PolySketchFormer: Fast Transformers via Sketches for Polynomial Kernels

The quadratic complexity of attention in transformer architectures remains a big bottleneck in scaling up large foundation models for long context. In fact, recent theoretical results show the hardness of approximating the output of softmax attention mechanism in sub-quadratic time assuming Strong Exponential Time Hypothesis. In this paper, we show how to break this theoretical barrier by replacing softmax with a polynomial function and polynomial sketching. In particular we show that sketches for Polynomial Kernel from the randomized numerical linear algebra literature can be used to approximate the polynomial attention which leads to a significantly faster attention mechanism without assuming any sparse structure for the attention matrix that has been done in many previous works. In addition, we propose an efficient block-based algorithm that lets us apply the causal mask to the attention matrix without explicitly realizing the $n \times n$ attention matrix and compute the output of the polynomial attention mechanism in time linear in the context length. The block-based algorithm gives significant speedups over the \emph{cumulative sum} algorithm used by Performer to apply the causal mask to the attention matrix. These observations help us design \emph{PolySketchFormer}, a practical linear-time transformer architecture for language modeling with provable guarantees. We validate our design empirically by training language models with long context lengths. We first show that the eval perplexities of our models are comparable to that of models trained with softmax attention. We then show that for large context lengths our training times are significantly faster than FlashAttention.

Differentially Private Clustering in Data Streams

The streaming model is an abstraction of computing over massive data streams, which is a popular way of dealing with large-scale modern data analysis. In this model, there is a stream of data points, one after the other. A streaming algorithm is only allowed one pass over the data stream, and the goal is to perform some analysis during the stream while using as small space as possible. Clustering problems (such as $k$-means and $k$-median) are fundamental unsupervised machine learning primitives, and streaming clustering algorithms have been extensively studied in the past. However, since data privacy becomes a central concern in many real-world applications, non-private clustering algorithms are not applicable in many scenarios. In this work, we provide the first differentially private streaming algorithms for $k$-means and $k$-median clustering of $d$-dimensional Euclidean data points over a stream with length at most $T$ using $poly(k,d,\log(T))$ space to achieve a {\it constant} multiplicative error and a $poly(k,d,\log(T))$ additive error. In particular, we present a differentially private streaming clustering framework which only requires an offline DP coreset algorithm as a blackbox. By plugging in existing DP coreset results via Ghazi, Kumar, Manurangsi 2020 and Kaplan, Stemmer 2018, we achieve (1) a $(1+\gamma)$-multiplicative approximation with $\tilde{O}_\gamma(poly(k,d,\log(T)))$ space for any $\gamma>0$, and the additive error is $poly(k,d,\log(T))$ or (2) an $O(1)$-multiplicative approximation with $\tilde{O}(k \cdot poly(d,\log(T)))$ space and $poly(k,d,\log(T))$ additive error. In addition, our algorithmic framework is also differentially private under the continual release setting, i.e., the union of outputs of our algorithms at every timestamp is always differentially private.

Measuring Re-identification Risk

Compact user representations (such as embeddings) form the backbone of personalization services. In this work, we present a new theoretical framework to measure re-identification risk in such user representations. Our framework, based on hypothesis testing, formally bounds the probability that an attacker may be able to obtain the identity of a user from their representation. As an application, we show how our framework is general enough to model important real-world applications such as the Chrome's Topics API for interest-based advertising. We complement our theoretical bounds by showing provably good attack algorithms for re-identification that we use to estimate the re-identification risk in the Topics API. We believe this work provides a rigorous and interpretable notion of re-identification risk and a framework to measure it that can be used to inform real-world applications.

Stars: Tera-Scale Graph Building for Clustering and Graph Learning

A fundamental procedure in the analysis of massive datasets is the construction of similarity graphs. Such graphs play a key role for many downstream tasks, including clustering, classification, graph learning, and nearest neighbor search. For these tasks, it is critical to build graphs which are sparse yet still representative of the underlying data. The benefits of sparsity are twofold: firstly, constructing dense graphs is infeasible in practice for large datasets, and secondly, the runtime of downstream tasks is directly influenced by the sparsity of the similarity graph. In this work, we present $\textit{Stars}$: a highly scalable method for building extremely sparse graphs via two-hop spanners, which are graphs where similar points are connected by a path of length at most two. Stars can construct two-hop spanners with significantly fewer similarity comparisons, which are a major bottleneck for learning based models where comparisons are expensive to evaluate. Theoretically, we demonstrate that Stars builds a graph in nearly-linear time, where approximate nearest neighbors are contained within two-hop neighborhoods. In practice, we have deployed Stars for multiple data sets allowing for graph building at the $\textit{Tera-Scale}$, i.e., for graphs with tens of trillions of edges. We evaluate the performance of Stars for clustering and graph learning, and demonstrate 10~1000-fold improvements in pairwise similarity comparisons compared to different baselines, and 2~10-fold improvement in running time without quality loss.

