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We study a new two-time-scale stochastic gradient method for solving optimization problems, where the gradients are computed with the aid of an auxiliary variable under samples generated by time-varying Markov random processes parameterized by the underlying optimization variable. These time-varying samples make gradient directions in our update biased and dependent, which can potentially lead to the divergence of the iterates. In our two-time-scale approach, one scale is to estimate the true gradient from these samples, which is then used to update the estimate of the optimal solution. While these two iterates are implemented simultaneously, the former is updated "faster" (using bigger step sizes) than the latter (using smaller step sizes). Our first contribution is to characterize the finite-time complexity of the proposed two-time-scale stochastic gradient method. In particular, we provide explicit formulas for the convergence rates of this method under different structural assumptions, namely, strong convexity, convexity, the Polyak-Lojasiewicz condition, and general non-convexity. We apply our framework to two problems in control and reinforcement learning. First, we look at the standard online actor-critic algorithm over finite state and action spaces and derive a convergence rate of O(k^(-2/5)), which recovers the best known rate derived specifically for this problem. Second, we study an online actor-critic algorithm for the linear-quadratic regulator and show that a convergence rate of O(k^(-2/3)) is achieved. This is the first time such a result is known in the literature. Finally, we support our theoretical analysis with numerical simulations where the convergence rates are visualized.

We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to unity, as is the case with Fokker-Planck equations. Our approach involves reparameterizing the solution as the exponential of a neural network appropriately normalized to ensure both requirements are satisfied. This then gives rise to nonlinear a partial integro-differential equation (PIDE) where the integral appearing in the equation is handled by a novel application of importance sampling. Secondly, we tackle a number of Hamilton-Jacobi-Bellman (HJB) equations that appear in stochastic optimal control problems. The key contribution is that these equations are approached in their unsimplified primal form which includes an optimization problem as part of the equation. We extend the DGM algorithm to solve for the value function and the optimal control \simultaneously by characterizing both as deep neural networks. Training the networks is performed by taking alternating stochastic gradient descent steps for the two functions, a technique inspired by the policy improvement algorithms (PIA).

Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. The novelty, in this paper, lies in letting the discretization parameter (time step-size) vary from layer to layer, which needs to be learned, in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional-DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. Stability of some of the existing continuous DNNs such as Fractional-DNN is also studied. The proposed approach is applied to an ill-posed 3D-Maxwell's equation.

In this paper, we introduce reduced-bias estimators for the estimation of the tail index of a Pareto-type distribution. This is achieved through the use of a regularised weighted least squares with an exponential regression model for log-spacings of top order statistics. The asymptotic properties of the proposed estimators are investigated analytically and found to be asymptotically unbiased, consistent and normally distributed. Also, the finite sample behaviour of the estimators are studied through a simulations theory. The proposed estimators were found to yield low bias and MSE. In addition, the proposed estimators are illustrated through the estimation of the tail index of the underlying distribution of claims from the insurance industry.

We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.

This paper proposes a numerical method based on the Adomian decomposition approach for the time discretization, applied to Euler equations. A recursive property is demonstrated that allows to formulate the method in an appropriate and efficient way. To obtain a fully numerical scheme, the space discretization is achieved using the classical DG techniques. The efficiency of the obtained numerical scheme is demonstrated through numerical tests by comparison to exact solution and the popular Runge-Kutta DG method results.

Autonomous marine vessels are expected to avoid inter-vessel collisions and comply with the international regulations for safe voyages. This paper presents a stepwise path planning method using stream functions. The dynamic flow of fluids is used as a guidance model, where the collision avoidance in static environments is achieved by applying the circular theorem in the sink flow. We extend this method to dynamic environments by adding vortex flows in the flow field. The stream function is recursively updated to enable on the fly waypoint decisions. The vessel avoids collisions and also complies with several rules of the Convention on the International Regulations for Preventing Collisions at Sea. The method is conceptually and computationally simple and convenient to tune, and yet versatile to handle complex and dense marine traffic with multiple dynamic obstacles. The ship dynamics are taken into account, by using B\'{e}zier curves to generate a sufficiently smooth path with feasible curvature. Numerical simulations are conducted to verify the proposed method.

Policy gradient (PG) estimation becomes a challenge when we are not allowed to sample with the target policy but only have access to a dataset generated by some unknown behavior policy. Conventional methods for off-policy PG estimation often suffer from either significant bias or exponentially large variance. In this paper, we propose the double Fitted PG estimation (FPG) algorithm. FPG can work with an arbitrary policy parameterization, assuming access to a Bellman-complete value function class. In the case of linear value function approximation, we provide a tight finite-sample upper bound on policy gradient estimation error, that is governed by the amount of distribution mismatch measured in feature space. We also establish the asymptotic normality of FPG estimation error with a precise covariance characterization, which is further shown to be statistically optimal with a matching Cramer-Rao lower bound. Empirically, we evaluate the performance of FPG on both policy gradient estimation and policy optimization, using either softmax tabular or ReLU policy networks. Under various metrics, our results show that FPG significantly outperforms existing off-policy PG estimation methods based on importance sampling and variance reduction techniques.

The best neural architecture for a given machine learning problem depends on many factors: not only the complexity and structure of the dataset, but also on resource constraints including latency, compute, energy consumption, etc. Neural architecture search (NAS) for tabular datasets is an important but under-explored problem. Previous NAS algorithms designed for image search spaces incorporate resource constraints directly into the reinforcement learning rewards. In this paper, we argue that search spaces for tabular NAS pose considerable challenges for these existing reward-shaping methods, and propose a new reinforcement learning (RL) controller to address these challenges. Motivated by rejection sampling, when we sample candidate architectures during a search, we immediately discard any architecture that violates our resource constraints. We use a Monte-Carlo-based correction to our RL policy gradient update to account for this extra filtering step. Results on several tabular datasets show TabNAS, the proposed approach, efficiently finds high-quality models that satisfy the given resource constraints.

One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.