arXiv subjects

Maxim Sviridenko

Publications and source records attributed to Maxim Sviridenko.

At least 19 recordsLinked to original sources

arXiv2023

Gradient Descent Converges Linearly for Logistic Regression on Separable Data

We show that running gradient descent with variable learning rate guarantees loss $f(x) \leq 1.1 \cdot f(x^*) + ε$ for the logistic regression objective, where the error $ε$ decays exponentially with the number of iterations and polynomially with the magnitude of the entries of an arbitrary fixed solution $x^*$. This is in contrast to the common intuition that the absence of strong convexity precludes linear convergence of first-order methods, and highlights the importance of variable learning rates for gradient descent. We also apply our ideas to sparse logistic regression, where they lead to an exponential improvement of the sparsity-error tradeoff.

cs.LG↗

arXiv2022

Fast Parallel Hypertree Decompositions in Logarithmic Recursion Depth

Modern trends in data collection are bringing current mainstream techniques for database query processing to their limits. Consequently, various novel approaches for efficient query processing are being actively studied. One such approach is based on hypertree decompositions (HDs), which have been shown to carry great potential to process complex queries more efficiently and with stronger theoretical guarantees. However, using HDs for query execution relies on the difficult task of computing decompositions of the query structure, which guides the efficient execution of the query. From theoretical results we know that the performance of purely sequential methods is inherently limited, yet the problem is susceptible to parallelisation. In this paper we propose the first algorithm for computing hypertree decompositions that is well-suited for parallelisation. The proposed algorithm log-k-decomp requires only a logarithmic number of recursion levels and additionally allows for highly parallelised pruning of the search space by restriction to balanced separators. We provide detailed experimental evaluation over the HyperBench benchmark and demonstrate that our approach is highly effective especially for complex queries.

cs.DB↗

arXiv2022

Iterative Hard Thresholding with Adaptive Regularization: Sparser Solutions Without Sacrificing Runtime

We propose a simple modification to the iterative hard thresholding (IHT) algorithm, which recovers asymptotically sparser solutions as a function of the condition number. When aiming to minimize a convex function $f(x)$ with condition number $κ$ subject to $x$ being an $s$-sparse vector, the standard IHT guarantee is a solution with relaxed sparsity $O(sκ^2)$, while our proposed algorithm, regularized IHT, returns a solution with sparsity $O(sκ)$. Our algorithm significantly improves over ARHT which also finds a solution of sparsity $O(sκ)$, as it does not require re-optimization in each iteration (and so is much faster), is deterministic, and does not require knowledge of the optimal solution value $f(x^*)$ or the optimal sparsity level $s$. Our main technical tool is an adaptive regularization framework, in which the algorithm progressively learns the weights of an $\ell_2$ regularization term that will allow convergence to sparser solutions. We also apply this framework to low rank optimization, where we achieve a similar improvement of the best known condition number dependence from $κ^2$ to $κ$.

math.OC↗

arXiv2021

TSI: an Ad Text Strength Indicator using Text-to-CTR and Semantic-Ad-Similarity

Coming up with effective ad text is a time consuming process, and particularly challenging for small businesses with limited advertising experience. When an inexperienced advertiser onboards with a poorly written ad text, the ad platform has the opportunity to detect low performing ad text, and provide improvement suggestions. To realize this opportunity, we propose an ad text strength indicator (TSI) which: (i) predicts the click-through-rate (CTR) for an input ad text, (ii) fetches similar existing ads to create a neighborhood around the input ad, (iii) and compares the predicted CTRs in the neighborhood to declare whether the input ad is strong or weak. In addition, as suggestions for ad text improvement, TSI shows anonymized versions of superior ads (higher predicted CTR) in the neighborhood. For (i), we propose a BERT based text-to-CTR model trained on impressions and clicks associated with an ad text. For (ii), we propose a sentence-BERT based semantic-ad-similarity model trained using weak labels from ad campaign setup data. Offline experiments demonstrate that our BERT based text-to-CTR model achieves a significant lift in CTR prediction AUC for cold start (new) advertisers compared to bag-of-words based baselines. In addition, our semantic-textual-similarity model for similar ads retrieval achieves a precision@1 of 0.93 (for retrieving ads from the same product category); this is significantly higher compared to unsupervised TF-IDF, word2vec, and sentence-BERT baselines. Finally, we share promising online results from advertisers in the Yahoo (Verizon Media) ad platform where a variant of TSI was implemented with sub-second end-to-end latency.

