arXiv subjects

Surbhi Goel

Publications and source records attributed to Surbhi Goel.

At least 55 records · Page 3Linked to original sources

arXiv2022

Understanding Contrastive Learning Requires Incorporating Inductive Biases

Contrastive learning is a popular form of self-supervised learning that encourages augmentations (views) of the same input to have more similar representations compared to augmentations of different inputs. Recent attempts to theoretically explain the success of contrastive learning on downstream classification tasks prove guarantees depending on properties of {\em augmentations} and the value of {\em contrastive loss} of representations. We demonstrate that such analyses, that ignore {\em inductive biases} of the function class and training algorithm, cannot adequately explain the success of contrastive learning, even {\em provably} leading to vacuous guarantees in some settings. Extensive experiments on image and text domains highlight the ubiquity of this problem -- different function classes and algorithms behave very differently on downstream tasks, despite having the same augmentations and contrastive losses. Theoretical analysis is presented for the class of linear representations, where incorporating inductive biases of the function class allows contrastive learning to work with less stringent conditions compared to prior analyses.

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arXiv2021

Gone Fishing: Neural Active Learning with Fisher Embeddings

There is an increasing need for effective active learning algorithms that are compatible with deep neural networks. This paper motivates and revisits a classic, Fisher-based active selection objective, and proposes BAIT, a practical, tractable, and high-performing algorithm that makes it viable for use with neural models. BAIT draws inspiration from the theoretical analysis of maximum likelihood estimators (MLE) for parametric models. It selects batches of samples by optimizing a bound on the MLE error in terms of the Fisher information, which we show can be implemented efficiently at scale by exploiting linear-algebraic structure especially amenable to execution on modern hardware. Our experiments demonstrate that BAIT outperforms the previous state of the art on both classification and regression problems, and is flexible enough to be used with a variety of model architectures.

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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.}

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arXiv2021

Investigating the Role of Negatives in Contrastive Representation Learning

Noise contrastive learning is a popular technique for unsupervised representation learning. In this approach, a representation is obtained via reduction to supervised learning, where given a notion of semantic similarity, the learner tries to distinguish a similar (positive) example from a collection of random (negative) examples. The success of modern contrastive learning pipelines relies on many parameters such as the choice of data augmentation, the number of negative examples, and the batch size; however, there is limited understanding as to how these parameters interact and affect downstream performance. We focus on disambiguating the role of one of these parameters: the number of negative examples. Theoretically, we show the existence of a collision-coverage trade-off suggesting that the optimal number of negative examples should scale with the number of underlying concepts in the data. Empirically, we scrutinize the role of the number of negatives in both NLP and vision tasks. In the NLP task, we find that the results broadly agree with our theory, while our vision experiments are murkier with performance sometimes even being insensitive to the number of negatives. We discuss plausible explanations for this behavior and suggest future directions to better align theory and practice.

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arXiv2021

Acceleration via Fractal Learning Rate Schedules

In practical applications of iterative first-order optimization, the learning rate schedule remains notoriously difficult to understand and expensive to tune. We demonstrate the presence of these subtleties even in the innocuous case when the objective is a convex quadratic. We reinterpret an iterative algorithm from the numerical analysis literature as what we call the Chebyshev learning rate schedule for accelerating vanilla gradient descent, and show that the problem of mitigating instability leads to a fractal ordering of step sizes. We provide some experiments to challenge conventional beliefs about stable learning rates in deep learning: the fractal schedule enables training to converge with locally unstable updates which make negative progress on the objective.

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arXiv2020

Tight Hardness Results for Training Depth-2 ReLU Networks

We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of $k$ ReLUs minimizing the squared error (for $k>1$) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error $ε$. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest $κ$-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in $1/ε^2$. Together with a previous work regarding improperly learning a ReLU (Goel et al., COLT'17), this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on $ε$.

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arXiv2020

Superpolynomial Lower Bounds for Learning One-Layer Neural Networks using Gradient Descent

We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions.

