arXiv subjects

Daniel Vial

Publications and source records attributed to Daniel Vial.

12 recordsLinked to original sources

arXiv2023

Collaborative Multi-Agent Heterogeneous Multi-Armed Bandits

The study of collaborative multi-agent bandits has attracted significant attention recently. In light of this, we initiate the study of a new collaborative setting, consisting of $N$ agents such that each agent is learning one of $M$ stochastic multi-armed bandits to minimize their group cumulative regret. We develop decentralized algorithms which facilitate collaboration between the agents under two scenarios. We characterize the performance of these algorithms by deriving the per agent cumulative regret and group regret upper bounds. We also prove lower bounds for the group regret in this setting, which demonstrates the near-optimal behavior of the proposed algorithms.

cs.LG↗

arXiv2022

Minimax Regret for Cascading Bandits

Cascading bandits is a natural and popular model that frames the task of learning to rank from Bernoulli click feedback in a bandit setting. For the case of unstructured rewards, we prove matching upper and lower bounds for the problem-independent (i.e., gap-free) regret, both of which strictly improve the best known. A key observation is that the hard instances of this problem are those with small mean rewards, i.e., the small click-through rates that are most relevant in practice. Based on this, and the fact that small mean implies small variance for Bernoullis, our key technical result shows that variance-aware confidence sets derived from the Bernstein and Chernoff bounds lead to optimal algorithms (up to log terms), whereas Hoeffding-based algorithms suffer order-wise suboptimal regret. This sharply contrasts with the standard (non-cascading) bandit setting, where the variance-aware algorithms only improve constants. In light of this and as an additional contribution, we propose a variance-aware algorithm for the structured case of linear rewards and show its regret strictly improves the state-of-the-art.

cs.LG↗

arXiv2022

Robust Multi-Agent Bandits Over Undirected Graphs

We consider a multi-agent multi-armed bandit setting in which $n$ honest agents collaborate over a network to minimize regret but $m$ malicious agents can disrupt learning arbitrarily. Assuming the network is the complete graph, existing algorithms incur $O( (m + K/n) \log (T) / \Delta )$ regret in this setting, where $K$ is the number of arms and $\Delta$ is the arm gap. For $m \ll K$, this improves over the single-agent baseline regret of $O(K\log(T)/\Delta)$. In this work, we show the situation is murkier beyond the case of a complete graph. In particular, we prove that if the state-of-the-art algorithm is used on the undirected line graph, honest agents can suffer (nearly) linear regret until time is doubly exponential in $K$ and $n$. In light of this negative result, we propose a new algorithm for which the $i$-th agent has regret $O( ( d_{\text{mal}}(i) + K/n) \log(T)/\Delta)$ on any connected and undirected graph, where $d_{\text{mal}}(i)$ is the number of $i$'s neighbors who are malicious. Thus, we generalize existing regret bounds beyond the complete graph (where $d_{\text{mal}}(i) = m$), and show the effect of malicious agents is entirely local (in the sense that only the $d_{\text{mal}}(i)$ malicious agents directly connected to $i$ affect its long-term regret).

cs.LG↗

arXiv2021

Improved Algorithms for Misspecified Linear Markov Decision Processes

For the misspecified linear Markov decision process (MLMDP) model of Jin et al. [2020], we propose an algorithm with three desirable properties. (P1) Its regret after $K$ episodes scales as $K \max \{ \varepsilon_{\text{mis}}, \varepsilon_{\text{tol}} \}$, where $\varepsilon_{\text{mis}}$ is the degree of misspecification and $\varepsilon_{\text{tol}}$ is a user-specified error tolerance. (P2) Its space and per-episode time complexities remain bounded as $K \rightarrow \infty$. (P3) It does not require $\varepsilon_{\text{mis}}$ as input. To our knowledge, this is the first algorithm satisfying all three properties. For concrete choices of $\varepsilon_{\text{tol}}$, we also improve existing regret bounds (up to log factors) while achieving either (P2) or (P3) (existing algorithms satisfy neither). At a high level, our algorithm generalizes (to MLMDPs) and refines the Sup-Lin-UCB algorithm, which Takemura et al. [2021] recently showed satisfies (P3) for contextual bandits. We also provide an intuitive interpretation of their result, which informs the design of our algorithm.

cs.LG↗

arXiv2021

Regret Bounds for Stochastic Shortest Path Problems with Linear Function Approximation

We propose an algorithm that uses linear function approximation (LFA) for stochastic shortest path (SSP). Under minimal assumptions, it obtains sublinear regret, is computationally efficient, and uses stationary policies. To our knowledge, this is the first such algorithm in the LFA literature (for SSP or other formulations). Our algorithm is a special case of a more general one, which achieves regret square root in the number of episodes given access to a certain computation oracle.

cs.LG↗

arXiv2020

One-bit feedback is sufficient for upper confidence bound policies

We consider a variant of the traditional multi-armed bandit problem in which each arm is only able to provide one-bit feedback during each pull based on its past history of rewards. Our main result is the following: given an upper confidence bound policy which uses full-reward feedback, there exists a coding scheme for generating one-bit feedback, and a corresponding decoding scheme and arm selection policy, such that the ratio of the regret achieved by our policy and the regret of the full-reward feedback policy asymptotically approaches one.

