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Hassan Ashtiani

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11 papers

2 author rows

NeurIPS Conference 2024 Conference Paper

Sample-Efficient Private Learning of Mixtures of Gaussians

Hassan Ashtiani
Mahbod Majid
Shyam Narayanan

We study the problem of learning mixtures of Gaussians with approximate differential privacy. We prove that roughly $kd^2 + k^{1. 5} d^{1. 75} + k^2 d$ samples suffice to learn a mixture of $k$ arbitrary $d$-dimensional Gaussians up to low total variation distance, with differential privacy. Our work improves over the previous best result (which required roughly $k^2 d^4$ samples) and is provably optimal when $d$ is much larger than $k^2$. Moreover, we give the first optimal bound for privately learning mixtures of $k$ univariate (i. e. , $1$-dimensional) Gaussians. Importantly, we show that the sample complexity for learning mixtures of univariate Gaussians is linear in the number of components $k$, whereas the previous best sample complexity was quadratic in $k$. Our algorithms utilize various techniques, including the inverse sensitivity mechanism, sample compression for distributions, and methods for bounding volumes of sumsets.

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NeurIPS Conference 2023 Conference Paper

On the Role of Noise in the Sample Complexity of Learning Recurrent Neural Networks: Exponential Gaps for Long Sequences

Alireza F. Pour
Hassan Ashtiani

We consider the class of noisy multi-layered sigmoid recurrent neural networks with $w$ (unbounded) weights for classification of sequences of length $T$, where independent noise distributed according to $\mathcal{N}(0, \sigma^2)$ is added to the output of each neuron in the network. Our main result shows that the sample complexity of PAC learning this class can be bounded by $O (w\log(T/\sigma))$. For the non-noisy version of the same class (i. e. , $\sigma=0$), we prove a lower bound of $\Omega (wT)$ for the sample complexity. Our results indicate an exponential gap in the dependence of sample complexity on $T$ for noisy versus non-noisy networks. Moreover, given the mild logarithmic dependence of the upper bound on $1/\sigma$, this gap still holds even for numerically negligible values of $\sigma$.

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ICML Conference 2023 Conference Paper

Polynomial Time and Private Learning of Unbounded Gaussian Mixture Models

Jamil Arbas
Hassan Ashtiani
Christopher Liaw

We study the problem of privately estimating the parameters of $d$-dimensional Gaussian Mixture Models (GMMs) with $k$ components. For this, we develop a technique to reduce the problem to its non-private counterpart. This allows us to privatize existing non-private algorithms in a blackbox manner, while incurring only a small overhead in the sample complexity and running time. As the main application of our framework, we develop an $(\varepsilon, \delta)$-differentially private algorithm to learn GMMs using the non-private algorithm of Moitra and Valiant (2010) as a blackbox. Consequently, this gives the first sample complexity upper bound and first polynomial time algorithm for privately learning GMMs without any boundedness assumptions on the parameters. As part of our analysis, we prove a tight (up to a constant factor) lower bound on the total variation distance of high-dimensional Gaussians which can be of independent interest.

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NeurIPS Conference 2022 Conference Paper

Benefits of Additive Noise in Composing Classes with Bounded Capacity

Alireza Fathollah Pour
Hassan Ashtiani

We observe that given two (compatible) classes of functions $\mathcal{F}$ and $\mathcal{H}$ with small capacity as measured by their uniform covering numbers, the capacity of the composition class $\mathcal{H} \circ \mathcal{F}$ can become prohibitively large or even unbounded. We then show that adding a small amount of Gaussian noise to the output of $\mathcal{F}$ before composing it with $\mathcal{H}$ can effectively control the capacity of $\mathcal{H} \circ \mathcal{F}$, offering a general recipe for modular design. To prove our results, we define new notions of uniform covering number of random functions with respect to the total variation and Wasserstein distances. We instantiate our results for the case of multi-layer sigmoid neural networks. Preliminary empirical results on MNIST dataset indicate that the amount of noise required to improve over existing uniform bounds can be numerically negligible (i. e. , element-wise i. i. d. Gaussian noise with standard deviation $10^{-240}$)

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NeurIPS Conference 2021 Conference Paper

Privately Learning Mixtures of Axis-Aligned Gaussians

Ishaq Aden-Ali
Hassan Ashtiani
Christopher Liaw

We consider the problem of learning multivariate Gaussians under the constraint of approximate differential privacy. We prove that $\widetilde{O}(k^2 d \log^{3/2}(1/\delta) / \alpha^2 \varepsilon)$ samples are sufficient to learn a mixture of $k$ axis-aligned Gaussians in $\mathbb{R}^d$ to within total variation distance $\alpha$ while satisfying $(\varepsilon, \delta)$-differential privacy. This is the first result for privately learning mixtures of unbounded axis-aligned (or even unbounded univariate) Gaussians. If the covariance matrices of each of the Gaussians is the identity matrix, we show that $\widetilde{O}(kd/\alpha^2 + kd \log(1/\delta) / \alpha \varepsilon)$ samples are sufficient. To prove our results, we design a new technique for privately learning mixture distributions. A class of distributions $\mathcal{F}$ is said to be list-decodable if there is an algorithm that, given "heavily corrupted" samples from $f \in \mathcal{F}$, outputs a list of distributions one of which approximates $f$. We show that if $\mathcal{F}$ is privately list-decodable then we can learn mixtures of distributions in $\mathcal{F}$. Finally, we show axis-aligned Gaussian distributions are privately list-decodable, thereby proving mixtures of such distributions are privately learnable.

