Learning Linear-Quadratic Regulators Efficiently with only $\sqrt{T}$ Regret

We present the first computationally-efficient algorithm with $\widetilde O(\sqrt{T})$ regret for learning in Linear Quadratic Control systems with unknown dynamics. By that, we resolve an open question of Abbasi-Yadkori and Szepesv\'ari (2011) and Dean, Mania, Matni, Recht, and Tu (2018).

Differentially Private Learning of Geometric Concepts

We present differentially private efficient algorithms for learning union of polygons in the plane (which are not necessarily convex). Our algorithms achieve $(\alpha,\beta)$-PAC learning and $(\epsilon,\delta)$-differential privacy using a sample of size $\tilde{O}\left(\frac{1}{\alpha\epsilon}k\log d\right)$, where the domain is $[d]\times[d]$ and $k$ is the number of edges in the union of polygons.

Learning and Generalization for Matching Problems

We study a classic algorithmic problem through the lens of statistical learning. That is, we consider a matching problem where the input graph is sampled from some distribution. This distribution is unknown to the algorithm; however, an additional graph which is sampled from the same distribution is given during a training phase (preprocessing). More specifically, the algorithmic problem is to match $k$ out of $n$ items that arrive online to $d$ categories ($d\ll k \ll n$). Our goal is to design a two-stage online algorithm that retains a small subset of items in the first stage which contains an offline matching of maximum weight. We then compute this optimal matching in a second stage. The added statistical component is that before the online matching process begins, our algorithms learn from a training set consisting of another matching instance drawn from the same unknown distribution. Using this training set, we learn a policy that we apply during the online matching process. We consider a class of online policies that we term \emph{thresholds policies}. For this class, we derive uniform convergence results both for the number of retained items and the value of the optimal matching. We show that the number of retained items and the value of the offline optimal matching deviate from their expectation by $O(\sqrt{k})$. This requires usage of less-standard concentration inequalities (standard ones give deviations of $O(\sqrt{n})$). Furthermore, we design an algorithm that outputs the optimal offline solution with high probability while retaining only $O(k\log \log n)$ items in expectation.

Adversarial Online Learning with noise

We present and study models of adversarial online learning where the feedback observed by the learner is noisy, and the feedback is either full information feedback or bandit feedback. Specifically, we consider binary losses xored with the noise, which is a Bernoulli random variable. We consider both a constant noise rate and a variable noise rate. Our main results are tight regret bounds for learning with noise in the adversarial online learning model.

Bayesian Exploration: Incentivizing Exploration in Bayesian Games

We consider a ubiquitous scenario in the Internet economy when individual decision-makers (henceforth, agents) both produce and consume information as they make strategic choices in an uncertain environment. This creates a three-way tradeoff between exploration (trying out insufficiently explored alternatives to help others in the future), exploitation (making optimal decisions given the information discovered by other agents), and incentives of the agents (who are myopically interested in exploitation, while preferring the others to explore). We posit a principal who controls the flow of information from agents that came before, and strives to coordinate the agents towards a socially optimal balance between exploration and exploitation, not using any monetary transfers. The goal is to design a recommendation policy for the principal which respects agents' incentives and minimizes a suitable notion of regret. We extend prior work in this direction to allow the agents to interact with one another in a shared environment: at each time step, multiple agents arrive to play a Bayesian game, receive recommendations, choose their actions, receive their payoffs, and then leave the game forever. The agents now face two sources of uncertainty: the actions of the other agents and the parameters of the uncertain game environment. Our main contribution is to show that the principal can achieve constant regret when the utilities are deterministic (where the constant depends on the prior distribution, but not on the time horizon), and logarithmic regret when the utilities are stochastic. As a key technical tool, we introduce the concept of explorable actions, the actions which some incentive-compatible policy can recommend with non-zero probability. We show how the principal can identify (and explore) all explorable actions, and use the revealed information to perform optimally.

Online Linear Quadratic Control

We study the problem of controlling linear time-invariant systems with known noisy dynamics and adversarially chosen quadratic losses. We present the first efficient online learning algorithms in this setting that guarantee $O(\sqrt{T})$ regret under mild assumptions, where $T$ is the time horizon. Our algorithms rely on a novel SDP relaxation for the steady-state distribution of the system. Crucially, and in contrast to previously proposed relaxations, the feasible solutions of our SDP all correspond to "strongly stable" policies that mix exponentially fast to a steady state.

Are All Experts Equally Good? A Study of Analyst Earnings Estimates

We investigate whether experts possess differential expertise when making predictions. We note that this would make it possible to aggregate multiple predictions into a result that is more accurate than their consensus average, and that the improvement prospects grow with the amount of differentiation. Turning this argument on its head, we show how differentiation can be measured by how much weighted aggregation improves on simple averaging. Taking stock-market analysts as experts in their domain, we do a retrospective study using historical quarterly earnings forecasts and actual results for large publicly traded companies. We use it to shed new light on the Sinha et al. (1997) result, showing that analysts indeed possess individual expertise, but that their differentiation is modest. On the other hand, they have significant individual bias. Together, these enable a 20%-30% accuracy improvement over consensus average.

Planning and Learning with Stochastic Action Sets

In many practical uses of reinforcement learning (RL) the set of actions available at a given state is a random variable, with realizations governed by an exogenous stochastic process. Somewhat surprisingly, the foundations for such sequential decision processes have been unaddressed. In this work, we formalize and investigate MDPs with stochastic action sets (SAS-MDPs) to provide these foundations. We show that optimal policies and value functions in this model have a structure that admits a compact representation. From an RL perspective, we show that Q-learning with sampled action sets is sound. In model-based settings, we consider two important special cases: when individual actions are available with independent probabilities; and a sampling-based model for unknown distributions. We develop poly-time value and policy iteration methods for both cases; and in the first, we offer a poly-time linear programming solution.

Hierarchical Reinforcement Learning: Approximating Optimal Discounted TSP Using Local Policies

In this work, we provide theoretical guarantees for reward decomposition in deterministic MDPs. Reward decomposition is a special case of Hierarchical Reinforcement Learning, that allows one to learn many policies in parallel and combine them into a composite solution. Our approach builds on mapping this problem into a Reward Discounted Traveling Salesman Problem, and then deriving approximate solutions for it. In particular, we focus on approximate solutions that are local, i.e., solutions that only observe information about the current state. Local policies are easy to implement and do not require substantial computational resources as they do not perform planning. While local deterministic policies, like Nearest Neighbor, are being used in practice for hierarchical reinforcement learning, we propose three stochastic policies that guarantee better performance than any deterministic policy.

Discriminative Learning of Prediction Intervals

In this work we consider the task of constructing prediction intervals in an inductive batch setting. We present a discriminative learning framework which optimizes the expected error rate under a budget constraint on the interval sizes. Most current methods for constructing prediction intervals offer guarantees for a single new test point. Applying these methods to multiple test points can result in a high computational overhead and degraded statistical guarantees. By focusing on expected errors, our method allows for variability in the per-example conditional error rates. As we demonstrate both analytically and empirically, this flexibility can increase the overall accuracy, or alternatively, reduce the average interval size. While the problem we consider is of a regressive flavor, the loss we use is combinatorial. This allows us to provide PAC-style, finite-sample guarantees. Computationally, we show that our original objective is NP-hard, and suggest a tractable convex surrogate. We conclude with a series of experimental evaluations.