arXiv subjects

Sahil Singla

Publications and source records attributed to Sahil Singla.

At least 73 records · Page 4Linked to original sources

arXiv2020

Random-Order Models

This chapter introduces the \emph{random-order model} in online algorithms. In this model, the input is chosen by an adversary, then randomly permuted before being presented to the algorithm. This reshuffling often weakens the power of the adversary and allows for improved algorithmic guarantees. We show such improvements for two broad classes of problems: packing problems where we must pick a constrained set of items to maximize total value, and covering problems where we must satisfy given requirements at minimum total cost. We also discuss how random-order model relates to other stochastic models used for non-worst-case competitive analysis.

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arXiv2019

Robust Algorithms for the Secretary Problem

In classical secretary problems, a sequence of $n$ elements arrive in a uniformly random order, and we want to choose a single item, or a set of size $K$. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin's popular $1/e$-secretary algorithm fails with even a single adversarial arrival. We investigate a robust version of the secretary problem. In the Byzantine Secretary model, we have two kinds of elements: green (good) and red (rogue). The values of all elements are chosen by the adversary. The green elements arrive at times uniformly randomly drawn from $[0,1]$. The red elements, however, arrive at adversarially chosen times. Naturally, the algorithm does not see these colors: how well can it solve secretary problems? We give algorithms which get value comparable to the value of the optimal green set minus the largest green item. Specifically, we give an algorithm to pick $K$ elements that gets within $(1-\varepsilon)$ factor of the above benchmark, as long as $K \geq \mathrm{poly}(\varepsilon^{-1} \log n)$. We extend this to the knapsack secretary problem, for large knapsack size $K$. For the single-item case, an analogous benchmark is the value of the second-largest green item. For value-maximization, we give a $\mathrm{poly} \log^* n$-competitive algorithm, using a multi-layered bucketing scheme that adaptively refines our estimates of second-max over time. For probability-maximization, we show the existence of a good randomized algorithm, using the minimax principle.

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arXiv2019

Faster Matroid Intersection

In this paper we consider the classic matroid intersection problem: given two matroids $\M_{1}=(V,\I_{1})$ and $\M_{2}=(V,\I_{2})$ defined over a common ground set $V$, compute a set $S\in\I_{1}\cap\I_{2}$ of largest possible cardinality, denoted by $r$. We consider this problem both in the setting where each $\M_{i}$ is accessed through an independence oracle, i.e. a routine which returns whether or not a set $S\in\I_{i}$ in $\indep$ time, and the setting where each $\M_{i}$ is accessed through a rank oracle, i.e. a routine which returns the size of the largest independent subset of $S$ in $\M_{i}$ in $\rank$ time. In each setting we provide faster exact and approximate algorithms. Given an independence oracle, we provide an exact $O(nr\log r \indep)$ time algorithm. This improves upon the running time of $O(nr^{1.5} \indep)$ due to Cunningham in 1986 and $\tilde{O}(n^{2} \indep+n^{3})$ due to Lee, Sidford, and Wong in 2015. We also provide two algorithms which compute a $(1-ε)$-approximate solution to matroid intersection running in times $\tilde{O}(n^{1.5}/\eps^{1.5} \indep)$ and $\tilde{O}((n^{2}r^{-1}ε^{-2}+r^{1.5}ε^{-4.5}) \indep)$, respectively. These results improve upon the $O(nr/\eps \indep)$-time algorithm of Cunningham as noted recently by Chekuri and Quanrud. Given a rank oracle, we provide algorithms with even better dependence on $n$ and $r$. We provide an $O(n\sqrt{r}\log n \rank)$-time exact algorithm and an $O(nε^{-1}\log n \rank)$-time algorithm which obtains a $(1-\eps)$-approximation to the matroid intersection problem. The former result improves over the $\tilde{O}(nr \rankt+n^{3})$-time algorithm by Lee, Sidford, and Wong. The rank oracle is of particular interest as the matroid intersection problem with this oracle is a special case of the submodular function minimization problem with an evaluation oracle.

