Optimal Rates for Random Order Online Optimization

We study online convex optimization in the random order model, recently proposed by \citet{garber2020online}, where the loss functions may be chosen by an adversary, but are then presented to the online algorithm in a uniformly random order. Focusing on the scenario where the cumulative loss function is (strongly) convex, yet individual loss functions are smooth but might be non-convex, we give algorithms that achieve the optimal bounds and significantly outperform the results of \citet{garber2020online}, completely removing the dimension dependence and improving their scaling with respect to the strong convexity parameter. Our analysis relies on novel connections between algorithmic stability and generalization for sampling without-replacement analogous to those studied in the with-replacement i.i.d.~setting, as well as on a refined average stability analysis of stochastic gradient descent.

Agnostic Reinforcement Learning with Low-Rank MDPs and Rich Observations

There have been many recent advances on provably efficient Reinforcement Learning (RL) in problems with rich observation spaces. However, all these works share a strong realizability assumption about the optimal value function of the true MDP. Such realizability assumptions are often too strong to hold in practice. In this work, we consider the more realistic setting of agnostic RL with rich observation spaces and a fixed class of policies $\Pi$ that may not contain any near-optimal policy. We provide an algorithm for this setting whose error is bounded in terms of the rank $d$ of the underlying MDP. Specifically, our algorithm enjoys a sample complexity bound of $\widetilde{O}\left((H^{4d} K^{3d} \log |\Pi|)/\epsilon^2\right)$ where $H$ is the length of episodes, $K$ is the number of actions and $\epsilon>0$ is the desired sub-optimality. We also provide a nearly matching lower bound for this agnostic setting that shows that the exponential dependence on rank is unavoidable, without further assumptions.

Differentially Private Multi-Armed Bandits in the Shuffle Model

We give an $(\varepsilon,\delta)$-differentially private algorithm for the multi-armed bandit (MAB) problem in the shuffle model with a distribution-dependent regret of $O\left(\left(\sum_{a\in [k]:\Delta_a>0}\frac{\log T}{\Delta_a}\right)+\frac{k\sqrt{\log\frac{1}{\delta}}\log T}{\varepsilon}\right)$, and a distribution-independent regret of $O\left(\sqrt{kT\log T}+\frac{k\sqrt{\log\frac{1}{\delta}}\log T}{\varepsilon}\right)$, where $T$ is the number of rounds, $\Delta_a$ is the suboptimality gap of the arm $a$, and $k$ is the total number of arms. Our upper bound almost matches the regret of the best known algorithms for the centralized model, and significantly outperforms the best known algorithm in the local model.

Stochastic Multi-Armed Bandits with Unrestricted Delay Distributions

We study the stochastic Multi-Armed Bandit (MAB) problem with random delays in the feedback received by the algorithm. We consider two settings: the reward-dependent delay setting, where realized delays may depend on the stochastic rewards, and the reward-independent delay setting. Our main contribution is algorithms that achieve near-optimal regret in each of the settings, with an additional additive dependence on the quantiles of the delay distribution. Our results do not make any assumptions on the delay distributions: in particular, we do not assume they come from any parametric family of distributions and allow for unbounded support and expectation; we further allow for infinite delays where the algorithm might occasionally not observe any feedback.

Minimax Regret for Stochastic Shortest Path

We study the Stochastic Shortest Path (SSP) problem in which an agent has to reach a goal state in minimum total expected cost. In the learning formulation of the problem, the agent has no prior knowledge about the costs and dynamics of the model. She repeatedly interacts with the model for $K$ episodes, and has to learn to approximate the optimal policy as closely as possible. In this work we show that the minimax regret for this setting is $\widetilde O(B_\star \sqrt{|S| |A| K})$ where $B_\star$ is a bound on the expected cost of the optimal policy from any state, $S$ is the state space, and $A$ is the action space. This matches the lower bound of Rosenberg et al. (2020) up to logarithmic factors, and improves their regret bound by a factor of $\sqrt{|S|}$. Our algorithm runs in polynomial-time per episode, and is based on a novel reduction to reinforcement learning in finite-horizon MDPs. To that end, we provide an algorithm for the finite-horizon setting whose leading term in the regret depends only logarithmically on the horizon, yielding the same regret guarantees for SSP.

Online Markov Decision Processes with Aggregate Bandit Feedback

We study a novel variant of online finite-horizon Markov Decision Processes with adversarially changing loss functions and initially unknown dynamics. In each episode, the learner suffers the loss accumulated along the trajectory realized by the policy chosen for the episode, and observes aggregate bandit feedback: the trajectory is revealed along with the cumulative loss suffered, rather than the individual losses encountered along the trajectory. Our main result is a computationally efficient algorithm with $O(\sqrt{K})$ regret for this setting, where $K$ is the number of episodes. We establish this result via an efficient reduction to a novel bandit learning setting we call Distorted Linear Bandits (DLB), which is a variant of bandit linear optimization where actions chosen by the learner are adversarially distorted before they are committed. We then develop a computationally-efficient online algorithm for DLB for which we prove an $O(\sqrt{T})$ regret bound, where $T$ is the number of time steps. Our algorithm is based on online mirror descent with a self-concordant barrier regularization that employs a novel increasing learning rate schedule.

Learning Adversarial Markov Decision Processes with Delayed Feedback

Reinforcement learning typically assumes that the agent observes feedback from the environment immediately, but in many real-world applications (like recommendation systems) the feedback is observed in delay. Thus, we consider online learning in episodic Markov decision processes (MDPs) with unknown transitions, adversarially changing costs and unrestricted delayed feedback. That is, the costs and trajectory of episode $k$ are only available at the end of episode $k + d^k$, where the delays $d^k$ are neither identical nor bounded, and are chosen by an adversary. We present novel algorithms based on policy optimization that achieve near-optimal high-probability regret of $\widetilde O ( \sqrt{K} + \sqrt{D} )$ under full-information feedback, where $K$ is the number of episodes and $D = \sum_{k} d^k$ is the total delay. Under bandit feedback, we prove similar $\widetilde O ( \sqrt{K} + \sqrt{D} )$ regret assuming that the costs are stochastic, and $\widetilde O ( K^{2/3} + D^{2/3} )$ regret in the general case. To our knowledge, we are the first to consider the important setting of delayed feedback in adversarial MDPs.

Adversarial Dueling Bandits

We introduce the problem of regret minimization in Adversarial Dueling Bandits. As in classic Dueling Bandits, the learner has to repeatedly choose a pair of items and observe only a relative binary `win-loss' feedback for this pair, but here this feedback is generated from an arbitrary preference matrix, possibly chosen adversarially. Our main result is an algorithm whose $T$-round regret compared to the \emph{Borda-winner} from a set of $K$ items is $\tilde{O}(K^{1/3}T^{2/3})$, as well as a matching $\Omega(K^{1/3}T^{2/3})$ lower bound. We also prove a similar high probability regret bound. We further consider a simpler \emph{fixed-gap} adversarial setup, which bridges between two extreme preference feedback models for dueling bandits: stationary preferences and an arbitrary sequence of preferences. For the fixed-gap adversarial setup we give an $\smash{ \tilde{O}((K/\Delta^2)\log{T}) }$ regret algorithm, where $\Delta$ is the gap in Borda scores between the best item and all other items, and show a lower bound of $\Omega(K/\Delta^2)$ indicating that our dependence on the main problem parameters $K$ and $\Delta$ is tight (up to logarithmic factors).