Motivated by stochastic optimization, we introduce the problem of learning from samples of contextual value distributions. A contextual value distribution can be understood as a family of real-valued distributions, where each sample consists of a context $x$ and a random variable drawn from the corresponding real-valued distribution $D_x$. By minimizing a convex surrogate loss, we learn an empirical distribution $D'_x$ for each context, ensuring a small Lévy distance to $D_x$. We apply this result to obtain the sample complexity bounds for the learning of an $ε$-optimal policy for stochastic optimization problems defined on an unknown contextual value distribution. The sample complexity is shown to be polynomial for the general case of strongly monotone and stable optimization problems, including Single-item Revenue Maximization, Pandora's Box and Optimal Stopping.

We consider Inverse Optimal Stopping (IOS) problem where, based on stopped expert trajectories, one aims to recover the optimal stopping region through continuation and stopping gain functions approximation. The uniqueness of the stopping region allows the use of IOS in real-world applications with safety concerns. While current state-of-the-art inverse reinforcement learning methods recover both a Q-function and the corresponding optimal policy, they fail to account for specific challenges posed by optimal stopping problems. These include data sparsity near the stopping region, non-Markovian nature of the continuation gain, a proper treatment of boundary conditions, the need for a stable offline approach for risk-sensitive applications, and a lack of a quality evaluation metric. These challenges are addressed with the proposed Dynamics-Aware Offline Inverse Q-Learning for Optimal Stopping (DO-IQS), which incorporates temporal information by approximating the cumulative continuation gain together with the world dynamics and the Q-function without querying to the environment. Moreover, a confidence-based oversampling approach is proposed to treat the data sparsity problem. We demonstrate the performance of our models on real and artificial data including an optimal intervention for critical events problem.

When the underlying conditional density is known, conditional expectations can be computed analytically or numerically. When, however, such knowledge is not available and instead we are given a collection of training data, the goal of this work is to propose simple and purely data-driven means for estimating directly the desired conditional expectation. Because conditional expectations appear in the description of a number of stochastic optimization problems with the corresponding optimal solution satisfying a system of nonlinear equations, we extend our data-driven method to cover such cases as well. We test our methodology by applying it to Optimal Stopping and Optimal Action Policy in Reinforcement Learning.

Oberman gave a stochastic control formulation of the problem of estimating the convex envelope of a non-convex function. Based on this, we develop a reinforcement learning scheme to approximate the convex envelope, using a variant of Q-learning for controlled optimal stopping. It shows very promising results on a standard library of test problems.

We study tltt' problem of when to stop If'arning a class of feedforward networks - networks with linear outputs I1PUrOIl and fixed input weights - when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there a.re in general three distinct phases in the generalization performance in the learning process, and in particular, the network has hetter gt'neralization pPTformance when learning is stopped at a certain time before til(' global miniIl111lu of the empirical error is reachert.

Functional data analysis has long been used in modeling time series enabling long term predictions with the ability to We propose a novel group of Gaussian Process based algorithms work with irregularly sampled data [7]. In time series modeling, for fast approximate optimal stopping of time series with specific approaches based on Gaussian Processes (GPs) allow long term applications to financial markets. We show that structural properties forecasting in settings with small quantities of data for calibration commonly exhibited by financial time series (e.g., the tendency and those with a need to estimate the covariance of predictions [30, to mean-revert) allow the use of Gaussian and Deep Gaussian Process 17]. GPs also come up in finance when studying mean reverting models that further enable us to analytically evaluate optimal processes called Ornstein-Uhlenbeck (OU) processes which are GPs stopping value functions and policies. We additionally quantify with an exponential kernel [29].

