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Carnegie Mellon University

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Abstract:The $\ell_p$ subspace approximation problem is an NP-hard low rank approximation problem that generalizes the median hyperplane problem ($p = 1$), principal component analysis ($p = 2$), and the center hyperplane problem ($p = \infty$). A popular approach to cope with the NP-hardness of this problem is to compute a strong coreset, which is a small weighted subset of the input points which simultaneously approximates the cost of every $k$-dimensional subspace, typically to $(1+\varepsilon)$ relative error for a small constant $\varepsilon$. We obtain the first algorithm for constructing a strong coreset for $\ell_p$ subspace approximation with a nearly optimal dependence on the rank parameter $k$, obtaining a nearly linear bound of $\tilde O(k)\mathrm{poly}(\varepsilon^{-1})$ for $p<2$ and $\tilde O(k^{p/2})\mathrm{poly}(\varepsilon^{-1})$ for $p>2$. Prior constructions either achieved a similar size bound but produced a coreset with a modification of the original points [SW18, FKW21], or produced a coreset of the original points but lost $\mathrm{poly}(k)$ factors in the coreset size [HV20, WY23]. Our techniques also lead to the first nearly optimal online strong coresets for $\ell_p$ subspace approximation with similar bounds as the offline setting, resolving a problem of [WY23]. All prior approaches lose $\mathrm{poly}(k)$ factors in this setting, even when allowed to modify the original points.

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Abstract:Recently, the question of adversarially robust streaming, where the stream is allowed to depend on the randomness of the streaming algorithm, has gained a lot of attention. In this work, we consider a strong white-box adversarial model (Ajtai et al. PODS 2022), in which the adversary has access to all past random coins and the parameters used by the streaming algorithm. We focus on the sparse recovery problem and extend our result to other tasks such as distinct element estimation and low-rank approximation of matrices and tensors. The main drawback of previous work is that it requires a random oracle, which is especially problematic in the streaming model since the amount of randomness is counted in the space complexity of a streaming algorithm. Also, the previous work suffers from large update time. We construct a near-optimal solution for the sparse recovery problem in white-box adversarial streams, based on the subexponentially secure Learning with Errors assumption. Importantly, our solution does not require a random oracle and has a polylogarithmic per item processing time. We also give results in a related white-box adversarially robust distributed model. Our constructions are based on homomorphic encryption schemes satisfying very mild structural properties that are currently satisfied by most known schemes.

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Abstract:Weighted low rank approximation (WLRA) is an important yet computationally challenging primitive with applications ranging from statistical analysis, model compression, and signal processing. To cope with the NP-hardness of this problem, prior work considers heuristics, bicriteria, or fixed parameter tractable algorithms to solve this problem. In this work, we introduce a new relaxed solution to WLRA which outputs a matrix that is not necessarily low rank, but can be stored using very few parameters and gives provable approximation guarantees when the weight matrix has low rank. Our central idea is to use the weight matrix itself to reweight a low rank solution, which gives an extremely simple algorithm with remarkable empirical performance in applications to model compression and on synthetic datasets. Our algorithm also gives nearly optimal communication complexity bounds for a natural distributed problem associated with this problem, for which we show matching communication lower bounds. Together, our communication complexity bounds show that the rank of the weight matrix provably parameterizes the communication complexity of WLRA. We also obtain the first relative error guarantees for feature selection with a weighted objective.