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arXiv2019

Approximation Schemes for a Unit-Demand Buyer with Independent Items via Symmetries

We consider a revenue-maximizing seller with $n$ items facing a single buyer. We introduce the notion of symmetric menu complexity of a mechanism, which counts the number of distinct options the buyer may purchase, up to permutations of the items. Our main result is that a mechanism of quasi-polynomial symmetric menu complexity suffices to guarantee a $(1-\varepsilon)$-approximation when the buyer is unit-demand over independent items, even when the value distribution is unbounded, and that this mechanism can be found in quasi-polynomial time. Our key technical result is a polynomial time, (symmetric) menu-complexity-preserving black-box reduction from achieving a $(1-\varepsilon)$-approximation for unbounded valuations that are subadditive over independent items to achieving a $(1-O(\varepsilon))$-approximation when the values are bounded (and still subadditive over independent items). We further apply this reduction to deduce approximation schemes for a suite of valuation classes beyond our main result. Finally, we show that selling separately (which has exponential menu complexity) can be approximated up to a $(1-\varepsilon)$ factor with a menu of efficient-linear $(f(\varepsilon) \cdot n)$ symmetric menu complexity.

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arXiv2019

Improved Truthful Mechanisms for Combinatorial Auctions with Submodular Bidders

A longstanding open problem in Algorithmic Mechanism Design is to design computationally-efficient truthful mechanisms for (approximately) maximizing welfare in combinatorial auctions with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and Schapira [STOC'06] who gave an $O(\log^2{m})$-approximation where $m$ is the number of items. This problem has been studied extensively since, culminating in an $O(\sqrt{\log{m}})$-approximation mechanism by Dobzinski [STOC'16]. We present a computationally-efficient truthful mechanism with approximation ratio that improves upon the state-of-the-art by an exponential factor. In particular, our mechanism achieves an $O((\log\log{m})^3)$-approximation in expectation, uses only $O(n)$ demand queries, and has universal truthfulness guarantee. This settles an open question of Dobzinski on whether $Θ(\sqrt{\log{m}})$ is the best approximation ratio in this setting in negative.

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arXiv2019

Algorithms and Adaptivity Gaps for Stochastic $k$-TSP

Given a metric $(V,d)$ and a $\textsf{root} \in V$, the classic $\textsf{$k$-TSP}$ problem is to find a tour originating at the $\textsf{root}$ of minimum length that visits at least $k$ nodes in $V$. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of $\textsf{$k$-TSP}$. In Stoch-Reward $k$-TSP, originally defined by Ene-Nagarajan-Saket [ENS17], each vertex $v$ in the given metric $(V,d)$ contains a stochastic reward $R_v$. The goal is to adaptively find a tour of minimum expected length that collects at least reward $k$; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an $O(\log k)$-approximation adaptive algorithm for this problem, and left open if there is an $O(1)$-approximation algorithm. We totally resolve their open question and even give an $O(1)$-approximation \emph{non-adaptive} algorithm for this problem. We also introduce and obtain similar results for the Stoch-Cost $k$-TSP problem. In this problem each vertex $v$ has a stochastic cost $C_v$, and the goal is to visit and select at least $k$ vertices to minimize the expected \emph{sum} of tour length and cost of selected vertices. This problem generalizes the Price of Information framework [Singla18] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for our problems, if we truncate the random variables at an ideal threshold and repeat, then their expected values form a good surrogate. Unfortunately, this ideal threshold is adaptive as it depends on how far we are from achieving our target $k$, so we truncate at various different scales and identify a "critical" scale.

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arXiv2019

Certifiably Robust Interpretation in Deep Learning

Deep learning interpretation is essential to explain the reasoning behind model predictions. Understanding the robustness of interpretation methods is important especially in sensitive domains such as medical applications since interpretation results are often used in downstream tasks. Although gradient-based saliency maps are popular methods for deep learning interpretation, recent works show that they can be vulnerable to adversarial attacks. In this paper, we address this problem and provide a certifiable defense method for deep learning interpretation. We show that a sparsified version of the popular SmoothGrad method, which computes the average saliency maps over random perturbations of the input, is certifiably robust against adversarial perturbations. We obtain this result by extending recent bounds for certifiably robust smooth classifiers to the interpretation setting. Experiments on ImageNet samples validate our theory.