cs.CL↗

arXiv2021

Local Search Algorithms for Rank-Constrained Convex Optimization

We propose greedy and local search algorithms for rank-constrained convex optimization, namely solving $\underset{\mathrm{rank}(A)\leq r^*}{\min}\, R(A)$ given a convex function $R:\mathbb{R}^{m\times n}\rightarrow \mathbb{R}$ and a parameter $r^*$. These algorithms consist of repeating two steps: (a) adding a new rank-1 matrix to $A$ and (b) enforcing the rank constraint on $A$. We refine and improve the theoretical analysis of Shalev-Shwartz et al. (2011), and show that if the rank-restricted condition number of $R$ is $κ$, a solution $A$ with rank $O(r^*\cdot \min\{κ\log \frac{R(\mathbf{0})-R(A^*)}ε, κ^2\})$ and $R(A) \leq R(A^*) + ε$ can be recovered, where $A^*$ is the optimal solution. This significantly generalizes associated results on sparse convex optimization, as well as rank-constrained convex optimization for smooth functions. We then introduce new practical variants of these algorithms that have superior runtime and recover better solutions in practice. We demonstrate the versatility of these methods on a wide range of applications involving matrix completion and robust principal component analysis.

cs.LG↗

arXiv2020

Sparse Convex Optimization via Adaptively Regularized Hard Thresholding

The goal of Sparse Convex Optimization is to optimize a convex function $f$ under a sparsity constraint $s\leq s^*γ$, where $s^*$ is the target number of non-zero entries in a feasible solution (sparsity) and $γ\geq 1$ is an approximation factor. There has been a lot of work to analyze the sparsity guarantees of various algorithms (LASSO, Orthogonal Matching Pursuit (OMP), Iterative Hard Thresholding (IHT)) in terms of the Restricted Condition Number $κ$. The best known algorithms guarantee to find an approximate solution of value $f(x^*)+ε$ with the sparsity bound of $γ= O\left(κ\min\left\{\log \frac{f(x^0)-f(x^*)}ε, κ\right\}\right)$, where $x^*$ is the target solution. We present a new Adaptively Regularized Hard Thresholding (ARHT) algorithm that makes significant progress on this problem by bringing the bound down to $γ=O(κ)$, which has been shown to be tight for a general class of algorithms including LASSO, OMP, and IHT. This is achieved without significant sacrifice in the runtime efficiency compared to the fastest known algorithms. We also provide a new analysis of OMP with Replacement (OMPR) for general $f$, under the condition $s > s^* \frac{κ^2}{4}$, which yields Compressed Sensing bounds under the Restricted Isometry Property (RIP). When compared to other Compressed Sensing approaches, it has the advantage of providing a strong tradeoff between the RIP condition and the solution sparsity, while working for any general function $f$ that meets the RIP condition.

cs.LG↗

arXiv2019

On the computational complexity of the probabilistic label tree algorithms

Label tree-based algorithms are widely used to tackle multi-class and multi-label problems with a large number of labels. We focus on a particular subclass of these algorithms that use probabilistic classifiers in the tree nodes. Examples of such algorithms are hierarchical softmax (HSM), designed for multi-class classification, and probabilistic label trees (PLTs) that generalize HSM to multi-label problems. If the tree structure is given, learning of PLT can be solved with provable regret guaranties [Wydmuch et.al. 2018]. However, to find a tree structure that results in a PLT with a low training and prediction computational costs as well as low statistical error seems to be a very challenging problem, not well-understood yet. In this paper, we address the problem of finding a tree structure that has low computational cost. First, we show that finding a tree with optimal training cost is NP-complete, nevertheless there are some tractable special cases with either perfect approximation or exact solution that can be obtained in linear time in terms of the number of labels $m$. For the general case, we obtain $O(\log m)$ approximation in linear time too. Moreover, we prove an upper bound on the expected prediction cost expressed in terms of the expected training cost. We also show that under additional assumptions the prediction cost of a PLT is $O(\log m)$.

