General policy improvement (GPI) and trust-region learning (TRL) are the predominant frameworks within contemporary reinforcement learning (RL), which serve as the core models for solving Markov decision processes (MDPs). Unfortunately, in their mathematical form, they are sensitive to modifications, and thus, the practical instantiations that implement them do not automatically inherit their improvement guarantees. As a result, the spectrum of available rigorous MDP-solvers is narrow. Indeed, many state-of-the-art (SOTA) algorithms, such as TRPO and PPO, are not proven to converge. In this paper, we propose \textsl{mirror learning} -- a general solution to the RL problem. We reveal GPI and TRL to be but small points within this far greater space of algorithms which boasts the monotonic improvement property and converges to the optimal policy. We show that virtually all SOTA algorithms for RL are instances of mirror learning, and thus suggest that their empirical performance is a consequence of their theoretical properties, rather than of approximate analogies. Excitingly, we show that mirror learning opens up a whole new space of policy learning methods with convergence guarantees.
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Applications of Reinforcement Learning (RL), in which agents learn to make a sequence of decisions despite lacking complete information about the latent states of the controlled system, that is, they act under partial observability of the states, are ubiquitous. Partially observable RL can be notoriously difficult -- well-known information-theoretic results show that learning partially observable Markov decision processes (POMDPs) requires an exponential number of samples in the worst case. Yet, this does not rule out the existence of large subclasses of POMDPs over which learning is tractable. In this paper we identify such a subclass, which we call weakly revealing POMDPs. This family rules out the pathological instances of POMDPs where observations are uninformative to a degree that makes learning hard. We prove that for weakly revealing POMDPs, a simple algorithm combining optimism and Maximum Likelihood Estimation (MLE) is sufficient to guarantee polynomial sample complexity. To the best of our knowledge, this is the first provably sample-efficient result for learning from interactions in overcomplete POMDPs, where the number of latent states can be larger than the number of observations.
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.
The naive importance sampling (IS) estimator generally does not work well in examples involving simultaneous inference on several targets, as the importance weights can take arbitrarily large values, making the estimator highly unstable. In such situations, alternative multiple IS estimators involving samples from multiple proposal distributions are preferred. Just like the naive IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. The selection of these proposal distributions is the focus of this article. We propose three methods: (i) a geometric space filling approach, (ii) a minimax variance approach, and (iii) a maximum entropy approach. The first two methods are applicable to any IS estimator, whereas the third approach is described in the context of Doss's (2010) two-stage IS estimator. For the first method, we propose a suitable measure of 'closeness' based on the symmetric Kullback-Leibler divergence, while the second and third approaches use estimates of asymptotic variances of Doss's (2010) IS estimator and Geyer's (1994) reverse logistic regression estimator, respectively. Thus, when samples from the proposal distributions are obtained by running Markov chains, we provide consistent spectral variance estimators for these asymptotic variances. The proposed methods for selecting proposal densities are illustrated using various detailed examples.
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.
In this paper, we propose a PAC-Bayesian \textit{a posteriori} parameter selection scheme for adaptive regularized regression in Hilbert scales under general, unknown source conditions. We demonstrate that our approach is adaptive to misspecification, and achieves the optimal learning rate under subgaussian noise. Unlike existing parameter selection schemes, the computational complexity of our approach is independent of sample size. We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems under general, misspecified source conditions, that notably do not require any conventional a priori assumptions on kernel eigendecay. Using the theory of interpolation, we demonstrate that the spectrum of the Mercer operator can be inferred in the presence of "tight" $L^{\infty}$ embeddings of suitable Hilbert scales. Finally, we prove, that under a $\Delta_2$ condition on the smoothness index functions, our PAC-Bayesian scheme can indeed achieve minimax rates. We discuss applications of our approach to statistical inverse problems and oracle-efficient contextual bandit algorithms.
We introduce a novel methodology for particle filtering in dynamical systems where the evolution of the signal of interest is described by a SDE and observations are collected instantaneously at prescribed time instants. The new approach includes the discretisation of the SDE and the design of efficient particle filters for the resulting discrete-time state-space model. The discretisation scheme converges with weak order 1 and it is devised to create a sequential dependence structure along the coordinates of the discrete-time state vector. We introduce a class of space-sequential particle filters that exploits this structure to improve performance when the system dimension is large. This is numerically illustrated by a set of computer simulations for a stochastic Lorenz 96 system with additive noise. The new space-sequential particle filters attain approximately constant estimation errors as the dimension of the Lorenz 96 system is increased, with a computational cost that increases polynomially, rather than exponentially, with the system dimension. Besides the new numerical scheme and particle filters, we provide in this paper a general framework for discrete-time filtering in continuous-time dynamical systems described by a SDE and instantaneous observations. Provided that the SDE is discretised using a weakly-convergent scheme, we prove that the marginal posterior laws of the resulting discrete-time state-space model converge to the posterior marginal posterior laws of the original continuous-time state-space model under a suitably defined metric. This result is general and not restricted to the numerical scheme or particle filters specifically studied in this manuscript.
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.
We demonstrate that merely analog transmissions and match filtering can realize the function of an edge server in federated learning (FL). Therefore, a network with massively distributed user equipments (UEs) can achieve large-scale FL without an edge server. We also develop a training algorithm that allows UEs to continuously perform local computing without being interrupted by the global parameter uploading, which exploits the full potential of UEs' processing power. We derive convergence rates for the proposed schemes to quantify their training efficiency. The analyses reveal that when the interference obeys a Gaussian distribution, the proposed algorithm retrieves the convergence rate of a server-based FL. But if the interference distribution is heavy-tailed, then the heavier the tail, the slower the algorithm converges. Nonetheless, the system run time can be largely reduced by enabling computation in parallel with communication, whereas the gain is particularly pronounced when communication latency is high. These findings are corroborated via excessive simulations.
Knowledge graph (KG) representation learning aims to encode entities and relations into dense continuous vector spaces such that knowledge contained in a dataset could be consistently represented. Dense embeddings trained from KG datasets benefit a variety of downstream tasks such as KG completion and link prediction. However, existing KG embedding methods fell short to provide a systematic solution for the global consistency of knowledge representation. We developed a mathematical language for KG based on an observation of their inherent algebraic structure, which we termed as Knowledgebra. By analyzing five distinct algebraic properties, we proved that the semigroup is the most reasonable algebraic structure for the relation embedding of a general knowledge graph. We implemented an instantiation model, SemE, using simple matrix semigroups, which exhibits state-of-the-art performance on standard datasets. Moreover, we proposed a regularization-based method to integrate chain-like logic rules derived from human knowledge into embedding training, which further demonstrates the power of the developed language. As far as we know, by applying abstract algebra in statistical learning, this work develops the first formal language for general knowledge graphs, and also sheds light on the problem of neural-symbolic integration from an algebraic perspective.
Reinforcement learning (RL) has shown great success in solving many challenging tasks via use of deep neural networks. Although using deep learning for RL brings immense representational power, it also causes a well-known sample-inefficiency problem. This means that the algorithms are data-hungry and require millions of training samples to converge to an adequate policy. One way to combat this issue is to use action advising in a teacher-student framework, where a knowledgeable teacher provides action advice to help the student. This work considers how to better leverage uncertainties about when a student should ask for advice and if the student can model the teacher to ask for less advice. The student could decide to ask for advice when it is uncertain or when both it and its model of the teacher are uncertain. In addition to this investigation, this paper introduces a new method to compute uncertainty for a deep RL agent using a secondary neural network. Our empirical results show that using dual uncertainties to drive advice collection and reuse may improve learning performance across several Atari games.
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