aggregation algorithm for improving generalization in a class of learning problems

In this brief paper, we present a naive aggregation algorithm for a typical learning problem with expert advice setting, in which the task of improving generalization, i.e., model validation, is embedded in the learning process as a sequential decision-making problem. In particular, we consider a class of learning problem of point estimations for modeling high-dimensional nonlinear functions, where a group of experts update their parameter estimates using the discrete-time version of gradient systems, with small additive noise term, guided by the corresponding subsample datasets obtained from the original dataset. Here, our main objective is to provide conditions under which such an algorithm will sequentially determine a set of mixing distribution strategies used for aggregating the experts' estimates that ultimately leading to an optimal parameter estimate, i.e., as a consensus solution for all experts, which is better than any individual expert's estimate in terms of improved generalization or learning performances. Finally, as part of this work, we present some numerical results for a typical case of nonlinear regression problem.