arXiv subjects

David Durfee

Publications and source records attributed to David Durfee.

At least 19 recordsLinked to original sources

arXiv2025

Unifying Laplace Mechanism with Instance Optimality in Differential Privacy

We adapt the canonical Laplace mechanism, widely used in differentially private data analysis, to achieve near instance optimality with respect to the hardness of the underlying dataset. In particular, we construct a piecewise Laplace distribution whereby we defy traditional assumptions and show that Laplace noise can in fact be drawn proportional to the local sensitivity when done in a piecewise manner. While it may initially seem counterintuitive that this satisfies (pure) differential privacy and can be sampled, we provide both through a simple connection to the exponential mechanism and inverse sensitivity along with the fact that the Laplace distribution is a two-sided exponential distribution. As a result, we prove that in the continuous setting our \textit{piecewise Laplace mechanism} strictly dominates the inverse sensitivity mechanism, which was previously shown to both be nearly instance optimal and uniformly outperform the smooth sensitivity framework. Furthermore, in the worst-case where all local sensitivities equal the global sensitivity, our method simply reduces to a Laplace mechanism. We also complement this with an approximate local sensitivity variant to potentially ease the computational cost, which can also extend to higher dimensions.

cs.DS↗

arXiv2023

Instance-Specific Asymmetric Sensitivity in Differential Privacy

We provide a new algorithmic framework for differentially private estimation of general functions that adapts to the hardness of the underlying dataset. We build upon previous work that gives a paradigm for selecting an output through the exponential mechanism based upon closeness of the inverse to the underlying dataset, termed the inverse sensitivity mechanism. Our framework will slightly modify the closeness metric and instead give a simple and efficient application of the sparse vector technique. While the inverse sensitivity mechanism was shown to be instance optimal, it was only with respect to a class of unbiased mechanisms such that the most likely outcome matches the underlying data. We break this assumption in order to more naturally navigate the bias-variance tradeoff, which will also critically allow for extending our method to unbounded data. In consideration of this tradeoff, we provide strong intuition and empirical validation that our technique will be particularly effective when the distances to the underlying dataset are asymmetric. This asymmetry is inherent to a range of important problems including fundamental statistics such as variance, as well as commonly used machine learning performance metrics for both classification and regression tasks. We efficiently instantiate our method in $O(n)$ time for these problems and empirically show that our techniques will give substantially improved differentially private estimations.

cs.DS↗

arXiv2023

Unbounded Differentially Private Quantile and Maximum Estimation

In this work we consider the problem of differentially private computation of quantiles for the data, especially the highest quantiles such as maximum, but with an unbounded range for the dataset. We show that this can be done efficiently through a simple invocation of $\texttt{AboveThreshold}$, a subroutine that is iteratively called in the fundamental Sparse Vector Technique, even when there is no upper bound on the data. In particular, we show that this procedure can give more accurate and robust estimates on the highest quantiles with applications towards clipping that is essential for differentially private sum and mean estimation. In addition, we show how two invocations can handle the fully unbounded data setting. Within our study, we show that an improved analysis of $\texttt{AboveThreshold}$ can improve the privacy guarantees for the widely used Sparse Vector Technique that is of independent interest. We give a more general characterization of privacy loss for $\texttt{AboveThreshold}$ which we immediately apply to our method for improved privacy guarantees. Our algorithm only requires one $O(n)$ pass through the data, which can be unsorted, and each subsequent query takes $O(1)$ time. We empirically compare our unbounded algorithm with the state-of-the-art algorithms in the bounded setting. For inner quantiles, we find that our method often performs better on non-synthetic datasets. For the maximal quantiles, which we apply to differentially private sum computation, we find that our method performs significantly better.

