arXiv subjects

Lior Kamma

Publications and source records attributed to Lior Kamma.

13 recordsLinked to original sources

arXiv2024

A New Approach for Approximating Directed Rooted Networks

We consider the k-outconnected directed Steiner tree problem (k-DST). Given a directed edge-weighted graph $G=(V,E,w)$, where $V=\{r\}\cup S \cup T$, and an integer $k$, the goal is to find a minimum cost subgraph of $G$ in which there are $k$ edge-disjoint $rt$-paths for every terminal $t\in T$. The problem is know to be NP-hard. Furthermore, the question on whether a polynomial time, subpolynomial approximation algorithm exists for $k$-DST was answered negatively by Grandoni et al. (2018), by proving an approximation hardness of $Ω(|T|/\log |T|)$ under $NP\neq ZPP$. Inspired by modern day applications, we focus on developing efficient algorithms for $k$-DST in graphs where terminals have out-degree $0$, and furthermore constitute the vast majority in the graph. We provide the first approximation algorithm for $k$-DST on such graphs, in which the approximation ratio depends (primarily) on the size of $S$. We present a randomized algorithm that finds a solution of weight at most $\mathcal O(k|S|\log |T|)$ times the optimal weight, and with high probability runs in polynomial time.

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arXiv2021

Compression Implies Generalization

Explaining the surprising generalization performance of deep neural networks is an active and important line of research in theoretical machine learning. Influential work by Arora et al. (ICML'18) showed that, noise stability properties of deep nets occurring in practice can be used to provably compress model representations. They then argued that the small representations of compressed networks imply good generalization performance albeit only of the compressed nets. Extending their compression framework to yield generalization bounds for the original uncompressed networks remains elusive. Our main contribution is the establishment of a compression-based framework for proving generalization bounds. The framework is simple and powerful enough to extend the generalization bounds by Arora et al. to also hold for the original network. To demonstrate the flexibility of the framework, we also show that it allows us to give simple proofs of the strongest known generalization bounds for other popular machine learning models, namely Support Vector Machines and Boosting.

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arXiv2020

Margins are Insufficient for Explaining Gradient Boosting

Boosting is one of the most successful ideas in machine learning, achieving great practical performance with little fine-tuning. The success of boosted classifiers is most often attributed to improvements in margins. The focus on margin explanations was pioneered in the seminal work by Schapire et al. (1998) and has culminated in the $k$'th margin generalization bound by Gao and Zhou (2013), which was recently proved to be near-tight for some data distributions (Gronlund et al. 2019). In this work, we first demonstrate that the $k$'th margin bound is inadequate in explaining the performance of state-of-the-art gradient boosters. We then explain the short comings of the $k$'th margin bound and prove a stronger and more refined margin-based generalization bound for boosted classifiers that indeed succeeds in explaining the performance of modern gradient boosters. Finally, we improve upon the recent generalization lower bound by Grønlund et al. (2019).

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arXiv2020

Near-Tight Margin-Based Generalization Bounds for Support Vector Machines

Support Vector Machines (SVMs) are among the most fundamental tools for binary classification. In its simplest formulation, an SVM produces a hyperplane separating two classes of data using the largest possible margin to the data. The focus on maximizing the margin has been well motivated through numerous generalization bounds. In this paper, we revisit and improve the classic generalization bounds in terms of margins. Furthermore, we complement our new generalization bound by a nearly matching lower bound, thus almost settling the generalization performance of SVMs in terms of margins.

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arXiv2020

Margin-Based Generalization Lower Bounds for Boosted Classifiers

Boosting is one of the most successful ideas in machine learning. The most well-accepted explanations for the low generalization error of boosting algorithms such as AdaBoost stem from margin theory. The study of margins in the context of boosting algorithms was initiated by Schapire, Freund, Bartlett and Lee (1998) and has inspired numerous boosting algorithms and generalization bounds. To date, the strongest known generalization (upper bound) is the $k$th margin bound of Gao and Zhou (2013). Despite the numerous generalization upper bounds that have been proved over the last two decades, nothing is known about the tightness of these bounds. In this paper, we give the first margin-based lower bounds on the generalization error of boosted classifiers. Our lower bounds nearly match the $k$th margin bound and thus almost settle the generalization performance of boosted classifiers in terms of margins.

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arXiv2019

Lower Bounds for Multiplication via Network Coding

Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by Fürer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg n \cdot 4^{\lg^*n})$, where $\lg^*n$ is the very slowly growing iterated logarithm. In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size $Ω(n \lg n)$, thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant's conjectures.

