A more general formulation of the linear bandit problem is considered to allow for dependencies over time. Specifically, it is assumed that there exists an unknown $\mathbb{R}^d$-valued stationary $\varphi$-mixing sequence of parameters $(\theta_t,~t \in \mathbb{N})$ which gives rise to pay-offs. This instance of the problem can be viewed as a generalization of both the classical linear bandits with iid noise, and the finite-armed restless bandits. In light of the well-known computational hardness of optimal policies for restless bandits, an approximation is proposed whose error is shown to be controlled by the $\varphi$-dependence between consecutive $\theta_t$. An optimistic algorithm, called LinMix-UCB, is proposed for the case where $\theta_t$ has an exponential mixing rate. The proposed algorithm is shown to incur a sub-linear regret of $\mathcal{O}\left(\sqrt{d n\mathrm{polylog}(n) }\right)$ with respect to an oracle that always plays a multiple of $\mathbb{E}\theta_t$. The main challenge in this setting is to ensure that the exploration-exploitation strategy is robust against long-range dependencies. The proposed method relies on Berbee's coupling lemma to carefully select near-independent samples and construct confidence ellipsoids around empirical estimates of $\mathbb{E}\theta_t$.

We propose methods to estimate the individual $\beta$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path $X_0,X_1, \dots,X_n$. Under standard smoothness conditions on the densities, namely, that the joint density of the pair $(X_0,X_m)$ for each $m$ lies in a Besov space $B^s_{1,\infty}(\mathbb R^2)$ for some known $s>0$, we obtain a rate of convergence of order $\mathcal{O}(\log(n) n^{-[s]/(2[s]+2)})$ for the expected error of our estimator in this case\footnote{We use $[s]$ to denote the integer part of the decomposition $s=[s]+\{s\}$ of $s \in (0,\infty)$ into an integer term and a {\em strictly positive} remainder term $\{s\} \in (0,1]$.}. 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 order $\mathcal O(\log(n) n^{-1/2})$.

We introduce PyChEst, a Python package which provides tools for the simultaneous estimation of multiple changepoints in the distribution of piece-wise stationary time series. The nonparametric algorithms implemented are provably consistent in a general framework: when the samples are generated by unknown piece-wise stationary processes. In this setting, samples may have long-range dependencies of arbitrary form and the finite-dimensional marginals of any (unknown) fixed size before and after the changepoints may be the same. The strength of the algorithms included in the package is in their ability to consistently detect the changes without imposing any assumptions beyond stationarity on the underlying process distributions. We illustrate this distinguishing feature by comparing the performance of the package against state-of-the-art models designed for a setting where the samples are independently and identically distributed.

The problem of time-series clustering is considered in the case where each data-point is a sample generated by a piecewise stationary ergodic process. Stationary processes are perhaps the most general class of processes considered in non-parametric statistics and allow for arbitrary long-range dependence between variables. Piecewise stationary processes studied here for the first time in the context of clustering, relax the last remaining assumption in this model: stationarity. A natural formulation is proposed for this problem and a notion of consistency is introduced which requires the samples to be placed in the same cluster if and only if the piecewise stationary distributions that generate them have the same set of stationary distributions. Simple, computationally efficient algorithms are proposed and are shown to be consistent without any additional assumptions beyond piecewise stationarity.

The multi-armed restless bandit problem is studied in the case where the pay-offs are not necessarily independent over time nor across the arms. Even though this version of the problem provides a more realistic model for most real-world applications, it cannot be optimally solved in practice since it is known to be PSPACE-hard. The objective of this paper is to characterize special sub-classes of the problem where good approximate solutions can be found using tractable approaches. Specifically, it is shown that in the case where the joint distribution over the arms is $\varphi$-mixing, and under some conditions on the $\varphi$-mixing coefficients, a modified version of UCB can prove optimal. On the other hand, it is shown that when the pay-off distributions are strongly dependent, simple switching strategies may be devised which leverage the strong inter-dependencies. To this end, an example is provided using Gaussian Processes. The techniques developed in this paper apply, more generally, to the problem of online sampling under dependence.

The problem of multiple change point estimation is considered for sequences with unknown number of change points. A consistency framework is suggested that is suitable for highly dependent time-series, and an asymptotically consistent algorithm is proposed. In order for the consistency to be established the only assumption required is that the data is generated by stationary ergodic time-series distributions. No modeling, independence or parametric assumptions are made; the data are allowed to be dependent and the dependence can be of arbitrary form. The theoretical results are complemented with experimental evaluations.