Optimal stopping is the problem of deciding the right time at which to take a particular action in a stochastic system, in order to maximize an expected reward. It has many applications in areas such as finance, healthcare, and statistics. In this paper, we employ deep Reinforcement Learning (RL) to learn optimal stopping policies in two financial engineering applications: namely option pricing, and optimal option exercise. We present for the first time a comprehensive empirical evaluation of the quality of optimal stopping policies identified by three state of the art deep RL algorithms: double deep Q-learning (DDQN), categorical distributional RL (C51), and Implicit Quantile Networks (IQN). In the case of option pricing, our findings indicate that in a theoretical Black-Schole environment, IQN successfully identifies nearly optimal prices. On the other hand, it is slightly outperformed by C51 when confronted to real stock data movements in a put option exercise problem that involves assets from the S&P500 index. More importantly, the C51 algorithm is able to identify an optimal stopping policy that achieves 8% more out-of-sample returns than the best of four natural benchmark policies. We conclude with a discussion of our findings which should pave the way for relevant future research.

This paper presents new machine learning approaches to approximate the solution of optimal stopping problems. The key idea of these methods is to use neural networks, where the hidden layers are generated randomly and only the last layer is trained, in order to approximate the continuation value. Our approaches are applicable for high dimensional problems where the existing approaches become increasingly impractical. In addition, since our approaches can be optimized using a simple linear regression, they are very easy to implement and theoretical guarantees can be provided. In Markovian examples our randomized reinforcement learning approach and in non-Markovian examples our randomized recurrent neural network approach outperform the state-of-the-art and other relevant machine learning approaches.

We study tltt' problem of when to stop If'arning a class of feedforward networks - networks with linear outputs I1PUrOIl and fixed input weights - when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there a.re in general three distinct phases in the generalization performance in the learning process, and in particular, the network has hetter gt'neralization pPTformance when learning is stopped at a certain time before til(' global miniIl111lu of the empirical error is reachert. A notion of effective size of a machine is rtefil1e i and used to explain the tradeoff betwf'en the complexity of the marhine and the training error ill the learning process. The study leads nat.urally to a network size selection critt'rion, which turns Ol1t to be a generalization of Akaike's Information Criterioll for the It'arning process. It if; shown that stopping Iparning before tiJt' global minimum of the empirical error has the effect of network size splectioll. 1 INTRODUCTION The primary goal of learning in neural nets is to find a network that gives valid generalization. In achieving this goal, a central issue is the tradeoff between the training error and network complexity. This usually reduces to a problem of network size selection, which has drawn much research effort in recent years. Various principles, theories, and intuitions, including Occam's razor, statistical model selection criteria such as Akaike's Information Criterion (AIC) [11 and many others [5, 1, 10,3,111 all quantitatively support the following PAC prescription: between two machines which have the same empirical error, the machine with smaller VC-dimf'nsion generalizes better. However, it is noted that these methods or criteria do not npcpssarily If'ad to optimal (or llearly optimal) generalization performance.

We study tltt' problem of when to stop If'arning a class of feedforward networks - networks with linear outputs I1PUrOIl and fixed input weights - when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there a.re in general three distinct phases in the generalization performance in the learning process, and in particular, the network has hetter gt'neralization pPTformance when learning is stopped at a certain time before til(' global miniIl111lu of the empirical error is reachert. A notion of effective size of a machine is rtefil1e i and used to explain the tradeoff betwf'en the complexity of the marhine and the training error ill the learning process. The study leads nat.urally to a network size selection critt'rion, which turns Ol1t to be a generalization of Akaike's Information Criterioll for the It'arning process. It if; shown that stopping Iparning before tiJt' global minimum of the empirical error has the effect of network size splectioll. 1 INTRODUCTION The primary goal of learning in neural nets is to find a network that gives valid generalization. In achieving this goal, a central issue is the tradeoff between the training error and network complexity. This usually reduces to a problem of network size selection, which has drawn much research effort in recent years. Various principles, theories, and intuitions, including Occam's razor, statistical model selection criteria such as Akaike's Information Criterion (AIC) [11 and many others [5, 1, 10,3,111 all quantitatively support the following PAC prescription: between two machines which have the same empirical error, the machine with smaller VC-dimf'nsion generalizes better. However, it is noted that these methods or criteria do not npcpssarily If'ad to optimal (or llearly optimal) generalization performance.