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We study streaming algorithms for two fundamental geometric problems: computing the cost of a Minimum Spanning Tree (MST) of an $n$-point set $X \subset \{1,2,\dots,\Delta\}^d$, and computing the Earth Mover Distance (EMD) between two multi-sets $A,B \subset \{1,2,\dots,\Delta\}^d$ of size $n$. We consider the turnstile model, where points can be added and removed. We give a one-pass streaming algorithm for MST and a two-pass streaming algorithm for EMD, both achieving an approximation factor of $\tilde{O}(\log n)$ and using polylog$(n,d,\Delta)$-space only. Furthermore, our algorithm for EMD can be compressed to a single pass with a small additive error. Previously, the best known sublinear-space streaming algorithms for either problem achieved an approximation of $O(\min\{ \log n , \log (\Delta d)\} \log n)$ [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20]. For MST, we also prove that any constant space streaming algorithm can only achieve an approximation of $\Omega(\log n)$, analogous to the $\Omega(\log n)$ lower bound for EMD of [Andoni-Indyk-Krauthgamer '08]. Our algorithms are based on an improved analysis of a recursive space partitioning method known generically as the Quadtree. Specifically, we show that the Quadtree achieves an $\tilde{O}(\log n)$ approximation for both EMD and MST, improving on the $O(\min\{ \log n , \log (\Delta d)\} \log n)$ approximation of [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20].

We consider an important generalization of the Steiner tree problem, the \emph{Steiner forest problem}, in the Euclidean plane: the input is a multiset $X \subseteq \mathbb{R}^2$, partitioned into $k$ color classes $C_1, C_2, \ldots, C_k \subseteq X$. The goal is to find a minimum-cost Euclidean graph $G$ such that every color class $C_i$ is connected in $G$. We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to $X$. Each input point $x\in X$ arrives with its color $\textsf{color}(x) \in [k]$, and as usual for dynamic geometric streams, the input points are restricted to the discrete grid $\{0, \ldots, \Delta\}^2$. We design a single-pass streaming algorithm that uses $\mathrm{poly}(k \cdot \log\Delta)$ space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio $\alpha_2$ (currently $1.1547 \le \alpha_2 \le 1.214$). This approximation guarantee matches the state of the art bound for streaming Steiner tree, i.e., when $k=1$. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and has so far not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite approximation requires $\Omega(k)$ bits of space.

Online allocation problems with resource constraints are central problems in revenue management and online advertising. In these problems, requests arrive sequentially during a finite horizon and, for each request, a decision maker needs to choose an action that consumes a certain amount of resources and generates reward. The objective is to maximize cumulative rewards subject to a constraint on the total consumption of resources. In this paper, we consider a data-driven setting in which the reward and resource consumption of each request are generated using an input model that is unknown to the decision maker. We design a general class of algorithms that attain good performance in various input models without knowing which type of input they are facing. In particular, our algorithms are asymptotically optimal under independent and identically distributed inputs as well as various non-stationary stochastic input models, and they attain an asymptotically optimal fixed competitive ratio when the input is adversarial. Our algorithms operate in the Lagrangian dual space: they maintain a dual multiplier for each resource that is updated using online mirror descent. By choosing the reference function accordingly, we recover the dual sub-gradient descent and dual multiplicative weights update algorithm. The resulting algorithms are simple, fast, and do not require convexity in the revenue function, consumption function and action space, in contrast to existing methods for online allocation problems. We discuss applications to network revenue management, online bidding in repeated auctions with budget constraints, online proportional matching with high entropy, and personalized assortment optimization with limited inventory.

We study the revenue guarantees and approximability of item pricing. Recent work shows that with $n$ heterogeneous items, item-pricing guarantees an $O(\log n)$ approximation to the optimal revenue achievable by any (buy-many) mechanism, even when buyers have arbitrarily combinatorial valuations. However, finding good item prices is challenging -- it is known that even under unit-demand valuations, it is NP-hard to find item prices that approximate the revenue of the optimal item pricing better than $O(\sqrt{n})$. Our work provides a more fine-grained analysis of the revenue guarantees and computational complexity in terms of the number of item ``categories'' which may be significantly fewer than $n$. We assume the items are partitioned in $k$ categories so that items within a category are totally-ordered and a buyer's value for a bundle depends only on the best item contained from every category. We show that item-pricing guarantees an $O(\log k)$ approximation to the optimal (buy-many) revenue and provide a PTAS for computing the optimal item-pricing when $k$ is constant. We also provide a matching lower bound showing that the problem is (strongly) NP-hard even when $k=1$. Our results naturally extend to the case where items are only partially ordered, in which case the revenue guarantees and computational complexity depend on the width of the partial ordering, i.e. the largest set for which no two items are comparable.

A search engine maintains local copies of different web pages to provide quick search results. This local cache is kept up-to-date by a web crawler that frequently visits these different pages to track changes in them. Ideally, the local copy should be updated as soon as a page changes on the web. However, finite bandwidth availability and server restrictions limit how frequently different pages can be crawled. This brings forth the following optimization problem: maximize the freshness of the local cache subject to the crawling frequencies being within prescribed bounds. While tractable algorithms do exist to solve this problem, these either assume the knowledge of exact page change rates or use inefficient methods such as MLE for estimating the same. We address this issue here. We provide three novel schemes for online estimation of page change rates, all of which have extremely low running times per iteration. The first is based on the law of large numbers and the second on stochastic approximation. The third is an extension of the second and includes a heavy-ball momentum term. All these schemes only need partial information about the page change process, i.e., they only need to know if the page has changed or not since the last crawled instance. Our main theoretical results concern asymptotic convergence and convergence rates of these three schemes. In fact, our work is the first to show convergence of the original stochastic heavy-ball method when neither the gradient nor the noise variance is uniformly bounded. We also provide some numerical experiments (based on real and synthetic data) to demonstrate the superiority of our proposed estimators over existing ones such as MLE. We emphasize that our algorithms are also readily applicable to the synchronization of databases and network inventory management.

We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems. In this work, we show that the answer is positive for various important streaming problems in the insertion-only model, including distinct elements and more generally $F_p$-estimation, $F_p$-heavy hitters, entropy estimation, and others. For all of these problems, we develop adversarially robust $(1+\varepsilon)$-approximation algorithms whose required space matches that of the best known non-robust algorithms up to a $\text{poly}(\log n, 1/\varepsilon)$ multiplicative factor (and in some cases even up to a constant factor). Towards this end, we develop several generic tools allowing one to efficiently transform a non-robust streaming algorithm into a robust one in various scenarios.

Non-convex optimization is ubiquitous in modern machine learning. Researchers devise non-convex objective functions and optimize them using off-the-shelf optimizers such as stochastic gradient descent and its variants, which leverage the local geometry and update iteratively. Even though solving non-convex functions is NP-hard in the worst case, the optimization quality in practice is often not an issue -- optimizers are largely believed to find approximate global minima. Researchers hypothesize a unified explanation for this intriguing phenomenon: most of the local minima of the practically-used objectives are approximately global minima. We rigorously formalize it for concrete instances of machine learning problems.

When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.

We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known randomized online coordinate descent algorithms. Furthermore, the proposed algorithms for stochastic optimization exhibit as good convergence rates as the best known randomized coordinate descent algorithms. We also show simulation results to demonstrate performance of the proposed algorithms.

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.