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We consider Bayesian best arm identification in the multi-armed bandit problem. Assuming certain continuity conditions of the prior, we characterize the rate of the Bayesian simple regret. Differing from Bayesian regret minimization (Lai, 1987), the leading factor in Bayesian simple regret derives from the region where the gap between optimal and sub-optimal arms is smaller than $\sqrt{\frac{\log T}{T}}$. We propose a simple and easy-to-compute algorithm with its leading factor matches with the lower bound up to a constant factor; simulation results support our theoretical findings.

We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every subset of $I$ of size $d$ is the $d$-dimensional cube. We show that determining the largest cardinality of a set with a given $\mathbb{U}_d$ dimension is equivalent to a Tur\'an-type problem related to the total number of cliques in a $d$-uniform hypergraph. This allows us to beat the Sauer--Shelah lemma for this notion of dimension. We use this to obtain several results on $\Sigma_3^k$-circuits, i.e., depth-$3$ circuits with top gate OR and bottom fan-in at most $k$: * Tight relationship between the number of satisfying assignments of a $2$-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex. '00). * Improved $\Sigma_3^3$-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. * We make progress towards settling the $\Sigma_3^2$ complexity of the inner product function and all degree-$2$ polynomials over $\mathbb{F}_2$ in general. The question of determining the $\Sigma_3^3$ complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).

Mutual exclusion is a classical problem in distributed computing that provides isolation among concurrent action executions that may require access to the same shared resources. Inspired by algorithmic research on distributed systems of weakly capable entities whose connections change over time, we address the local mutual exclusion problem that tasks each node with acquiring exclusive locks for itself and the maximal subset of its "persistent" neighbors that remain connected to it over the time interval of the lock request. Using the established time-varying graphs model to capture adversarial topological changes, we propose and rigorously analyze a local mutual exclusion algorithm for nodes that are anonymous and communicate via asynchronous message passing. The algorithm satisfies mutual exclusion (non-intersecting lock sets) and lockout freedom (eventual success) under both semi-synchronous and asynchronous concurrency. It requires $\mathcal{O}(\Delta\log\Delta)$ memory per node and messages of size $\mathcal{O}(\log\Delta)$, where $\Delta$ is the maximum number of connections per node. For systems of weak entities, $\Delta$ is often a small constant, reducing the memory and message size requirements to $\mathcal{O}(1)$. We conclude by describing how our algorithm can be used to implement the schedulers assumed by population protocols and the concurrency control operations assumed by the canonical amoebot model, demonstrating its utility in both passively and actively dynamic distributed systems.

We give an algorithm to find a minimum cut in an edge-weighted directed graph with $n$ vertices and $m$ edges in $\tilde O(n\cdot \max(m^{2/3}, n))$ time. This improves on the 30 year old bound of $\tilde O(nm)$ obtained by Hao and Orlin for this problem. Our main technique is to reduce the directed mincut problem to $\tilde O(\min(n/m^{1/3}, \sqrt{n}))$ calls of {\em any} maxflow subroutine. Using state-of-the-art maxflow algorithms, this yields the above running time. Our techniques also yield fast {\em approximation} algorithms for finding minimum cuts in directed graphs. For both edge and vertex weighted graphs, we give $(1+\epsilon)$-approximation algorithms that run in $\tilde O(n^2 / \epsilon^2)$ time.

We study fast rates of convergence in the setting of nonparametric online regression, namely where regret is defined with respect to an arbitrary function class which has bounded complexity. Our contributions are two-fold: - In the realizable setting of nonparametric online regression with the absolute loss, we propose a randomized proper learning algorithm which gets a near-optimal mistake bound in terms of the sequential fat-shattering dimension of the hypothesis class. In the setting of online classification with a class of Littlestone dimension $d$, our bound reduces to $d \cdot {\rm poly} \log T$. This result answers a question as to whether proper learners could achieve near-optimal mistake bounds; previously, even for online classification, the best known mistake bound was $\tilde O( \sqrt{dT})$. Further, for the real-valued (regression) setting, the optimal mistake bound was not even known for improper learners, prior to this work. - Using the above result, we exhibit an independent learning algorithm for general-sum binary games of Littlestone dimension $d$, for which each player achieves regret $\tilde O(d^{3/4} \cdot T^{1/4})$. This result generalizes analogous results of Syrgkanis et al. (2015) who showed that in finite games the optimal regret can be accelerated from $O(\sqrt{T})$ in the adversarial setting to $O(T^{1/4})$ in the game setting. To establish the above results, we introduce several new techniques, including: a hierarchical aggregation rule to achieve the optimal mistake bound for real-valued classes, a multi-scale extension of the proper online realizable learner of Hanneke et al. (2021), an approach to show that the output of such nonparametric learning algorithms is stable, and a proof that the minimax theorem holds in all online learnable games.

