arXiv subjects

Yuval Dagan

Publications and source records attributed to Yuval Dagan.

At least 37 records · Page 2Linked to original sources

arXiv2022

Dispersion of Free-Falling Saliva Droplets by Two-Dimensional Vortical Flows

The dispersion of respiratory saliva droplets by indoor wake structures may enhance the transmission of various infectious diseases, as the wake spreads virus-laden droplets across the room. Thus, this study analyses the interaction between vortical wake structures and exhaled multi-component saliva droplets. A self-propelling analytically-described dipolar vortex is chosen as a model wake flow, passing through a cloud of micron-sized evaporating saliva droplets. The droplets' spatial location, velocity, diameter, and temperature are traced and coupled to their local flow field. For the first time, the wake structure decay is incorporated and analyzed, which is proved essential for accurately predicting the settling distances of the dispersed droplets. The model also considers the non-volatile saliva components, adequately capturing the essence of droplet-aerosol transition and predicting the equilibrium diameter of the residual aerosols. Our analytic model reveals non-intuitive interactions between wake flows, droplet relaxation time, gravity, and transport phenomena. We reveal that given the right conditions, a virus-laden saliva droplet might translate to distances two orders of magnitude larger than the carrier-flow characteristic size. Moreover, accounting for the non-volatile contents inside the droplet may lead to fundamentally different dispersion and settling behavior compared to non-evaporating particles or pure water droplets. Ergo, we suggest that the implementation of more complex evaporation models might be critical in high-fidelity simulations aspiring to assess the spread of airborne respiratory droplets.

physics.flu-dyn↗

arXiv2022

EM's Convergence in Gaussian Latent Tree Models

We study the optimization landscape of the log-likelihood function and the convergence of the Expectation-Maximization (EM) algorithm in latent Gaussian tree models, i.e. tree-structured Gaussian graphical models whose leaf nodes are observable and non-leaf nodes are unobservable. We show that the unique non-trivial stationary point of the population log-likelihood is its global maximum, and establish that the expectation-maximization algorithm is guaranteed to converge to it in the single latent variable case. Our results for the landscape of the log-likelihood function in general latent tree models provide support for the extensive practical use of maximum likelihood based-methods in this setting. Our results for the EM algorithm extend an emerging line of work on obtaining global convergence guarantees for this celebrated algorithm. We show our results for the non-trivial stationary points of the log-likelihood by arguing that a certain system of polynomial equations obtained from the EM updates has a unique non-trivial solution. The global convergence of the EM algorithm follows by arguing that all trivial fixed points are higher-order saddle points.

cs.LG↗

arXiv2022

Score-Guided Intermediate Layer Optimization: Fast Langevin Mixing for Inverse Problems

We prove fast mixing and characterize the stationary distribution of the Langevin Algorithm for inverting random weighted DNN generators. This result extends the work of Hand and Voroninski from efficient inversion to efficient posterior sampling. In practice, to allow for increased expressivity, we propose to do posterior sampling in the latent space of a pre-trained generative model. To achieve that, we train a score-based model in the latent space of a StyleGAN-2 and we use it to solve inverse problems. Our framework, Score-Guided Intermediate Layer Optimization (SGILO), extends prior work by replacing the sparsity regularization with a generative prior in the intermediate layer. Experimentally, we obtain significant improvements over the previous state-of-the-art, especially in the low measurement regime.

cs.LG↗

arXiv2022

Smoothed Online Learning is as Easy as Statistical Learning

Much of modern learning theory has been split between two regimes: the classical offline setting, where data arrive independently, and the online setting, where data arrive adversarially. While the former model is often both computationally and statistically tractable, the latter requires no distributional assumptions. In an attempt to achieve the best of both worlds, previous work proposed the smooth online setting where each sample is drawn from an adversarially chosen distribution, which is smooth, i.e., it has a bounded density with respect to a fixed dominating measure. We provide tight bounds on the minimax regret of learning a nonparametric function class, with nearly optimal dependence on both the horizon and smoothness parameters. Furthermore, we provide the first oracle-efficient, no-regret algorithms in this setting. In particular, we propose an oracle-efficient improper algorithm whose regret achieves optimal dependence on the horizon and a proper algorithm requiring only a single oracle call per round whose regret has the optimal horizon dependence in the classification setting and is sublinear in general. Both algorithms have exponentially worse dependence on the smoothness parameter of the adversary than the minimax rate. We then prove a lower bound on the oracle complexity of any proper learning algorithm, which matches the oracle-efficient upper bounds up to a polynomial factor, thus demonstrating the existence of a statistical-computational gap in smooth online learning. Finally, we apply our results to the contextual bandit setting to show that if a function class is learnable in the classical setting, then there is an oracle-efficient, no-regret algorithm for contextual bandits in the case that contexts arrive in a smooth manner.