cs.LG↗

arXiv2017

Determining Tournament Payout Structures for Daily Fantasy Sports

With an exploding global market and the recent introduction of online cash prize tournaments, fantasy sports contests are quickly becoming a central part of the social gaming and sports industries. For sports fans and online media companies, fantasy sports contests are an opportunity for large financial gains. However, they present a host of technical challenges that arise from the complexities involved in running a web-scale, prize driven fantasy sports platform. We initiate the study of these challenges by examining one concrete problem in particular: how to algorithmically generate contest payout structures that are 1) economically motivating and appealing to contestants and 2) reasonably structured and succinctly representable. We formalize this problem and present a general two-staged approach for producing satisfying payout structures given constraints on contest size, entry fee, prize bucketing, etc. We then propose and evaluate several potential algorithms for solving the payout problem efficiently, including methods based on dynamic programming, integer programming, and heuristic techniques. Experimental results show that a carefully designed heuristic scales very well, even to contests with over 100,000 prize winners. Our approach extends beyond fantasy sports -- it is suitable for generating engaging payout structures for any contest with a large number of entrants and a large number of prize winners, including other massive online games, poker tournaments, and real-life sports tournaments.

cs.DS↗

arXiv2015

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs. We study the approximability of JRP-D, the version of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, a stronger, computer-assisted lower bound of 1.245, as well as an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness.

cs.DS↗

arXiv2015

A bi-criteria approximation algorithm for $k$ Means

We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $βk > k$ clusters, and must produce a clustering with cost at most $α$ times the to the cost of the optimal set of $k$ clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor. We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee $α(β)$ depending on the number $βk$ of clusters that may be opened. Our gurantee $α(β)$ is always at most $9 + ε$ and improves rapidly with $β$ (for example: $α(2)<2.59$, and $α(3) < 1.4$). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.

cs.DS↗

arXiv2015

An Algorithm for Online K-Means Clustering

This paper shows that one can be competitive with the k-means objective while operating online. In this model, the algorithm receives vectors v_1,...,v_n one by one in an arbitrary order. For each vector the algorithm outputs a cluster identifier before receiving the next one. Our online algorithm generates ~O(k) clusters whose k-means cost is ~O(W*). Here, W* is the optimal k-means cost using k clusters and ~O suppresses poly-logarithmic factors. We also show that, experimentally, it is not much worse than k-means++ while operating in a strictly more constrained computational model.

cs.DS↗

arXiv2015

Greedy Minimization of Weakly Supermodular Set Functions

This paper defines weak-$α$-supermodularity for set functions. Many optimization objectives in machine learning and data mining seek to minimize such functions under cardinality constrains. We prove that such problems benefit from a greedy extension phase. Explicitly, let $S^*$ be the optimal set of cardinality $k$ that minimizes $f$ and let $S_0$ be an initial solution such that $f(S_0)/f(S^*) \le ρ$. Then, a greedy extension $S \supset S_0$ of size $|S| \le |S_0| + \lceil αk \ln(ρ/\varepsilon) \rceil$ yields $f(S)/f(S^*) \le 1+\varepsilon$. As example usages of this framework we give new bicriteria results for $k$-means, sparse regression, and columns subset selection.