In this paper, we give bounds on the dichromatic number $\vec{\chi}(\Sigma)$ of a surface $\Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $\Sigma$. We determine the asymptotic behaviour of $\vec{\chi}(\Sigma)$ by showing that there exist constants $a_1$ and $a_2$ such that, $a_1\frac{\sqrt{-c}}{\log(-c)} \leq \vec{\chi}(\Sigma) \leq a_2 \frac{\sqrt{-c}}{\log(-c)} $ for every surface $\Sigma$ with Euler characteristic $c\leq -2$. We then give more explicit bounds for some surfaces with high Euler characteristic. In particular, we show that the dichromatic numbers of the projective plane $\mathbb{N}_1$, the Klein bottle $\mathbb{N}_2$, the torus $\mathbb{S}_1$, and Dyck's surface $\mathbb{N}_3$ are all equal to $3$, and that the dichromatic numbers of the $5$-torus $\mathbb{S}_5$ and the $10$-cross surface $\mathbb{N}_{10}$ are equal to $4$. We also consider the complexity of deciding whether a given digraph or oriented graph embeddable on a fixed surface is $k$-dicolourable. In particular, we show that for any fixed surface, deciding whether a digraph embeddable on this surface is $2$-dicolourable is NP-complete, and that deciding whether a planar oriented graph is $2$-dicolourable is NP-complete unless all planar oriented graphs are $2$-dicolourable (which was conjectured by Neumann-Lara).

We study flow scheduling under node capacity constraints. We are given capacitated nodes and an online sequence of jobs, each with a release time and a demand to be routed between two nodes. A schedule specifies which jobs are routed in each step, guaranteeing that the total demand on a node in any step is at most its capacity. A key metric in this scenario is response time: the time between a job's release and its completion. Prior work shows no un-augmented algorithm is competitive for average response time, and that a constant factor competitive ratio is achievable with augmentation exceeding 2 (Dinitz-Moseley Infocom 2020). For maximum response time, the best known result is a 2-competitive algorithm with a augmentation 4 (Jahanjou et al SPAA 2020). We improve these bounds under various response time objectives. We show that, without resource augmentation, the best competitive ratio for maximum response time is $\Omega(n)$, where $n$ is the number of nodes. Our Proportional Allocation algorithm uses $(1+\varepsilon)$ resource augmentation to achieve a $(1/\varepsilon)$-competitive ratio in the setting with general demands and capacities, and splittable jobs. Our Batch Decomposition algorithm is $2$-competitive (resp., optimal) for maximum response time using resource augmentation 2 (resp., 4) in the setting with unit demands and capacities, and unsplittable jobs. We also derive bounds for the simultaneous approximation of average and maximum response time metrics.

In this work, we study the multi-agent decision problem where agents try to coordinate to optimize a given system-level objective. While solving for the global optimal is intractable in many cases, the greedy algorithm is a well-studied and efficient way to provide good approximate solutions - notably for submodular optimization problems. Executing the greedy algorithm requires the agents to be ordered and execute a local optimization based on the solutions of the previous agents. However, in limited information settings, passing the solution from the previous agents may be nontrivial, as some agents may not be able to directly communicate with each other. Thus the communication time required to execute the greedy algorithm is closely tied to the order that the agents are given. In this work, we characterize interplay between the communication complexity and agent orderings by showing that the complexity using the best ordering is O(n) and increases considerably to O(n^2) when using the worst ordering. Motivated by this, we also propose an algorithm that can find an ordering and execute the greedy algorithm quickly, in a distributed fashion. We also show that such an execution of the greedy algorithm is advantageous over current methods for distributed submodular maximization.

Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena. We train a Restricted Boltzmann Machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods. From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system. We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near criticality region of the Ising model. In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the Renormalization Group (RG) flow of the lattice model. Our results suggest an alternative explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points. Then from the reconstructed data we deduce the critical exponent associated to the magnetization to find satisfactory agreement with the actual physical value. We assume no prior knowledge about the criticality of the system and its Hamiltonian.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.