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Abstract:Recent works in dimensionality reduction for regression tasks have introduced the notion of sensitivity, an estimate of the importance of a specific datapoint in a dataset, offering provable guarantees on the quality of the approximation after removing low-sensitivity datapoints via subsampling. However, fast algorithms for approximating $\ell_p$ sensitivities, which we show is equivalent to approximate $\ell_p$ regression, are known for only the $\ell_2$ setting, in which they are termed leverage scores. In this work, we provide efficient algorithms for approximating $\ell_p$ sensitivities and related summary statistics of a given matrix. In particular, for a given $n \times d$ matrix, we compute $\alpha$-approximation to its $\ell_1$ sensitivities at the cost of $O(n/\alpha)$ sensitivity computations. For estimating the total $\ell_p$ sensitivity (i.e. the sum of $\ell_p$ sensitivities), we provide an algorithm based on importance sampling of $\ell_p$ Lewis weights, which computes a constant factor approximation to the total sensitivity at the cost of roughly $O(\sqrt{d})$ sensitivity computations. Furthermore, we estimate the maximum $\ell_1$ sensitivity, up to a $\sqrt{d}$ factor, using $O(d)$ sensitivity computations. We generalize all these results to $\ell_p$ norms for $p > 1$. Lastly, we experimentally show that for a wide class of matrices in real-world datasets, the total sensitivity can be quickly approximated and is significantly smaller than the theoretical prediction, demonstrating that real-world datasets have low intrinsic effective dimensionality.

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Abstract:Sketching algorithms have recently proven to be a powerful approach both for designing low-space streaming algorithms as well as fast polynomial time approximation schemes (PTAS). In this work, we develop new techniques to extend the applicability of sketching-based approaches to the sparse dictionary learning and the Euclidean $k$-means clustering problems. In particular, we initiate the study of the challenging setting where the dictionary/clustering assignment for each of the $n$ input points must be output, which has surprisingly received little attention in prior work. On the fast algorithms front, we obtain a new approach for designing PTAS's for the $k$-means clustering problem, which generalizes to the first PTAS for the sparse dictionary learning problem. On the streaming algorithms front, we obtain new upper bounds and lower bounds for dictionary learning and $k$-means clustering. In particular, given a design matrix $\mathbf A\in\mathbb R^{n\times d}$ in a turnstile stream, we show an $\tilde O(nr/\epsilon^2 + dk/\epsilon)$ space upper bound for $r$-sparse dictionary learning of size $k$, an $\tilde O(n/\epsilon^2 + dk/\epsilon)$ space upper bound for $k$-means clustering, as well as an $\tilde O(n)$ space upper bound for $k$-means clustering on random order row insertion streams with a natural "bounded sensitivity" assumption. On the lower bounds side, we obtain a general $\tilde\Omega(n/\epsilon + dk/\epsilon)$ lower bound for $k$-means clustering, as well as an $\tilde\Omega(n/\epsilon^2)$ lower bound for algorithms which can estimate the cost of a single fixed set of candidate centers.

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Abstract:We present an approximate attention mechanism named HyperAttention to address the computational challenges posed by the growing complexity of long contexts used in Large Language Models (LLMs). Recent work suggests that in the worst-case scenario, quadratic time is necessary unless the entries of the attention matrix are bounded or the matrix has low stable rank. We introduce two parameters which measure: (1) the max column norm in the normalized attention matrix, and (2) the ratio of row norms in the unnormalized attention matrix after detecting and removing large entries. We use these fine-grained parameters to capture the hardness of the problem. Despite previous lower bounds, we are able to achieve a linear time sampling algorithm even when the matrix has unbounded entries or a large stable rank, provided the above parameters are small. HyperAttention features a modular design that easily accommodates integration of other fast low-level implementations, particularly FlashAttention. Empirically, employing Locality Sensitive Hashing (LSH) to identify large entries, HyperAttention outperforms existing methods, giving significant speed improvements compared to state-of-the-art solutions like FlashAttention. We validate the empirical performance of HyperAttention on a variety of different long-context length datasets. For example, HyperAttention makes the inference time of ChatGLM2 50\% faster on 32k context length while perplexity increases from 5.6 to 6.3. On larger context length, e.g., 131k, with causal masking, HyperAttention offers 5-fold speedup on a single attention layer.

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Authors:Hai Pham, Young Jin Kim, Subhabrata Mukherjee, David P. Woodruff, Barnabas Poczos, Hany Hassan Awadalla

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Abstract:Mixture-of-experts (MoE) architecture has been proven a powerful method for diverse tasks in training deep models in many applications. However, current MoE implementations are task agnostic, treating all tokens from different tasks in the same manner. In this work, we instead design a novel method that incorporates task information into MoE models at different granular levels with shared dynamic task-based adapters. Our experiments and analysis show the advantages of our approaches over the dense and canonical MoE models on multi-task multilingual machine translations. With task-specific adapters, our models can additionally generalize to new tasks efficiently.