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ICML Conference 2020 Conference Paper

Black-box Certification and Learning under Adversarial Perturbations

Hassan Ashtiani
Vinayak Pathak
Ruth Urner

We formally study the problem of classification under adversarial perturbations from a learner’s perspective as well as a third-party who aims at certifying the robustness of a given black-box classifier. We analyze a PAC-type framework of semi-supervised learning and identify possibility and impossibility results for proper learning of VC-classes in this setting. We further introduce a new setting of black-box certification under limited query budget, and analyze this for various classes of predictors and perturbation. We also consider the viewpoint of a black-box adversary that aims at finding adversarial examples, showing that the existence of an adversary with polynomial query complexity can imply the existence of a sample efficient robust learner.

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NeurIPS Conference 2019 Conference Paper

Disentangled behavioural representations

Amir Dezfouli
Hassan Ashtiani
Omar Ghattas
Richard Nock
Peter Dayan
Cheng Soon Ong

Individual characteristics in human decision-making are often quantified by fitting a parametric cognitive model to subjects' behavior and then studying differences between them in the associated parameter space. However, these models often fit behavior more poorly than recurrent neural networks (RNNs), which are more flexible and make fewer assumptions about the underlying decision-making processes. Unfortunately, the parameter and latent activity spaces of RNNs are generally high-dimensional and uninterpretable, making it hard to use them to study individual differences. Here, we show how to benefit from the flexibility of RNNs while representing individual differences in a low-dimensional and interpretable space. To achieve this, we propose a novel end-to-end learning framework in which an encoder is trained to map the behavior of subjects into a low-dimensional latent space. These low-dimensional representations are used to generate the parameters of individual RNNs corresponding to the decision-making process of each subject. We introduce terms into the loss function that ensure that the latent dimensions are informative and disentangled, i. e. , encouraged to have distinct effects on behavior. This allows them to align with separate facets of individual differences. We illustrate the performance of our framework on synthetic data as well as a dataset including the behavior of patients with psychiatric disorders.

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NeurIPS Conference 2018 Conference Paper

Nearly tight sample complexity bounds for learning mixtures of Gaussians via sample compression schemes

Hassan Ashtiani
Shai Ben-David
Nicholas Harvey
Christopher Liaw
Abbas Mehrabian
Yaniv Plan

We prove that ϴ(k d^2 / ε^2) samples are necessary and sufficient for learning a mixture of k Gaussians in R^d, up to error ε in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that O(k d / ε^2) samples suffice, matching a known lower bound. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in R^d has an efficient sample compression.

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AAAI Conference 2018 Conference Paper

Sample-Efficient Learning of Mixtures

Hassan Ashtiani
Shai Ben-David
Abbas Mehrabian

We consider PAC learning of probability distributions (a. k. a. density estimation), where we are given an i. i. d. sample generated from an unknown target distribution, and want to output a distribution that is close to the target in total variation distance. Let F be an arbitrary class of probability distributions, and let Fk denote the class of k-mixtures of elements of F. Assuming the existence of a method for learning F with sample complexity mF (ε), we provide a method for learning Fk with sample complexity O(k log k · mF (ε)/ε2 ). Our mixture learning algorithm has the property that, if the Flearner is proper and agnostic, then the Fk -learner would be proper and agnostic as well. This general result enables us to improve the best known sample complexity upper bounds for a variety of important mixture classes. First, we show that the class of mixtures of k axis-aligned Gaussians in Rd is PAC-learnable in the agnostic setting with O(kd/ε4 ) samples, which is tight in k and d up to logarithmic factors. Second, we show that the class of mixtures of k Gaussians in Rd is PAC-learnable in the agnostic setting with sample complexity O(kd2 /ε4 ), which improves the previous known bounds of O(k3 d2 /ε4 ) and O(k4 d4 /ε2 ) in its dependence on k and d. Finally, we show that the class of mixtures of k log-concave distributions over Rd is PAClearnable using O(d(d+5)/2 ε−(d+9)/2 k) samples.

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NeurIPS Conference 2016 Conference Paper

Clustering with Same-Cluster Queries

Hassan Ashtiani
Shrinu Kushagra
Shai Ben-David

We propose a framework for Semi-Supervised Active Clustering framework (SSAC), where the learner is allowed to interact with a domain expert, asking whether two given instances belong to the same cluster or not. We study the query and computational complexity of clustering in this framework. We consider a setting where the expert conforms to a center-based clustering with a notion of margin. We show that there is a trade off between computational complexity and query complexity; We prove that for the case of $k$-means clustering (i. e. , when the expert conforms to a solution of $k$-means), having access to relatively few such queries allows efficient solutions to otherwise NP hard problems. In particular, we provide a probabilistic polynomial-time (BPP) algorithm for clustering in this setting that asks $O\big(k^2\log k + k\log n)$ same-cluster queries and runs with time complexity $O\big(kn\log n)$ (where $k$ is the number of clusters and $n$ is the number of instances). The success of the algorithm is guaranteed for data satisfying the margin condition under which, without queries, we show that the problem is NP hard. We also prove a lower bound on the number of queries needed to have a computationally efficient clustering algorithm in this setting.

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UAI Conference 2015 Conference Paper

Representation Learning for Clustering: A Statistical Framework

Hassan Ashtiani
Shai Ben-David

We address the problem of communicating domain knowledge from a user to the designer of a clustering algorithm. We propose a protocol in which the user provides a clustering of a relatively small random sample of a data set. The algorithm designer then uses that sample to come up with a data representation under which kmeans clustering results in a clustering (of the full data set) that is aligned with the user’s clustering. We provide a formal statistical model for analyzing the sample complexity of learning a clustering representation with this paradigm. We then introduce a notion of capacity of a class of possible representations, in the spirit of the VCdimension, showing that classes of representations that have finite such dimension can be successfully learned with sample size error bounds, and end our discussion with an analysis of that dimension for classes of representations induced by linear embeddings.

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