Deep Q-Learning for Nash Equilibria: Nash-DQN

Model-free learning for multi-agent stochastic games is an active area of research. Existing reinforcement learning algorithms, however, are often restricted to zero-sum games, and are applicable only in small state-action spaces or other simplified settings. Here, we develop a new data efficient Deep-Q-learning methodology for model-free learning of Nash equilibria for general-sum stochastic games. The algorithm uses a local linear-quadratic expansion of the stochastic game, which leads to analytically solvable optimal actions. The expansion is parametrized by deep neural networks to give it sufficient flexibility to learn the environment without the need to experience all state-action pairs. We study symmetry properties of the algorithm stemming from label-invariant stochastic games and as a proof of concept, apply our algorithm to learning optimal trading strategies in competitive electronic markets.Comment: 16 pages, 4 figure

Stochastic Answer Networks for Machine Reading Comprehension

We propose a simple yet robust stochastic answer network (SAN) that simulates multi-step reasoning in machine reading comprehension. Compared to previous work such as ReasoNet which used reinforcement learning to determine the number of steps, the unique feature is the use of a kind of stochastic prediction dropout on the answer module (final layer) of the neural network during the training. We show that this simple trick improves robustness and achieves results competitive to the state-of-the-art on the Stanford Question Answering Dataset (SQuAD), the Adversarial SQuAD, and the Microsoft MAchine Reading COmprehension Dataset (MS MARCO).Comment: 11 pages, 5 figures, Accepted to ACL 201

RSA: Byzantine-Robust Stochastic Aggregation Methods for Distributed Learning from Heterogeneous Datasets

In this paper, we propose a class of robust stochastic subgradient methods for distributed learning from heterogeneous datasets at presence of an unknown number of Byzantine workers. The Byzantine workers, during the learning process, may send arbitrary incorrect messages to the master due to data corruptions, communication failures or malicious attacks, and consequently bias the learned model. The key to the proposed methods is a regularization term incorporated with the objective function so as to robustify the learning task and mitigate the negative effects of Byzantine attacks. The resultant subgradient-based algorithms are termed Byzantine-Robust Stochastic Aggregation methods, justifying our acronym RSA used henceforth. In contrast to most of the existing algorithms, RSA does not rely on the assumption that the data are independent and identically distributed (i.i.d.) on the workers, and hence fits for a wider class of applications. Theoretically, we show that: i) RSA converges to a near-optimal solution with the learning error dependent on the number of Byzantine workers; ii) the convergence rate of RSA under Byzantine attacks is the same as that of the stochastic gradient descent method, which is free of Byzantine attacks. Numerically, experiments on real dataset corroborate the competitive performance of RSA and a complexity reduction compared to the state-of-the-art alternatives.Comment: To appear in AAAI 201

A Parameter-Free Learning Automaton Scheme

For a learning automaton, a proper configuration of its learning parameters, which are crucial for the automaton's performance, is relatively difficult due to the necessity of a manual parameter tuning before real applications. To ensure a stable and reliable performance in stochastic environments, parameter tuning can be a time-consuming and interaction-costing procedure in the field of LA. Especially, it is a fatal limitation for LA-based applications where the interactions with environments are expensive. In this paper, we propose a parameter-free learning automaton scheme to avoid parameter tuning by a Bayesian inference method. In contrast to existing schemes where the parameters should be carefully tuned according to the environment, the performance of this scheme is not sensitive to external environments because a set of parameters can be consistently applied to various environments, which dramatically reduce the difficulty of applying a learning automaton to an unknown stochastic environment. A rigorous proof of $\epsilon$-optimality for the proposed scheme is provided and numeric experiments are carried out on benchmark environments to verify its effectiveness. The results show that, without any parameter tuning cost, the proposed parameter-free learning automaton (PFLA) can achieve a competitive performance compared with other well-tuned schemes and outperform untuned schemes on consistency of performance

Extended Distributed Learning Automata:A New Method for Solving Stochastic Graph Optimization Problems