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arXiv2020

Statistical-Query Lower Bounds via Functional Gradients

We give the first statistical-query lower bounds for agnostically learning any non-polynomial activation with respect to Gaussian marginals (e.g., ReLU, sigmoid, sign). For the specific problem of ReLU regression (equivalently, agnostically learning a ReLU), we show that any statistical-query algorithm with tolerance $n^{-(1/ε)^b}$ must use at least $2^{n^c} ε$ queries for some constant $b, c > 0$, where $n$ is the dimension and $ε$ is the accuracy parameter. Our results rule out general (as opposed to correlational) SQ learning algorithms, which is unusual for real-valued learning problems. Our techniques involve a gradient boosting procedure for "amplifying" recent lower bounds due to Diakonikolas et al. (COLT 2020) and Goel et al. (ICML 2020) on the SQ dimension of functions computed by two-layer neural networks. The crucial new ingredient is the use of a nonstandard convex functional during the boosting procedure. This also yields a best-possible reduction between two commonly studied models of learning: agnostic learning and probabilistic concepts.

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arXiv2020

Approximation Schemes for ReLU Regression

We consider the fundamental problem of ReLU regression, where the goal is to output the best fitting ReLU with respect to square loss given access to draws from some unknown distribution. We give the first efficient, constant-factor approximation algorithm for this problem assuming the underlying distribution satisfies some weak concentration and anti-concentration conditions (and includes, for example, all log-concave distributions). This solves the main open problem of Goel et al., who proved hardness results for any exact algorithm for ReLU regression (up to an additive $ε$). Using more sophisticated techniques, we can improve our results and obtain a polynomial-time approximation scheme for any subgaussian distribution. Given the aforementioned hardness results, these guarantees can not be substantially improved. Our main insight is a new characterization of surrogate losses for nonconvex activations. While prior work had established the existence of convex surrogates for monotone activations, we show that properties of the underlying distribution actually induce strong convexity for the loss, allowing us to relate the global minimum to the activation's Chow parameters.

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arXiv2020

From Boltzmann Machines to Neural Networks and Back Again

Graphical models are powerful tools for modeling high-dimensional data, but learning graphical models in the presence of latent variables is well-known to be difficult. In this work we give new results for learning Restricted Boltzmann Machines, probably the most well-studied class of latent variable models. Our results are based on new connections to learning two-layer neural networks under $\ell_{\infty}$ bounded input; for both problems, we give nearly optimal results under the conjectured hardness of sparse parity with noise. Using the connection between RBMs and feedforward networks, we also initiate the theoretical study of $supervised~RBMs$ [Hinton, 2012], a version of neural-network learning that couples distributional assumptions induced from the underlying graphical model with the architecture of the unknown function class. We then give an algorithm for learning a natural class of supervised RBMs with better runtime than what is possible for its related class of networks without distributional assumptions.

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arXiv2020

Efficiently Learning Adversarially Robust Halfspaces with Noise

We study the problem of learning adversarially robust halfspaces in the distribution-independent setting. In the realizable setting, we provide necessary and sufficient conditions on the adversarial perturbation sets under which halfspaces are efficiently robustly learnable. In the presence of random label noise, we give a simple computationally efficient algorithm for this problem with respect to any $\ell_p$-perturbation.

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arXiv2020

Recovering the Lowest Layer of Deep Networks with High Threshold Activations

Giving provable guarantees for learning neural networks is a core challenge of machine learning theory. Most prior work gives parameter recovery guarantees for one hidden layer networks, however, the networks used in practice have multiple non-linear layers. In this work, we show how we can strengthen such results to deeper networks -- we address the problem of uncovering the lowest layer in a deep neural network under the assumption that the lowest layer uses a high threshold before applying the activation, the upper network can be modeled as a well-behaved polynomial and the input distribution is Gaussian.