cs.DS↗

arXiv2015

Optimization Problems with Diseconomies of Scale via Decoupling

We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as $x^q$, $q\ge 1$, with the amount $x$ of resources used. We define a novel linear programming relaxation for such problems, and then show that the integrality gap of the relaxation is $A_q$, where $A_q$ is the $q$-th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that $$\big \|\sum_{i=1}^n X_i\big\|_{q} \leq C_q \,\big \|\sum_{i=1}^n Y_i\big\|_{q},$$ where $X_i$ are independent nonnegative random variables, $Y_i$ are possibly dependent nonnegative random variable, and each $Y_i$ has the same distribution as $X_i$. The inequality was proved by de la Peña in 1990. De la Peña, Ibragimov, and Sharakhmetov (2003) showed that $C_q\leq 2$ for $q\in (1,2)$ and $C_q\leq A_q^{1/q}$ for $q\geq 2$. We show that the optimal constant is $C_q=A_q^{1/q}$ for any $q\geq 1$. We then prove a more general inequality: For every convex function $φ$, $$\mathbb{E}[φ\Big(\sum_{i=1}^n X_i\Big)]\leq \mathbb{E}[φ\Big(P\sum_{i=1}^n Y_i\Big)],$$ and, for every concave function $ψ$, $$\mathbb{E}[ψ\Big(\sum_{i=1}^n X_i\Big)] \geq \mathbb{E}[ψ\Big(P\sum_{i=1}^n Y_i\Big)],$$ where $P$ is a Poisson random variable with parameter 1 independent of the random variables $Y_i$.

cs.DS↗

arXiv2014

Polynomial-Time Approximation Schemes for Circle and Other Packing Problems

We give an asymptotic approximation scheme (APTAS) for the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height $1+γ$, for some arbitrarily small number $γ> 0$. Our algorithm is polynomial on $\log 1/γ$, and thus $γ$ is part of the problem input. For the special case that $γ$ is constant, we give a (one dimensional) resource augmentation scheme, that is, we obtain a packing into bins of unit width and height $1+γ$ using no more than the number of bins in an optimal packing. Additionally, we obtain an APTAS for the circle strip packing problem, whose goal is to pack a set of circles into a strip of unit width and minimum height. These are the first approximation and resource augmentation schemes for these problems. Our algorithm is based on novel ideas of iteratively separating small and large items, and may be extended to a wide range of packing problems that satisfy certain conditions. These extensions comprise problems with different kinds of items, such as regular polygons, or with bins of different shapes, such as circles and spheres. As an example, we obtain APTAS's for the problems of packing d-dimensional spheres into hypercubes under the $L_p$-norm.

cs.DS↗

arXiv2014

Optimal approximation for submodular and supermodular optimization with bounded curvature

We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a nondecreasing submodular function or minimize a nonincreasing supermodular function in the setting of bounded total curvature $c$. In the case of submodular maximization with curvature $c$, we obtain a $(1-c/e)$-approximation --- the first improvement over the greedy $(1-e^{-c})/c$-approximation of Conforti and Cornuejols from 1984, which holds for a cardinality constraint, as well as recent approaches that hold for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and non-oblivious local search, and allows us to approximately maximize the sum of a nonnegative, nondecreasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show $(1-c)$-approximation for maximization and $1/(1-c)$-approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem.

cs.DS↗

arXiv2014

Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm

We show that for every positive $ε> 0$, unless NP $\subset$ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than $2^{\log^{1-ε} n}$ by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is in fact, $1 - ε$ vs $ε$ hard assuming the Unique Games Conjecture. Then, we present an $O(\sqrt{n})$-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in the state of the art exact algorithms.

cs.CC↗

arXiv2014

Energy Efficient Scheduling and Routing via Randomized Rounding

We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speed-scaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same quality as for the homogeneous environment. By exploiting the result for the preemptive-non-migratory variant, we are able to improve the best known approximation ratio for the single processor non-preemptive problem. Furthermore, we show that our approach allows to obtain a constant-factor approximation algorithm for the power-aware preemptive job shop scheduling problem. Finally, we consider the min-power routing problem where we are given a network modeled by an undirected graph and a set of uniform demands that have to be routed on integral routes from their sources to their destinations so that the energy consumption is minimized. We improve the best known approximation ratio for this problem.

cs.DS↗