stat.ML↗

arXiv2022

Generalized stability theory of polydisperse particle-laden flows. Part1. Channel flow

We present a generalized hydrodynamic stability theory for interacting particles in polydisperse particle-laden flows. The addition of dispersed particulate matter to a clean flow can either stabilize or destabilize the flow, depending on the particles' relaxation time-scale relative to the carrier flow time scales and the particle loading. To study the effects of polydispersity and particle interactions on the hydrodynamic stability of shear flows, we propose a new mathematical framework by combining a linear stability analysis and a discrete Eulerian sectional formulation to describe the flow and the dispersed particulate matter. In this formulation, multiple momentum and transport equations are written for each size-section of the dispersed phase, where interphase and inter-particle mass and momentum transfer are modelled as source terms in the governing equations. A new modal linear stability framework is derived by linearizing the coupled equations. Using this approach, particle-flow interactions, such as polydispersity, droplet vaporization, condensation, and coalescence, may be modelled. The method is validated with linear stability analyses of clean and monodisperse particle-laden flows. We show that the stability characteristics of a channel flow laden with particles drastically change due to polydispersity. While relatively large monodisperse particles tend to stabilize the flow, adding a second size section of a very small mass fraction of low-to-moderate Stokes number particles may significantly increase the growth rates, and for high-Reynolds numbers may destabilize flows that might have been regarded as linearly stable in the monodisperse case. These findings may apply to a vast number of fluid mechanics applications involving particle-laden flows such as atmospheric flows, environmental flows, medical applications, propulsion, and energy systems.

physics.flu-dyn↗

arXiv2022

Dynamics of Evaporating Respiratory Droplets in the Vicinity of Vortex Dipoles

A new mathematical analysis of exhaled respiratory droplet dynamics and settling distances in the vicinity of vortical environments is presented. Recent experimental and theoretical studies suggest that vortical flow structures may enhance the settling distances of exhaled respiratory droplets beyond the two-meter distancing rule recommended by health authorities lately. We propose a mathematical framework to study the underlying physical mechanism responsible for the entrapment and subsequently delayed settling times of evaporating droplets and solid particles. A dipolar vortex is considered self-propelling through a cloud of micron-sized evaporating droplets. This configuration might be utilized to approximate an indoor environment in which similar unsteady vortical flow structures interact with exhaled respiratory droplets. We demonstrate the vortex dipole effect on droplet and solid particles settling distances, depending on the evaporation rate, the vorticity of the dipole, and the droplet's initial diameter and location relative to the vortex core. Our theoretical analysis reveals non-intuitive interactions between the vortex dipole, droplet relaxation time, gravity, and mass transfer. The existence of optimal conditions for maximum displacement is suggested, where the droplet entrainment reaches up to an order of magnitude larger than the vortex core length scale. We present a basic model that may be applied for evaluating the spread of exhaled respiratory droplets in vortical environments. Our theoretical study suggests that exhaled respiratory droplets initially at rest can translate to significant distances, hence implying that vortical flow might enhance the transmission of airborne pathogens.

physics.flu-dyn↗

arXiv2021

A bounded-noise mechanism for differential privacy

We present an asymptotically optimal $(ε,δ)$ differentially private mechanism for answering multiple, adaptively asked, $Δ$-sensitive queries, settling the conjecture of Steinke and Ullman [2020]. Our algorithm has a significant advantage that it adds independent bounded noise to each query, thus providing an absolute error bound. Additionally, we apply our algorithm in adaptive data analysis, obtaining an improved guarantee for answering multiple queries regarding some underlying distribution using a finite sample. Numerical computations show that the bounded-noise mechanism outperforms the Gaussian mechanism in many standard settings.

cs.DS↗

arXiv2021

Statistical Estimation from Dependent Data

We consider a general statistical estimation problem wherein binary labels across different observations are not independent conditioned on their feature vectors, but dependent, capturing settings where e.g. these observations are collected on a spatial domain, a temporal domain, or a social network, which induce dependencies. We model these dependencies in the language of Markov Random Fields and, importantly, allow these dependencies to be substantial, i.e do not assume that the Markov Random Field capturing these dependencies is in high temperature. As our main contribution we provide algorithms and statistically efficient estimation rates for this model, giving several instantiations of our bounds in logistic regression, sparse logistic regression, and neural network settings with dependent data. Our estimation guarantees follow from novel results for estimating the parameters (i.e. external fields and interaction strengths) of Ising models from a {\em single} sample. {We evaluate our estimation approach on real networked data, showing that it outperforms standard regression approaches that ignore dependencies, across three text classification datasets: Cora, Citeseer and Pubmed.}

cs.LG↗

arXiv2021

Majorizing Measures, Sequential Complexities, and Online Learning

We introduce the technique of generic chaining and majorizing measures for controlling sequential Rademacher complexity. We relate majorizing measures to the notion of fractional covering numbers, which we show to be dominated in terms of sequential scale-sensitive dimensions in a horizon-independent way, and, under additional complexity assumptions establish a tight control on worst-case sequential Rademacher complexity in terms of the integral of sequential scale-sensitive dimension. Finally, we establish a tight contraction inequality for worst-case sequential Rademacher complexity. The above constitutes the resolution of a number of outstanding open problems in extending the classical theory of empirical processes to the sequential case, and, in turn, establishes sharp results for online learning.

