Online weighted matching problem is a fundamental problem in machine learning due to its numerous applications. Despite many efforts in this area, existing algorithms are either too slow or don't take $\mathrm{deadline}$ (the longest time a node can be matched) into account. In this paper, we introduce a market model with $\mathrm{deadline}$ first. Next, we present our two optimized algorithms (\textsc{FastGreedy} and \textsc{FastPostponedGreedy}) and offer theoretical proof of the time complexity and correctness of our algorithms. In \textsc{FastGreedy} algorithm, we have already known if a node is a buyer or a seller. But in \textsc{FastPostponedGreedy} algorithm, the status of each node is unknown at first. Then, we generalize a sketching matrix to run the original and our algorithms on both real data sets and synthetic data sets. Let $\epsilon \in (0,0.1)$ denote the relative error of the real weight of each edge. The competitive ratio of original \textsc{Greedy} and \textsc{PostponedGreedy} is $\frac{1}{2}$ and $\frac{1}{4}$ respectively. Based on these two original algorithms, we proposed \textsc{FastGreedy} and \textsc{FastPostponedGreedy} algorithms and the competitive ratio of them is $\frac{1 - \epsilon}{2}$ and $\frac{1 - \epsilon}{4}$ respectively. At the same time, our algorithms run faster than the original two algorithms. Given $n$ nodes in $\mathbb{R} ^ d$, we decrease the time complexity from $O(nd)$ to $\widetilde{O}(\epsilon^{-2} \cdot (n + d))$.
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We study the online variant of the Min-Sum Set Cover (MSSC) problem, a generalization of the well-known list update problem. In the MSSC problem, an algorithm has to maintain the time-varying permutation of the list of $n$ elements, and serve a sequence of requests $R_1, R_2, \dots, R_t, \dots$. Each $R_t$ is a subset of elements of cardinality at most $r$. For a requested set $R_t$, an online algorithm has to pay the cost equal to the position of the first element from $R_t$ on its list. Then, it may arbitrarily permute its list, paying the number of swapped adjacent element pairs. We present the first constructive deterministic algorithm for this problem, whose competitive ratio does not depend on $n$. Our algorithm is $O(r^2)$-competitive, which beats both the existential upper bound of $O(r^4)$ by Bienkowski and Mucha [AAAI '23] and the previous constructive bound of $O(r^{3/2} \cdot \sqrt{n})$ by Fotakis et al. [ICALP '20]. Furthermore, we show that our algorithm attains an asymptotically optimal competitive ratio of $O(r)$ when compared to the best fixed permutation of elements.
In this paper, practically computable low-order approximations of potentially high-dimensional differential equations driven by geometric rough paths are proposed and investigated. In particular, equations are studied that cover the linear setting, but we allow for a certain type of dissipative nonlinearity in the drift as well. In a first step, a linear subspace is found that contains the solution space of the underlying rough differential equation (RDE). This subspace is associated to covariances of linear Ito-stochastic differential equations which is shown exploiting a Gronwall lemma for matrix differential equations. Orthogonal projections onto the identified subspace lead to a first exact reduced order system. Secondly, a linear map of the RDE solution (quantity of interest) is analyzed in terms of redundant information meaning that state variables are found that do not contribute to the quantity of interest. Once more, a link to Ito-stochastic differential equations is used. Removing such unnecessary information from the RDE provides a further dimension reduction without causing an error. Finally, we discretize a linear parabolic rough partial differential equation in space. The resulting large-order RDE is subsequently tackled with the exact reduction techniques studied in this paper. We illustrate the enormous complexity reduction potential in the corresponding numerical experiments.
We study a variant of online multiclass classification where the learner predicts a single label but receives a \textit{set of labels} as feedback. In this model, the learner is penalized for not outputting a label contained in the revealed set. We show that unlike online multiclass learning with single-label feedback, deterministic and randomized online learnability are \textit{not equivalent} even in the realizable setting with set-valued feedback. Accordingly, we give two new combinatorial dimensions, named the Set Littlestone and Measure Shattering dimension, that tightly characterize deterministic and randomized online learnability respectively in the realizable setting. In addition, we show that the Measure Shattering dimension tightly characterizes online learnability in the agnostic setting. Finally, we show that practical learning settings like online multilabel ranking, online multilabel classification, and online interval learning are specific instances of our general framework.
The cutting plane method is a key technique for successful branch-and-cut and branch-price-and-cut algorithms that find the exact optimal solutions for various vehicle routing problems (VRPs). Among various cuts, the rounded capacity inequalities (RCIs) are the most fundamental. To generate RCIs, we need to solve the separation problem, whose exact solution takes a long time to obtain; therefore, heuristic methods are widely used. We design a learning-based separation heuristic algorithm with graph coarsening that learns the solutions of the exact separation problem with a graph neural network (GNN), which is trained with small instances of 50 to 100 customers. We embed our separation algorithm within the cutting plane method to find a lower bound for the capacitated VRP (CVRP) with up to 1,000 customers. We compare the performance of our approach with CVRPSEP, a popular separation software package for various cuts used in solving VRPs. Our computational results show that our approach finds better lower bounds than CVRPSEP for large-scale problems with 400 or more customers, while CVRPSEP shows strong competency for problems with less than 400 customers.