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Abstract:We consider the randomized communication complexity of the distributed $\ell_p$-regression problem in the coordinator model, for $p\in (0,2]$. In this problem, there is a coordinator and $s$ servers. The $i$-th server receives $A^i\in\{-M, -M+1, \ldots, M\}^{n\times d}$ and $b^i\in\{-M, -M+1, \ldots, M\}^n$ and the coordinator would like to find a $(1+\epsilon)$-approximate solution to $\min_{x\in\mathbb{R}^n} \|(\sum_i A^i)x - (\sum_i b^i)\|_p$. Here $M \leq \mathrm{poly}(nd)$ for convenience. This model, where the data is additively shared across servers, is commonly referred to as the arbitrary partition model. We obtain significantly improved bounds for this problem. For $p = 2$, i.e., least squares regression, we give the first optimal bound of $\tilde{\Theta}(sd^2 + sd/\epsilon)$ bits. For $p \in (1,2)$,we obtain an $\tilde{O}(sd^2/\epsilon + sd/\mathrm{poly}(\epsilon))$ upper bound. Notably, for $d$ sufficiently large, our leading order term only depends linearly on $1/\epsilon$ rather than quadratically. We also show communication lower bounds of $\Omega(sd^2 + sd/\epsilon^2)$ for $p\in (0,1]$ and $\Omega(sd^2 + sd/\epsilon)$ for $p\in (1,2]$. Our bounds considerably improve previous bounds due to (Woodruff et al. COLT, 2013) and (Vempala et al., SODA, 2020).

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Abstract:We consider sketching algorithms which first compress data by multiplication with a random sketch matrix, and then apply the sketch to quickly solve an optimization problem, e.g., low-rank approximation and regression. In the learning-based sketching paradigm proposed by~\cite{indyk2019learning}, the sketch matrix is found by choosing a random sparse matrix, e.g., CountSketch, and then the values of its non-zero entries are updated by running gradient descent on a training data set. Despite the growing body of work on this paradigm, a noticeable omission is that the locations of the non-zero entries of previous algorithms were fixed, and only their values were learned. In this work, we propose the first learning-based algorithms that also optimize the locations of the non-zero entries. Our first proposed algorithm is based on a greedy algorithm. However, one drawback of the greedy algorithm is its slower training time. We fix this issue and propose approaches for learning a sketching matrix for both low-rank approximation and Hessian approximation for second order optimization. The latter is helpful for a range of constrained optimization problems, such as LASSO and matrix estimation with a nuclear norm constraint. Both approaches achieve good accuracy with a fast running time. Moreover, our experiments suggest that our algorithm can still reduce the error significantly even if we only have a very limited number of training matrices.

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Abstract:We introduce efficient $(1+\varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem, where the inputs are a matrix $\mathbf{A}\in\{0,1\}^{n\times d}$, a rank parameter $k>0$, as well as an accuracy parameter $\varepsilon>0$, and the goal is to approximate $\mathbf{A}$ as a product of low-rank factors $\mathbf{U}\in\{0,1\}^{n\times k}$ and $\mathbf{V}\in\{0,1\}^{k\times d}$. Equivalently, we want to find $\mathbf{U}$ and $\mathbf{V}$ that minimize the Frobenius loss $\|\mathbf{U}\mathbf{V} - \mathbf{A}\|_F^2$. Before this work, the state-of-the-art for this problem was the approximation algorithm of Kumar et. al. [ICML 2019], which achieves a $C$-approximation for some constant $C\ge 576$. We give the first $(1+\varepsilon)$-approximation algorithm using running time singly exponential in $k$, where $k$ is typically a small integer. Our techniques generalize to other common variants of the BMF problem, admitting bicriteria $(1+\varepsilon)$-approximation algorithms for $L_p$ loss functions and the setting where matrix operations are performed in $\mathbb{F}_2$. Our approach can be implemented in standard big data models, such as the streaming or distributed models.

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