Linear Bandit Algorithms with Sublinear Time Complexity

We propose to accelerate existing linear bandit algorithms to achieve per-step time complexity sublinear in the number of arms $K$. The key to sublinear complexity is the realization that the arm selection in many linear bandit algorithms reduces to the maximum inner product search (MIPS) problem. Correspondingly, we propose an algorithm that approximately solves the MIPS problem for a sequence of adaptive queries yielding near-linear preprocessing time complexity and sublinear query time complexity. Using the proposed MIPS solver as a sub-routine, we present two bandit algorithms (one based on UCB, and the other based on TS) that achieve sublinear time complexity. We explicitly characterize the tradeoff between the per-step time complexity and regret, and show that our proposed algorithms can achieve $O(K^{1-alpha(T)})$ per-step complexity for some $alpha(T) > 0$ and $widetilde O(sqrt{T})$ regret, where $T$ is the time horizon. Further, we present the theoretical limit of the tradeoff, which provides a lower bound for the per-step time complexity. We also discuss other choices of approximate MIPS algorithms and other applications to linear bandit problems.

How Does the Task Landscape Affect MAML Performance?

Model-Agnostic Meta-Learning (MAML) has become increasingly popular for training models that can quickly adapt to new tasks via one or few stochastic gradient descent steps. However, the MAML objective is significantly more difficult to optimize compared to standard Empirical Risk Minimization (ERM), and little is understood about how much MAML improves over ERM in terms of the fast adaptability of their solutions in various scenarios. We analytically address this issue in a linear regression setting consisting of a mixture of easy and hard tasks, where hardness is related to the condition number of the tasks loss function. Specifically, we prove that in order for MAML to achieve substantial gain over ERM, (i) there must be some discrepancy in hardness among the tasks, and (ii) the optimal solutions of the hard tasks must be closely packed with the center far from the center of the easy tasks optimal solutions. We also give numerical and analytical results suggesting that these insights also apply to two-layer neural networks. Finally, we provide few-shot image classification experiments that support our insights for when MAML should be used and emphasize the importance of training MAML on hard tasks in practice.

Multi-Agent Low-Dimensional Linear Bandits

We study a multi-agent stochastic linear bandit with side information, parameterized by an unknown vector $theta^* in mathbb{R}^d$. The side information consists of a finite collection of low-dimensional subspaces, one of which contains $theta^*$. In our setting, agents can collaborate to reduce regret by sending recommendations across a communication graph connecting them. We present a novel decentralized algorithm, where agents communicate subspace indices with each other and each agent plays a projected variant of LinUCB on the corresponding (low-dimensional) subspace. By distributing the search for the optimal subspace across users and learning of the unknown vector by each agent in the corresponding low-dimensional subspace, we show that the per-agent finite-time regret is much smaller than the case when agents do not communicate. We finally complement these results through simulations.

Applications of Common Entropy for Causal Inference

We study the problem of discovering the simplest latent variable that can make two observed discrete variables conditionally independent. The minimum entropy required for such a latent is known as common entropy in information theory. We extend this notion to Renyi common entropy by minimizing the Renyi entropy of the latent variable. To efficiently compute common entropy, we propose an iterative algorithm that can be used to discover the trade-off between the entropy of the latent variable and the conditional mutual information of the observed variables. We show two applications of common entropy in causal inference: First, under the assumption that there are no low-entropy mediators, it can be used to distinguish causation from spurious correlation among almost all joint distributions on simple causal graphs with two observed variables. Second, common entropy can be used to improve constraint-based methods such as PC or FCI algorithms in the small-sample regime, where these methods are known to struggle. We propose a modification to these constraint-based methods to assess if a separating set found by these algorithms is valid using common entropy. We finally evaluate our algorithms on synthetic and real data to establish their performance.