Motivated by online decision-making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability. We show that the resulting online algorithms have $O(\sqrt{T})$ (approximate) regret under the full information setting. We further introduce a bandit extension of Blackwell approachability that we call Bandit Blackwell approachability. We leverage this notion to transform greedy robust offline algorithms into a $O(T^{2/3})$ (approximate) regret in the bandit setting. Demonstrating the flexibility of our framework, we apply our offline-to-online transformation to several problems at the intersection of revenue management, market design, and online optimization, including product ranking optimization in online platforms, reserve price optimization in auctions, and submodular maximization. We show that our transformation, when applied to these applications, leads to new regret bounds or improves the current known bounds.

Nowadays, combinatorial optimization (CO) is an interdisciplinary field spanning optimization, operations research, discrete mathematics, and computer science, with many critical real-world applications such as vehicle routing or scheduling; see [71] for a general overview. Intuitively, CO deals with selecting a subset from a finite set that optimizes a cost or objective function. Although many CO problems are hard from a complexity theory standpoint due to their discrete nature, many of them are routinely solved in practice. Historically, the optimization and theoretical computer science communities have been focusing on finding optimal [71], heuristic [12], or approximative [130] solutions for individual problem instances. However, in many practical situations of interest, one often needs to solve problem instances which share patterns and characteristics repeatedly.

Minimax optimization has recently gained a lot of attention as adversarial architectures and algorithms proliferate. Often, smooth minimax games proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice for many applications (e.g., GAN training), the majority of existing theoretical analyses focus on simultaneous algorithms. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart (Sim-GDA) in many settings. In particular, we prove that Alt-GDA achieves a near-optimal local convergence rate for strongly-convex strongly-concave problems while Sim-GDA converges with a much slower rate. Moreover, we show that the acceleration effect of alternating updates remains when the minimax problem has only strong concavity in the dual variables. Numerical experiments on quadratic minimax games validate our claims. Additionally, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms.

We consider the problem of routing a large fleet of drones to deliver packages simultaneously across broad urban areas. Besides flying directly, drones can use public transit vehicles such as buses and trams as temporary modes of transportation to conserve energy. Adding this capability to our formulation augments effective drone travel range and the space of possible deliveries but also increases problem input size due to the large transit networks. We present a comprehensive algorithmic framework that strives to minimize the maximum time to complete any delivery and addresses the multifaceted computational challenges of our problem through a two-layer approach. First, the upper layer assigns drones to package delivery sequences with an approximately optimal polynomial time allocation algorithm. Then, the lower layer executes the allocation by periodically routing the fleet over the transit network, using efficient, bounded suboptimal multi-agent pathfinding techniques tailored to our setting. We demonstrate the efficiency of our approach on simulations with up to 200 drones, 5000 packages, and transit networks with up to 8000 stops in San Francisco and the Washington DC Metropolitan Area. Our framework computes solutions for most settings within a few seconds on commodity hardware and enables drones to extend their effective range by a factor of nearly four using transit.

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time an adversary chooses an input distribution with density function bounded above by $\tfrac{1}{\sigma}$ times that of the uniform distribution; nature then samples an input from this distribution. Crucially, our results hold for {\em adaptive} adversaries that can choose an input distribution based on the decisions of the algorithm and the realizations of the inputs in the previous time steps. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of adaptive adversaries to the simpler case of oblivious adversaries. We apply this technique to prove strong smoothed guarantees for three problems: -Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of $\sigma$-smooth distributions and the hypothesis class has VC dimension $d$. We bound the regret by $\tilde{O}\big(\sqrt{T d\ln(1/\sigma)} + d\sqrt{\ln(T/\sigma)}\big)$. This answers open questions of [RST11,Hag18]. -Online discrepancy minimization: We consider the online Koml\'os problem, where the input is generated from an adaptive sequence of $\sigma$-smooth and isotropic distributions on the $\ell_2$ unit ball. We bound the $\ell_\infty$ norm of the discrepancy vector by $\tilde{O}\big(\ln^2\!\big( \frac{nT}{\sigma}\big) \big)$. -Dispersion in online optimization: We consider online optimization of piecewise Lipschitz functions where functions with $\ell$ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is $\big( {\sigma}/{\sqrt{T\ell}}, \tilde O\big(\sqrt{T\ell} \big)\big)$-dispersed. This matches the parameters of [BDV18] for oblivious adversaries, up to log factors.