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arXiv2019

Nearly Optimal Time Bounds for kPath in Hypergraphs

We give almost tight conditional lower bounds on the running time of the kHyperPath problem. Given an $r$-uniform hypergraph for some integer $r$, kHyperPath seeks a tight path of length $k$. That is, a sequence of $k$ nodes such that every consecutive $r$ of them constitute a hyperedge in the graph. This problem is a natural generalization of the extensively-studied kPath problem in graphs. We show that solving kHyperPath in time $O^*(2^{(1-γ)k})$ where $γ>0$ is independent of $r$ is probably impossible. Specifically, it implies that Set Cover on $n$ elements can be solved in time $O^*(2^{(1 - δ)n})$ for some $δ>0$. The only known lower bound for the kPath problem is $2^{Ω(k)} poly(n)$ where $n$ is the number of nodes assuming the Exponential Time Hypothesis (ETH), and finding any conditional lower bound with an explicit constant in the exponent has been an important open problem. We complement our lower bound with an almost tight upper bound. Formally, for every integer $r\geq 3$ we give algorithms that solve kHyperPath and kHyperCycle on $r$-uniform hypergraphs with $n$ nodes and $m$ edges in time $2^k m \cdot poly(n)$ and $2^k m^2 poly(n)$ respectively, and that is even for the directed version of these problems. To the best of our knowledge, this is the first algorithm for kHyperPath. The fastest algorithms known for kPath run in time $2^k poly(n)$ for directed graphs (Williams, 2009), and in time $1.66^k poly(n)$ for undirected graphs (Björklund \etal, 2014).

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arXiv2018

Batch Sparse Recovery, or How to Leverage the Average Sparsity

We introduce a \emph{batch} version of sparse recovery, where the goal is to report a sequence of vectors $A_1',\ldots,A_m' \in \mathbb{R}^n$ that estimate unknown signals $A_1,\ldots,A_m \in \mathbb{R}^n$ using a few linear measurements, each involving exactly one signal vector, under an assumption of \emph{average sparsity}. More precisely, we want to have \newline $(1) \;\;\; \sum_{j \in [m]}{\|A_j- A_j'\|_p^p} \le C \cdot \min \Big\{ \sum_{j \in [m]}{\|A_j - A_j^*\|_p^p} \Big\}$ for predetermined constants $C \ge 1$ and $p$, where the minimum is over all $A_1^*,\ldots,A_m^*\in\mathbb{R}^n$ that are $k$-sparse on average. We assume $k$ is given as input, and ask for the minimal number of measurements required to satisfy $(1)$. The special case $m=1$ is known as stable sparse recovery and has been studied extensively. We resolve the question for $p =1$ up to polylogarithmic factors, by presenting a randomized adaptive scheme that performs $\tilde{O}(km)$ measurements and with high probability has output satisfying $(1)$, for arbitrarily small $C > 1$. Finally, we show that adaptivity is necessary for every non-trivial scheme.

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arXiv2018

Fully Understanding the Hashing Trick

Feature hashing, also known as {\em the hashing trick}, introduced by Weinberger et al. (2009), is one of the key techniques used in scaling-up machine learning algorithms. Loosely speaking, feature hashing uses a random sparse projection matrix $A : \mathbb{R}^n \to \mathbb{R}^m$ (where $m \ll n$) in order to reduce the dimension of the data from $n$ to $m$ while approximately preserving the Euclidean norm. Every column of $A$ contains exactly one non-zero entry, equals to either $-1$ or $1$. Weinberger et al. showed tail bounds on $\|Ax\|_2^2$. Specifically they showed that for every $\varepsilon, δ$, if $\|x\|_{\infty} / \|x\|_2$ is sufficiently small, and $m$ is sufficiently large, then $$\Pr[ \; | \;\|Ax\|_2^2 - \|x\|_2^2\; | < \varepsilon \|x\|_2^2 \;] \ge 1 - δ\;.$$ These bounds were later extended by Dasgupta \etal (2010) and most recently refined by Dahlgaard et al. (2017), however, the true nature of the performance of this key technique, and specifically the correct tradeoff between the pivotal parameters $\|x\|_{\infty} / \|x\|_2, m, \varepsilon, δ$ remained an open question. We settle this question by giving tight asymptotic bounds on the exact tradeoff between the central parameters, thus providing a complete understanding of the performance of feature hashing. We complement the asymptotic bound with empirical data, which shows that the constants "hiding" in the asymptotic notation are, in fact, very close to $1$, thus further illustrating the tightness of the presented bounds in practice.

