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Bayesian optimization (BO) with Gaussian processes (GP) as surrogate models is widely used to optimize analytically unknown and expensive-to-evaluate functions. In this paper, we propose Prior-mean-RObust Bayesian Optimization (PROBO) that outperforms classical BO on specific problems. First, we study the effect of the Gaussian processes' prior specifications on classical BO's convergence. We find the prior's mean parameters to have the highest influence on convergence among all prior components. In response to this result, we introduce PROBO as a generalization of BO that aims at rendering the method more robust towards prior mean parameter misspecification. This is achieved by explicitly accounting for GP imprecision via a prior near-ignorance model. At the heart of this is a novel acquisition function, the generalized lower confidence bound (GLCB). We test our approach against classical BO on a real-world problem from material science and observe PROBO to converge faster. Further experiments on multimodal and wiggly target functions confirm the superiority of our method.

For the problem of maximizing a monotone, submodular function with respect to a cardinality constraint $k$ on a ground set of size $n$, we provide an algorithm that achieves the state-of-the-art in both its empirical performance and its theoretical properties, in terms of adaptive complexity, query complexity, and approximation ratio; that is, it obtains, with high probability, query complexity of $O(n)$ in expectation, adaptivity of $O(\log(n))$, and approximation ratio of nearly $1-1/e$. The main algorithm is assembled from two components which may be of independent interest. The first component of our algorithm, LINEARSEQ, is useful as a preprocessing algorithm to improve the query complexity of many algorithms. Moreover, a variant of LINEARSEQ is shown to have adaptive complexity of $O( \log (n / k) )$ which is smaller than that of any previous algorithm in the literature. The second component is a parallelizable thresholding procedure THRESHOLDSEQ for adding elements with gain above a constant threshold. Finally, we demonstrate that our main algorithm empirically outperforms, in terms of runtime, adaptive rounds, total queries, and objective values, the previous state-of-the-art algorithm FAST in a comprehensive evaluation with six submodular objective functions.

In this paper, we develop a novel virtual-queue-based online algorithm for online convex optimization (OCO) problems with long-term and time-varying constraints and conduct a performance analysis with respect to the dynamic regret and constraint violations. We design a new update rule of dual variables and a new way of incorporating time-varying constraint functions into the dual variables. To the best of our knowledge, our algorithm is the first parameter-free algorithm to simultaneously achieve sublinear dynamic regret and constraint violations. Our proposed algorithm also outperforms the state-of-the-art results in many aspects, e.g., our algorithm does not require the Slater condition. Meanwhile, for a group of practical and widely-studied constrained OCO problems in which the variation of consecutive constraints is smooth enough across time, our algorithm achieves $O(1)$ constraint violations. Furthermore, we extend our algorithm and analysis to the case when the time horizon $T$ is unknown. Finally, numerical experiments are conducted to validate the theoretical guarantees of our algorithm, and some applications of our proposed framework will be outlined.

We consider the rooted prize-collecting walks (PCW) problem, wherein we seek a collection $C$ of rooted walks having minimum prize-collecting cost, which is the (total cost of walks in $C$) + (total node-reward of nodes not visited by any walk in $C$). This problem arises naturally as the Lagrangian relaxation of both orienteering, where we seek a length-bounded walk of maximum reward, and the $\ell$-stroll problem, where we seek a minimum-length walk covering at least $\ell$ nodes. Our main contribution is to devise a simple, combinatorial algorithm for the PCW problem in directed graphs that returns a rooted tree whose prize-collecting cost is at most the optimum value of the prize-collecting walks problem. We utilize our algorithm to develop combinatorial approximation algorithms for two fundamental vehicle-routing problems (VRPs): (1) orienteering; and (2) $k$-minimum-latency problem ($k$-MLP), wherein we seek to cover all nodes using $k$ paths starting at a prescribed root node, so as to minimize the sum of the node visiting times. Our combinatorial algorithm allows us to sidestep the part where we solve a preflow-based LP in the LP-rounding algorithms of Friggstand and Swamy (2017) for orienteering, and in the state-of-the-art $7.183$-approximation algorithm for $k$-MP in Post and Swamy (2015). Consequently, we obtain combinatorial implementations of these algorithms with substantially improved running times compared with the current-best approximation factors. We report computational results for our resulting (combinatorial implementations of) orienteering algorithms, which show that the algorithms perform quite well in practice, both in terms of the quality of the solution they return, as also the upper bound they yield on the orienteering optimum (which is obtained by leveraging the workings of our PCW algorithm).

We study sparse linear regression over a network of agents, modeled as an undirected graph (with no centralized node). The estimation problem is formulated as the minimization of the sum of the local LASSO loss functions plus a quadratic penalty of the consensus constraint -- the latter being instrumental to obtain distributed solution methods. While penalty-based consensus methods have been extensively studied in the optimization literature, their statistical and computational guarantees in the high dimensional setting remain unclear. This work provides an answer to this open problem. Our contribution is two-fold. First, we establish statistical consistency of the estimator: under a suitable choice of the penalty parameter, the optimal solution of the penalized problem achieves near optimal minimax rate $\mathcal{O}(s \log d/N)$ in $\ell_2$-loss, where $s$ is the sparsity value, $d$ is the ambient dimension, and $N$ is the total sample size in the network -- this matches centralized sample rates. Second, we show that the proximal-gradient algorithm applied to the penalized problem, which naturally leads to distributed implementations, converges linearly up to a tolerance of the order of the centralized statistical error -- the rate scales as $\mathcal{O}(d)$, revealing an unavoidable speed-accuracy dilemma.Numerical results demonstrate the tightness of the derived sample rate and convergence rate scalings.

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.

Influence maximization is the task of selecting a small number of seed nodes in a social network to maximize the spread of the influence from these seeds, and it has been widely investigated in the past two decades. In the canonical setting, the whole social network as well as its diffusion parameters is given as input. In this paper, we consider the more realistic sampling setting where the network is unknown and we only have a set of passively observed cascades that record the set of activated nodes at each diffusion step. We study the task of influence maximization from these cascade samples (IMS), and present constant approximation algorithms for this task under mild conditions on the seed set distribution. To achieve the optimization goal, we also provide a novel solution to the network inference problem, that is, learning diffusion parameters and the network structure from the cascade data. Comparing with prior solutions, our network inference algorithm requires weaker assumptions and does not rely on maximum-likelihood estimation and convex programming. Our IMS algorithms enhance the learning-and-then-optimization approach by allowing a constant approximation ratio even when the diffusion parameters are hard to learn, and we do not need any assumption related to the network structure or diffusion parameters.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.