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arXiv2020

Learning Mixtures of Graphs from Epidemic Cascades

We consider the problem of learning the weighted edges of a balanced mixture of two undirected graphs from epidemic cascades. While mixture models are popular modeling tools, algorithmic development with rigorous guarantees has lagged. Graph mixtures are apparently no exception: until now, very little is known about whether this problem is solvable. To the best of our knowledge, we establish the first necessary and sufficient conditions for this problem to be solvable in polynomial time on edge-separated graphs. When the conditions are met, i.e., when the graphs are connected with at least three edges, we give an efficient algorithm for learning the weights of both graphs with optimal sample complexity (up to log factors). We give complimentary results and provide sample-optimal (up to log factors) algorithms for mixtures of directed graphs of out-degree at least three, for mixture of undirected graphs of unbalanced and/or unknown priors.

cs.SI↗

arXiv2019

Time/Accuracy Tradeoffs for Learning a ReLU with respect to Gaussian Marginals

We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\mathsf{opt} < 1$ be the population loss of the best-fitting ReLU. We prove: 1. Finding a ReLU with square-loss $\mathsf{opt} + ε$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph unconditionally}- that gradient descent cannot converge to the global minimum in polynomial time. 2. There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error $O(\mathsf{opt}^{2/3})$. The algorithm uses a novel reduction to noisy halfspace learning with respect to $0/1$ loss. Prior work due to Soltanolkotabi [Sol17] showed that gradient descent can find the best-fitting ReLU with respect to Gaussian marginals, if the training set is exactly labeled by a ReLU.

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arXiv2019

Learning Restricted Boltzmann Machines with Arbitrary External Fields

We study the problem of learning graphical models with latent variables. We give the first algorithm for learning locally consistent (ferromagnetic or antiferromagnetic) Restricted Boltzmann Machines (or RBMs) with {\em arbitrary} external fields. Our algorithm has optimal dependence on dimension in the sample complexity and run time however it suffers from a sub-optimal dependency on the underlying parameters of the RBM. Prior results have been established only for {\em ferromagnetic} RBMs with {\em consistent} external fields (signs must be same)\cite{bresler2018learning}. The proposed algorithm strongly relies on the concavity of magnetization which does not hold in our setting. We show the following key structural property: even in the presence of arbitrary external field, for any two observed nodes that share a common latent neighbor, the covariance is high. This enables us to design a simple greedy algorithm that maximizes covariance to iteratively build the neighborhood of each vertex.

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arXiv2019

Quantifying Perceptual Distortion of Adversarial Examples

Recent work has shown that additive threat models, which only permit the addition of bounded noise to the pixels of an image, are insufficient for fully capturing the space of imperceivable adversarial examples. For example, small rotations and spatial transformations can fool classifiers, remain imperceivable to humans, but have large additive distance from the original images. In this work, we leverage quantitative perceptual metrics like LPIPS and SSIM to define a novel threat model for adversarial attacks. To demonstrate the value of quantifying the perceptual distortion of adversarial examples, we present and employ a unifying framework fusing different attack styles. We first prove that our framework results in images that are unattainable by attack styles in isolation. We then perform adversarial training using attacks generated by our framework to demonstrate that networks are only robust to classes of adversarial perturbations they have been trained against, and combination attacks are stronger than any of their individual components. Finally, we experimentally demonstrate that our combined attacks retain the same perceptual distortion but induce far higher misclassification rates when compared against individual attacks.

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arXiv2019

Learning Ising Models with Independent Failures

We give the first efficient algorithm for learning the structure of an Ising model that tolerates independent failures; that is, each entry of the observed sample is missing with some unknown probability p. Our algorithm matches the essentially optimal runtime and sample complexity bounds of recent work for learning Ising models due to Klivans and Meka (2017). We devise a novel unbiased estimator for the gradient of the Interaction Screening Objective (ISO) due to Vuffray et al. (2016) and apply a stochastic multiplicative gradient descent algorithm to minimize this objective. Solutions to this minimization recover the neighborhood information of the underlying Ising model on a node by node basis.

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arXiv2018

Improved Learning of One-hidden-layer Convolutional Neural Networks with Overlaps

We propose a new algorithm to learn a one-hidden-layer convolutional neural network where both the convolutional weights and the outputs weights are parameters to be learned. Our algorithm works for a general class of (potentially overlapping) patches, including commonly used structures for computer vision tasks. Our algorithm draws ideas from (1) isotonic regression for learning neural networks and (2) landscape analysis of non-convex matrix factorization problems. We believe these findings may inspire further development in designing provable algorithms for learning neural networks and other complex models.

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