We develop algorithms for online linear regression which achieve optimal static and dynamic regret guarantees \emph{even in the complete absence of prior knowledge}. We present a novel analysis showing that a discounted variant of the Vovk-Azoury-Warmuth forecaster achieves dynamic regret of the form $R_{T}(\vec{u})\le O\left(d\log(T)\vee \sqrt{dP_{T}^{\gamma}(\vec{u})T}\right)$, where $P_{T}^{\gamma}(\vec{u})$ is a measure of variability of the comparator sequence, and show that the discount factor achieving this result can be learned on-the-fly. We show that this result is optimal by providing a matching lower bound. We also extend our results to \emph{strongly-adaptive} guarantees which hold over every sub-interval $[a,b]\subseteq[1,T]$ simultaneously.

Machine Learning requires a large amount of training data in order to build accurate models. Sometimes the data arrives over time, requiring significant storage space and recalculating the model to account for the new data. On-line learning addresses these issues by incrementally modifying the model as data is encountered, and then discarding the data. In this study we introduce a new online linear regression approach. Our approach combines newly arriving data with a previously existing model to create a new model. The introduced model, named OLR-WA (OnLine Regression with Weighted Average) uses user-defined weights to provide flexibility in the face of changing data to bias the results in favor of old or new data. We have conducted 2-D and 3-D experiments comparing OLR-WA to a static model using the entire data set. The results show that for consistent data, OLR-WA and the static batch model perform similarly and for varying data, the user can set the OLR-WA to adapt more quickly or to resist change.

Endogeneity, i.e. the dependence between noise and covariates, is a common phenomenon in real data due to omitted variables, strategic behaviours, measurement errors etc. In contrast, the existing analyses of stochastic online linear regression with unbounded noise and linear bandits depend heavily on exogeneity, i.e. the independence between noise and covariates. Motivated by this gap, we study the over-and just-identified Instrumental Variable (IV) regression for stochastic online learning. IV regression and the Two-Stage Least Squares approach to it are widely deployed in economics and causal inference to identify the underlying model from an endogenous dataset. Thus, we propose to use an online variant of Two-Stage Least Squares approach, namely O2SLS, to tackle endogeneity in stochastic online learning. Our analysis shows that O2SLS achieves $\mathcal{O}\left(d_x d_z \log ^2 T\right)$ identification and $\tilde{\mathcal{O}}\left(\gamma \sqrt{d_x T}\right)$ oracle regret after $T$ interactions, where $d_x$ and $d_z$ are the dimensions of covariates and IVs, and $\gamma$ is the bias due to endogeneity. For $\gamma=0$, i.e. under exogeneity, O2SLS achieves $\mathcal{O}\left(d_x^2 \log ^2 T\right)$ oracle regret, which is of the same order as that of the stochastic online ridge. Then, we leverage O2SLS as an oracle to design OFUL-IV, a stochastic linear bandit algorithm that can tackle endogeneity and achieves $\widetilde{\mathcal{O}}\left(\sqrt{d_x d_z T}\right)$ regret. For different datasets with endogeneity, we experimentally show efficiencies of O2SLS and OFUL-IV in terms of regrets.

We provide a provably efficient algorithm for learning Markov Decision Processes (MDPs) with continuous state and action spaces in the online setting. Specifically, we take a model-based approach and show that a special type of online linear regression allows us to learn MDPs with (possibly kernalized) linearly parameterized dynamics. This result builds on Kearns and Singh's work that provides a provably efficient algorithm for finite state MDPs. Our approach is not restricted to the linear setting, and is applicable to other classes of continuous MDPs.

We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for $\sigma$-sub-Gaussian label noise, our analysis provides a regret upper bound of $O(\sigma^2 d \log T) + o(\log T)$, where $d$ is the dimension of the input vector, $T$ is the total number of rounds. We also prove a $\Omega(\sigma^2d\log(T/d))$ lower bound for stochastic online linear regression, which indicates that our upper bound is nearly optimal. In addition, we extend our analysis to a more refined Bernstein noise condition. As an application, we study generalized linear bandits with heteroscedastic noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.

We consider the problem of nonstochastic control with a sequence of quadratic losses, i.e., LQR control. We provide an efficient online algorithm that achieves an optimal dynamic (policy) regret of $\tilde{O}(\text{max}\{n^{1/3} \mathcal{TV}(M_{1:n})^{2/3}, 1\})$, where $\mathcal{TV}(M_{1:n})$ is the total variation of any oracle sequence of Disturbance Action policies parameterized by $M_1,...,M_n$ -- chosen in hindsight to cater to unknown nonstationarity. The rate improves the best known rate of $\tilde{O}(\sqrt{n (\mathcal{TV}(M_{1:n})+1)} )$ for general convex losses and we prove that it is information-theoretically optimal for LQR. Main technical components include the reduction of LQR to online linear regression with delayed feedback due to Foster and Simchowitz (2020), as well as a new proper learning algorithm with an optimal $\tilde{O}(n^{1/3})$ dynamic regret on a family of ``minibatched'' quadratic losses, which could be of independent interest.