stat.ML↗

arXiv2021

Adversarial Laws of Large Numbers and Optimal Regret in Online Classification

Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed sub-population is well-estimated by its frequency in the sample. We study laws of large numbers in sampling processes that can affect the environment they are acting upon and interact with it. Specifically, we consider the sequential sampling model proposed by Ben-Eliezer and Yogev (2020), and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are \emph{online learnable}. Our characterization may be interpreted as an online analogue to the equivalence between learnability and uniform convergence in statistical (PAC) learning. The sample-complexity bounds we obtain are tight for many parameter regimes, and as an application, we determine the optimal regret bounds in online learning, stated in terms of \emph{Littlestone's dimension}, thus resolving the main open question from Ben-David, Pál, and Shalev-Shwartz (2009), which was also posed by Rakhlin, Sridharan, and Tewari (2015).

cs.LG↗

arXiv2020

Learning Ising models from one or multiple samples

There have been two separate lines of work on estimating Ising models: (1) estimating them from multiple independent samples under minimal assumptions about the model's interaction matrix; and (2) estimating them from one sample in restrictive settings. We propose a unified framework that smoothly interpolates between these two settings, enabling significantly richer estimation guarantees from one, a few, or many samples. Our main theorem provides guarantees for one-sample estimation, quantifying the estimation error in terms of the metric entropy of a family of interaction matrices. As corollaries of our main theorem, we derive bounds when the model's interaction matrix is a (sparse) linear combination of known matrices, or it belongs to a finite set, or to a high-dimensional manifold. In fact, our main result handles multiple independent samples by viewing them as one sample from a larger model, and can be used to derive estimation bounds that are qualitatively similar to those obtained in the afore-described multiple-sample literature. Our technical approach benefits from sparsifying a model's interaction network, conditioning on subsets of variables that make the dependencies in the resulting conditional distribution sufficiently weak. We use this sparsification technique to prove strong concentration and anti-concentration results for the Ising model, which we believe have applications beyond the scope of this paper.

math.ST↗

arXiv2020

Interaction is necessary for distributed learning with privacy or communication constraints

Local differential privacy (LDP) is a model where users send privatized data to an untrusted central server whose goal it to solve some data analysis task. In the non-interactive version of this model the protocol consists of a single round in which a server sends requests to all users then receives their responses. This version is deployed in industry due to its practical advantages and has attracted significant research interest. Our main result is an exponential lower bound on the number of samples necessary to solve the standard task of learning a large-margin linear separator in the non-interactive LDP model. Via a standard reduction this lower bound implies an exponential lower bound for stochastic convex optimization and specifically, for learning linear models with a convex, Lipschitz and smooth loss. These results answer the questions posed in \citep{SmithTU17,DanielyF18}. Our lower bound relies on a new technique for constructing pairs of distributions with nearly matching moments but whose supports can be nearly separated by a large margin hyperplane. These lower bounds also hold in the model where communication from each user is limited and follow from a lower bound on learning using non-adaptive \emph{statistical queries}.

cs.LG↗

arXiv2020

PAC learning with stable and private predictions

We study binary classification algorithms for which the prediction on any point is not too sensitive to individual examples in the dataset. Specifically, we consider the notions of uniform stability (Bousquet and Elisseeff, 2001) and prediction privacy (Dwork and Feldman, 2018). Previous work on these notions shows how they can be achieved in the standard PAC model via simple aggregation of models trained on disjoint subsets of data. Unfortunately, this approach leads to a significant overhead in terms of sample complexity. Here we demonstrate several general approaches to stable and private prediction that either eliminate or significantly reduce the overhead. Specifically, we demonstrate that for any class $C$ of VC dimension $d$ there exists a $γ$-uniformly stable algorithm for learning $C$ with excess error $α$ using $\tilde O(d/(αγ) + d/α^2)$ samples. We also show that this bound is nearly tight. For $ε$-differentially private prediction we give two new algorithms: one using $\tilde O(d/(α^2ε))$ samples and another one using $\tilde O(d^2/(αε) + d/α^2)$ samples. The best previously known bounds for these problems are $O(d/(α^2γ))$ and $O(d/(α^3ε))$, respectively.

