We propose methods to estimate the β-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-pathX0,X1, . . .,Xn−1. Under standard smoothness conditions on the densities, namely, that the joint density of the pair (X0,Xm) for each m lies in a Besov spaceB1s,∞(R2) for some knowns> 0, we obtain a rate of convergence of orderO(log(n)n-[s]/(2[s]+2)) for the expected error of our estimator in this case; we use [s] to denote the integer part of the decompositions= [s] + }s{ ofs∈ (0, ∞) into an integer term and a strictly positive remainder term }s{ ∈ (0, 1]. We complement this result with a high-probability bound on the estimation error, and further obtain analogues of these bounds in the case where the state-space is finite. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of orderO(log(n)n−1/2). The theoretical results are complemented with empirical evaluations.

We investigate the problem of algorithmic fairness in the case where sensitive and non-sensitive features are available and one aims to generate new, `oblivious', features that closely approximate the non-sensitive features, and are only minimally dependent on the sensitive ones. We study this question in the context of kernel methods. We analyze a relaxed version of the Maximum Mean Discrepancy criterion which does not guarantee full independence but makes the optimization problem tractable. We derive a closed-form solution for this relaxed optimization problem and complement the result with a study of the dependencies between the newly generated features and the sensitive ones. Our key ingredient for generating such oblivious features is a Hilbert-space-valued conditional expectation, which needs to be estimated from data. We propose a plug-in approach and demonstrate how the estimation errors can be controlled. While our techniques help reduce the bias, we would like to point out that no post-processing of any dataset could possibly serve as an alternative to well-designed experiments.

We study the recovering bandits problem, a variant of the stochastic multi-armed bandit problem where the expected reward of each arm varies according to some unknown function of the time since the arm was last played. While being a natural extension of the classical bandit problem that arises in many real-world settings, this variation is accompanied by significant difficulties. In particular, methods need to plan ahead and estimate many more quantities than in the classical bandit setting. In this work, we explore the use of Gaussian processes to tackle the estimation and planing problem. We also discuss different regret definitions that let us quantify the performance of the methods. To improve computational efficiency of the methods, we provide an optimistic planning approximation. We complement these discussions with regret bounds and empirical studies.

The multi-armed restless bandit problem is studied in the case where the pay-off distributions are stationary $\varphi$-mixing. This version of the problem provides a more realistic model for most real-world applications, but cannot be optimally solved in practice, since it is known to be PSPACE-hard. The objective of this paper is to characterize a sub-class of the problem where good approximate solutions can be found using tractable approaches. Specifically, it is shown that under some conditions on the $\varphi$-mixing coefficients, a modified version of UCB can prove effective. The main challenge is that, unlike in the i.i.d. setting, the distributions of the sampled pay-offs may not have the same characteristics as those of the original bandit arms. In particular, the $\varphi$-mixing property does not necessarily carry over. This is overcome by carefully controlling the effect of a sampling policy on the pay-off distributions. Some of the proof techniques developed in this paper can be more generally used in the context of online sampling under dependence. Proposed algorithms are accompanied with corresponding regret analysis.

We study kernel least-squares estimation under a norm constraint. This form of regularisation is known as Ivanov regularisation and it provides better control of the norm of the estimator than the well-established Tikhonov regularisation. Ivanov regularisation can be studied under minimal assumptions. In particular, we assume only that the RKHS is separable with a bounded and measurable kernel. We provide rates of convergence for the expected squared $L^2$ error of our estimator under the weak assumption that the variance of the response variables is bounded and the unknown regression function lies in an interpolation space between $L^2$ and the RKHS. We then obtain faster rates of convergence when the regression function is bounded by clipping the estimator. In fact, we attain the optimal rate of convergence. Furthermore, we provide a high-probability bound under the stronger assumption that the response variables have subgaussian errors and that the regression function lies in an interpolation space between $L^\infty$ and the RKHS. Finally, we derive adaptive results for the settings in which the regression function is bounded.

We study plug-in estimators of conditional expectations and probabilities, and we provide a systematic analysis of their rates of convergence. The plug-in approach is particularly useful in this setting since it introduces a natural link to VC- and empirical process theory. We make use of this link to derive rates of convergence that hold uniformly over large classes of functions and sets, and under various conditions. For instance, we demonstrate that elementary conditional probabilities are estimated by these plug-in estimators with a rate of $n^{α-1/2}$ if one conditions with a VC-class of sets and where $α∈[0,1/2)$ controls a lower bound on the size of sets we can estimate given n samples. We gain similar results for Kolmogorov’s conditional expectation and probability which generalize the elementary forms of conditioning. Due to their simplicity, plug-in estimators can be evaluated in linear time and there is no up-front cost for inference.

We study a variant of the stochastic $K$-armed bandit problem, which we call "bandits with delayed, aggregated anonymous feedback”. In this problem, when the player pulls an arm, a reward is generated, however it is not immediately observed. Instead, at the end of each round the player observes only the sum of a number of previously generated rewards which happen to arrive in the given round. The rewards are stochastically delayed and due to the aggregated nature of the observations, the information of which arm led to a particular reward is lost. The question is what is the cost of the information loss due to this delayed, aggregated anonymous feedback? Previous works have studied bandits with stochastic, non-anonymous delays and found that the regret increases only by an additive factor relating to the expected delay. In this paper, we show that this additive regret increase can be maintained in the harder delayed, aggregated anonymous feedback setting when the expected delay (or a bound on it) is known. We provide an algorithm that matches the worst case regret of the non-anonymous problem exactly when the delays are bounded, and up to logarithmic factors or an additive variance term for unbounded delays.