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arXiv2019

Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization

Consider a unit interval $[0,1]$ in which $n$ points arrive one-by-one independently and uniformly at random. On arrival of a point, the problem is to immediately and irrevocably color it in $\{+1,-1\}$ while ensuring that every interval $[a,b] \subseteq [0,1]$ is nearly-balanced. We define \emph{discrepancy} as the largest imbalance of any interval during the entire process. If all the arriving points were known upfront then we can color them alternately to achieve a discrepancy of $1$. What is the minimum possible expected discrepancy when we color the points online? We show that the discrepancy of the above problem is sub-polynomial in $n$ and that no algorithm can achieve a constant discrepancy. This is a substantial improvement over the trivial random coloring that only gets an $\widetilde{O}(\sqrt n)$ discrepancy. We then obtain similar results for a natural generalization of this problem to $2$-dimensions where the points arrive uniformly at random in a unit square. This generalization allows us to improve recent results of Benade et al.\cite{BenadeKPP-EC18} for the online envy minimization problem when the arrivals are stochastic.

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arXiv2019

Prepare for the Expected Worst: Algorithms for Reconfigurable Resources Under Uncertainty

In this paper we study how to optimally balance cheap inflexible resources with more expensive, reconfigurable resources despite uncertainty in the input problem. Specifically, we introduce the MinEMax model to study "build versus rent" problems. In our model different scenarios appear independently. Before knowing which scenarios appear, we may build rigid resources that cannot be changed for different scenarios. Once we know which scenarios appear, we are allowed to rent reconfigurable but expensive resources to use across scenarios. Although computing the objective in our model might seem to require enumerating exponentially-many possibilities, we show it is well-estimated by a surrogate objective which is representable by a polynomial-size LP. In this surrogate objective we pay for each scenario only to the extent that it exceeds a certain threshold. Using this objective we design algorithms that approximately-optimally balance inflexible and reconfigurable resources for several NP-hard covering problems. For example, we study minimum spanning and Steiner trees, minimum cuts and facility location variants. Up to constants our approximation guarantees match those of previous algorithms for the previously-studied demand-robust and stochastic two-stage models. Lastly, we demonstrate that our problem is sufficiently general to smoothly interpolate between previous demand-robust and stochastic two-stage problems.

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arXiv2019

Understanding Impacts of High-Order Loss Approximations and Features in Deep Learning Interpretation

Current methods to interpret deep learning models by generating saliency maps generally rely on two key assumptions. First, they use first-order approximations of the loss function neglecting higher-order terms such as the loss curvatures. Second, they evaluate each feature's importance in isolation, ignoring their inter-dependencies. In this work, we study the effect of relaxing these two assumptions. First, by characterizing a closed-form formula for the Hessian matrix of a deep ReLU network, we prove that, for a classification problem with a large number of classes, if an input has a high confidence classification score, the inclusion of the Hessian term has small impacts in the final solution. We prove this result by showing that in this case the Hessian matrix is approximately of rank one and its leading eigenvector is almost parallel to the gradient of the loss function. Our empirical experiments on ImageNet samples are consistent with our theory. This result can have implications in other related problems such as adversarial examples as well. Second, we compute the importance of group-features in deep learning interpretation by introducing a sparsity regularization term. We use the $L_0-L_1$ relaxation technique along with the proximal gradient descent to have an efficient computation of group feature importance scores. Our empirical results indicate that considering group features can improve deep learning interpretation significantly.

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arXiv2019

Non-clairvoyant Precedence Constrained Scheduling

We consider the online problem of scheduling jobs on identical machines, where jobs have precedence constraints. We are interested in the demanding setting where the jobs sizes are not known up-front, but are revealed only upon completion (the non-clairvoyant setting). Such precedence-constrained scheduling problems routinely arise in map-reduce and large-scale optimization. In this paper, we make progress on this problem. For the objective of total weighted completion time, we give a constant-competitive algorithm. And for total weighted flow-time, we give an $O(1/ε^2)$-competitive algorithm under $(1+ε)$-speed augmentation and a natural ``no-surprises'' assumption on release dates of jobs (which we show is necessary in this context). Our algorithm proceeds by assigning {\em virtual rates} to all the waiting jobs, including the ones which are dependent on other uncompleted jobs, and then use these virtual rates to decide on the actual rates of minimal jobs (i.e., jobs which do not have dependencies and hence are eligible to run). Interestingly, the virtual rates are obtained by allocating time in a fair manner, using a Eisenberg-Gale-type convex program (which we can also solve optimally using a primal-dual scheme). The optimality condition of this convex program allows us to show dual-fitting proofs more easily, without having to guess and hand-craft the duals. We feel that this idea of using fair virtual rates should have broader applicability in scheduling problems.