We study reinforcement learning in an infinite-horizon average-reward setting with linear function approximation, where the transition probability function of the underlying Markov Decision Process (MDP) admits a linear form over a feature mapping of the current state, action, and next state. We propose a new algorithm UCRL2-VTR, which can be seen as an extension of the UCRL2 algorithm with linear function approximation. We show that UCRL2-VTR with Bernstein-type bonus can achieve a regret of $\tilde{O}(d\sqrt{DT})$, where $d$ is the dimension of the feature mapping, $T$ is the horizon, and $\sqrt{D}$ is the diameter of the MDP. We also prove a matching lower bound $\tilde{\Omega}(d\sqrt{DT})$, which suggests that the proposed UCRL2-VTR is minimax optimal up to logarithmic factors. To the best of our knowledge, our algorithm is the first nearly minimax optimal RL algorithm with function approximation in the infinite-horizon average-reward setting.

High-dimensional black-box optimisation remains an important yet notoriously challenging problem. Despite the success of Bayesian optimisation methods on continuous domains, domains that are categorical, or that mix continuous and categorical variables, remain challenging. We propose a novel solution -- we combine local optimisation with a tailored kernel design, effectively handling high-dimensional categorical and mixed search spaces, whilst retaining sample efficiency. We further derive convergence guarantee for the proposed approach. Finally, we demonstrate empirically that our method outperforms the current baselines on a variety of synthetic and real-world tasks in terms of performance, computational costs, or both.

In collaborative privacy preserving planning (CPPP), a group of agents jointly creates a plan to achieve a set of goals while preserving each others' privacy. During planning, agents often reveal the private dependencies between their public actions to other agents, that is, which public action facilitates the preconditions of another public action. Previous work in CPPP does not limit the disclosure of such dependencies. In this paper, we explicitly limit the amount of disclosed dependencies, allowing agents to publish only a part of their private dependencies. We investigate different strategies for deciding which dependencies to publish, and how they affect the ability to find solutions. We evaluate the ability of two solvers -- distribute forward search and centralized planning based on a single-agent projection -- to produce plans under this constraint. Experiments over standard CPPP domains show that the proposed dependency-sharing strategies enable generating plans while sharing only a small fraction of all private dependencies.

In this paper, we revisit the problem of smoothed online learning, in which the online learner suffers both a hitting cost and a switching cost, and target two performance metrics: competitive ratio and dynamic regret with switching cost. To bound the competitive ratio, we assume the hitting cost is known to the learner in each round, and investigate the greedy algorithm which simply minimizes the weighted sum of the hitting cost and the switching cost. Our theoretical analysis shows that the greedy algorithm, although straightforward, is $1+ \frac{2}{\alpha}$-competitive for $\alpha$-polyhedral functions, $1+O(\frac{1}{\lambda})$-competitive for $\lambda$-quadratic growth functions, and $1 + \frac{2}{\sqrt{\lambda}}$-competitive for convex and $\lambda$-quadratic growth functions. To bound the dynamic regret with switching cost, we follow the standard setting of online convex optimization, in which the hitting cost is convex but hidden from the learner before making predictions. We modify Ader, an existing algorithm designed for dynamic regret, slightly to take into account the switching cost when measuring the performance. The proposed algorithm, named as Smoothed Ader, attains an optimal $O(\sqrt{T(1+P_T)})$ bound for dynamic regret with switching cost, where $P_T$ is the path-length of the comparator sequence. Furthermore, if the hitting cost is accessible in the beginning of each round, we obtain a similar guarantee without the bounded gradient condition.

The Clustered Shortest-Path Tree Problem (CluSPT) plays an important role in various types of optimization problems in real-life. Recently, some Multifactorial Evolutionary Algorithm (MFEA) have been introduced to deal with the CluSPT, however these researches still have some shortcomings such as evolution operators only perform on complete graphs, huge resource consumption for finding the solution on large search spaces. To overcome these limitations, this paper describes a MFEA-based approach to solve the CluSPT. The proposed algorithm utilizes Dijkstra's algorithm to construct the spanning trees in clusters while using evolutionary operators for building the spanning tree connecting clusters. This approach takes advantage of both exact and approximate algorithms so it enables the algorithm to function efficiently on complete and sparse graphs alike. Furthermore, evolutionary operators such as individual encoding and decoding methods are also designed with great consideration regarding performance and memory usage. We have included a proof on the repairing method's efficacy in ensuring all solutions are valid. We have conducted tests on various types of Euclidean instances to assess the effectiveness of the proposed algorithm and methods. Experiment results point out the effectiveness of the proposed algorithm existing heuristic algorithms in most of the test cases. The impact of the proposed MFEA was analyzed and a possible influential factor that may be useful for further study was also pointed out.