The stochastic knapsack problem is a stochastic resource allocation problem that arises frequently and yet is exceptionally hard to solve. We derive and study an optimistic planning algorithm specifically designed for the stochastic knapsack problem. Unlike other optimistic planning algorithms for MDPs, our algorithm, OpStoK, avoids the use of discounting and is adaptive to the amount of resources available. We achieve this behavior by means of a concentration inequality that simultaneously applies to capacity and reward estimates. Crucially, we are able to guarantee that the aforementioned confidence regions hold collectively over all time steps by an application of Doob’s inequality. We demonstrate that the method returns an $ε$-optimal solution to the stochastic knapsack problem with high probability. To the best of our knowledge, our algorithm is the first which provides such guarantees for the stochastic knapsack problem. Furthermore, our algorithm is an anytime algorithm and will return a good solution even if stopped prematurely. This is particularly important given the difficulty of the problem. We also provide theoretical conditions to guarantee OpStoK does not expand all policies and demonstrate favorable performance in a simple experimental setting.

We develop a generic approach to form smooth versions of basic mathematical operations like multiplication, composition, change of measure, and conditional expectation, among others. Operations which result in functions outside the reproducing kernel Hilbert space (such as the product of two RKHS functions) are approximated via a natural cost function, such that the solution is guaranteed to be in the targeted RKHS. This approximation problem is reduced to a regression problem using an adjoint trick, and solved in a vector-valued RKHS, consisting of continuous, linear, smooth operators which map from an input, real-valued RKHS to the desired target RKHS. Important constraints, such as an almost everywhere positive density, can be enforced or approximated naturally in this framework, using convex constraints on the operators. Finally, smooth operators can be composed to accomplish more complex machine learning tasks, such as the sum rule and kernelized approximate Bayesian inference, where state-of-the-art convergence rates are obtained.

The linear correlation coefficient is typically used to characterize and analyze dependencies of neural spike counts. Here, we show that the correlation coefficient is in general insufficient to characterize these dependencies. We construct two neuron spike count models with Poisson-like marginals and vary their dependence structure using copulas. To this end, we construct a copula that allows to keep the spike counts uncorrelated while varying their dependence strength. Moreover, we employ a network of leaky integrate-and-fire neurons to investigate whether weakly correlated spike counts with strong dependencies are likely to occur in real networks. We find that the entropy of uncorrelated but dependent spike count distributions can deviate from the corresponding distribution with independent components by more than 25% and that weakly correlated but strongly dependent spike counts are very likely to occur in biological networks. Finally, we introduce a test for deciding whether the dependence structure of distributions with Poisson-like marginals is well characterized by the linear correlation coefficient and verify it for different copula-based models.

Simultaneous spike-counts of neural populations are typically modeled by a Gaussian distribution. On short time scales, however, this distribution is too restrictive to describe and analyze multivariate distributions of discrete spike-counts. We present an alternative that is based on copulas and can account for arbitrary marginal distributions, including Poisson and negative binomial distributions as well as second and higher-order interactions. We describe maximum likelihood-based procedures for fitting copula-based models to spike-count data, and we derive a so-called flashlight transformation which makes it possible to move the tail dependence of an arbitrary copula into an arbitrary orthant of the multivariate probability distribution. Mixtures of copulas that combine different dependence structures and thereby model different driving processes simultaneously are also introduced. First, we apply copula-based models to populations of integrate-and-fire neurons receiving partially correlated input and show that the best fitting copulas provide information about the functional connectivity of coupled neurons which can be extracted using the flashlight transformation. We then apply the new method to data which were recorded from macaque prefrontal cortex using a multi-tetrode array. We find that copula-based distributions with negative binomial marginals provide an appropriate stochastic model for the multivariate spike-count distributions rather than the multivariate Poisson latent variables distribution and the often used multivariate normal distribution. The dependence structure of these distributions provides evidence for common inhibitory input to all recorded stimulus encoding neurons. Finally, we show that copula-based models can be successfully used to evaluate neural codes, e.g., to characterize stimulus-dependent spike-count distributions with information measures. This demonstrates that copula-based models are not only a versatile class of models for multivariate distributions of spike-counts, but that those models can be exploited to understand functional dependencies.

Correlations between spike counts are often used to analyze neural coding. The noise is typically assumed to be Gaussian. Yet, this assumption is often inappropriate, especially for low spike counts. In this study, we present copulas as an alternative approach. With copulas it is possible to use arbitrary marginal distributions such as Poisson or negative binomial that are better suited for modeling noise distributions of spike counts. Furthermore, copulas place a wide range of dependence structures at the disposal and can be used to analyze higher order interactions. We develop a framework to analyze spike count data by means of copulas. Methods for parameter inference based on maximum likelihood estimates and for computation of Shannon entropy are provided. We apply the method to our data recorded from macaque prefrontal cortex. The data analysis leads to three significant findings: (1) copula-based distributions provide better fits than discretized multivariate normal distributions; (2) negative binomial margins fit the data better than Poisson margins; and (3) a dependence model that includes only pairwise interactions overestimates the information entropy by at least 19% compared to the model with higher order interactions.

In this analytical study we compare the risk of the Monte Carlo (MC) and the least-squares TD (LSTD) estimator. We prove that for the case of acyclic Markov Reward Processes (MRPs) LSTD has minimal risk for any convex loss function in the class of unbiased estimators. When comparing the Monte Carlo estimator, which does not assume a Markov structure, and LSTD, we find that the Monte Carlo estimator is equivalent to LSTD if both estimators have the same amount of information. Theoretical results are supported by an empirical evaluation of the estimators.