cs.DS↗

arXiv2023

mSAM: Micro-Batch-Averaged Sharpness-Aware Minimization

Modern deep learning models are over-parameterized, where different optima can result in widely varying generalization performance. The Sharpness-Aware Minimization (SAM) technique modifies the fundamental loss function that steers gradient descent methods toward flatter minima, which are believed to exhibit enhanced generalization prowess. Our study delves into a specific variant of SAM known as micro-batch SAM (mSAM). This variation involves aggregating updates derived from adversarial perturbations across multiple shards (micro-batches) of a mini-batch during training. We extend a recently developed and well-studied general framework for flatness analysis to theoretically show that SAM achieves flatter minima than SGD, and mSAM achieves even flatter minima than SAM. We provide a thorough empirical evaluation of various image classification and natural language processing tasks to substantiate this theoretical advancement. We also show that contrary to previous work, mSAM can be implemented in a flexible and parallelizable manner without significantly increasing computational costs. Our implementation of mSAM yields superior generalization performance across a wide range of tasks compared to SAM, further supporting our theoretical framework.

stat.ML↗

arXiv2022

Improved Deep Neural Network Generalization Using m-Sharpness-Aware Minimization

Modern deep learning models are over-parameterized, where the optimization setup strongly affects the generalization performance. A key element of reliable optimization for these systems is the modification of the loss function. Sharpness-Aware Minimization (SAM) modifies the underlying loss function to guide descent methods towards flatter minima, which arguably have better generalization abilities. In this paper, we focus on a variant of SAM known as mSAM, which, during training, averages the updates generated by adversarial perturbations across several disjoint shards of a mini-batch. Recent work suggests that mSAM can outperform SAM in terms of test accuracy. However, a comprehensive empirical study of mSAM is missing from the literature -- previous results have mostly been limited to specific architectures and datasets. To that end, this paper presents a thorough empirical evaluation of mSAM on various tasks and datasets. We provide a flexible implementation of mSAM and compare the generalization performance of mSAM to the performance of SAM and vanilla training on different image classification and natural language processing tasks. We also conduct careful experiments to understand the computational cost of training with mSAM, its sensitivity to hyperparameters and its correlation with the flatness of the loss landscape. Our analysis reveals that mSAM yields superior generalization performance and flatter minima, compared to SAM, across a wide range of tasks without significantly increasing computational costs.

cs.LG↗

arXiv2022

Heterogeneous Calibration: A post-hoc model-agnostic framework for improved generalization

We introduce the notion of heterogeneous calibration that applies a post-hoc model-agnostic transformation to model outputs for improving AUC performance on binary classification tasks. We consider overconfident models, whose performance is significantly better on training vs test data and give intuition onto why they might under-utilize moderately effective simple patterns in the data. We refer to these simple patterns as heterogeneous partitions of the feature space and show theoretically that perfectly calibrating each partition separately optimizes AUC. This gives a general paradigm of heterogeneous calibration as a post-hoc procedure by which heterogeneous partitions of the feature space are identified through tree-based algorithms and post-hoc calibration techniques are applied to each partition to improve AUC. While the theoretical optimality of this framework holds for any model, we focus on deep neural networks (DNNs) and test the simplest instantiation of this paradigm on a variety of open-source datasets. Experiments demonstrate the effectiveness of this framework and the future potential for applying higher-performing partitioning schemes along with more effective calibration techniques.

stat.ML↗

arXiv2020

LinkedIn's Audience Engagements API: A Privacy Preserving Data Analytics System at Scale

We present a privacy system that leverages differential privacy to protect LinkedIn members' data while also providing audience engagement insights to enable marketing analytics related applications. We detail the differentially private algorithms and other privacy safeguards used to provide results that can be used with existing real-time data analytics platforms, specifically with the open sourced Pinot system. Our privacy system provides user-level privacy guarantees. As part of our privacy system, we include a budget management service that enforces a strict differential privacy budget on the returned results to the analyst. This budget management service brings together the latest research in differential privacy into a product to maintain utility given a fixed differential privacy budget.

cs.CR↗

arXiv2019

Optimal Differential Privacy Composition for Exponential Mechanisms and the Cost of Adaptivity