In this paper, a new structure of cooperative learning automata so-called extended learning automata (eDLA) is introduced. Based on the proposed structure, a new iterative randomized heuristic algorithm for finding optimal sub-graph in a stochastic edge-weighted graph through sampling is proposed. It has been shown that the proposed algorithm based on new networked-structure can be to solve the optimization problems on stochastic graph through less number of sampling in compare to standard sampling. Stochastic graphs are graphs in which the edges have an unknown distribution probability weights. Proposed algorithm uses an eDLA to find a policy that leads to an induced sub-graph that satisfies some restrictions such as minimum or maximum weight (length). At each stage of the proposed algorithm, eDLA determines which edges to be sampled. This eDLA-based proposed sampling method may result in decreasing unnecessary samples and hence decreasing the time that algorithm requires for finding the optimal sub-graph. It has been shown that proposed method converge to optimal solution, furthermore the probability of this convergence can be made arbitrarily close to 1 by using a sufficiently small learning rate. A new variance-aware threshold value was proposed that can be improving significantly convergence rate of the proposed eDLA-based algorithm. It has been shown that the proposed algorithm is competitive in terms of the quality of the solutio

Learning Rate Adaptation for Federated and Differentially Private Learning

We propose an algorithm for the adaptation of the learning rate for stochastic gradient descent (SGD) that avoids the need for validation set use. The idea for the adaptiveness comes from the technique of extrapolation: to get an estimate for the error against the gradient flow which underlies SGD, we compare the result obtained by one full step and two half-steps. The algorithm is applied in two separate frameworks: federated and differentially private learning. Using examples of deep neural networks we empirically show that the adaptive algorithm is competitive with manually tuned commonly used optimisation methods for differentially privately training. We also show that it works robustly in the case of federated learning unlike commonly used optimisation methods.Comment: 17 pages, 9 figure

Fast Quantization of Stochastic Volatility Models

Recursive Marginal Quantization (RMQ) allows fast approximation of solutions to stochastic differential equations in one-dimension. When applied to two factor models, RMQ is inefficient due to the fact that the optimization problem is usually performed using stochastic methods, e.g., Lloyd's algorithm or Competitive Learning Vector Quantization. In this paper, a new algorithm is proposed that allows RMQ to be applied to two-factor stochastic volatility models, which retains the efficiency of gradient-descent techniques. By margining over potential realizations of the volatility process, a significant decrease in computational effort is achieved when compared to current quantization methods. Additionally, techniques for modelling the correct zero-boundary behaviour are used to allow the new algorithm to be applied to cases where the previous methods would fail. The proposed technique is illustrated for European options on the Heston and Stein-Stein models, while a more thorough application is considered in the case of the popular SABR model, where various exotic options are also priced

Conditional Generative Moment-Matching Networks

Maximum mean discrepancy (MMD) has been successfully applied to learn deep generative models for characterizing a joint distribution of variables via kernel mean embedding. In this paper, we present conditional generative moment- matching networks (CGMMN), which learn a conditional distribution given some input variables based on a conditional maximum mean discrepancy (CMMD) criterion. The learning is performed by stochastic gradient descent with the gradient calculated by back-propagation. We evaluate CGMMN on a wide range of tasks, including predictive modeling, contextual generation, and Bayesian dark knowledge, which distills knowledge from a Bayesian model by learning a relatively small CGMMN student network. Our results demonstrate competitive performance in all the tasks.Comment: 12 page

Projective simulation for classical learning agents: a comprehensive investigation

We study the model of projective simulation (PS), a novel approach to artificial intelligence based on stochastic processing of episodic memory which was recently introduced [H.J. Briegel and G. De las Cuevas. Sci. Rep. 2, 400, (2012)]. Here we provide a detailed analysis of the model and examine its performance, including its achievable efficiency, its learning times and the way both properties scale with the problems' dimension. In addition, we situate the PS agent in different learning scenarios, and study its learning abilities. A variety of new scenarios are being considered, thereby demonstrating the model's flexibility. Furthermore, to put the PS scheme in context, we compare its performance with those of Q-learning and learning classifier systems, two popular models in the field of reinforcement learning. It is shown that PS is a competitive artificial intelligence model of unique properties and strengths.Comment: Accepted for publication in New Generation Computing. 23 pages, 23 figure

Sample Complexity of Estimating the Policy Gradient for Nearly Deterministic Dynamical Systems

Reinforcement learning is a promising approach to learning robot controllers. It has recently been shown that algorithms based on finite-difference estimates of the policy gradient are competitive with algorithms based on the policy gradient theorem. We propose a theoretical framework for understanding this phenomenon. Our key insight is that many dynamical systems (especially those of interest in robot control tasks) are \emph{nearly deterministic}---i.e., they can be modeled as a deterministic system with a small stochastic perturbation. We show that for such systems, finite-difference estimates of the policy gradient can have substantially lower variance than estimates based on the policy gradient theorem. We interpret these results in the context of counterfactual estimation. Finally, we empirically evaluate our insights in an experiment on the inverted pendulum