Minimax problems arise in a wide range of important applications including robust adversarial learning and Generative Adversarial Network (GAN) training. Recently, algorithms for minimax problems in the Federated Learning (FL) paradigm have received considerable interest. Existing federated algorithms for general minimax problems require the full aggregation (i.e., aggregation of local model information from all clients) in each training round. Thus, they are inapplicable to an important setting of FL known as the cross-device setting, which involves numerous unreliable mobile/IoT devices. In this paper, we develop the first practical algorithm named CDMA for general minimax problems in the cross-device FL setting. CDMA is based on a Start-Immediately-With-Enough-Responses mechanism, in which the server first signals a subset of clients to perform local computation and then starts to aggregate the local results reported by clients once it receives responses from enough clients in each round. With this mechanism, CDMA is resilient to the low client availability. In addition, CDMA is incorporated with a lightweight global correction in the local update steps of clients, which mitigates the impact of slow network connections. We establish theoretical guarantees of CDMA under different choices of hyperparameters and conduct experiments on AUC maximization, robust adversarial network training, and GAN training tasks. Theoretical and experimental results demonstrate the efficiency of CDMA.
The extremal theory of forbidden 0--1 matrices studies the asymptotic growth of the function $\mathrm{Ex}(P,n)$, which is the maximum weight of a matrix $A\in\{0,1\}^{n\times n}$ whose submatrices avoid a fixed pattern $P\in\{0,1\}^{k\times l}$. This theory has been wildly successful at resolving problems in combinatorics, discrete and computational geometry, structural graph theory, and the analysis of data structures, particularly corollaries of the dynamic optimality conjecture. All these applications use acyclic patterns, meaning that when $P$ is regarded as the adjacency matrix of a bipartite graph, the graph is acyclic. The biggest open problem in this area is to bound $\mathrm{Ex}(P,n)$ for acyclic $P$. Prior results have only ruled out the strict $O(n\log n)$ bound conjectured by Furedi and Hajnal. It is consistent with prior results that $\forall P. \mathrm{Ex}(P,n)\leq n\log^{1+o(1)} n$, and also consistent that $\forall \epsilon>0.\exists P. \mathrm{Ex}(P,n) \geq n^{2-\epsilon}$. In this paper we establish a stronger lower bound on the extremal functions of acyclic $P$. Specifically, we give a new construction of relatively dense 0--1 matrices with $\Theta(n(\log n/\log\log n)^t)$ 1s that avoid an acyclic $X_t$. Pach and Tardos have conjectured that this type of result is the best possible, i.e., no acyclic $P$ exists for which $\mathrm{Ex}(P,n)\geq n(\log n)^{\omega(1)}$.
Uniform sampling of bipartite graphs and hypergraphs with given degree sequences is necessary for building null models to statistically evaluate their topology. Because these graphs can be represented as binary matrices, the problem is equivalent to uniformly sampling $r \times c$ binary matrices with fixed row and column sums. The trade algorithm, which includes both the curveball and fastball implementations, is the state-of-the-art for performing such sampling. Its mixing time is currently unknown, although $5r$ is currently used as a heuristic. In this paper we propose a new distribution-based approach that not only provides an estimation of the mixing time, but also actually returns a sample of matrices that are guaranteed (within a user-chosen error tolerance) to be uniformly randomly sampled. In numerical experiments on matrices that vary by size, fill, and row and column sum distributions, we find that the upper bound on mixing time is at least $10r$, and that it increases as a function of both $c$ and the fraction of cells containing a 1.
We prove the first polynomial separation between randomized and deterministic time-space tradeoffs of multi-output functions. In particular, we present a total function that on the input of $n$ elements in $[n]$, outputs $O(n)$ elements, such that: (1) There exists a randomized oblivious algorithm with space $O(\log n)$, time $O(n\log n)$ and one-way access to randomness, that computes the function with probability $1-O(1/n)$; (2) Any deterministic oblivious branching program with space $S$ and time $T$ that computes the function must satisfy $T^2S\geq\Omega(n^{2.5}/\log n)$. This implies that logspace randomized algorithms for multi-output functions cannot be black-box derandomized without an $\widetilde{\Omega}(n^{1/4})$ overhead in time. Since previously all the polynomial time-space tradeoffs of multi-output functions are proved via the Borodin-Cook method, which is a probabilistic method that inherently gives the same lower bound for randomized and deterministic branching programs, our lower bound proof is intrinsically different from previous works. We also examine other natural candidates for proving such separations, and show that any polynomial separation for these problems would resolve the long-standing open problem of proving $n^{1+\Omega(1)}$ time lower bound for decision problems with $\mathrm{polylog}(n)$ space.
We study generalizations of online bipartite matching in which each arriving vertex (customer) views a ranked list of offline vertices (products) and matches to (purchases) the first one they deem acceptable. The number of products that the customer has patience to view can be stochastic and dependent on the products seen. We develop a framework that views the interaction with each customer as an abstract resource consumption process, and derive new results for these online matching problems under the adversarial, non-stationary, and IID arrival models, assuming we can (approximately) solve the product ranking problem for each single customer. To that end, we show new results for product ranking under two cascade-click models: an optimal algorithm when each item has its own hazard rate for making the customer depart, and a 1/2-approximate algorithm when the customer has a general item-independent patience distribution. We also present a constant-factor 0.027-approximate algorithm in a new model where items are not initially available and arrive over time. We complement these positive results by presenting three additional negative results relating to these problems.
We employ a toolset -- dubbed Dr. Frankenstein -- to analyse the similarity of representations in deep neural networks. With this toolset, we aim to match the activations on given layers of two trained neural networks by joining them with a stitching layer. We demonstrate that the inner representations emerging in deep convolutional neural networks with the same architecture but different initializations can be matched with a surprisingly high degree of accuracy even with a single, affine stitching layer. We choose the stitching layer from several possible classes of linear transformations and investigate their performance and properties. The task of matching representations is closely related to notions of similarity. Using this toolset, we also provide a novel viewpoint on the current line of research regarding similarity indices of neural network representations: the perspective of the performance on a task.
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