Composition is one of the most important properties of differential privacy (DP), as it allows algorithm designers to build complex private algorithms from DP primitives. We consider precise composition bounds of the overall privacy loss for exponential mechanisms, one of the fundamental classes of mechanisms in DP. We give explicit formulations of the optimal privacy loss for both the adaptive and non-adaptive settings. For the non-adaptive setting in which each mechanism has the same privacy parameter, we give an efficiently computable formulation of the optimal privacy loss. Furthermore, we show that there is a difference in the privacy loss when the exponential mechanism is chosen adaptively versus non-adaptively. To our knowledge, it was previously unknown whether such a gap existed for any DP mechanisms with fixed privacy parameters, and we demonstrate the gap for a widely used class of mechanism in a natural setting. We then improve upon the best previously known upper bounds for adaptive composition of exponential mechanisms with efficiently computable formulations and show the improvement.

cs.CR↗

arXiv2019

Parallel Batch-Dynamic Graphs: Algorithms and Lower Bounds

In this paper we study the problem of dynamically maintaining graph properties under batches of edge insertions and deletions in the massively parallel model of computation. In this setting, the graph is stored on a number of machines, each having space strongly sublinear with respect to the number of vertices, that is, $n^\epsilon$ for some constant $0 < \epsilon < 1$. Our goal is to handle batches of updates and queries where the data for each batch fits onto one machine in constant rounds of parallel computation, as well as to reduce the total communication between the machines. This objective corresponds to the gradual buildup of databases over time, while the goal of obtaining constant rounds of communication for problems in the static setting has been elusive for problems as simple as undirected graph connectivity. We give an algorithm for dynamic graph connectivity in this setting with constant communication rounds and communication cost almost linear in terms of the batch size. Our techniques combine a new graph contraction technique, an independent random sample extractor from correlated samples, as well as distributed data structures supporting parallel updates and queries in batches. We also illustrate the power of dynamic algorithms in the MPC model by showing that the batched version of the adaptive connectivity problem is $\mathsf{P}$-complete in the centralized setting, but sub-linear sized batches can be handled in a constant number of rounds. Due to the wide applicability of our approaches, we believe it represents a practically-motivated workaround to the current difficulties in designing more efficient massively parallel static graph algorithms.

cs.DS↗

arXiv2019

Fully Dynamic Spectral Vertex Sparsifiers and Applications

We study \emph{dynamic} algorithms for maintaining spectral vertex sparsifiers of graphs with respect to a set of terminals $T$ of our choice. Such objects preserve pairwise resistances, solutions to systems of linear equations, and energy of electrical flows between the terminals in $T$. We give a data structure that supports insertions and deletions of edges, and terminal additions, all in sublinear time. Our result is then applied to the following problems. (1) A data structure for maintaining solutions to Laplacian systems $\mathbf{L} \mathbf{x} = \mathbf{b}$, where $\mathbf{L}$ is the Laplacian matrix and $\mathbf{b}$ is a demand vector. For a bounded degree, unweighted graph, we support modifications to both $\mathbf{L}$ and $\mathbf{b}$ while providing access to $\epsilon$-approximations to the energy of routing an electrical flow with demand $\mathbf{b}$, as well as query access to entries of a vector $\tilde{\mathbf{x}}$ such that $\left\lVert \tilde{\mathbf{x}}-\mathbf{L}^{\dagger} \mathbf{b} \right\rVert_{\mathbf{L}} \leq \epsilon \left\lVert \mathbf{L}^{\dagger} \mathbf{b} \right\rVert_{\mathbf{L}}$ in $\tilde{O}(n^{11/12}\epsilon^{-5})$ expected amortized update and query time. (2) A data structure for maintaining All-Pairs Effective Resistance. For an intermixed sequence of edge insertions, deletions, and resistance queries, our data structure returns $(1 \pm \epsilon)$-approximation to all the resistance queries against an oblivious adversary with high probability. Its expected amortized update and query times are $\tilde{O}(\min(m^{3/4},n^{5/6} \epsilon^{-2}) \epsilon^{-4})$ on an unweighted graph, and $\tilde{O}(n^{5/6}\epsilon^{-6})$ on weighted graphs. These results represent the first data structures for maintaining key primitives from the Laplacian paradigm for graph algorithms in sublinear time without assumptions on the underlying graph topologies.

cs.DS↗

arXiv2019

Practical Differentially Private Top-$k$ Selection with Pay-what-you-get Composition