Differentially Private Graph Learning via Sensitivity-Bounded Personalized PageRank

Personalized PageRank (PPR) is a fundamental tool in unsupervised learning of graph representations such as node ranking, labeling, and graph embedding. However, while data privacy is one of the most important recent concerns, existing PPR algorithms are not designed to protect user privacy. PPR is highly sensitive to the input graph edges: the difference of only one edge may cause a big change in the PPR vector, potentially leaking private user data. In this work, we propose an algorithm which outputs an approximate PPR and has provably bounded sensitivity to input edges. In addition, we prove that our algorithm achieves similar accuracy to non-private algorithms when the input graph has large degrees. Our sensitivity-bounded PPR directly implies private algorithms for several tools of graph learning, such as, differentially private (DP) PPR ranking, DP node classification, and DP node embedding. To complement our theoretical analysis, we also empirically verify the practical performances of our algorithms.

Average Case Column Subset Selection for Entrywise $\ell_1$-Norm Loss

We study the column subset selection problem with respect to the entrywise $\ell_1$-norm loss. It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $n^{\Omega(1)}$ number of columns to obtain a $(1+\epsilon)$-approximation to the best entrywise $\ell_1$-norm low rank approximation of an $n \times n$ matrix. Nevertheless, we show that under certain minimal and realistic distributional settings, it is possible to obtain a $(1+\epsilon)$-approximation with a nearly linear running time and poly$(k/\epsilon)+O(k\log n)$ columns. Namely, we show that if the input matrix $A$ has the form $A = B + E$, where $B$ is an arbitrary rank-$k$ matrix, and $E$ is a matrix with i.i.d. entries drawn from any distribution $\mu$ for which the $(1+\gamma)$-th moment exists, for an arbitrarily small constant $\gamma > 0$, then it is possible to obtain a $(1+\epsilon)$-approximate column subset selection to the entrywise $\ell_1$-norm in nearly linear time. Conversely we show that if the first moment does not exist, then it is not possible to obtain a $(1+\epsilon)$-approximate subset selection algorithm even if one chooses any $n^{o(1)}$ columns. This is the first algorithm of any kind for achieving a $(1+\epsilon)$-approximation for entrywise $\ell_1$-norm loss low rank approximation.

Efficient Symmetric Norm Regression via Linear Sketching

We provide efficient algorithms for overconstrained linear regression problems with size $n \times d$ when the loss function is a symmetric norm (a norm invariant under sign-flips and coordinate-permutations). An important class of symmetric norms are Orlicz norms, where for a function $G$ and a vector $y \in \mathbb{R}^n$, the corresponding Orlicz norm $\|y\|_G$ is defined as the unique value $\alpha$ such that $\sum_{i=1}^n G(|y_i|/\alpha) = 1$. When the loss function is an Orlicz norm, our algorithm produces a $(1 + \varepsilon)$-approximate solution for an arbitrarily small constant $\varepsilon > 0$ in input-sparsity time, improving over the previously best-known algorithm which produces a $d \cdot \mathrm{polylog} n$-approximate solution. When the loss function is a general symmetric norm, our algorithm produces a $\sqrt{d} \cdot \mathrm{polylog} n \cdot \mathrm{mmc}(\ell)$-approximate solution in input-sparsity time, where $\mathrm{mmc}(\ell)$ is a quantity related to the symmetric norm under consideration. To the best of our knowledge, this is the first input-sparsity time algorithm with provable guarantees for the general class of symmetric norm regression problem. Our results shed light on resolving the universal sketching problem for linear regression, and the techniques might be of independent interest to numerical linear algebra problems more broadly.

Resisting Adversarial Attacks by $k$-Winners-Take-All

We propose a simple change to the current neural network structure for defending against gradient-based adversarial attacks. Instead of using popular activation functions (such as ReLU), we advocate the use of $k$-Winners-Take-All ($k$-WTA) activation, a $C^0$ discontinuous function that purposely invalidates the neural network model's gradient at densely distributed input data points. Our proposal is theoretically rationalized. We show why the discontinuities in $k$-WTA networks can largely prevent gradient-based search of adversarial examples and why they at the same time remain innocuous to the network training. This understanding is also empirically backed. Even without notoriously expensive adversarial training, the robustness performance of our networks is comparable to conventional ReLU networks optimized by adversarial training. Furthermore, after also optimized through adversarial training, our networks outperform the state-of-the-art methods under white-box attacks on various datasets that we experimented with.