cs.LG↗

arXiv2020

Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression

In this paper, we study two problems: (1) estimation of a $d$-dimensional log-concave distribution and (2) bounded multivariate convex regression with random design with an underlying log-concave density or a compactly supported distribution with a continuous density. First, we show that for all $d \ge 4$ the maximum likelihood estimators of both problems achieve an optimal risk of $Θ_d(n^{-2/(d+1)})$ (up to a logarithmic factor) in terms of squared Hellinger distance and $L_2$ squared distance, respectively. Previously, the optimality of both these estimators was known only for $d\le 3$. We also prove that the $ε$-entropy numbers of the two aforementioned families are equal up to logarithmic factors. We complement these results by proving a sharp bound $Θ_d(n^{-2/(d+4)})$ on the minimax rate (up to logarithmic factors) with respect to the total variation distance. Finally, we prove that estimating a log-concave density - even a uniform distribution on a convex set - up to a fixed accuracy requires the number of samples \emph{at least} exponential in the dimension. We do that by improving the dimensional constant in the best known lower bound for the minimax rate from $2^{-d}\cdot n^{-2/(d+1)}$ to $c\cdot n^{-2/(d+1)}$ (when $d\geq 2$).

math.ST↗

arXiv2019

Learning from weakly dependent data under Dobrushin's condition

Statistical learning theory has largely focused on learning and generalization given independent and identically distributed (i.i.d.) samples. Motivated by applications involving time-series data, there has been a growing literature on learning and generalization in settings where data is sampled from an ergodic process. This work has also developed complexity measures, which appropriately extend the notion of Rademacher complexity to bound the generalization error and learning rates of hypothesis classes in this setting. Rather than time-series data, our work is motivated by settings where data is sampled on a network or a spatial domain, and thus do not fit well within the framework of prior work. We provide learning and generalization bounds for data that are complexly dependent, yet their distribution satisfies the standard Dobrushin's condition. Indeed, we show that the standard complexity measures of Gaussian and Rademacher complexities and VC dimension are sufficient measures of complexity for the purposes of bounding the generalization error and learning rates of hypothesis classes in our setting. Moreover, our generalization bounds only degrade by constant factors compared to their i.i.d. analogs, and our learnability bounds degrade by log factors in the size of the training set.

cs.LG↗

arXiv2019

Space lower bounds for linear prediction in the streaming model

We show that fundamental learning tasks, such as finding an approximate linear separator or linear regression, require memory at least \emph{quadratic} in the dimension, in a natural streaming setting. This implies that such problems cannot be solved (at least in this setting) by scalable memory-efficient streaming algorithms. Our results build on a memory lower bound for a simple linear-algebraic problem -- finding orthogonal vectors -- and utilize the estimates on the packing of the Grassmannian, the manifold of all linear subspaces of fixed dimension.

cs.LG↗

arXiv2018

The entropy of lies: playing twenty questions with a liar

`Twenty questions' is a guessing game played by two players: Bob thinks of an integer between $1$ and $n$, and Alice's goal is to recover it using a minimal number of Yes/No questions. Shannon's entropy has a natural interpretation in this context. It characterizes the average number of questions used by an optimal strategy in the distributional variant of the game: let $μ$ be a distribution over $[n]$, then the average number of questions used by an optimal strategy that recovers $x\sim μ$ is between $H(μ)$ and $H(μ)+1$. We consider an extension of this game where at most $k$ questions can be answered falsely. We extend the classical result by showing that an optimal strategy uses roughly $H(μ) + k H_2(μ)$ questions, where $H_2(μ) = \sum_x μ(x)\log\log\frac{1}{μ(x)}$. This also generalizes a result by Rivest et al. for the uniform distribution. Moreover, we design near optimal strategies that only use comparison queries of the form `$x \leq c$?' for $c\in[n]$. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.

cs.DS↗

arXiv2018

Detecting Correlations with Little Memory and Communication

We study the problem of identifying correlations in multivariate data, under information constraints: Either on the amount of memory that can be used by the algorithm, or the amount of communication when the data is distributed across several machines. We prove a tight trade-off between the memory/communication complexity and the sample complexity, implying (for example) that to detect pairwise correlations with optimal sample complexity, the number of required memory/communication bits is at least quadratic in the dimension. Our results substantially improve those of Shamir [2014], which studied a similar question in a much more restricted setting. To the best of our knowledge, these are the first provable sample/memory/communication trade-offs for a practical estimation problem, using standard distributions, and in the natural regime where the memory/communication budget is larger than the size of a single data point. To derive our theorems, we prove a new information-theoretic result, which may be relevant for studying other information-constrained learning problems.

cs.LG↗