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arXiv2017

Tight Bounds for Gomory-Hu-like Cut Counting

By a classical result of Gomory and Hu (1961), in every edge-weighted graph $G=(V,E,w)$, the minimum $st$-cut values, when ranging over all $s,t\in V$, take at most $|V|-1$ distinct values. That is, these $\binom{|V|}{2}$ instances exhibit redundancy factor $Ω(|V|)$. They further showed how to construct from $G$ a tree $(V,E',w')$ that stores all minimum $st$-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum $st$-cut problem. 1. Group-Cut: Consider the minimum $(A,B)$-cut, ranging over all subsets $A,B\subseteq V$ of given sizes $|A|=α$ and $|B|=β$. The redundancy factor is $Ω_{α,β}(|V|)$. 2. Multiway-Cut: Consider the minimum cut separating every two vertices of $S\subseteq V$, ranging over all subsets of a given size $|S|=k$. The redundancy factor is $Ω_{k}(|V|)$. 3. Multicut: Consider the minimum cut separating every demand-pair in $D\subseteq V\times V$, ranging over collections of $|D|=k$ demand pairs. The redundancy factor is $Ω_{k}(|V|^k)$. This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.

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arXiv2017

Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and Williams [SODA'17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in $\tilde{O}(n^{7/4})$ time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional $\tilde{O}(n^{7/4})$ bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time $\tilde{O}(n^{3/2})$ storing just $\tilde{O}(n^{3/2})$ bits on the side. We then complement our data structure with a lower bound showing that any data structure storing $r$ bits on the side, with $n < r < n^2$ must have query time $t$ satisfying $t r = \tildeΩ(n^3)$. For $r \leq n$, any data structure must have $t = \tildeΩ(n^2)$. Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over $\mathbb{F}_2$.

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arXiv2016

Metric Decompositions of Path-Separable Graphs

A prominent tool in many problems involving metric spaces is a notion of randomized low-diameter decomposition. Loosely speaking, $β$-decomposition refers to a probability distribution over partitions of the metric into sets of low diameter, such that nearby points (parameterized by $β>0$) are likely to be "clustered" together. Applying this notion to the shortest-path metric in edge-weighted graphs, it is known that $n$-vertex graphs admit an $O(\ln n)$-padded decomposition (Bartal, 1996), and that excluded-minor graphs admit $O(1)$-padded decomposition (Klein, Plotkin and Rao 1993, Fakcharoenphol and Talwar 2003, Abraham et al. 2014). We design decompositions to the family of $p$-path-separable graphs, which was defined by Abraham and Gavoille (2006). and refers to graphs that admit vertex-separators consisting of at most $p$ shortest paths in the graph. Our main result is that every $p$-path-separable $n$-vertex graph admits an $O(\ln (p \ln n))$-decomposition, which refines the $O(\ln n)$ bound for general graphs, and provides new bounds for families like bounded-treewidth graphs. Technically, our clustering process differs from previous ones by working in (the shortest-path metric of) carefully chosen subgraphs.

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arXiv2015

Cutting corners cheaply, or how to remove Steiner points

Our main result is that the Steiner Point Removal (SPR) problem can always be solved with polylogarithmic distortion, which answers in the affirmative a question posed by Chan, Xia, Konjevod, and Richa (2006). Specifically, we prove that for every edge-weighted graph $G = (V,E,w)$ and a subset of terminals $T \subseteq V$, there is a graph $G'=(T,E',w')$ that is isomorphic to a minor of $G$, such that for every two terminals $u,v\in T$, the shortest-path distances between them in $G$ and in $G'$ satisfy $d_{G,w}(u,v) \le d_{G',w'}(u,v) \le O(\log^5|T|) \cdot d_{G,w}(u,v)$. Our existence proof actually gives a randomized polynomial-time algorithm. Our proof features a new variant of metric decomposition. It is well-known that every $n$-point metric space $(X,d)$ admits a $β$-separating decomposition for $β=O(\log n)$, which roughly means for every desired diameter bound $Δ>0$ there is a randomized partitioning of $X$, which satisfies the following separation requirement: for every $x,y \in X$, the probability they lie in different clusters of the partition is at most $β\,d(x,y)/Δ$. We introduce an additional requirement, which is the following tail bound: for every shortest-path $P$ of length $d(P) \leq Δ/β$, the number of clusters of the partition that meet the path $P$, denoted $Z_P$, satisfies $\Pr[Z_P > t] \le 2e^{-Ω(t)}$ for all $t>0$.

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