We study the problem of top-$k$ selection over a large domain universe subject to user-level differential privacy. Typically, the exponential mechanism or report noisy max are the algorithms used to solve this problem. However, these algorithms require querying the database for the count of each domain element. We focus on the setting where the data domain is unknown, which is different than the setting of frequent itemsets where an apriori type algorithm can help prune the space of domain elements to query. We design algorithms that ensures (approximate) $(\epsilon,\delta>0)$-differential privacy and only needs access to the true top-$\bar{k}$ elements from the data for any chosen $\bar{k} \geq k$. This is a highly desirable feature for making differential privacy practical, since the algorithms require no knowledge of the domain. We consider both the setting where a user's data can modify an arbitrary number of counts by at most 1, i.e. unrestricted sensitivity, and the setting where a user's data can modify at most some small, fixed number of counts by at most 1, i.e. restricted sensitivity. Additionally, we provide a pay-what-you-get privacy composition bound for our algorithms. That is, our algorithms might return fewer than $k$ elements when the top-$k$ elements are queried, but the overall privacy budget only decreases by the size of the outcome set.

cs.CR↗

arXiv2019

Efficient Second-Order Shape-Constrained Function Fitting

We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted-$L_{\infty}$ norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first- and/or second-order differences. Our algorithm computes an approximation with additive error $\varepsilon$ in $O\left(n \log \frac{U}{\varepsilon} \right)$ time, where $U$ captures the range of input values. We also give a simple greedy algorithm that runs in $O(n)$ time for the special case of unweighted $L_{\infty}$ convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming.

cs.DS↗

arXiv2018

Individual Sensitivity Preprocessing for Data Privacy

The sensitivity metric in differential privacy, which is informally defined as the largest marginal change in output between neighboring databases, is of substantial significance in determining the accuracy of private data analyses. Techniques for improving accuracy when the average sensitivity is much smaller than the worst-case sensitivity have been developed within the differential privacy literature, including tools such as smooth sensitivity, Sample-and-Aggregate, Propose-Test-Release, and Lipschitz extensions. In this work, we provide a new and general Sensitivity-Preprocessing framework for reducing sensitivity, where efficient application gives state-of-the-art accuracy for privately outputting the important statistical metrics median and mean when no underlying assumptions are made about the database. In particular, our framework compares favorably to smooth sensitivity for privately outputting median, in terms of both running time and accuracy. Furthermore, because our framework is a preprocessing step, it can also be complementary to smooth sensitivity and any other private mechanism, where applying both can achieve further gains in accuracy. We additionally introduce a new notion of individual sensitivity and show that it is an important metric in the variant definition of personalized differential privacy. We show that our algorithm can extend to this context and serve as a useful tool for this variant definition and its applications in markets for privacy.

cs.DS↗

arXiv2018

Fully Dynamic Effective Resistances

In this paper we consider the \emph{fully-dynamic} All-Pairs Effective Resistance problem, where the goal is to maintain effective resistances on a graph $G$ among any pair of query vertices under an intermixed sequence of edge insertions and deletions in $G$. The effective resistance between a pair of vertices is a physics-motivated quantity that encapsulates both the congestion and the dilation of a flow. It is directly related to random walks, and it has been instrumental in the recent works for designing fast algorithms for combinatorial optimization problems, graph sparsification, and network science. We give a data-structure that maintains $(1+\epsilon)$-approximations to all-pair effective resistances of a fully-dynamic unweighted, undirected multi-graph $G$ with $\tilde{O}(m^{4/5}\epsilon^{-4})$ expected amortized update and query time, against an oblivious adversary. Key to our result is the maintenance of a dynamic \emph{Schur complement}~(also known as vertex resistance sparsifier) onto a set of terminal vertices of our choice. This maintenance is obtained (1) by interpreting the Schur complement as a sum of random walks and (2) by randomly picking the vertex subset into which the sparsifier is constructed. We can then show that each update in the graph affects a small number of such walks, which in turn leads to our sub-linear update time. We believe that this local representation of vertex sparsifiers may be of independent interest.