cs.LG↗

arXiv2020

Robust Multi-Agent Multi-Armed Bandits

Recent works have shown that agents facing independent instances of a stochastic $K$-armed bandit can collaborate to decrease regret. However, these works assume that each agent always recommends their individual best-arm estimates to other agents, which is unrealistic in envisioned applications (machine faults in distributed computing or spam in social recommendation systems). Hence, we generalize the setting to include $n$ honest and $m$ malicious agents who recommend best-arm estimates and arbitrary arms, respectively. We first show that even with a single malicious agent, existing collaboration-based algorithms fail to improve regret guarantees over a single-agent baseline. We propose a scheme where honest agents learn who is malicious and dynamically reduce communication with (i.e., "block") them. We show that collaboration indeed decreases regret for this algorithm, assuming $m$ is small compared to $K$ but without assumptions on malicious agents' behavior, thus ensuring that our algorithm is robust against any malicious recommendation strategy.

cs.LG↗

arXiv2020

Empirical Policy Evaluation with Supergraphs

We devise and analyze algorithms for the empirical policy evaluation problem in reinforcement learning. Our algorithms explore backward from high-cost states to find high-value ones, in contrast to forward approaches that work forward from all states. While several papers have demonstrated the utility of backward exploration empirically, we conduct rigorous analyses which show that our algorithms can reduce average-case sample complexity from $O(S \log S)$ to as low as $O(\log S)$.

cs.LG↗

arXiv2019

Restart perturbations for reversible Markov chains: trichotomy and pre-cutoff equivalence

Given a reversible Markov chain $P_n$ on $n$ states, and another chain $\tilde{P}_n$ obtained by perturbing each row of $P_n$ by at most $\alpha_n$ in total variation, we study the total variation distance between the two stationary distributions, $\| \pi_n - \tilde{\pi}_n \|$. We show that for chains with cutoff, $\| \pi_n - \tilde{\pi}_n \|$ converges to $0$, $e^{-c}$, and $1$, respectively, if the product of $\alpha_n$ and the mixing time of $P_n$ converges to $0$, $c$, and $\infty$, respectively. This echoes recent results for specific random walks that exhibit cutoff, suggesting that cutoff is the key property underlying such results. Moreover, we show $\| \pi_n - \tilde{\pi}_n \|$ is maximized by restart perturbations, for which $\tilde{P}_n$ "restarts" $P_n$ at a random state with probability $\alpha_n$ at each step. Finally, we show that pre-cutoff is (almost) equivalent to a notion of "sensitivity to restart perturbations," suggesting that chains with sharper convergence to stationarity are inherently less robust.

math.PR↗

arXiv2019

Local non-Bayesian social learning with stubborn agents

We study a social learning model in which agents iteratively update their beliefs about the true state of the world using private signals and the beliefs of other agents in a non-Bayesian manner. Some agents are stubborn, meaning they attempt to convince others of an erroneous true state (modeling fake news). We show that while agents learn the true state on short timescales, they "forget" it and believe the erroneous state to be true on longer timescales. Using these results, we devise strategies for seeding stubborn agents so as to disrupt learning, which outperform intuitive heuristics and give novel insights regarding vulnerabilities in social learning.

cs.SI↗

arXiv2018

Personalized PageRank dimensionality and algorithmic implications

Many systems, including the Internet, social networks, and the power grid, can be represented as graphs. When analyzing graphs, it is often useful to compute scores describing the relative importance or distance between nodes. One example is Personalized PageRank (PPR), which assigns to each node $v$ a vector whose $i$-th entry describes the importance of the $i$-th node from the perspective of $v$. PPR has proven useful in many applications, such as recommending who users should follow on social networks (if this $i$-th entry is large, $v$ may be interested in following the $i$-th user). Unfortunately, computing $n$ such PPR vectors (where $n$ is the number of nodes) is infeasible for many graphs of interest. In this work, we argue that the situation is not so dire. Our main result shows that the dimensionality of the set of PPR vectors scales sublinearly in $n$ with high probability, for a certain class of random graphs and for a notion of dimensionality similar to rank. Put differently, we argue that the effective dimension of this set is much less than $n$, despite the fact that the matrix containing these vectors has rank $n$. Furthermore, we show this dimensionality measure relates closely to the complexity of a PPR estimation scheme that was proposed (but not analyzed) by Jeh and Widom. This allows us to argue that accurately estimating all $n$ PPR vectors amounts to computing a vanishing fraction of the $n^2$ vector elements (when the technical assumptions of our main result are satisfied). Finally, we demonstrate empirically that similar conclusions hold when considering real-world networks, despite the assumptions of our theory not holding.

cs.SI↗

arXiv2017

On the role of clustering in Personalized PageRank estimation

Personalized PageRank (PPR) is a measure of the importance of a node from the perspective of another (we call these nodes the $\textit{target}$ and the $\textit{source}$, respectively). PPR has been used in many applications, such as offering a Twitter user (the source) recommendations of who to follow (targets deemed important by PPR); additionally, PPR has been used in graph-theoretic problems such as community detection. However, computing PPR is infeasible for large networks like Twitter, so efficient estimation algorithms are necessary. In this work, we analyze the relationship between PPR estimation complexity and clustering. First, we devise algorithms to estimate PPR for many source/target pairs. In particular, we propose an enhanced version of the existing single pair estimator $\texttt{Bidirectional-PPR}$ that is more useful as a primitive for many pair estimation. We then show that the common underlying graph can be leveraged to efficiently and jointly estimate PPR for many pairs, rather than treating each pair separately using the primitive algorithm. Next, we show the complexity of our joint estimation scheme relates closely to the degree of clustering among the sources and targets at hand, indicating that estimating PPR for many pairs is easier when clustering occurs. Finally, we consider estimating PPR when several machines are available for parallel computation, devising a method that leverages our clustering findings, specifically the quantities computed $\textit{in situ}$, to assign tasks to machines in a manner that reduces computation time. This demonstrates that the relationship between complexity and clustering has important consequences in a practical distributed setting.

cs.SI↗