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arXiv2019

The Markovian Price of Information

Suppose there are $n$ Markov chains and we need to pay a per-step \emph{price} to advance them. The "destination" states of the Markov chains contain rewards; however, we can only get rewards for a subset of them that satisfy a combinatorial constraint, e.g., at most $k$ of them, or they are acyclic in an underlying graph. What strategy should we choose to advance the Markov chains if our goal is to maximize the total reward \emph{minus} the total price that we pay? In this paper we introduce a Markovian price of information model to capture settings such as the above, where the input parameters of a combinatorial optimization problem are given via Markov chains. We design optimal/approximation algorithms that jointly optimize the value of the combinatorial problem and the total paid price. We also study \emph{robustness} of our algorithms to the distribution parameters and how to handle the \emph{commitment} constraint. Our work brings together two classical lines of investigation: getting optimal strategies for Markovian multi-armed bandits, and getting exact and approximation algorithms for discrete optimization problems using combinatorial as well as linear-programming relaxation ideas.

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arXiv2019

(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge $e$ exists, we need to perform an expensive test, in which case the edge $e$ appears independently with a \emph{known} probability $p_e$. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is \emph{adaptive}---decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the \emph{adaptivity gap} is small, i.e., the best \emph{non-adaptive} strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most $2$ and provide a matching lower bound. For a weighted rank function of a $k$-extendible system (a generalization of intersection of $k$ matroids), we show the adaptivity gap is between $O(k\log k)$ and $k$. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al.

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arXiv2019

Robustness Certificates Against Adversarial Examples for ReLU Networks

While neural networks have achieved high performance in different learning tasks, their accuracy drops significantly in the presence of small adversarial perturbations to inputs. Defenses based on regularization and adversarial training are often followed by new attacks to defeat them. In this paper, we propose attack-agnostic robustness certificates for a multi-label classification problem using a deep ReLU network. Although computing the exact distance of a given input sample to the classification decision boundary requires solving a non-convex optimization, we characterize two lower bounds for such distances, namely the simplex certificate and the decision boundary certificate. These robustness certificates leverage the piece-wise linear structure of ReLU networks and use the fact that in a polyhedron around a given sample, the prediction function is linear. In particular, the proposed simplex certificate has a closed-form, is differentiable and is an order of magnitude faster to compute than the existing methods even for deep networks. In addition to theoretical bounds, we provide numerical results for our certificates over MNIST and compare them with some existing upper bounds.

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arXiv2018

Optimal Online Contention Resolution Schemes via Ex-Ante Prophet Inequalities

Online contention resolution schemes (OCRSs) were proposed by Feldman, Svensson, and Zenklusen as a generic technique to round a fractional solution in the matroid polytope in an online fashion. It has found applications in several stochastic combinatorial problems where there is a commitment constraint: on seeing the value of a stochastic element, the algorithm has to immediately and irrevocably decide whether to select it while always maintaining an independent set in the matroid. Although OCRSs immediately lead to prophet inequalities, these prophet inequalities are not optimal. Can we instead use prophet inequalities to design optimal OCRSs? We design the first optimal $1/2$-OCRS for matroids by reducing the problem to designing a matroid prophet inequality where we compare to the stronger benchmark of an ex-ante relaxation. We also introduce and design optimal $(1-1/e)$-random order CRSs for matroids, which are similar to OCRSs but the arrival is chosen uniformly at random.

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arXiv2018

Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

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arXiv2018

Online Matroid Intersection: Beating Half for Random Arrival

For two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ defined on the same ground set $E$, the online matroid intersection problem is to design an algorithm that constructs a large common independent set in an online fashion. The algorithm is presented with the ground set elements one-by-one in a uniformly random order. At each step, the algorithm must irrevocably decide whether to pick the element, while always maintaining a common independent set. While the natural greedy algorithm---pick an element whenever possible---is half competitive, nothing better was previously known; even for the special case of online bipartite matching in the edge arrival model. We present the first randomized online algorithm that has a $\frac12 + δ$ competitive ratio in expectation, where $δ>0$ is a constant. The expectation is over the random order and the coin tosses of the algorithm. As a corollary, we also obtain the first linear time algorithm that beats half competitiveness for offline matroid intersection.

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arXiv2017

The Price of Information in Combinatorial Optimization

Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility? A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of $n$ independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.

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