cs.DS↗

arXiv2017

$\ell_1$ Regression using Lewis Weights Preconditioning and Stochastic Gradient Descent

We present preconditioned stochastic gradient descent (SGD) algorithms for the $\ell_1$ minimization problem $\min_{x}\|A x - b\|_1$ in the overdetermined case, where there are far more constraints than variables. Specifically, we have $A \in \mathbb{R}^{n \times d}$ for $n \gg d$. Commonly known as the Least Absolute Deviations problem, $\ell_1$ regression can be used to solve many important combinatorial problems, such as minimum cut and shortest path. SGD-based algorithms are appealing for their simplicity and practical efficiency. Our primary insight is that careful preprocessing can yield preconditioned matrices $\tilde{A}$ with strong properties (besides good condition number and low-dimension) that allow for faster convergence of gradient descent. In particular, we precondition using Lewis weights to obtain an isotropic matrix with fewer rows and strong upper bounds on all row norms. We leverage these conditions to find a good initialization, which we use along with recent smoothing reductions and accelerated stochastic gradient descent algorithms to achieve $\epsilon$ relative error in $\tilde{O}(nnz(A) + d^{2.5} \epsilon^{-2})$ time with high probability, where $nnz(A)$ is the number of non-zeros in $A$. This improves over the previous best result using gradient descent for $\ell_1$ regression. We also match the best known running times for interior point methods in several settings. Finally, we also show that if our original matrix $A$ is approximately isotropic and the row norms are approximately equal, we can give an algorithm that avoids using fast matrix multiplication and obtains a running time of $\tilde{O}(nnz(A) + s d^{1.5}\epsilon^{-2} + d^2\epsilon^{-2})$, where $s$ is the maximum number of non-zeros in a row of $A$. In this setting, we beat the best interior point methods for certain parameter regimes.

cs.DS↗

arXiv2017

Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees

We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statistical leverage scores / effective resistances and the analysis of random graphs by [Janson, Combinatorics, Probability and Computing `94]. This leads to a routine that in quadratic time, sparsifies a graph down to about $n^{1.5}$ edges in ways that preserve both the determinant and the distribution of spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximate Choleksy factorizations leads to algorithms for counting and sampling spanning trees which are nearly optimal for dense graphs. We give an algorithm that computes a $(1 \pm \delta)$ approximation to the determinant of any SDDM matrix with constant probability in about $n^2 \delta^{-2}$ time. This is the first routine for graphs that outperforms general-purpose routines for computing determinants of arbitrary matrices. We also give an algorithm that generates in about $n^2 \delta^{-2}$ time a spanning tree of a weighted undirected graph from a distribution with total variation distance of $\delta$ from the $w$-uniform distribution .

cs.DS↗

arXiv2017

Nearly Tight Bounds for Sandpile Transience on the Grid

We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an $O(n^4\log^4{n})$ upper bound and an $\Omega(n^4)$ lower bound for the transience class of the $n \times n$ grid. Our methods naturally extend to $n^d$-sized $d$-dimensional grids to give $O(n^{3d - 2}\log^{d+2}{n})$ upper bounds and $\Omega(n^{3d -2})$ lower bounds.

cs.DS↗

arXiv2016

Sampling Random Spanning Trees Faster than Matrix Multiplication

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in $\tilde{O}(n^{4/3}m^{1/2}+n^{2})$ time (The $\tilde{O}(\cdot)$ notation hides $\operatorname{polylog}(n)$ factors). The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, $O(n^\omega)$. For the special case of unweighted graphs, this improves upon the best previously known running time of $\tilde{O}(\min\{n^{\omega},m\sqrt{n},m^{4/3}\})$ for $m \gg n^{5/3}$ (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute $\epsilon$-approximate effective resistances for a set $S$ of vertex pairs via approximate Schur complements in $\tilde{O}(m+(n + |S|)\epsilon^{-2})$ time, without using the Johnson-Lindenstrauss lemma which requires $\tilde{O}( \min\{(m + |S|)\epsilon^{-2}, m+n\epsilon^{-4} +|S|\epsilon^{-2}\})$ time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn't sufficiently accurate.

cs.DS↗